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I

PRACTICAL ESSAYS

OH

MILL WORK

AND OTHER

MACHINERY.

Q. WOODFALL Am9 SON* amqkl coubT. bkimnbk vtkbbt, lokooii.

PRACTICAL ESSAYS

ON

MILL WORK

AND OTHER

MACHINERY.

BY ROBERTSON BUCHANAN, Engineer.

WITH

NOTES AND ADDITIONAL ARTICLES,

CONTAININO NEW BBSEARCHE8 ON VARIOUS MECHANICAL SUBJECTS^

BY THOMAS TREDGOLD, C.E.,

MBMBBB OP THE INSTITUTION OF CIVIL EN0INESB8.

AND NOW

REVISED INTO A THIRD EDITION WITH ADDITIONS,

BY GEORGE RENNIE, ESQ. C.E. F.R.S.

ETC.

1-

lUiU8TRATED BY UPWARDS OF SEVENTY PLATES,

AND NUMEROUS FIGURES.

A' /-• \\Z\0 ^ V^ LONDON :

^ JOHN WEALE,

ABCHITECTURAL LIBRARY, 59, HIGH HOLBORN.

1841.

publisher's address.

lumeH S and 4 ; 1750 copies of the two editions of Comte
dc5 Panibour's Practical Treatise on Locomotive Engines ;
870 copies of the work on Bridges ; 500 copies, in a few
months, of the work of Mr. Clegg, Jan., on Coal Gaa ;
and within one month upwards of 500 of Mr. Wicksteed's
Experiments on the Cornish Engine were sold, which, with
niimy others, are testimonies of the esteem in which such
works are held at the present time.

In the present instance it affords me much pleasure
gratefully to acknowledge and publicly state the liberality
of Mr. George Rennie, the editor of this work, who, al-
though having multitudinous professional engagements,
has (anient in the love of his art) found the necessary
time for the arrangement, the addition to, and editing of
this new edition. This has been done gratuitously,
and it is hoped that the Subscribers, in receiving the work
iH^nstHiuently so much cheaper, will, in the acknowledg-
ment tvf its utility, respond to the Publisher's thanks now
oxpn^ssoil for the kindness conferred.

IkttmKfr 1.1841. JOHN WEALE.

i

PUBLISHER'S ADDRESS.

The production of Works specially devoted to Engineering
is, in this country, frequently attended with difficulty, not
arising from the scarcity of subject-matter or the disin-
clination of practical men to facilitate its arrangement, but
from their inability to find time to render their willing
aid. Delay, as in the instance of the present work, is in
consequence unavoidable. A publisher's risk is increased
in no trifling degree, when he ventures upon publications of
a scientific character unaided by an author or editor of ex-
perience in the matters of which they treat ; but it is his
duty to select such useful and novel subjects as shall not
only be of practical help to the engineer, but afibrd a clear
view of elementary principles to the student ; and as there
are now amateurs in Engineering as weU as in other de-
partments of art, such works are peculiarly acceptable.

Keeping this twofold object steadily before me, it has been
my oanstant endeavour for many years to render works of
practical reference as complete as possible, especially by an
adequate number of engraved illustrations of examples ;
and it is with grateful feelings that I acknowledge the
libenility of many gentlemen, whose names appear as con-
trflmtOFB of drawings in my numerous published works,
loaie few of which are here mentioned, together with an
aoooimt ct their sales : viz. — ^Tredgold on the Steam En-
gDDe, 2S00 copies since October, 1838 ; Public Works of
Great Britain, 97^ copies within the same period ; Papers
if tibe Royal EngineerB, 1000 copies of each of the vo-

VI GENERAL PREFACE.

corrected the Essay on the Teeth of Wheels, and supplied
some additional tahles and a second appendix.

With a view to practical utility, I have endeavoured to
adapt the style of these Essays to the comprehension of
such operative mechanics as have not had the advantage
of mathematical instruction ; but at the same time I have
given reference to authors for the demonstrations of such
propositions as I found it necessary to introduce, in order
to give such workmen some notion of the principles on
which their work should be conducted.

For any repetitions, want of unity, and other imperfec-
tions, which will doubtless too readily appear in these
Essays, I may offer the same apology which I did on a
former occasion, that they were written at many Afferent
and distant intervals, occasioned by intemipticms from
professional and other engagements.

Note. — The Second Edition was superintended by the hite Mr. Tred-
gold ; and the principal facts noticed in it are incorporated in the following
Preface.

PREFACE

TO THB

THIRD EDITION.

The Essays of Robertson Buchanan on Practical Me-
chanics have been long known, and duly appreciated by
the public

They consist of a series of treatises, seven in number,
on several of the elementary parts of machinery ; such as
the Teeth of Wheels, published in 1808 ; on the Shafts of
Mflls, in 1809 ; and on Millwork and other Machinery, in
1814. The copious Index of the contents of the present
edition, drawn up by Dr. Jamieson, sufficiently explains
the nature of the work.

In perusing the Essay on the Configuration of the Teeth
of Wheels, we are at once struck with its resemblance to
the admirable treatise of Camus, published in 1782% and
which the author duly acknowledges. The subject is
divided into two parts ; firstly, the principles as laid down
by Camus ; secondly, the application of these principles to
difEsreiit kinds of spur and bevel gear.

* Coon de Math^matique.

TUl PREFACE TO THE

The first application of the epicvcloidal curve to the
teeth of wheels is generally ascrihed to Roemer, a Danisii
mathematician, in 1674^% although De la Hire^ claimed the
merit, and demonstrated that if a tooth of either a wheel
or pinion be formed by a portion of an exterior epicycloid,
described by a generating circle of any dimensions what-
ever, the tooth of its follower will be properly formed by a
portion of an interior epicycloid, described by the same
generating circle. The object he had in view, was to se-
cure a perfect uniformity of pressure and velocity to the
machine, so that, in all positions, the wheels which trans-
mit the power should act equally and similarly, and (hat
the surfaces of the teeth, by touching in a point, should
roll over each other when in motion, and thus avoid all
friction, a desideratum hitherto impracticable to ac-
complish.

The general properties of the cycloid and epicycloid,
and the modes of generating these curves, both geometric-
ally and mechanically, have been given by various authors,
and Buchanan applied the principles of Camus to the forms
of the surfaces of the teeth of wheels and pinions acting
against each other imder different circumstances ; such as
the wheel and trundle, the wheel and pinion, the rack and
pinion, conductor or conducted, external or internal, spur
or bevel gear, so as to render them comprehensible by the
general reader. The author next investigates the action
of conical, or bevel wheels, under the different circum-
stances of the inclination of their axles, applying the same
exterior forms of the teeth of spur wheels to the teeth of
bevel wheels, with the exception that the curves should be
a spherical epicycloid. The principles of bevel wheels had
lieen already pointed out by De la Hire in the year 1666 %

* Wolfii Opera Mathcmaticis.

^ Traite des Epicycloidcs, 1694.

e Mcmoires de rAcadcmie, 1666. 1669.

THIRD EDITION. ix

igh long knonn and applied previously*, in the
p of a conical tnmdlc.

De la Hire'' was not only among the first to apply the
«?picycloid to the configuration of the teeth of wheels, but
le considered the involute of a circle, as the best of the
«xterior epicycloids, and which it may be proved to be,
if we consider the generating straight line as a curve of
I^H infinite radins, and which would strictly apply to a pinion
^Baeting on a rack, and vice versa to the teeth of a rack.
^HEoler, in I76O, treated the case of the involute very
^Bgcoerally '.

^B Kaestner*, in 1771. shewed a method of describing
1^^ and applying the involute to the teetli of wheels. Pro-
fessor Kobison applied the involute to the wheels of a
miH near Edinburgh, but the result was any thing but
■Satisfactory. The same principle has since been advo-
by Ferguson ", Professor Airey ', and Professor
I WiUis, whose valuable paper appears in the Appendix.

We are also indebted to several continental writers, but
^■^particularly to M. Hachctte', for his elaborate investigation
^Bnf the curves most applicable to the teeth of wheels. Pro-
^Bfessor Airey states, " That in order that the mechanical effect
which one wheel will produce upon another, may in all
positions be the same, it is necessary that the line perpen-
dicular to the surfaces of the teeth at the point of contact,
intersect the line joining the centres at a fixed point, which
divides that line into two parts, the ratio of which is the me-

• Beswni, 1582. Tlicatnim Mafliinnrum.

* Traite Jm Epicycloids, and Novo ConuneDt. Petropol. 1754, 1755.
■ Comment. Petro|>o]. 1T5+, 1755.

* De Denubus Botorum Reg. Soc. Goltingensis.

• Sir Dnvid Brewster, edition of 1807.

' Cambridge Pliilosojiliicai TransnctionB, Vol. U.
» Tniite felemcntaire dcs Machines, 181 1.

X PREFACE TO THE

chanical power ; when this holds, the proposition of the an-
gular velocity will be constant." Mr. Airey then deni(Hi-
strates the case mathematically, and advises tiiat the teeth
be made to work a litde before and after tiie line of cen-
tres, and thinks that a tooth formed by the union of an
epicycloid and hypocycloid is preferable to any form what-
ever, for tiie line of action is always very nearly perpen-
dicular to the radius, by which means, not only is the fric-
tion made much less, but also the strain upon tiie axles is
considerably diminished. The same applies to bevel wheels
and rack-work, with reference to uniformity of motion and
action, which he conceives to be of far greater consequence
than any diminution of friction, which can never be re-
duced to nothing, except the part of contact be always in
the line of centres, a condition which may be satisfied only
by means of an infinite number of curves, and amongst
others by two logarithmic spirals, but the mechanical ac-
tion, and the motion would be dreadfully irr^ular.

This question is now littie more than one of mere cu-
riosity, arising from the smallness of the teeth of wheels
now made, and the greater perfection of workmanship in
the materials, in consequence of the use of iron wheels,
and the accuracy with which the teeth are formed and ad-
justed by the most simple method of templates and com-
passes ; and the approximation of the form thus generated
to the form presented by theory is very close. The deter-
mination of the strength of the teeth to the power to be
transmitted, is given in the fourth chapter on the principles
of proportioning the strength of the teeth of wheels.

The rule adopted by millwrights for finding the depth of
the teeth from the bottom to the pitch line, and from the
pitch line to the top of the tooth, is simply to multiply the
pitch by 5, and divide by 9, and vice versd.

The curves of the exterior and interior surfaces of the

THIRD EDITION. XI

a are sometimes traced by means of a tracer fixed in
the radius line of one of two pieces of board, the edges of
IJL which arc cut out to suit the primitive circles of the wheels
^■required ; then, by fixing a template to one edge, and di-
^^riiUng the teeth accurately, the tracer will, by the rolling
I of the two circles, describe the curves required.

The usual mode of describing the teeth of wheels by
arcs of circles, is admitted to approximate to the true
cur\'e, if the centre and radius of the wheel be deter-
mined correctly. In the best establishments, this is invari-
ably done, and the result of many years' practice has proved
the goodness of the system.

With respect to Buchanan's tables on the pitches of
wheels, and the strenjrth of the teeth with the correspond-
ing numbers and horses' power moving at the ])itch line at
different velocities, it will be observed that the prevailing
proportion is, that the pitch is about double the thickness
of the tooth, and the length rather longer than the thickness,
but three to four times the thickness is more usually
^K^diopted.

^B The investigation of the pr()i)er curves to be given to the

^■teeth of wheels, by Professor Willis, has been added by way

of an appendix to the concluding chapter on the Teeth of

Wheels. The first section gives a succinct account of the

curves adapted to practice, and shews, by way of corollary,

that if for a set of wheels of the same pitch a constant de-

^^ Jcribing circle be taken, and employed to trace those por-

^Btions of the teeth which project beyond each pitch line by

^Vmlling on an exterior circumference, and those which lie

within it, by rolling on its interior circumference; then

any Iwo wheels of this set will work correctly together.

I Profesgor Willis then shews how this can be accomplished,

^'ind then gives a form of increased strength to the backs of

tlie teeth, but which arc only suited to move in one diroc

The second section of this paj)cr shews how the

XU PREFACE TO THE

practical approximation to the true form can be accom-
plished by arcs of circles, a form which approaches to mo-
dem practice. An instrument termed an odontagraph,
together with tables for facilitating its use, and for forming
cutters for shaping the teeth, is proposed, and a theory given
of the nature of the motion which is produced by the pres-
sure of one circular arc upon another, when disposed so as
to work in the manner of teeth.

The Essay on the Shafts of Mills is divided into five
chapters, containing a general description of shafts most
employed in miU-work, and the strains to which they are
subject from lateral stress and torsion ; the strength and
stiffening of shafts, journals, and gudgeons, with refer-
ence to the strength of materials, according to the ex-
periments of different authors. The subject of torsion is
briefly examined, in conjunction with lateral stress. It is
shewn that, in general, the strength of a cylinder or solid
axle to resist the force of torsion, is as the cube of its
diameter, and that the length of a cast iron shaft has no in-
fluence on its resistance to torsion, whatever may be the
exception with wooden shafts. The power of a cast iron
shaft to resist torsion is calculated firom Mr. Tredgold*s
formula, which considers the resistance the same as firom
the lateral stress. The Table of Shafts, at the end of the
fifth Chapter, takes into consideration the two kinds of
resistance.

The use of iron in machinery previously to its adoption
in England, is evidenced by referring to the works of
RameUi, Bockler, and Bessoni, where there are repre-
sentations of iron wheels, and portable miUs and cranks
of the same metal, but it was only used in this country
about the \^^ar 1550.

Iron pipotsi canio into use in France about the year I672,
aiul fKun the UHur 178^2 to 17S4, cast iron was used in
machinery at Culobrook Dalt\ Rotherfaam, and at most of

THIRD EDITION. "XIU

e gtetA iron works in England. Cast iron wheels were
silso in iise at Manchester, Liverpool, Nethcrhy, and several

k other places.
The theory of torsion has been investigated by several
writers, but with very little effect. Coulomb was the first
lo direct the attention of mathematicians to this kind of
Stsistance. If a cylindrical body, such as a line or series
i, of lines or fibres be suspended vertically, but having its
upper end fixed, be turned round through any angle by
1^^ the existence of some lateral force, and if its elasticity
^B<bc not impaired, it will, after the deranging influence has
^Vccftsed, return to its former position, and may perform this
retrocession in a certain time. Professor Leslie ', in con-
sidering this question, conceived the cylinder to consist of
^_ a series of thin discs, which, when twisted, each successive
^^^sc will make an angular advance till this accumulation
^Vst a certain distance amounts to a complete circuit, and
'T such revolutions will be repeated at every like interval
The torsion is thus proportionate to the angle of deviation
and the fourth power of the diameter of the cylinder, and
inversely as its height.

If the exterior particles of a line or series of lines be

lOverstrained, then their cohesion will be destroyed, and the

ine return no longer to its pristine state. If the strain be

rithin the limits of the elasticity of the exterior particles,

[the oscillations will be perfectly isochronous.

The application of this principle by Coulomb'' led to the
istmction of an exquisite balance for detecting and
tneasuring the smallest forces. Thus, by establishing a
connexion between the elastic and passive resistance of
metals, and the resistance &om torsion, we are led to an
approximation to the true theory. The experiments of

• Eienieots of Nntuml Pliilosophy.

Meinoircs de TAcademJc dcs Sciences, 178*.

•XIV PREFACE TO THE

Minard* and D^sormes, Lagerhjelm^, Bomet% Segum^ and
Ardant% however, effected much towards the solution of
the question.

The experiments of Savart* on the torsion of different
bars show the following results :

1st. That the angles of torsion are in every case pro-
portionate to the forces of torsion within the limits of
elasticity.

2dly. That in bars of the same section, the forces of tor-
sion are directly proportionate to the length of the bars,
the length having little other influence than increasing the
angle of torsion.

The experiments by Duleau, Banks, Dunlop, and by
the Editor of this work, approximate pretty nearly the
mean values.

On the dynamical effect of men and horses, Buchanan
quotes the estimates of Desaguliers, Emerson, Smeaton,
and Watt, as also a table in units of force from Dr. Young.
Buchanan's own experiments of the effective power of men
working in different positions, had already been published
in the fifteenth volume of the Repertory of Arts, for
1801. Of the authors who have written upon this sub-
ject, the names De la Hire ', Amontons ', Lambert \ Ber-
noulli ^ Coulomb \ and Schulze, are most distinguished.
Do la Hire considered the question almost entirely with re-

* Annales do Chimie.

^ Du Fer dans lee Fonts suspendus.

^ Poncelet, Mecanique Industrielle, 1839.

^ Des Fonts en Fils de Fer, 1826.

* Annales de Chimie.

' M^moires de TAcadeniie des Sciences, 1699.
K M^moires de I'Academie des Sciences, 1709.
^ Mcmoires de TAcademie de Berlin, 1776.

* Mcmoires de TAcademie des Sciences, 1760.
^ Prix do TAcademie, torn. vii. 1753.

THIRD EDITION. •jy

( to the muscular powers of animals. Amontons
with reference to their velocity. Deparcicux their weight
and muscular action conjoiutly, and Lambert reduced the
problem to an equation, which gives the relation between
the weight, velocity, burthen, and path of inclination
pursued by the animal. Bernoulli maintained that the
quantitj- of fatigue was always proportional to the quan-
tity of action, whatever be the nature of the work per-
fonoed.
Coulomb proved the absurdity of this position, by shew-

fing that the daily quantity of action is variable according
to the circumstances under which it is developed, and by a
iKries of valuable experiments, comprehending almost everj-
ease, be assigned values to each kind of action.
Schulzc's investigations on the absolute and mean effect
of human labour to lift, draw, and push weights, have fur-
oished us with standards of value which may be safely
■ taken as the average of continental labour.
^H The more modem investigations of Gerstner \ Morin,
^ Tourier, Devilliers, and others on the continent, and of
Trcdgold, Palmer, Sylvester, Bevan, M'Neill, and Field, in
this countrj", besides many experiments which have been
made by ourselves ^ on the extreme and mean powers of men
and horses, have established sufficient data whereon to found

' Treadse on ME^haoics, tronslBtcd from the Oerm&n, 1834.
'' TliB foUowing ia a. brief statement of the dynamical effect of human
1 liorec power aevcrelly dppUed to wulki&g whcel-croncs, cruba, cranes,
FpI^^Tifig engines, hor»e runs, &c., for raising iliffcrent materials.

MANUAL LABOUR.

BuiLDiNo Matkrials.

I iHlj. One man in 67 juunicys raised a weight of 16,342 lbs.

(incluiUtig Uie weight of liiniiSclF] to tlic height of 30 feot in Ibf).

lOhonis, oiual lo a weight of 817

niW I fuot high jwr niinute.

•Xvi PREFACE TO THE

tiprmaht suitable to <fverv cage of the apfdicatioD of animal
|Mnri$r Up perfirirm mechanical operaticMi&

S^J/. Ocme; nuMi in 47 journepi raised a wei^ of 11,374 lbs.

(imludUig bk own weij^t) to a hei^ of 50 feet in 10 honn, lbs.

Mltml Uf a wei^t of 974.8

nimi 1 foot ttijg^ per minate.

OKniNAfty Ciuifis. — Expsbimbnts madb at thb Wbbt India Docks.

'Idly. Tlio power of six men applied to a crane is capable of ^
rakitig 224,000 Ibt. 15 feet high in eight hours, equal to a

witiglitof 1166.6

rained 1 foot high per minute by 1 man.

4thly. Tlio power of 0 men applied to a crane is capable of
nuNJiig a weight of 262,080 lbs. 12 feet high in 8 hours, equal

to a weight of 1092

raiitefi 1 foot high |>er minute by 1 man.

N,li* Hy experiment, the friction of these cranes varied from
^Qth to j^^tli of tlio absolute weight.

OhDINAHY ChANKS.— ExPBRIMBNTS IfADB AT THB LONDON D0CK8.

Athly. Hy tlio ^idking whool-crane, worked by 6 men, a

wolglit of 787»920 lbs. >i-as raised 7 feet in 8 hours, equal to 1915

ndmfiA 1 foot high per minute by 1 man.
(tthly. Again, by tlio >i*alking-wheel crane, worked also by 6

mon, a >\*t>ight of 91 K680lb6. was raised 8 feet in 11 hours,

(M)ual to 1841

iniMHl \ foot high por minute by 1 man.
?thly« Hy t orabu >i^Mrkcd by 6 men each, a ^-cight of 728,000 lbs.

WHN miw^l to a height of 16 fe«l in 8 hours* equal to . 2012

nuii«Hl \ (\H\t higli |^>r mimiti!' by 1 man.

HORSE POWER.

SihK\ TW vKiMttiknJ cdixt of a horse power appBed to a pile-
\lri\u^ o^ue >{k\vrk«^ by i iMneiv was fNuad to be equal to
M \H\H$bt vxt' 4:(^A^ cKjk vaib«ol 1 jRwt bi^ ui ;^ MceaAi^ ar a
>m\^^t^a' »AS9

THIRD EDITION. *XVU

The Appendix to the Second Essay contains tables, by
iTrcdgold, on the properties of materials, and the influence
f alloys in increasinj!; the tenacity of metals. These data
lave been farther extended by the experiments of Messrs.
tairbaim and Hodgkinson on the relative strength of hot
od cold blast iron % and on the compression of cast iron
nlumns*. But our knowledge of the elastic properties, of
materials, the laws of the elongation and compression,
and the effect of temperature upon their cohesion, is as yet
hut imperfectly known. The experiments of Rondelet,
Dupiu, Tredgold, Barlow, Bramah, Gerstner, and Adam
Burg on the flexure and resilience of wood and iron have
Tiishcd some valuable facts on this subject, but it is to
Messrs. Minard, Desormes, and Ardant that we are prin-
Uy indebted, for determining the law of elongation by
direct tension of the fibres of wood and iron. Vicat
lowed, in the case of a cubical prism of lead, that the law
r compression ie constant from a constant augmentation
f pressure. Peclet proved that for c^ast iron the molecular
Uplaccment of the crystals did not, in the first instant
r compression, exactly follow the compression in propor-
to the resistance, and our own experiments in the
compression of several of the softer metals have shewn the

Btliljr. Again, the power of a horse applied to working runs for

fcninug earthwork up a riui or inclined plane, the hone of
nliidi was 60 feet, and the vertical height 10 feet, was equal
to a resistance of ilOlhs. trnvelling through a space of 72 feet lbs.

ID I nunatc by two horses, which ia equal to . . . 14,760
ntted 1 fool high per horse power per minute ; a result very
inferior to the laat, arising from the inconstant nature of the

» S«rtnth Report of the British Association for the Adi

k Bxpcrimental Researches on the Strength of Ptllora of Cast 1)
dF otber MctcriaJd.— Philosophical Transacdonii, 1840.

•xviii PREFACE TO THE

densities to have increased in a greater ratio than the com-
pression.

The influence of temperature, so far as the temperature
of the atmosphere is concerned, appears to exercise very
little influence, but when carried beyond the limit of at-
mospherical temperature, the experiments of Messrs. Tre-
mery and Poirier have shewn that, at a dull red heat, (450*"
Fahrenheit,) the tenacity of a bar of iron had lost one
sixth of its original strength.

M. Savart * proved, by means of a series of ingenious ex-
periments on the sonorous vibrations of difierent materials^
the influence of time in the aggregation of the particles in
cooling of substances, apparently homogeneous ; and Messrs.
Vicat, Minard, and Desormes, and ourselves ^ have, by
means of iron bars loaded to within the limits of their ab-
solute strength, shewn that permanent set or loss of elasti-
city, and even rupture, takes place when influenced by time.

On the subject of Shafts and Couplings, a new era had
arisen. The introduction of the textile fabrics in the
country, by Lombe and others, and the inventions of Wyatt»
Arkwright, and Watt, led to a new system of machinery.
The necessity of producing high velocities occasioned a cor-
responding diminution in the dimensions of shafts, and
those ponderous masses of wood, cast iron, and their enor-
mous bearings and couplings, gave place to slender rods of
wrought iron and light frames or hooks for suspending
them. In like manner, wheels and pulleys of large dia-
meters were replaced by pulleys and straps of moderate dia-
meters and dimensions, and by uniting the pulleys in series
of difierent diameters, and alternating their positions oppo-

* Annales de Ghimie et de Physique, snr les Vibrations longitadinales des
Corps, tome 65.

^ On the Effects of Temperature on the Arches of Bridges. Tnmsaotiinns
of the Institntion of Civil Engineers, Vol. III.

THIRD EDITION-. *XIX

I each other, a greater variety of velocities were ob-
ncd, and a great deal ttf friction and noise done away with,
rithout taking into conaideratioii the economy resulting
IVoin the lighter kind of machincrj' and the less quantity
of power than formerly required to put the wliole in motion.
"besB improvements are in a great measure due to
. Fairhaim and Lillie". To use the words of Dr.
" The method of increased velocities in the driving
i of factories is undoubtedly one of the most remarkable
nprovements in practical dynamics. It diminishes greatly
the inertia of the mass to be moved, by giving to much
lighter shafts and wheels the same momentum, and it per-
raitii the pulleys or drums which immediately impel the
machines by straps* to be reduced to a size much nearer to
lat of the steam pulleys fixed on the main axes of these

I The same improvements have taken place with regard
B the couplings, which are now reduced to simple rings
f wrought metal keyed to the circular ends of the abutting
ids of the shaiU.

• In a letter to the Editor of this pnblicatioo, Mr. F&irbaJm dates the

action of llie new system of gearing from the year 1815: at that

B, wyn he. " the ahufts of our cotton mills were moving at 40 and 50 re-

is per minute, whereas at the present day we hove none under 60,

u numy as 300 and 350. The same number of revolntionK are appli-

i now in use for flnl oud Bilk. The extensive use of wrought

Kbon for (bafts, and the slide lathe, bnve ^ven wonderful facilities to the

r p*iurtioii of ihofts, and increased velocities and reduced friction by the

•Bwrniwoti of great power through a comparatively small section. In

nnr rif the more recent mills of iny coiistrucdon, we have shafts only 2^

r overconiing the power of a iO-horse engine. Another

aDpnivinicnt n-iu' our system of coupling, and the mode of suspending

I A*Ai fmn tliG main beams and ceilings of rooms, &c. In the first instance

f never get loose, and In tlie second, the shafts ai'e strung like

' *)ns ibittg the celling, and with small iron pulleys transmit the motion to

I At udiincry without crowding the room or obstructing the light."

* PUlMOphy nf Manufactures.

Aft

•XX PREFACE TO THE

As respects iron, cast iron pipes and cranks and
pumps were used in the old London Bridge Water-
works, by Sorocold, in the time of Charles II., and
mention is made of a cast iron wheel, 4 feet in diameter,
which worked into a pinion 6 inches in diameter ; and he
adds, ** If the teeth of the wheel be of brass, and the teeth
of the leaves of the pinions of iron, the machine will work
more equally." It seems generally believed, however, that
Smeaton was the first to introduce cast iron wheel work in
machinery at the Carron Iron Works, for the purpose of
boring cannon, about the year 1769> although he had pre-
viously applied a cast iron axis for a windmill in 17^4 ;
but the founder's art was so imperfect, that Smeaton was
obliged to proceed cautiously: and it was not until the
years 1784 and 1785, when the Albion Mills were built,
that cast iron was applied to all parts of machinery, and
the late Mr. Rennie was the first to introduce accuracy in
the forms of the teeth of wheels, by turning and adjusting
the teeth, and causing the iron to work into wooden cogs.
The subsequent progress which has been made in the
later period of his life, introduced a new era in mill ma-
chinery, which, in point of accuracy and smoothness of
workmanship, has not been exceeded, even under the au-
tomatic system of self-acting tools. Arkwright used iron
bevel wheels and band pulleys, at the cotton spinning mills
at Cromford and Helper, in 1775.

The Fourth Essay of Buchanan, on the Method of Disen-
gaging and Re-engaging Machinery while in Motion, may
be fairly included in the Third Essay on Couplings, with the
exception of the fast and loose pidleys and friction clutches,
which are found to be the simplest and best for engaging
and disengaging machinery without shocks. The friction
plate inclosed between two other plates, introduced some
years back by ourselves, has been found to answer all the
conditions in point of simplicity and efiect required by a

THIRD EDITION. •xxi

friction pulley, and does away with all the ineonyeniences
of the cones.

The Fifth Essay on the Mechanism for equalizing the Mo-
lion of Mills, relates to the changes of velocity to which every
first mover is suhject, either from an increase or diminution
in the supply of power, or where the power is uniform, from
the increase or diminution of the resistances required to he
overcome. This is accomplished by means of double or
conical pendulums and balls, cither for regulating the sup-
ply of wind, water, or steam, according to the quantity of
action required.

The Appendix to the Fifth Essay is extracted from a
paper communicated by Buchanan, in the year 1799, to
the Philosophical Society of Edinburgh, and afterwards to
the Editor of the Philosophical Magazine, on the Velocity
of Water Wheels. The author negatives the conclusions
of Banks, viz. that the velocity of an overshot wheel is as
the cube root of the quantity of water it receives, by con-
trasdng his own experiments on water wheels moving with
their common velocity and half that velocity ; and the re-
sult was, that the last half required just half the quantity
that the first did ; and this he confirms bv two letters from
Mr. Robcrton, in which the author contrasts the maximum
velocities of Smeaton and Banks's water wheel ; and says that
while Smeaton, by his maximum velocity of throe feet, lost
only one-fourth of the original effect, Banks, at his maximum
velocity of one foot per second, reduced it to one half of
that velocity, thus making the same quantity of water pro-
duce four times the quantity of work, or twenty times the
quantity of work which Smeaton could perform with the
same quantity of water. The continuation of Buchanan's
Appendix shews that the mechanical effect depends on the
wheel's diameter, the height of the fall, and on the velo-
city of the circumference of the wheel ; and it is shewn
that a water wheel will produce the greatest effect when

•XXU PREFACE TO THE

the diameter of the wheel is proportioned to the height of
the fall, so that the water flows upon the wheel at a point
about 52f degrees distant from the summit of the wheel.

The subject of water wheels has been fully treated, both
theoretically and practically, by many authors both on the
continent and in this country ; suflSce it to mention the
names of Pitot, Deparcieux, Lambert, Borda, Bossut,
Eytelwein, Morosi, &c., &c., among the former, and of
Smeaton, Robison, Fenwick, and Banks among our own
countrymen ; and in more modem times by Navier, Ponce-
let, Morin, Foumeyron, &c., and by several eminent me-
chanicians in this country.

Of the several classes of overshot, breast, and under-
shot wheels, a great diversity of opinion prevailed* By
Pitot it was maintained that the float boards of undershot
wheals should be continued in the line of the radius. By
Deparcieux, that the floats should be inclined to an angle
of 15 or more degrees. Bossut was of a contrary opinion.
Borda, Bossut, and Robison considered that the maxi-
mum velocity of the wheel's circumference should be one
third of the velocity of the current Smeaton made the
maximum velocity of the wheel between one third and one
half of the current Banks difiers firom all the authorities.
Navier, Poncclet, and Morin % make it one half, whether
the floats are on the line of the radius of the wheel, or
curved. Again : as regards the diameter of the wheel, it
was maintained by some that the diameter of the wheel
should never exceed the height of the fall, and by others
that the diameter should in all cases exceed the height of
the fall, in which latter opinion Smeaton coincides ; for, says
he, ** the higher the wheel is in proportion to the whole
<lescent, the greater will be the eflect.** The same divers-

* Exporionccs snr los Roues Hydiuuliqnes a Aubes, Planes, et svr les
Roues Hydmuliques a Aujets^ 1836. Subsequent experiments haTe
the useful effect to 75 per cent of the abeolnte expanditure.

THIRD EDITION. *Xxiii

ity of opinion existed relative to the proper number of
floats ; Pitot maintaining that the number of floats should
be equal to 360** divided by the arc of the circle plunged
in the chamiel, and Fabre and others, that the number
should be as great as possible. Bossut found the best
number of floats to be forty-eight. Smeaton from 24 to
40, for a wheel of 20 feet diameter, and Poncelet 30 to 36
floats for a wheel of 1 6 to 18 feet diameter*.

In considering the action of the water on the vertical or
radial floats of the common undershot wheel, M. Poncelet
was of opinion that one of the causes of the small efiect
produced by undershot wheels, arose from the imperfect ac-
tion of the fluid by shocks in entering the wheel, and by
gravity in quitting it ; whereas, if the floats were so con-
structed as to admit of the water entering and quitting the
wheel quietly, the effect would be a maximum.

The inclination of the floats has long been a favourite
project with mechanicians, but in so far as our own expe-
rience goes, little or no benefit has been derived from that
arrangement ; and in this opinion we are confirmed by the
experiments of Bossut, who found the effect, at different
angles, to be rather disadvantageous than otherwise. But
M. Poncelefs curved floats produced the following results.

* The following experiment was mode in the year 1820, by the editor of
this work, upon two water wheels, each 19 feet 4 inches diameter, and 6
feet in width ; one wheel had 40 floats, and the other 48 floats, and tlie
&D 12 feet 2 inches in height. The machinery consisted of two Avheels,
Hid two pinions of cast iron, and tAVo pairs of 4 feet diameter French burr
■tones.

The result was, that the wheel with 40 floats, ground, in 31 hours, 359
Imiheils of whent ; and the wheel with 48 floats ground, in 32 hours, 392
hodielB of wheat. Hence, the wheel with 40 floats ground the same quantity
of wheat (by experiment) with 1*43 per cent, less water than the wheel with
48 floftts. It was also proved that 58*33 lbs. of water ground 12*74 lbs. of
wheet per minute, and 650*62 cubic feet of water, falling one foot, ground
one Imhel of 60 lbs. weight of wheat, for the wheel with 40 floats.

XXIV

PREFACE TO THE

Istly. That the TnaximuTn velocity of a wheel with cuired
floats, was 0*55 of the velocity of the stream.

Qdlj. That the dynamic effect for small falls and large
openings, and 0*65 for large falls and small openings, and,
generally speaking, the effect of the wheel with the curved
float, compared with the effect of the wheel with vertical
floats, was as 0*60 to 0*50 of the power expended.

M. Poncelet^s researches on this subject have placed it
upon its true basis. But it is to the experiments of the
committee of the Franklin Institution * that we are indebted
for the most detailed information we possess on the subject
of water wheels.

The great defect of all former experiments, is the small
scale upon which they have been made.

The experiments of the Franklin Institution were made
with wheels of 20, 16, 10, and 6 feet diameter, respectively)
and all the conditions of friction and other retarding forces
were strictly attended to. The results prove that the maxi*
mum and mean effects of large wheels are greater with mo^
derate velocities (double what Smcaton assigned as the
maximum velocity for the mean circumference of a water
wheel). The maximum and mean effects are diminished
with an increased velocity, as the wheels are diminished
in diameter.

* 20 foet ^ameter wheel :
128 Exi>crimeiit8

Maximum
effect.

Correroond.
veloaty.

Mean effect

Coireapond.
velodtj.

•800
•692
•643
•567

5-48

5-87
5'8S
7-59

•784
•609
•562
•484

601
5^73
7^90
8-18

15 feet diameter wheel :
88 Experiments

10 feet diameter wheel :
180 Experiments

6 foet diameter wheel :
178 Experiments

THIRD EDITION. •XXV

In the case of vertical water wheels, the water acts
either by its impulse or gravity. But with horizontal
wheels with inclined or curved floats, the motion is pro-
duced by the impulse and gravity of the water conjointly.
The experiments of Messrs. Piobert and Tardy • on several
wheels of this description, in the south of France, have
given very feeble results, seldom exceeding one fourth of
the power expended, and averaging much less. The reac-
tion of a column of water upon the curved floats of a hori-
zontal wheel has been found to bo more effective, and the
recent experiments of M. Morin^ upon the Turbine of
Foumeyron have shewn this new and curious machine,
when properly constructed and moving at its maximum ve-
locity, to be equally effective (if not more so) with the best
vertical wheels. The effect of the reaction of a column of
water had previously attracted the attention of Euler and
Bernoulli in 17^0% and a machine was proposed by Euler
in 1754, upon the principle of the steam wheel of Hero
of Syracuse. This machine was further improved by Man-
noury D'Hectot**, who constructed several in the neigh-
bourhood of Paris, with bent tubes, originally suggested
by Euler. The theor\' of the reaction of a column of
water against the sides or circumference of an upright
tube when allowed to flow through a hole or pipe fixed
in its base, has often been investigated by philosophers.
Daniel Bernoulli, in his Hydrodynamica in I788, and
John Bernoulli, in his Hydraulica, and in the St. Peters-
burg Transactions, proposed a very ingenious and elegant
method of determining the impulse of a column of fluid
fidling perpendicularly upon a plane surface infinitely cx-

* Ezp&iences ewt les Roues Hydrauliqucs a Axo vertical, Paris, 1840.

b Ibid., Turbines Mctz, 1838.

' Memoires de rAcademie de Berlin.

' Journal deB Mines^ 1813.

*XZT1 PREFACE TO THE

tended. The fonner considered the curve described by
every filament of fluid as a channel in which a body moves,
and which experiences at each point the action of a centri-
fugal and tangential force, which varies according to a
given law. He then calculated all these forces, and found
that the impulsion of a fluid against a horizontal plane is
equal to the weight of a column of fluid, whose base is
equal to the section of the fluid vein, and whose alti-
tude is equal to twice the height of the fall due to the ve-
locity of the fluid. The theory was afterwards very fully
verified by a series of experiments. Tlie question of
water flowing from a cylindrical or any other shaped
vessel was also treated by Madaurin in his Fluxions, pub-
lished in 1742. But the application of the principle of re-
action to produce motion in machines, is due to Segner %
professor of mathematic^^ at Gottingen, who first con-
structed the machme, commonly known as Barker's miU\
The celebrated Euler made this machine of Segner the
object of his investigation, in a paper published in the
Memoirs of the Academy of Berlin, in the years 17^0 and
17^1f and shewed that, in order to produce the greatest
eflFect, as weU from the pressure as from the centrifugal
force of the effluent water, it was necessary to curve the
horizontal arms or tubes of the machine, so that the aper-
tures should be in a line with the radius of the wheeL In
1754, he again turned his attention to the subject, and
constructed a machine with two systems of wheels, the
upper wheel or cylinder which received the water being
fiixed, and the lower one moveable and attached to a ver-
tical axis ; the water then flowed from the upper cylindrical
to the lower conical wheel, and from thence through twenty
small conical pipes fixed into its circumference, into the air,

* Exercitationes hydraulicsB, Qott 1 li^l.

^ It was called Segnersche Wassenad, in Geamianj.

THIRD EDITION.

^
¥

(d the machine to revolve *. Mather de la Coiir,
and 'W'arm^I^ proposed to introduce the column of water from
below at once into the horizontal arms ; and a patent for a
similar application of this principle has recently been taken
out in Scotland. As regards the effect of these machines,
Dpiniona are various ; Banks does not estimate it at above
one third. Waring concludes that the greatest effect will be
produced when the reloeitv of the orifice is half that of the
issuing water, and that this effect will be nearly the same
88 that of a well constructed undershot water wheel.

Mr. Ewart' estimates the maximum effect to he consider.
-ably greater than the same quantity of water applied to an
undershot wheel, but less than that which it would produce
if properly applied to an overshot wheel. In i82i M.
Burdiu invented a modification of Segner's machine, which
he termed turbine a riacfion*. It received the water in the
tipper part of a cylindrical drum, and allowed it to issue at
its base through a series of helical channels wound round
the outer surface of the drum, and from these through
three pyramidal pipes issued horizontally into the atmo-
(berc This machine was found to produce an effect of
65 to 75 per cent, of the power expended.

It was reserved, however, to M. Foumeyron to bring
tlh« turbine to its present perfection, and this he has ac-
iplished, after the most unremitting perseverance of

* Joamal de Bozier.
> TrsDnctioDa of the American Philosophical Society oF Pbiladelphio.

' On lie roeasure of Moving Force ; VoL H. Memoirs of the Literary
and FtiLlo«ophicBl Society of Manchester, 1 808.

* Aniulcsdes Mines, Tom. III., 1828. A more improved tnaehino of
Una doecnption erected by M. Burdiu at Pontgihuud in Fnuice, called a Tur-
Udo 1 £vacuittioD alternative, when submitted to the teat of llie iiictioD
lercr of IVony, produced an useful effect equal to 67 per cent, of the power
rapcnded, and pcrfonaed the same qusotity of work with ooe third of the
waMr formorly r«iuuod by n horizoatftl whe«l worked by the percusnoD

• • •

*XXY1U PREFACE TO THE

many years devoted to the subject. As before stated, the
turbine consists of a horizontal wheel with curved floats,
which are set in motion by the pressure of the water
issuing from the centre to the circumference, or vice versd,
and which, having performed its office, quits the floats hori-
zontally. But as the problem requires that the water
should enter the wheel without shocks, and leave it with-
out velocity, a peculiar kind of construction both of the
wheel and floats is necessary, and it is the practical deter-
mination of the curves, derived from experience alone,
which has led M. Foumeyron to the solution of the ques-
tion. Most of the turbines established by M. Foumey-
ron in France and Germany have been submitted to the
investigations of M. Morin, and the results have so far
exceeded the expectations of men of science, as must
eventually lead to a very considerable modification in hy-
draulic engines as first movers; and the report of the
Commissioners, Messrs. De Prony, Arago, Gambey, and
Savary, appointed by the Royal Academy of Sciences at
Paris, on the experiments of M. Morin, entirely adopts his
conclusions. M. Morin's experiments were made upon
the turbines erected at Moussay, Miilbach, Lupine, Inval,
and at St Blaise \

^ The first scries of experiments w»s made on the turbine of Moussay, m
1837. Tbo diameter of the wheel T«-as *085 metres, or 33^ inches; the
height of the fall was 7| metres, or 24 feet 8 inches; and the number of
turns made bv the wheel varied from 76 to 240 per minute, according to
the opening of the sluice ; the relation between the cffectiTe and theoretical
expenditure of T«*ater ii'as 0*910. The maximum effect corresponded to a
Telocity of 180 to 190 turns per minute^ and the useful effect mu 0*690,
or from 31 to 52} honaes' poTi-er; but at velocities of 140 and 230 turns
per minxite^ this illation varied only from 0*650 to 0*690 of the absolute
power expendeiU or a variation of only -jW^ thus showing that the effect
of the wheel was not altered materially by variations in its velocity.
The wheel also was not affected when working submerged in tail water.

The wheel at Miilbach of only 2 mecres^ or 6^ feet diameter, and a fidi
of 3| metit«k or about ll^ fwc^ with a volume expended of 2| cMc

THlttD EDITION. 'XlUt

The Sixth Essay relates to changing the velocity in
machinery hj' means of lathes, alternating pulleys, alter-

1, yielded a useful effect of 91 ioreea' power, or 78 per cent, of the
BnditDTe. In this cose tho number of revolutions of the ivbcel varied
1 00 per minute.

I Th* turbine of Lepine, with a fall of 2 metrca, or fij feel, and a velocity
Jfirami 60 to 100 revolutions per minute, yielded a power ofl2 horses,
f Filwily, the turbine at St. Blaixe, with a fall of 108 metres, or 354 feet,
■od n wheel nnder 22 inobes diameter, miulc 2300 turns iu a minute, and
traoainitied a force equal to 40 horses.
, M. Horin concJudes from bis experiments : —
I latly. That turbines are equally adapted to great as to small falls of

Sdly. That they are capable of tronsmitlmg an useful effect equal to

O-70 to 0-78 of the absolute power.
3dly. That their velocities may vary very considerably from tho mftxi-

cffpct, without differing very sensibly from it.
-Hhly. That they will work nearly as cffectuaUy when drowned to tho
itb of one or two metres, as when free.

SiUy, Tliat conBcquently, they will; make use of the whole of the fall
leu platod below the level of estreme low water,
ethly. That they may receive variable quantttiea of water without al-

tbe ratio of the power to the effect.
And if to thcee properties be added ^mplicity, economy, and compnct-
Ipgcther with the facility of communicating high velocities to iuB'
lery without the intervention of wheels or pulleys, the turbine, he
^ra, ought to rank among the best bydraiilic machinery' in use.
At SL Maur, near Paris, four turbines have been erected for the purpose
gnodiiig corn. Each turbine is 3 feet 2 inches in diameter, and 8 inches
ibickueris a""! makes 50 revolutions per minute, driving 10 pair of
ie« 3 feet * inches in diameter at the rate of 200 revolutions per minute,
each hirbliic doing tho work of 10 horses' power.

At Corbeil, about 16 miles from Paris, M, D'Arblay has recently re-
pkoi-d twu out of four vertical iron wheels upon the best principles, and re-
gihMWd tliem with two turbines of similar diameters as those at St. Maur,
and tluiT ore now working each IU pairs of stones with the greatest regu-
.hrttj and satisfaction.

For farthw information on this snhject, sec Experiences sur !es Roues a
rcrticn], par M. Anliur Morin, Metz, 1838. Also Versuche mit
DemoDUKlen Wosscrriiden von Herrn Wedding nnd Herni Carliczect,
BetOn, 183*.

*XXX PREFACE TO THE

nating cones, friction wheels, mules, and double speeds, as
applied to cotton spinning. The theory of mechanical
motions has been very little examined until recently. Some
of the early writers, such as Ramelli, Bessoni, Zonca, &c,
describe the various continuous or alternate motions used
in machines; but these motions were scarcely classi-
fied until 1794(, when Monge produced his Elements of
Machines for the use of the Polytechnic SchooL It was
afterwards treated by Hachette% Lanz and Betancourt\
Ampere' and Borgnis*, Whewell' and Willis'.

In the Trait6 de M6canique of Borgnis, mechanical
organs are divided into six classes. — 1st Receptors, under
which are classed every description of machine moved by
the power of animals. — ^Sdly. Hydraulic receptors, such as
vertical and horizontal wheels, machinery moved by the
reaction or pressure of water, or by heat, vapour, or wind.

Under the secondand third classes, or communicators and
modifiers, are machines for transmitting and modifying mo-
tion, such as toothed wheels, eccentrics, indinedor curvilinear
surfaces, chains, levers, pulleys, wheel eccentrics, screws,
cams, &c«, together with the machines for producing con-
tinuous, or variable, or alternate motions. The fourth daas
comprehends simple supports for maintaining vertical or
horizontal axles, and rotative supports for wiMTifaMTirng
motions of translation m one or more directions ; and under
the third class in this division are comprehended toob.
The fifth order relates to r^rulalcNrs, such as fly wheels,
governors counter weights, horological scapements, ec-
centric wheels, curvilinear motions, friction levers, and

^ Composauon dc« MMhines. ISOS.

^ Kmu Air k Philowpbie des ScMnee^ ISas.

* Tndu^ dc MMuq[Q««

THIRD EDITION. *XXX1

nScal pulleys and wheels, (alluded to by Buchanan.)
The sixth, or last class, termed operators, comprehends
'ery kind of machine for blowing air, for agitating
quids, or semifluids, or solids j for compressing substances
r means of rollers and presses, or for stretching or ex-
nding metals : again, for operating by friction, such as
nding, polishing and filing. Fourth sub-division,
by shocks, such as hammers, stampers, pile en-
, wedges, &c. And under the last or fifth sub-divi-
tioa, come the operators by separation, such as rakes,
scribbling and carding machines, knives, chisels, scrapers
and boring tools.
^H^ A new work, however, by Professor Willis, has just
^Hppeared, the object of which, to use his own words,
^^phns been to form a system that would embrace all the
^VuBmeDtary combinations of mechanism, and at the same
uptime admit of a mathematical investigation of the laws by
which their modifications of motion are governed. 1 have
therefore, says he, confined myself to the elements of pure me-
chanism, that is, to those contrivances by which motion is
commonicated purely by connexion of parts, without re-
qniring the essential intermixture of dynamical effects.
Instead of considering a machine to be an instrument by
means of which we may change the direction and velocity
of a git>en moHon, I have treated it as an instrument by
i of which we may produce any relations of motion
reen two jneces."'
The system adopted by Professor Willis is condensed,
I a ^'nopticai table of the elementary combinations of pure
xhouism, into five divisions and three sub-divisions:
The first class comprehends motion by rolling contact,
toothed wheeb, annular wheels, racks, sectors,
face gearing, hook gearing, and wheels in general for pro>
' Willis's Principlea of Meclianism, 1841.

*XXXU PB£FAC£ TO THE

ducing constant or variable velocities, or a combination of
both.

The second division includes motion produced by sliding
contact, such as cones, screws, and worms, pin and slit levers,
spiral, and other curved surfaces, under the different cir-
cumstances of constant or variable motion.

The third division shews how the same motion can be
produced by wrapping connectors, such as guide pulleys,
gearing chains, curvilinear pulleys, and fusees.

The fourth division includes the motion produceable by
link work, such as cranks, joints, ratchet wheels, and inter-
mittent link work.

And the fifth or last division includes reduplication by
means of tackle of ropes, either parallel or unparalleL

The aggregate combinations and velocities, and adjust-
ments of machinery, are treated with that ingenuity, pre-
cision, and order, which might be expected finom the au-
thor. As regards the practical application of the various
motions used in machinery, we need only adduce the
early inventions introduced into the texdle fabrics by Ark-
wright and Cromptim, Wyatt, Hargreaves, of Watt, of
Boulton, of Huddart, and others who have illustrated the
history of mechanical inventions, not to mention invidi-
ously inventors and men of science who in modem times
have carrioil the art to the highest perfi^^tion.

The Sewuth Kssav treats of the framing of mill work
and small maohim^r}% acconling to the principles of Robi-
sim and T>e<l^ Jd«

The obj^vt \4r framiugs in mill work» is to support and
uuuutain th^^ diffV^n^nt i>art$ i\f machines in their proper
ainl n^lative vU^taiH^ $a^ that all the wheels shall work as
$m\H>ihly as [¥^Wt\ and witUvHit shirks or vihraiicHis ; for
this }HirtH^8?e it is mn^Nssan that the framing be made in con-
f\>rmitY to the strn^t^xi^t ruW^j^ \rf ;f^*ieiKv ; that is* with refer-

THIRD EDITION'. 'MXIIl

QDce to the composition and resolution of forces, that the
resultants of these forces should he represented by ties or
struts ; in short, that all pressures should be so distributed
and resisted as to maintain a perfect state of equilibrium
throughout. In obtaining a knowledge of these principles,
it is necessary that we understand the properties of the
materials with which we have to deal j their strength and
stress in all positions, their durability, and their powers to
resist decay. These properties will be found in our table
of the strength of materials, and it is on the judicious distri-
bation of these materials that much, if not the whole of the
art of the mechanician dejiends. In all cases of tension, to
use wrought iron, and in those of compression, cast iron ;
to observe the proper forms best suited to the pressure or
tension they are to undcrfro, and to avoid as much as pos-
shle the use of framing in all heavy machinery, availing
lives of masses of materials, such as stone, brick,
icrete, or sand, in all cases where vibrations or shocks
to be resisted. For although cast iron, as a material,
combines the advantages of stiffness, strength, and dura-
biH^> and the facility of its being moulded into every pos-
sible form suited to the framing of small raachinerj-, yet it
is occasionally subject to break by unequal contraction in
the cooling, and by blows or changes of temperature.
Framing of wrought iron ia, therefore, much used in marine
steam engines.

The Eighth Essay, although not in the original edition
of Buchanan's Essavs, treats of the geometrical and prac-
tical methods for finding the centres of gravity of miU
lis, illustrated by examples of two, three, or four wheels
red upon the same shaft. This subject has been so
iplv illustrated by Dr. .lamicson, but particularly in his
Mechnnics for Practical Men, that further comment is

Irecssary.
series of tables of square and cube numbers and roots,

peat r
^■leels

•XXXIV PREFACE TO THE

taken from Hutton's Course of Mathematics, closes the
whole of Buchanan's work.

In the precedmg ohservations we have confined our at-
tention to a hrief outline of the past and present state of
our knowledge of the subjects treated by the Essays, and
an imperfect review of the labours of those to whom we
are so deeply indebted for the knowledge we possess of
mechanical science. The labours of Buchanan are con-
fined to the development of a few elementary principles
connected with practical mechanics, excellent in them-
selves, but defective both in the extent and arrangement
necessary to a complete system of mechanics.

The science of mechanics, which treats of the equilibrium
and motion of solid or fluid bodies, and which, under its
various divisions of statics, hydrostatics, dynamics, and
hydrodynamics, comprehends the theory of action and re-
action. Practical or technical mechanics, on the con-
trary, treats of forces as realities, and machines as material
objects, capable of transmitting, regulating, or modifying
motion. It also depends on a multitude of facts com-
bined together, and establishes, by way of experiment,
values to every element subservient to industry. Further-
more, it determines the value of animate and inanimate force,
such as the force of men and animals ; the force of gravity,
such as weight, water, or other fluids ; of elastic fluids, such
as wind, steam, gas, &c., all of which forces are made sens-
ible through the agency of machinery. By machinery we
understand an assemblage of materials, particles or parts
susceptible of receiving, communicating, or modifying
motion. A machine may consist of a simple or compound
lever, or assemblage of levers, revolving on a centre, such
as band wheels or rollers, or any of the mechanical powers ;
or it may be divided into three parts, — the parts which re-
ceive, the parts which transmit, and the parts which com-
municate or perform the woA : aU these motions are

THIRD EDITION.

•xxxv

I

I

by certain reaistances which we terra passive, such
as inertia and friction, but which deduct or abstract
from the absolute force in proportion to the perfection of
and mode of applying the machine. Machines may be
employed for displacing solid or fluid masses, for changing
the forms of ductile and compressible materials by pres-
sure, for separating masses of solid materials by friction,
for producing changes of volume in solids by percussion,
for separating solids into fragments by the same force, for
TniTcing solids together by penetration, for separating fila-
mentous substances from other extraneous substances with
which they are interlaced, and for rearranging and inter-

I lacing them.
I Whatever be the nature of the machine, it ought to be
%o combined that its useful effect be as great as possible ;
that it should be as simple in its construction as possible ;
that its parts shoidd combine strength, stifiness, lightness,
oniformity of action, and he as free as the nature of the re-
Mfltance will permit from passive resistance ; that it should act
without shocks or sudden changes of motion j and that the
comnmoication between the power and resistance should
be as simultaneous as possible. These important prin-
ciples exact an intimate knowledge of the properties of
materials, the modes of transmitting motion in all its
varieties, of contact by means of the teeth, cams, and other
mured surfaces, by bands and pulleys, or by direct or
oblique pressure. Machines are the implements of manu-
fiujture, a word which applies to every product of art
which is made by machinery, and with little or no aid
from human labour. It forms a separate section, or rather
a science, of automatic labour. It is the automatic science
which bos raised our country to its present elevated posi-
tion in the world, as displayed in its cotton, silk, woollen,
and flax manufactures ; in its multitudinous and beautiful

a 2

•XXXVl PREFACE TO THE

machines for shortening, multiplying, and even dispensing
with the labour of man, evinced in the construction of au-
tomatic machines for creating the instruments of power
whereby the elements are chained to perform their un-
remitting toil, — whereby the powers of wind and water, and
steam and gas are rendered subservient to our uses, and
ere long, let us hope, that mysterious power of electric
magnetism, which seems to govern alL What have we
not witnessed during the present century ? If we turn to
the triumphs of steam, we find that, whereas the duty of
the pumping engines in Cornwall in the year 1808 was
barely equal to 20 millions of pounds of water raised one
foot high by a bushel of coals ; in 1835, the duty per-
formed by Mr. Austen's engines at the Fowey Consols and
Lanescot mines, with an 80 inch cylinder, was upwards of
125 millions of pounds of water lifted by one bushel of
coals weighing 94 lbs., and this has been confirmed more
recently by the valuable experiments of Wicksteed \ Thus
carrying out the ideas and principles of the great Watt, so
fully detailed in his patent for 178^ and in the works of
Robison\ Tredgold', and Farey*. If we look to the
marvels which have been effected in locomotion*, both on
sea and land, no longer subject to the uncertainty of the
elements, the untiring machine impels the mighty fabric
against the wind and waves, annihilating almost time and
distance between the New and Old Worlds, while by its
stupendous energies, and the art of the engineer, dis-
tances ae no longer measured by space.

* Expenmental Enquiry concenimg the Rdatiye Power and Useful ESect
produced by the Cornish and Boulton and Watt Pumping Engine and
CyUudrical Waggon-bead Boi]«s. 1841.

^ Robison^ Article Steam Engine.

« Treilgold on the Steam Engine, 2 toIs. Weale, 1838-40.

"^ Farey H Treatise on the Steam Engine*

* Comte de Pkanbour s Practical Treatise cm Locomolhve K^es, 1840.

THIRD EDITION. 'xXXvil

Let us reckon upon the future, eays M. Arago, iu his
istorical eloge~of James Watt
' A time will come when the science of destruction
shall bend before the arts of peace ; when the genius which
multiplies our powers, which creates new products, which
diffuses comfort and happiness among the great mass of
people, shall occupy, in the general estimation of mankind,
that rank which reason and common sense now assign to it.
■ *' Then Watt will appem- before the grand jury of the in-
Btabitants of the two worlds. Every one will behold him,
with the help of his steam engine, penetrating in a few
weeks into the bowels of the earth, to depths which, before
his time, could not have been reached without an age of
the most toilsome labour, excavating vast mines, clearing
them in a few minutes of the immense volume of water
which daily inundates them, and extracting from a virgin
Boil the inexhaustible mineral treasures which nature has
deposited there. Combining delicacy with power. Watt
will twist, with equal success, the huge ropes of the gigantic
cable by which the man-of-war rides at anchor in the
midst of the raging ocean, and the microscopic filaments
of the aerial gauze and lace. A few strokes of the same
engine wiU bring vast swamps into cultivation, and fertile
countries will also thus be spared the periodical returns of
deadly pestilential fevers, caused in those places by the
beat of the summer sim.

" The great mechanical powers which had formerly to be
songbt for in mountainous districts, at the foot of rapid
cascades, will, thanks to Watt's invention, readily and
easily arise in the midst of towns, on any story of a house.
The extent of these powers will varj' at the will of the me-
chanician ; it will no longer deiMiud, as heretofore, on the
moit inconstant of natural causes, on atmospherical in-
fluences

" Installed in ships, the steam engine will exercise a power

aS

^XXXVm PREFACE TO THE

a ImnilredBald greater than the triple and quadruple ranks

of rovers and bjr the hdp of a few bushels of coal,

waa win Tanqnish the eLements ; he will play with calms
and cootrarr winds and storms.

" Lasdv : The steam engine drawing in its train thou-
sands of traTellerSy will ran on railroads with far greater
speed than the smiflest raoe-horse."*

** And, in condosion, let ns quote the opinion of Sir John
HorscheL On the importance of practical mechanics (he
sap^ in his admirable treatise on the Study of Natural Phi-
losophy,) *' Practical mechanics is in the most preeminent
sense, a scientiJU: art^ and it may be truly asserted, that
ahnosi all the great combinations of modern mechanism,
and many of its refinements and nicer improvements, are
creations of pore intdlect, grounding its exertion upon a
moderate number of elementary propositions in theoretical
mechanics and geometry. On this head we might dwell
long, and find ample matter both for reflection and wonder.
But it would require not volumes merely, but libraries, to
cnumarale and describe the prodigies of ingenuity which
have been lavished on every thing connected with machinery
and engineering. By these we are enabled to diffuse over
the whole earth the productions of any part of it, to fill
eveiy comer of it with miracles of art and labour in ex-
change for its peculiar commodities ; and to concentrate
around us, in our dwellings, apparel, and utensils, the skill
and workmanship not of a few expert individuals, but of all
who, in the present and past generations, have contributed
their improvements to the processes of our manufactures/' ^

• Tlje annals of racing record sereiml wonderful feats performed by race-
bone^— Eclipse once ran 2 mOes in 2 minntes, and on another occasion
4 vBw in « minntes and 2 flecrads. Fljing Childera ran over the New-
Miikec covrae. 7420 yards, in 7| minutes. Greyhounds have been known
ti» m aeariy as &8t as raoe-horses.

* IVe&MBarr DisconrBe on the Study of Natoial Philosophy, pages 63
awlCf.

THIRD EDITION.

ON TOOLS.

I

The subject of tools has been so amply illustrated bv
Mr. James Nasmyth, in the Appendix, that little remains
to be added. By tools, we understand instruments em-
ployed in the manual arts for facilitating mechanical ope-
rations by means of hammers, pmicbes, cbisels, axes, adzes,
jilanes, saws, driUs, files, &c., by means of percussion,
penetration, separation and abrasion of the substances ope-
rated upon ; for all of which operations various motions
are required to be given cither to the tool or to the work.
In handicraft work the tool receives motion, but in self-
acting or automatic tools, motion may be given to either.
In the case of the turning lathe, the tool remains fixed,
and the object moves. In that of the planing machine,
the tool may remain fixed, or be made to move accoi-ding
to the duty required to be performed. In almost all the
other machines, such as the slotting, the key-grooving,
the punching, the drilling, the nut-cutting, the teeth of
wheels cutting, the boring, the screw-cutting machines, the
tools receive motion. In the screw, bolt, and nut ma-
chines the tool is either moveable or fixed. The use of
handicraft tools is coeval with the earliest periods of
antiquity, and the recent researches of modem travellers
have proved the ancients to have beeu acquainted with
almost all the tools now in use'. The potter's wheel, the
axe, the chisel, the saw, &c, attest the perfection to which
the mechanical arts were carried by the Greeks and Ro-
mans, and subsequently in the arts of turning exhibited
by the Dondi family, in the construction of their clocks
and machines for spinning silk*", in the middle of the

* Muiners and Customs of the Ancient Egyptians, by Sir Gardner Wil-
kmaon, F.R.S.. 1837.
^ UiHoire dee Sciences Matliomatiques, par Guillaumo Libri, Vol. I., 1 838.

a 4.

•"xl PREFACE TO THE

ISth century in Italy, and afterwards by Bessoni% De la
Hire^ De la Condamine% Grand Jean^ Plumier, and
Morin*. The three plates of Bessoni shew the different
modes of turning and cutting screws of all sorts of fancy
work. De la Hire shews how all sorts of polygons may
be made by the lathe, and Condamine shews how a lathe
may be made to turn all sorts of irregular figures by means
of tracers moved over the surface of models and sculptures,
medals, &C, and this is perhaps the first idea of the
machine called the Tour a Portrait

The work of Plumier enters most extensively into the art
of turning, for he shews the construction of the lathe and
its difierent parts, the art of making, hardening, tempering,
and sharpening tools, the different kinds of motions which
may be given to the lathe by means of wheels, excentrics,
and models, and the difierent inventions relative to works of
art which have been performed by the lathe, among which
mav be mentioned the moveable or slide rest In the com-
mon rest which supports the tool, the idea of fixing the
tool and pushing it in the direction of the parallel bed of
the lathe, so as to cause the tool to traverse the work pa-
rallel to it, must have been obvious, and as this could have
been easily effected by means of the screw and handle, it
required little ingenuity to carry out the idea to its fullest
extent, by constructing a rest to allow of the slide traversing
the horizontal or vertical plane in any direction. The
machine described by Plumier is neither more nor less
than the slide-rest and planing machine combined: it
consists of two parallel bars of wood or iron connected to-
gether at both extrt^mities by bolts or keys of sufficient

• TKecUnim Mftchinarask 15S!2«

^ MnehiiM^ AppnMiTf«$ p«r rAcmdemie* 1719,

* IhKk 1733.

^ MiichiiM>« Ap|iiottTf«« )i«r rAc«de«ue« 1733.

C«]

Lint

or
for
by
bai

THIRD EurrioM. •ili

vridth to admit of the article required to be plaiied ; a
moveable frame being placed between the two bars, and
motion being given to it by a long cylindrical thread, is
capable of giving motion to any tool which may be put
into the sliding frame, and consequently either causing the
rew, by means of a handle at each end of it, to push or
aw the point or cutting edge of the tool either way. If
also motion be given to the tool by means of guides upwards
or downwards, it is evident that any kind of reticulated
form can be given to the work, as in the machine described
by Plumier, which was intended for ornamenting the
handles of knives, and which is called by Plumier, Machine
,d mnnvfie de Coutemi d" Angleterre', from its ha^ong been
English invention. The Machine d Conneler de-
scribed by Bergeron '', a mode of grooving columns, is ])ro-
bably derived from the same sowce, from its resemblance
to the English machine. We have given a plate and de-
acription of Nicolas Eorq's machine in Plate 45 of the
present work, and we have a drawing of a similar machine
which was used in (Jermany many years back. The origin
of the planing mac-hine, in more recent times, is said to
hare arisen from the grooving or fluting of the drawing roU-
ere used in cotton machines shortly after the introduction
of Arkwright's inventions. The patent of Sir Samuel
Bentham" in 1793, for various new methods for working
WfKxl, metal, and other materials, certainly contemplates
the working of tools similarly to the tools employed in the
planing macliine. The patent comprehends giving all
sorts of motion to tools, and the patent of Joseph Bramah'*,
taken out in 1 802, was for machinery for producing

■ See pngce 1.55, 15B, anil Pktes 5+, 5.5, 5fl, Plumier I'Art de Toumer.
iFuia, 1754.

> Mnroel du Toumeur, Paris, 1816.
' Repcrtorj' of Arte, 1793. Vol. X.
• Bq>«irMirf of Aftt, 1802.

a 5

♦xlii PREFACE TO THE

straight, parallel, and smooth surfaces and other materials
requiring truth, in a manner more expeditious and perfect
than can he performed hy the use of axes, screws, planes,
and other cutting instruments used hy hand in the usual way.

Billingshy% of Birkenshaw, took out a patent in 1802,
for horing cylinders in a vertical position, although hori-
zontal machines had their advantages. The horing of
large cylinders hy horizontal machines had long heen
practised hy Smeaton, Wilkinson, Walker, Darhy, and
Boulton and Watt, and at Butterley and other great iron
works ; hut it was only within the last few years that the
vertical horing machines came into use.

As respects the introduction of the first planing ma-
chines which have heen used during the present century,
opinions are at variance. Messrs. Fox, of Derhy, the
eminent tool makers, state that the first machine em-
]doyed for this purpose was constructed hy Mr. Fox,
senior, in the year 1891, for the purpose of planing the
wrought and cast iron hars used in the lace machines.
The machine was capaUe of planing an article 10 feet 6
inches in length, 22 inches in width, and 12 inches in
depth ; others give the credit of the invention to Man-
chester, and we ourselves put in our claim for constructing
a planing machine with a moveable bed, urged by an end-
less screw and rack, and furnished with a revolving tool,
so early as 1820, having several years previously employed
the principle for grooving and planing parallel bars.

Mr. Bramah, in 1811, employed the revolving cutter to
[dano iron. Mr. Clement ^ states that he made a planing
machine, for planing the sides of weaving looms and the
triangular bars of lathes, previously to 1 820. He afler-
wants ixmstruotoit a beautiful machine for planing large
and small wv>rk with the sn^^eatest accuracv. The bed

« RepertiMnr of Aft^ Vol. lU 1^0^

^ ieih tad i$di Y«4wuM« ciT a» Ti«MactiQ«s ^

^^ ant
Btnl

THIRD EDITION. •xliu

■moTed on rollers, and the tools cut both ways. The beau-
tiful work executed by this tool, for Mr. Babbage's'calcu-
Uting machine, evinces the perfection of its performance.
It is thus by the aid of automatic tools that we are enabled
to produce the greatest precision and identity of parts in
machinery, which could never before be attained by ma-
chines made by hand labour ; and it is hoped that, ere
long, the cbisel, the file, and the grindstone will be
banished from the factorj-, and that nicety of parts and
uniformity and silence of action, blended with the science
of construction, will eventually supersede the expensive
and imperfect construction of the handicraft system.
We might enlarge upon this subject, by detailing the
iture and properties of the materials required for tools ;
the forging, hardening, and tempering of them ; the velo-
cities at which they should be made to move through the
difierent materials, such as woimI, iron, brass, copper, and
tin. We might ^ve the principles of the action of the
different machines employed to produce different effects ' ;
but we have exceeded our limits, and it only remains to
express our great obligations to the several gentlemen who
have so liberally assisted us on the present occasion. To
Professor Willis, for bis article on the Teeth of Wheels.
To Mr. James Nasmyth, for his Paper on Tools, and his
numerous and beautiful drawings of the tools which bear
the name of Nasmyth and Gaskell. To the late lamented
Mr. I-'rancis Bramah, we owe the original drawing of
the first slide rest of his father, in 1794, the work of
the late Mr. Maudslay, and it is yet in use ; and also
for the drawing of the lathe for turning spheres. To
Mr. FairbaJm, for his plate riveting and punching machine,
and for his advice and assistance on several occasions. To
Mr. Wliiiworlh, for the information we have derived fr«m

' EipcriineiiM of M. Morin, on tie Measures of the Dynamic EtTeeta of
MinniU) ttnd Animal Power, and on Machines in general.

♦xliv PREFACE TO THE THIRD EDITION.

his various pamphlets on plane metallic surfaces, and on the
proper mode of preparing them ;. likewise, on an miiform sys-
tem of screw threads '. To Messrs. Fox, for their screw-
cutting machine, and other information. To Messrs.
Benjamin Hick and Son, of Bolton, for the liberal assist-
ance of the drawings for the plates which bear their name ;
and we take this opportunity of noticing their ingenious
machine for cutting the teeth of the largest sized wheel
used in mill-work, and their mandril for holding rings ;
their steel belts, as a substitute for leather bands, are used
very successfully. To Mr. Francis Lewis, of Manchester,
we are equally indebted for the drawings of the different
machines, placed by that gentleman at our disposal. To
Messrs. Maudslay and Field, for the liberal present of the
drawing of their self-acting punching machine, by which
accuracy is insured in the heretofore neglected art of boiler-
making i it is one among the proofe of the high state of
excellence to which those gentlemen have brought the me-
chanical arts in this country. A table of references, and an
ample description of the different tools, by our assistant,
Mr. George Pinchbeck, will, we trust, explain the different
details. With respect to the plates, it is suflRcient to state,
that they are engraved by Lowry, a name too well known
to need further comment. The liberality with which the
whole has been got up by its spirited publisher, will, it is
trusted, be acceptable to the public.

■ On an Uniform System of Screw Threads. 8vo, 1841.

We are indebted to the late Thomas Tredgold for whatever
is known of the life of Buchanan. It was furnished by his
friends, and though brief, the life of a man of genius is
always interesting.

Robertson Buchanan was bom on the I4th of July^
1769, at Glasgow, and was connected by birth with some
of its principal citizens. His father was nephew to Neil
Buchanan, who, in the year I7G8, represented Glasgow
m Parliament j and his mother was daughter of Arthur
Robertson, who for many years was chamberlain of that
city. Buchanan lost his mother at his birth, and his
Enther when he was only fifteen. His father had not been
fortunate in business, and the son was left unprovided for,
but lie had already shewn some talent for drawing and me-
chanics, whifh induced his maternal uncle to place him
with a house-carpenter at Glasgow. The genius of Bu-
chanan sought its native field in a short time, for we after-
wards find him working with a millwright, and subsequently
crossing the border for London. After a short time he
quitted the metropolis, returned to Glasgow, and com-
menced business there as a millwright; in the year 1791
he gave it up to take the management of the new cotton-mill
then building at Rothesay in the Isle of Bute. There
he invented his pump which is not liable to choke, and
for which he obtained a patent in the year 1796. In the
same year he wTOte some papers, which were published
Repertory of Arts and Manufactures j — one on

•xlvi LIPE OF BUCHANAN.

the improvement of cattle mills, another on preventing
carding machines from injuring the health of those em-
ployed to attend on them.

He left Bute in the year 1801, much impaired in health
by the anxiety of a responsible situation in a losing busi-
ness, and returned to London, with a view of deriving
some benefit from the pump he had invented ; but in this
he never succeeded. He was introduced, however, to
Count Rumford and Professor Pictet, by whom his atten-
tion was directed to the heating of rooms, and in the
year I8O7, he published an "Essay on Warming Build-
ings by Steam.** He had previously been engaged in
preparing the " Essay on the Teeth of Wheels,** but when
a considerable part of it was printed ofi^, an unfortunate oc-
currence to the printer and publisher delayed the publica-
tion until the year 1808. In the year 1810 he published
his work on heating buildings in an improved form, with
the title of "Practical and Descriptive Essays on the
Economy of Fuel and Management of Heat.** In the
year 1814 appeared the Six Essays on Mill Work, which,
with that on the Teeth of Wheels, constitute the present
work.

In the year 181 6 he published a practical essay on pro-
pelling vessels by steam, a work fiill of new views and
principles in that most important art, and the commence-
ment of a new era of civilization in the annals of society at
large. He also contributed the articles " Cotton-spinning **
and " Arkwright ** to the Edinburgh Encyclopsedia, besides
several papers on less important subjects.

He died in the 47th year of his age, at Creech St. Mi-
chael, in Somersetshire, on the 22d of July, I8I6.

He was a man of amiable character, with a strong
sense of religious and moral duty, and was greatly re-
spected by all that knew him. His knowledge in mecha-
nics was very extensive. He was a close and accu-

LIFE OF BUCHANAN. •xlvU

rate observer, extremely assiduous in collecting every fact
or experiment which came under his notice, and he was
unquestionably one of the few practical men who have
shewn inclination, or sought leisure, to reason on &cta in
general with accuracy and judgment, aad always with a
view of rendering the information thus acquired, an avail-
able source for unforeseen emergencies. Buchaaan was
happy in the choice of popular subjects, and he fully com-
pensates for want of system in haadling them by the va-
riety and utility of particulars no less interesting than
abundant^ whether learned from his predecessors, or de-
rived from contemporaries.

ANALYTICAL TABLE

or THS

CONTENTS OF THE WHOLE WORK.

Art. Pig»

OSNSRAL PbBFACB y

Preface to third Edition vii

Life of Buchanan *xly

ESSAY I.

On the teeth of wheels, comprehending pnnciples of their
i^plication in practice to mill-work and other machinery.

General definitions of wheels and pinions, trundles and teeth,
cogs, leaves and staves 1-4 1

Of the line of centres, and the proportionate radii . . 6, 7 2

Of proportional drdes or pitch lines, and real radii • . 8, 9 B

CHAPTER I.

Of the principles of the confignration of the teeth of wheels — 4

Of the proper formation of the teeth of wheels . . .10 4

Of one wheel conducting another as if they simply touched,

or their pitch lines have in every part of their revolution

equal velocities II 5

Notes illustrative of peculiar cases, and of the fundamental

proposition — 6

Demonstration that the pitch lines have in corresponding

places, equal velocities 12 6

Definitions that the epicycloid ^ves the property to wheels

whose pitch lines shall have in corresponding places equal

velocities 13 7

The generating circle of the epicycloid . . . .14 7

Of the exterior epicycloid 15 8

Of the interior epicycloid 15 8

Girollaries. — 1. Determining the points through which to

trace tbe ^ncycloidal curve 10 8

XIV CONTENTS.

Alt.

2. The generatbg circle revolTiog within the dicle of its
base 17 9

d. The generating circle touching the drcomference of its

base 18 10

4. Of the generation of the figure by means of three circles

moTeable round their centres only 19 11

5. Mechanical methods of efiecting this by three circles .20 12

6. Mechanical methods by two circles for determining the
best figure which can be given to the teeth of wheels,

when the pinion shall be a trundle composed of staves . 21 14

Properties of the epicycloid both curious and scientific .22 16

Rules for finding the lengths of epicycloidal curves, and the

areas they inclose 28 16

Halley's rule for the area of all cycloids and epicycloids .23 17

CHAPTER II.

Of the application of the principles of the configuration of

the teeth of wheels 24 18

Practical explanation of the epicycloid curve . • .24 18

Section I. — Of spur gear 25 18

Of the wheel and trundle 26 18

To find the figure of the teeth when the staves are indefi-
nitely small 27 19

To find the figures of the teeth of the wheel, when the staves

of the trundle are cylinders of a finite diameter . . 28 21

To describe the teeth ^f a wheel for a trundle by means of

circular arcs 29 23

Of the wheel and pinion. — To find thefi^re of the teeth and
leaves of a wheel and pinion^ when that part of the teeth
and leaveSy which lies within their respective proportional
circles are straight lines directed to the centres of these

circles 80 24

Remarks. — On friction 82 27

On what it depends 83 28

Friction of metal teeth ZZ 29

Friction at the line of centres, and receding from it . .83 30

Rule more general of easier application for describing teeth

than that of Camus 34 30

Demonstration that a pinion of 10 leaves may be moved uni-
formly by a wheel of 209 teeth 36 32

CONTENTS.

A tnindle with less than 8 staves cannot he moTed uniformly

hy a wheel with any number of t«etb whatever
The qticvcloid necessary on conductors only, whether wboel
or pinion .........

Stares prefcr«blo to teeth, if but few in the pinion

Advantage of a small trundle over a pinion

Wheels of cast iron .......

In sniail trundles of cast iron, toeth are preferable, lim-iiig

their a4;ling porta of the figure of a. stave
Smple method of describing teeth to resemhle staves for the
conducted wheel or pinion ......

Method of calculating the real radius when the wheel is the

conductor

^^ Hetbod of calculating the smallest real radius which a wheel

^^L adapted to a tnmdie can have

^^Blethod of calculating the smallest real radius which can be
^H^ given to a wheel adapted to the leaf of a pinion
■ Advantages of long teeth over short ones shown in case of
Asctnre and also in friction ......

BnJe to determine [he length of the teeth of wheels .
Obaemitions on the preceding rule .....

Of the internal pinion, and the cases in which it may be
adopted with advantage ......

IQuatmtian tliat it has less friction than the external one
Showing also that upon this principle bevelled wheels have
less friction than external spur wheels ....

Of the rack and pinion .......

Should be made upon the principles of spur gear
^^■jKgDfe representing the teeth of a rack and pinion fonned
^^B for cases in which great weight is attached to the rock
^^Hcorrect eonatructiou for the rack and pinion
^^^jf cthod of giving durability to the teeth of the rack, and lind-
iug the real nidiiis of the pinion .....

Of the farm of the face of tlie teeth of the rack when it im-
pels the pinion

Sbctioh n. — Of hovel gear

]t> action represented by cones rolling on the surfuce of each

other

K^rmnu illustrating their motions

Kzpbnstions of these revolutious of cones ....

XVI CONTENTS.

Alt.

How the epicycloid, which giTes the teeth of bevel gear, is

generated 56 5B

Illustration of the spherical epicycloid . ... 56 54

Practical method of laying down the lines necessary to the

right construction of boTol gear 57 55

To determine, firstly, the diameter at the pitch line of the
wheel; and, secondly, the length and breadth of the

teeth 57 56

Mode of drawing the section of the pinion ... 57 57

Specific definitions and illustrations for moving one another

uniformly 58 57

1st, When the wheel drives the pinion, having the acting
faces of the teeth spherical epicycloids ; 2dly, when the
teeth of the pinion are staves, or are formed to act as staves ;
ddly, when the face of the tooth is a spherical involute of

a circle 58 58

A new and general method of describing these curves for
wheels ; spur wheels, racks, &c., being only particular

cases of its application 59 58

Developement of the cone teeth upon a plane ... 60 58

Patterns for drawing these teeth 61 59

Rules for regulating the limit of the pitch, and breadth of

the teeth 61 60

CHAPTER III.

Carnprisififf a series of articks having direct application to
mill'tDork^ and the teeth of wheels, — Professor Robison's
mode of forming the teeth of spur wheels ... 62 62

Properties of involute teeth in wheel-work ... 63 63

Supplementary observations on the alignments of Professor

Robison, Camus, and Dr. Young 64 64

Dr. Young's remarks on the friction of bevel gear wheels,

extracted from the 1st vol. of his Natural Philosophy . 65 65

Dr. Young's opinion, by letter to Buchanan, on friction, and
the forms best suited for teeth

Supplementary definitions of geometrical figures

Geometrical constructions of plane figures .

Definitions of geometrical figures resumed

Definitions of some terms in science and mechanics

66-71

66

72-82

71

82-85

73

85, 86

75

87-91

75

mustmtions of the term power as used in mathematics ; of
force — momentum ,..,., 02-97

^^H Of moclianic&l power ... .93

^^■IhstiDctioa between the nicsaure of power and the meHsoro

^^P of effect OS

^^GnnmeroUon of works on mi!!- work and mecliaiiics, useful in

fitud\iag Buchanan's kboufs 99

CHAPTER IV.

i p»ctic&l inquir)' respecting the Btrength and durability of
e teeth of wheels used in mill-work ....

k On the Htrenglh of the teeth of wheels in relation to the re-
islnnce they have to overcome .....

Peneral observobons on the wbecl-work of mills

e mcaanrc below which the diameter of wheels ought not
f to be reduced ........

piEthods hy wliich a saving of power has been obtained
e praedcot limits of a fine pitch, ami also of breadth
indpleA of proportioning the teeth of wheels. — PBoroai-
Tio» I. Tie ttrenytk of any piece of tivJ>ery or metal, wAose
tertian it a reeiangle, i* in direet proportion to the hreadth,
omd at the tquare of the depth .....

The strength of tho teeth of wheels moving at the some velo-
city and under the same circumstances, is directly in pro-
portion to their breadth, and as the square of the thick-

ne«

PaoPOSiTioir n.—If any force be applied latercdly to a lever
or beam, the tirett upon any place, is directly as the force

and ittditlancc from that place

pKorosiTiON III. — The pitch being the saTne, the ttrets it ij
e neloeitt/
bivcrval opphcation of the propoBition by reducing tlie fii'st

o the same standard . . . 107

0 be horses" power IO7

d by the greatest number of horses' power ne-
» perform the work of assigned trains of ma-
chiuciy
ScimtiBc Hid practical values of horses' power illustrated .
Ueiagutier'B measure ; Smcalon's measure ; James Watt's

XVlll CONTENTS.

Alt.

Number of spindles of cotton t^ist driven by one bone . 109 89

Ditto of cotton mule yam 110 89

Ditto of flax yam Ill 89

Comparison of different estimates of the force of moving

powers . • • • • • • .112 89

Immediate force of men and horses vrithout deduction for

friction 113 89

Performance of men and horses by machines . . .114 90

Practical methods of calculating this force . . . .115 90

Mechanical power 116 90

Exposition of this power 116 90

Demonstration of the greatest advantage from men and
horses when moving with half the velocity they would
continue at work were the effective resistance of the ma-
chine nothing 117 90

Demonstration of that portion of the mechanical power which

is efficient in impelling the machine . . . .117 91

Practical illustrations of the strength of men and horses . 118 91

Strength of a man ascending vertically half the horizontal

velocity 118 91

Man will walk d| miles an hour for 10 hours a day . .118 92

His maximum of effect is then If miles an hour, or 2| feet

a second 118 93

Quantity of velocity lost in friction = one-fifth . * . maximum

of useful effect is 2 feet per second . . . .118 92

Smeaton's comparisons of animal power in man = 31*25 lbs.
moving with a velocity of 2 feet per second, or to ^ cubic
foot of water raised 2 feet per second ; a cubic foot of

water being 62| lbs 118 92

Bricklayers' labourers ascend 9 inches per second . .118 93

Ascent of stairs, &c 118 93

Force of a horse equal to that of 6 men, according to esti-
mates; exertion for 8 hours about 2^ feet per second .119 93
Or mechanical power of a horse := 187^ lbs., moving with a
velocity of 2| feet per second; or to 3 cubic feet of
water raised 2^ feet per second ; the day's work being 8

hours 119 93

Equal to 28,125 lbs. raised 1 foot per minute =: a mean of

Smeaton and Watt's estimate 119 94

French and American dynamical measure of power . .119 94

Table of pitches of wheels in actual use in mill-work, exhibit-

^^^^^^M

■

^

^^P tng by inspection, the kind of macliine, power, pitch iti

^^P belies, bnaidthoftectli in inches; of the wheel, its number

1

■ of teeth, revohitions per minute, and diameter; of the

1

ptninn, ita nnmber of teeth, revolutions per minute, dii^

1

■ meter; the breadth proportionate to the iiorscs' power

^^^J

^^ft Kod present velocity ; present velocity in feet per second,

^^^fl

^^Psnd brcodtli proportionnte to 10 horses' power at 3 feet

^^^H

^^Bpcr iecand; that is, reducing all the examples to the some

^^^H

^B deoomination

120

95

^ESxpbnntian of the tuble of wheels in actual use in mill-work

120

95

FfflMerratioiis on the table of wheels in actual uee in mill-work

ISI

96

Rnle 1. anil esomple for tlic construction of wheels to equal

the horses' power employed in the macliinery .

122

97

Description of six (ablee of pitches

122

98

Table I.— The velocity of the pitch line being 3 feet per

•ecottd, and the breadth of the teeth 9 inches .

123

99

Table II. — The velocity being 3 feet per second, and the

breadth of the teeth doable each pitch ....

124

99

Table III.— The breiwith of the teeth 8 inches, and velocity

1 1 foct per second

125

100

Table IV. — The velocity being constant, 1 1 feet per second.

ond the breadth constant

126

101

Tobic V. — At a velocity of 3 feet per second, the breadth

being coustantly B inches

127

101

Table VI.— Velocity at 3 feet, the breadth being double

tJie pitch

128

102

Comporisou of these tables

128

102

Ealo II. — For a pitch of 3 inches, with a velocity of 3 feet

per second, every inch of breadth being valued at 1 J horses"

power

129

102

Braidtli of teeth as made by the best millwrights now seems

to be about twice or thrice the pitch ....

130

103

Bobcnon's rule for the teeth of wheels ....

131

103

JEIetnents for the constructian of a table of pitches of wheels

132

104

^LZtble of pitches of wheels, with the breadth and thicknesa of

^m

^M

^^Lmoring Bt tlie pitch line at the rate of 3 feet, 4 feet, of 6

^^

^^nfect, and of 8 feet per second

133

V

1

^^Kmu! RAberton'e tables of pitches

134

bs

105 ■

K

1

XX CONTENTS.

Art. Pa^

Rule by Garmichael for calculating the proportionate Btrength

of the teeth of wheels 104 106

Table of pitches by Carmichael, founded on three cases of

Roberton 8 and three cases of Buchanan's tables . .134 106
Explanation of the table ; pitch, thickness, breadth, length
and strength of the teeth ; horses' power at 3, 6, and 11

feet per second 134 106

Remarks on this table 134 107

Table of pitches computed from the rule of Garmichael, with
the breadth and thickness of the teeth, and the correspond-
ing strength in horses' power 135 107

Method of determining from first principles the strength pro-
per for teeth of wheels, by Tredgold . . . .136 108
Rule for the thickness of cast iron teeth for wheels. — Find
the number of horses which are equivalent to the power
of the first mover of the train of machinery, and divide
that number by the velocity, in feet per second, of the
pitch line of the pinion or wheel ; extract the square root
of the quotient, and three fourths of this root will be the
least thickness of the tooth for the wheel or pinion, in

inches 137 109

Rule for the least quantity of pitch for a wheel or pinion
with teeth of cast iron. — If the thickness of the teeth of
the pinion be intended to be the same as those of the wheel,
multiply the thickness above determined by 2*1, the pro-
duct will be the pitch required 138 109

Example to illustrate practically the foregoing rule . « 139 110

Of the thickness of wooden teeth 140 110

How to determine the breadth of cast iron teeth, and to as-
certain what breadth is essential to strength • . .141 111
Rule for finding the breadth of cast iron teeth . . .142 111

Example illustrative of the rule 142 112

Of the breadth of wooden teeth ; — rule and example . . 143 112
Of the strength of staves for trundles ; — ^rule and practical

example 144 113

Table of the radii of wheels from 10 to 300 teeth, the pitch

being 2 inches 145 114

Of arranging the numbers of wheel-work .... 146 115
Rule I. For the number of teeth in one pinion, when the

wheels drive the pinions 147 116

CONTENTS. XXI

Art. Page

Rale II. For the some when the pinions drive the wheels . 148 116

Bole III. That the numher of teeth in a wheel should not
be divisible by the number of teeth in the pinion without
a remainder 149 116

Rule IV. To determine the exact ratio which should obtain
between the teeth of the wheel and that of the pinion . 150 117

Of calculating tlie numbers for wheel-work . .151 117

Of adapting the trains of machinery to produce different ve-
locities at the working points 151 118

Of the position of the first mover being as near as possible to

the resistance 151 118

Example iUustrative of the proportions between the wheels

Bud "pinionsj Bud vice versd 152 119

Friction, in the generality of combinations, balances two-
thirds of the power appHed 153 120

The resistance to be overcome at the working point expends

the remaining third 152 120

Dr. Jamieson's demonstrations of these facts . . Pages 120-123

Examples illustrative of the previous demonstrations, by the
same author 154 123

htcdcal observations with regard to the making of patterns
for cast iron wheels 155 124

Role for making the length of the teeth equal to the pitch,

deducting freedom 155 124

Hafcton on clock-work, his rule 155 125

Of the shrinking of metal in cooling 156 125

IVoportions that have been found to answer in practice fbr
cast iron wheels 157 126

Of casting wheels in parts, and oflcrwards bolting the parts
together 158 127

Of materials for patterns 159 128

CHAPTER V.

Of the use of charts, and some further explanation of tlio con-
stmction of the tables of pitches of wheel-work ; showing
the horses' power to which the teeth of wheels of certain
pttdiesi working under different circumstances, are equal

Qmotitiss incraasiiig or decreasing in arithmetical proportion
jpcrMsing in geometrical proportion .

160

129

161

129

162

129

163

129

164

130

165

130

166

130

167

131

168

131

XXll CONTENTS.

Art.

Examples illustrative of these cases, and of laying down such

proportions on a chart

Method of tracing on a chart accelerating motion
Of tracing curves from proportions on such tahles
Mechanical method of tracing these curves
Various uses of such charts ....
Comparative view of the tahles of pitches of wheel- work
Table I. — Velocity of the pitch line 3 feet per second, and

breadth of the teeth 9 inches, and value of strength in

horses' power . . . . . . . . 168 131

Table II. — Velocity 3 feet per second, breadth of the teeth

double each pitch, and value of strengtii in horses' power . 168 132
Table III. — Velocity 11 feet per second, breadth of the

teeth 8 inches, and value of strength in horses' power . 168 132
Table IV. — Velocity 1 1 feet per second, the breadth double

each pitch, and value of strength in horses' power . .168 133
Table V. — Velocity 3 feet per second, breadth of the teeth

8 inches, and value of strength in horses' power . .168 133
Table VI. — Velocity 3 feet per second, breadth double each

pitch, and value of strength in horses' power .
Reference and example explanatory of Table I. .
Reference and example explanatory of Table II.
Reference and example explanatory of Table III.
Reference and example explanatory of Table IV.
Reference and example explanatory of Table V.
Reference and example explanatory of Table VI.
Explanation of the chart, in regard to the scales representing

the line of pitches and horses' power ....
Observations on the intersection of the curves . . 17(

APPENDIX A.

Profesmtr Willis on the Teeth of Wheels, — Investigation of

the curves given to the teeth of wheels . . .180 139

Section I. — On the curves adapted to practice *. . . — 139

Instrument to illustrate the curve and furnish a practical so-
lution of the problem — 140

Remarks on epicycloid and involute curves . . . — 141

• The numeriod refiarence of 180 applies to the whole of this Appendix, and it is not
therafore repeated in these contents.

168

134

169

134

170

135

171

135

172

136

173

136

174

136

175

137

179

137

CONTENTS.

XXlll

Art

Proportions that obtain in epicycloidal teeth

Cbroflleirjr.— If for a set of wheels of the same pitch, a con-
stant describing drclo be taken, and employed to trace
those portions of the teeth which project beyond each
pitch line by rolling on the exterior circumference, and
those which lie within it by rolling on its interior circum-
ference : then any two wheels of this set will work cor-
rectly together

Application of the proportion to any pair of wheels .

Method of settling the proper diameter to be given to the
constant describing circle ......

Application of the proportion and corollary to racks .

Proofs that this system is more easy of practice for the work-
man than the old one .......

Of a form of increased strength

Analogous to the wheels of watches when the teeth and
pinion leaves are of a saw-tooth form ....

Section II. — On a practical approximation to the true form
by arcs of circles, and the identity of Professor Willis's
method with that of Euler, who first suggested the sub-
stitution of an arc of the circle of curvature for the real
curve •...•..

Of the Odontagraph, or tooth fashioner * .

Tables shewing the place of the centres upon a scale

Centres for the teeth within the pitch circle

Centres for the teeth outside the pitch circle

Table and rule for finding the radius of the wheel

Chsometrical construction of wheels .

Of teeth working virith trundles or radial flanks

On cntters — and the method of obtaining a correct form of
tooth by means of the Odontagraph, or tooth modelling
mstnuuent .........

Table of equidistant values for cutters ....

Table of cutters

SlCTiOH III. — Theory of the preceding constructions .

Method of describing teeth consisting of a single arc .

Method of describing teeth consisting of two arcs of circles

Constniction of the Odontagraph or tooth modeller

Example shewing how this instrument is connected i^ith the
profioiis demonstration

Page
142

— 143

— 143

— 144

— 145

— 145

— 14G

— 147

148
150
151
151
151
152
153
154

— 155

— 156

— 157

— 157

— 163

— 166

— 168

— 168

* HoUMpiBlj of CSuring Croa, mikei thb iKUtmment

XXIV CONTENTS.

Alt. Pace

Results of calculations for obtaining a principle for the num-
ber and arrangement of the wheds selected . . . 180 171

ESSAY II.

On the shafts of mills, gudgeons, journals, the kinds of stress
to which they are subject, their strength, stiffiiess, and pro-
portion — 173

CHAPTER L

Introductory remarks on mill-work . . » .181 173

Manner in which this essay is treated .... 181 175
Remarks on the proportion which the diameters of axles

ought to have to the stress they are to bear . . 182 170

On the use of cast iron for shafts 183 176

CHAPTER II.

General description of shafts — 178

Distinction between shafts and spindles . . . .184 178

Horizontal and Tcrtical shafts 185 178

Materiab of which they are usually made . . . 186 178

Wooden shaft laid in gudgeons 187 178

Wooden shaft ^ith cross^taikd gudgeons . .188 179

Of hoops on shafts 189 179

An improved method of fixing gudgeons invented by Robert

Hughes 190 180

Of hollow cast iron shafts 191 181

Cast iron cylindrical shaft, which npiy be variously con-
structed according to circumstances .... 192 181

Of the feathered arrow shaft 193 181

Section II. — Of the kinds of stress to which shafts are sub-
ject — 182

Of lateral stress and torsion 194 182

Horiiontal shafts liable to lateral stress . .194 182

Upright shafts liable to torsion 194 183

Stress compouudoil of lateral pressure . . .194 183

Roborton's means of avoidiog stress and friciioii 194 183

Kxample to illiHtfate tiieae mmn by an oreidMii

CONTENTS. XXV

Art. Page

wheel, shewing that the stress and friction of the gudgeon
must depend, in a great measure, on the size of the toothed
wheel attached to the water-wheel, and to the situation of
the pinion 195 183

In a single pair of wheels of whatever form or construction,
the tendency to hreak or hend the shaft, or cause friction,
18 the same as the action of the teeth on each other . 196 184

In the case of an intervening wheel, the force or tendency
to break the shaft depends on the situation of such inter-
vening wheel 197 184

A wheel placed betwixt two others, the forces being equal

and opposite, removes the strain from the shaft .198 185

Of the place on the shaft on which the wheels are fixed.

Example to illustrate this case 199 186

Of the strains upon journals, and placing wheels and pinions,
so that their action on each other may be in contrary di-
rections; to avoid the strain on the shaft or journals . 200 186

Of lying shafts having the heaviest shaft on the lift of the

wheels to take off the friction on the journals . . 201 187

Roberton's observations on the foregoing subjects . . 202 187

By increasing the size both of wheels and pinions, the force,
strain and firiction on the shafts and journals are dimi-
nished in the same ratio 203 188

ExAMPLB I. — Illustrative of Roberton's views . . 204 190

ExAMPLB II.— Illustrative of the same matter . . . 204 190

In a horse-gin, where the pinion is driven by a toothed wheel
on the gin, the friction, or strain on the journals, depends
on the situation of the horse beam .... 204 190

In a water-wheel turning machinery, the strain on the shaft
and teeth are the same 205 190

Example to illustrate this 205 190

Roberton's judgment of Fenwick's opinion that the most
perfect machine is that which operates with the fewest
moving parts 206 192

When wheels differ considerably in size, the gudgeon next
the nnaller wheel will have to sustain the greater part of
tfaeatjesB 207 193

PiMgnro downward on one gudgeon, and upward on another 208 193

When the poaanre at the gudgeon is wholly in a lateral di-

209 193

XXVI CONTENTS.

Art.

Methods of computing and comparing the pressore in these

different cases 209 194

Gasb I. — The power and resistance being at opposite sides

of the shaft 209 194

Gasb II. — The power and resistance being at the same side

of the shaft 209 194

When the power and the weight are oblique in respect to

one another . . 210 195

Methods of operation by the resolution of forces . .210 196

CHAPTER III.

Section I. — Of the strength of gudgeons where the stress is

produced by lateral pressure only — 197

Of the size and strength of gudgeons . . . .211 197

Pbop. I. — Solid cylinders of the same length have their lateral
strength as the cube of their diameters^ for^ in general^ the
lateral strength of any pieces of iron or timber^ whose sec^
tions are similar figures^ are as the cubes of the similar
sides of the sections 212 197

Strength of gudgeon limited by the strain it will bear, with-
out permanent derangement .....

Investigation of a new rule for the strength of gudgeons •

Practical rule for finding the diameters of gudgeons .

Comparison of the rule that the diameter of the gudgeon
should be equal to the cube root supported in cwts.

Sbction II. — Of gudgeons of water-wheels

Introductory remarks on water-wheels of various weights,
and the diameters of the gudgeons in actual use

Description of the first table of gudgeons

Notes upon empirical rules 210

Table I. — Gudgeons of water-wheels of different materials

Observations on the first table of gudgeons

Proof that the cube root of the weight in cwts. is nearly

equal to the diameter in inches of the gudgeon . .218 203

Rule for finding the diameter of the gudgeon of a water-
wheel. — The cube root of the weight of a water-wheel^ in
hundredweightSy is nearly equal to the diameter in inches if
a oast iron gudgeon sufficiently strong to support such wheel 219 203

Example illustrative of the rule 219 204

213

198

213

198

214

199

214

199

215

200

215

200

216

201

216

201

216

202

217

202

CONTENTS. XXVU

Art Fife

The weights of OTenhot or bucket water-wheels will be to
one another nearly as their circumferences or diameters

and breadth 220 204

Rnlc for the diameter of the gudgeons. — For wooden water^

wheds^ mtikiplif the diameter in feet hy the width also in

feeti to which add the sgtiare of half of the diameter. The

cube root of the sum will be nearly equal to the diameter

of the ^tufyeon in inches 220 204

Example illustratiTe of the rule 220 204

Explanation of Table II., of water-wheels . . .221 205
Table II. — Gudgeons of water-wheels . . . .221 205
Sbction III. — Of cast iron gudgeons for various purposes . — - 205

Introductory remarks 222 205

Explanation of the table of cast iron gudgeons . . . 223 206

Table of cast iron gudgeons 223 206

Use of the table shewn by Example I., and also Example

II 224 207

Section IV.-— Of malleable and cast iron gudgeons . . — 207
I^ofessor Robison s remarks on the strength of cast and

wrought iron 225 207

Buchanan's results of experiments on gudgeons of cast and

wrought iron 225 208

Method of finding the diameter which any cast iron gudgeon

should have to sustain any given pressure . . . 226 208
Note, — Tredgold 8 experiments on the stiffness of cast and

malleable iron 226 208

Distinction between strength and stiffness .... 226 208
Example shewing the method of finding the diameter of a
wrought iron gudgeon, having given the lateral pressure
aod the diameter of the cast iron gudgeon . . . 226 209
Explanation of a table of cast and wrought iron gudgeons . 226 209
Table of cast and wrought iron gudgeons, shewing their
lespective diameters, and the weights they can sustain :
the diameters of the cast iron swelling from 1 inch to 11
inches, and the wrought from 1 to 9 inches, and the
wei^ts expressed in the cubes of these nmnbers . . 226 210
Use of the table shewn by a practical example . . 227 211

CHAPTER IV.
Bmtniom Lf— Of the itrength of journals, when the stress

XXTUl CONTENTS.

An. Plge

arises from torgion and twisting in addition to lateral

stress — 212

Horses' power used as the measure for the strain brought on

shafts by torsion or twisting 228 212

Buchanan's idea that wrought iron will not remst torsion

equal to cast iron 228 212

JVofe.— Definition of a journal 228 212

Section II.— Of proportioning, journals to the stress which
they haTe to sustain — 213

Illustrations of the proportion between journals and the
stress tiiey have to sustain 229 213

In all cases where the horses' power divided by the revolu-
tions per minute produces the same quotient, the sta^ess is
the same 229 213

A resistance equal to 50 horses' power making 50 revc^n-
tions per minute, produces the same stress as 10 horses'
power making 10 revolutions per minute . . . 229 213

Bules for calculating the strength in proportion to the resist-
ance 230 214

ExAMPLS I.F— When the horses' power and the revolutions
per minute are the same number 230 214

Example II.— To find the diameter of the journal propor-
tionate to the velocity 230 215

Description of a table of journals proportionate to d having

420 as a multiplier 231 215

Table of journals, shewing, 1st, the horses' power : 2dly,
the revolutions of the journal per minute ; 3dly, the pro-
duct of the povirer divided by the revolution of the
journal ; 4thly, the proportionate strain on the journal ;
and lasdy, the diameter of the journal from observation . 231 216

Observations on the journals of fly-wheel shafts 232 216

And on secondary shafts 232 217

Multipliers for journals of steam-engine fly-wheel shafts 233 217

Example illustrative of the table 233 217

Note. — When Buchanan uses the word journal, he supposes
it subject to torsion ; where there is lateral pressure only,
and no torsion, he uses the word gudgeon . . 233 217

Rules for calculating the resistance of a journal, or its dia-
meter as regards the twisting strain .... 234 217

Rule I.— To find the number of horses' power tiie journal
is sufficient to resist 234 218

CONTENTS. XXIX

Alt. Page

RuLB 11. — To find the diameter of the jotumal in inches . 234 218

Example illasirative of the rules 234 218

General rule for the diameter in inches of the journals . 235 219
Section 1X1^— When the diameter of a joiumal and its re-
volutions per minute are giyen, to find the horses' power

to which it is equal 236 219

Rule for determining this case 236 219

EzAUPLB I.«»Shewing the horses' power to which the

journal is equal 236 219

Example II. — ^When the journal is connected with heavy

machinery 236 220

Example III. — The same journal for internal work of the

ordinary kind 236 220

CHAPTER V.

Section I. — On the hodics of shafts . . . . — 221

Preliminary remarks on the distinction hetweon stiffness and
strength 237 221

Shewing that the limit of stiffness is flexure ; and the limit

of strength is fracture 237 222

The laws which govern stifihess, and those which determine
strength 237 222

Application of those laws 237 222

Of lateral stiffness, and lateral strength .... 238 222

Proposition II. — Any beams of equal length have their lateral
d^ness^ ^to bear a load at any point in the length^ as the
breadth and cube of the depths and have their latercd strength,
as the breadth and square of the depth .... 239 223

Example I. — Illustrative of the comparative stiffness of dif-
ferent beams or shafts 239 223

Example II. — Illustrative of the comparative strength of dif-
ferent beams 239 224

Pbofosition III. — Any beams of different lengths have their
diffness ^ bear a load at any point in the length'] directly
as the breadth and the cube of the depth, and inversely as
Ike cube of the length, and have their strength directly as the
hreadthy and as the square of the depth, and inversely as
tketatgik 240 224

Ifate upon this proposition as applied to practical pur-

240 224

XXX CONTENTS.

Art.

Example I. — To determine the comparatiYe tft^ffnen of

beams or shafts of a given length and thickness . . 240 225
ExAMPLB II. — To determine the comparatiye sbreng^ of

beams or shafts of a given length and thickness . 240 225

PaoposiTiON IV. — SuppoHng a tube^ indefinitefy t&tft, to be
expanded into a similar tube of a greater diameter y but of
equal lengths, the quantity of matter remaining the same^
the STiFFNSss will be inereasedy in the ratio of the square of
the diameter,, and the stbbnoth in the ratio of the diameter 241 226
Example I. — ^With a given length and thickness to deter-
mine the comparative stiffness of cylindrical beams or

shafts 241 226

Example II.— To determine the comparative strength of

cylindrical beams of a given length and thickness . . 241 227
The strength of shafts is increased in proportion to the areas

of their ends and diameters 242 227

Professor Robison's remarks on cylindrical beams 243 227

Galileo's observations on cylindrical hollow bodies . . 243 228

Section II.— Of lateral stress — 228

Definitions and explanations of the terms stress and strain^

and of lateral stress in particular 244 228

Proposition V. — The stress on a beam arising from one
weight hung upon t^ is proportional to the rectangle of the
parts of the beam^ and is greatest when the load is laid on

the middle of the beam 245 229

Definition of the rectangle of the parts .... 245 229

Illustration of the proposition 246 229

Of shafts loaded in the middle 247 229

Of the load united in the centre of gravity . . . 248 229
Every shaft should be able to resist the strain excited at that

centre 249 230

Shafb subjected to lateral stress should swell in the middle . 250 230
Shafts of the form of a cubical or semicubical parabola . 251 230

Section III. — Of torsion — 231

Proposition VI. — In general the strength of a cylinder or
solid axle by which it resists being wrenched asunder by
twisting is as the cube of its diameter .... 252 231

Of hollow axles 253 231

Method of estimating their strength 253 231

The superiority of strength of hollow tubes over solid

CONTENTS. XXXI

Art. Pace

cylinders is mucli greater in resisting torsion than trans-

Terse or lateral stress 254 232

Notes and illustrations shewing the general ratio that ohtoins
between the strength of a solid cylinder, and that of a
tube containing the same quantity of matter . . . 254 232

Professor Bobison's observations on the adhesion of the
fibres in wood, and the molecules of metal in iron shafts . 254 233

Of the excess of force in lateral stress and twisting — or when

one of these forces exceeds the other .... 255 233

How to measure the resistance of a lateral stress . . 255 234

One hundredth part of an inch the quantity of flexure that
may be allowed without sensibly affecting the regularity of
motion in a shaft 255 234

Method of calculating the stress when referred to the middle

of a cast iron shaft 255 234

Circumstances when the diameter of the shaft must bo de-
termined by the rule for lateral stress, and when by the
rule for torsion 256 234

Practical illustrations, shewing that the bodies of shafts need

not be greater than tihe journals 256 235

Section IV. — Application of the foregoing laws practically,

with r^ard to the proportions of shafts ... — 235

Preliminary observations regarding the stress upon gudgeons

or journals 257 235

Construction of water-wheels without shafts, the gudgeons

being fixed to the arms at each side of the wheel . .258 236

Practical example at Cartside mill, in the Note ... — 237

Cast iron shafts. — Ist. Shaft 8 feet long, with gudgeons of 4
inches, is weakest in the middle ; but 5 inches in the mid-
dle, it would be as strong as one of 4 feet long, and 4
inches in the middle 259 237

Of the stiffness of this shaft =: that of one 6 feet long, and

4 inches throughout 259 237

2nd. When the point of greatest lateral pressure is 2 feet
from one end 260 237

From the properties of the lever, the gudgeon next the point

of greatest pressure has three-fourths of the whole to sustain 260 238

Bxamples illostrative of these cases, and of the strength and
•tiflbesB when the shaft is reduced to a given section, in
order to enable the millwright to judge how the shafts
dMmld amil at the place of the greatest stress . 260 238

-.J

XXXll CONTENTS.

Art. P^

A cylinder is stiffer than any figure that can be inscribed
within it 261 239

Rules for computing the diameters of different forms of cast

iron shafts to resist lateral stress 261 239

Istly. If the stress be in the middle, the fourth root of half
the stress in cwts. multiplied by the square root of the
length in feet, is equal to the diameter in inches . . 261 239

2dly. If a cylindrical shaft has no other lateral weight to
sustain but its own weight, multiply the cube of the length
by *007, and the square root of ibis product is the diameter
in inches 262 239

This rule enables us to include the effect of the weight of
the shaft itself. Hence, a table of shafts of cast iron to
resist lateral pressure ; shewing, Istly, the length of the
shaft; 2dly, its diameter in inches when it bears only its
own weight ; ddly, its diameter in inches when the stress is
equal to its own weight; 4thly, its diameter in indies when
the stress is double its own weight ; 5thly, its diameter in
inches when the stress is three times its own wraght ; and
lastly, when the stress is four times its own weight . • 263 240

Of hollow cylindrical shafts of cast iron^F— The cube of the
length in feet multiplied by *009, and also by the number
of times the weight of the shaft is contained in the stress,
then the square root of this product is the diameter in
mchee 264 240

Table of hollow shafts of cast iron to resist lateral stress, ex-
hibiting, Istly, the length from 4 to 16 feet; 2dly, the ex-
terior and interior diameter in indies, when the stress is
four times the weight of the diaft ; ddly, the same dimen-
sions when the stress is six times the weight of the diaft;
4thly« when the stress is eight times the weight of the
shaft ; and StUy, when it is tm timea the wei^t of the
diaft 265 241

Of wrought iron dudb to rcdst lateral stress . 266 241

Of wooden sludb of oak to hare the same stroigih with cmst

iron shafW fite udchce square 267 242

Of the comparatiTe stiftMca of good oak sludb as compared
to thoeo made of iron 268 242

Example lo ilhislnile the fdatire propottioii of an oak to a
c«sl in^ shaft 269 242

Of the slilbMt of iW or ysUew fir as coHfttRa to casl iron 270 843

CONTENTS. XXXlll

Art. Pnge

Remarks on the foregoing data and examples . .271 243

Phurtical case, showing the possibility of failure when excess

of strength seemed to obtain ..... 272 243

Hollow cylindrical shafts equal in size throughout . .273 243
Of shafts subject to torsion, especially those made of wood 274 243
Practical case given by Buchanan, wherein the shaft was only

equal in strength to the gudgeon 274 244

Of cross-tailed gudgeons of wooden shafts . . .274 244

Of cylindrical shafts of cast iron to resist torsion . 275 244

Table of cylindrical shafts of cast iron to resist torsion ; com-
prising, 1st, their diameter in inches ; 2dly, the number of
revolutions from 5 to 50, under a given horses' power . 276 245
Application of the table to other cases in which the shafts

are either hollow or solid cylinders . . . .277 246
Bemark upon vertical and horizontal shafts loaded with

wheels 278 246

Example illustrating the foregoing table .... 278 246
Of a shaft of cast iron, when the number of revolutions is 20
per minute, and the power of the first mover equal to 18

horses 279 246

Of an oak abaft, and the method of determining its diameter

when the number of revolutions is 40 ; horses' power 18 . 279 246
When fir is used for a shaft, its diameter should be 2*06

times that of one of cast iron to do the same work . .281 247
EzAJCPLB. — Power, 7 horses; turns, 11| per minute; to
find the diameter == 5-8 inches, being of cast iron, or 11

inches if of fir 281 247

Allowance should be made in fir shafts to resist torsion, when

the abaft has to sustain both lateral strength and torsion . 281 247
EzAicPLXS — ^illustrative of the sum of the straining forces for
acjlindrical shaft of cast iron to determine their diameter 281 247

Ofthe patterns of cast iron shafts 283 247

Of the dimensions of shafts subject to torsion . . 284 248
Ofthe diameters of jonmals, and a table of shafts of cast
and malleable iron 284 249

APPENDIX.

CohMm itrangth of different metals 285 250

itrangth of different woods 285 251

of fiweigiierBy philosophers, and engmeers on the

c

287

254

287

254

287

254

288

255

288

259

289

259

XXXIV CONTENTS.

Art.

Strength of materials. — Table of MnsdienbroSk's ex-
periments on the strength of materials • • . • 285 252
Emerson's table of the load which may be safely suspended

to an inch square of various materials . . • • 285 253
Banks takes iron to be 4 times as strong as oak, and 5|

times as strong as deal or fir 286 253

Results of yarious authors on the cohesiye strength of ma-
terials

Strength of materials in resisting compression • ^
Iron more liable than wood to accidental imperfections
Table of the experiments of Brown, Buffon, Muschenbro^k,
Perronet, Rondelet, Morveau, Rennie, Rumford, Tredgold,
Telford, Sickingen, from the Philosophical Magazine
Remarks upon this table, the most extensiye of its kind

Experiments on alloys of the metals 289

Copper and tin, by Muschenbroek ; gun metal and brass, by
Rennie ; English tin and lead, Muschenbroek ; Banca tin
and antimony, by the same ; Banca tin and bismuth, by
the same ; Banca tin and Indian sine, by the same ; Eng-
lish tin and zinc, by the same ; English tin and antimony,
by the same ; Dutch lead and bismuth, by the same • 289 260

Observations on the composition of these alloys . . . 289 261
Authorities for the cohesive force of woods of various kinds,

are Tredgold, Muschenbroek, and Barlow . . . 289 261

ESSAY III.

On the construction and durability of the longitudinal con-
nexions of shafts, denominated couplings ... — 262

Preface.— Different methods employed in coupling shafts . — 262

Remarks on the fallacies of some eminent men in applica-
tions of favourite theories — 264

Dr. Robison's strictures on the blimders of practical men who

disregard entirely scientific knowledge . . . . -i- 265

CHAPTER I.

On the longitudinal connexions of shafts, denominated cou-
plings 290 266

Class I. — Of couplings with two bearings . . . 291 266

Coupling I.^^f the square coupling .... 292 267

Of the oblong coupling 292 ^67

CONTENTS. XXXV

Art. Page

Remarks on these coupliDgs, and on the imperfection called

Ali/i 293 268

Coupling of rollers as mules 293 268

Couplings with donhle hearings, how made . .294 268
Coupling II. — Of the round coupling .... 295 268
Oheervations on the effects of round coupling . . 296 269
CovPLiNe III. — Of clutches or glands, having douhle hear-
ings 297 269

OhservationB on glands or couplings for douhle hearings 298 269

Methods of adjusting the arms and points of glands . 299 270

Coupling IV. — Of the horing mill clutch ; first construction 300 270
Oheervations relating to the application of this coupling to

slow work 301 271

Coupling V. — Second construction of the horing mill clutch 303 271
Observatioiis showing this to he a stronger and hotter

clutch 304,305 272

Coupling VI. — Having two round plates that serve to engage

theahafls 306 272

Ohservations to show the durahility of this clutch, or species

ofglands 307 272

Naie^ upon making the circular heads toothed ... — 272

Coupling VII.— Boulton and Watt's coupling link • . 308 273
Ohservadons showing the durahility of this coupling, and

that the axes move without twisting .... 309 273

Of the length of the crank 309 273

Coupling VIII. — For conveying motion to a fly wheel . 310 273
Observations on the durahility of this coupling, and its ap-

plicalnlity to thrashing mills 311 274

OufKBAL Obsbbvations. — Firstly, on friction, and couplings

with one bearing 312 274

Secondly, when heavy drums or wheels are placed near the

ends of the shaAs, two hearings must he used . . .313 275

CHAPTER II.

CSlajm IIh— Of couplings having one hearing ... — 276

Smxion L— Hook's universal joint 314 276

Rojperiy of lihe universal joint for communicating angular

aotkn 315 276

lb i1iwilwntig,n of the universal joint .316 277

e 2

XXXVl CONTENTS.

Art. P«ce

Couplings described in Chapter I. may be converted into

couplings haying one bearing . . . . .317 277

Coupling IX. — The square coupling . . . .318 277
Observations showing the efficiency of this coupling in con-
veying motion through a great length of shafts . .319 277
This coupling liable to lifting or straining . . . .319 278

Mules, having only one bearing 320 278

Accuracy required in mtde and throstle rollers . . . 320 278
Notes, — Samuel Crompton, the inventor of the mule ; Ark-
wright's patent for preparing cotton by machinery ; inven-
tion of the throstle — 278

Coupling X.— Of the round coupling, having only one

bearing 321 279

Observations on round coupling 322 279

Coupling XI. — With a scarfed joint .... 323 279

Observations on this contrivance, showing that it has all

the defects of a solid shaft 324 279

Coupling XII.— A variety of Coupling XI. . . . 325 280
Coupling XIII. — Another modification of Coupling XL,
having one shaft firmly fixed to the other by flanches and

bolts 326 280

Coupling XIV. — Has the bearing and the joint of the
coupling at the same parts of the shaft, the ends of which

are quadrants 327 280

Observations showing this coupling to be attended with '

trouble and expense 328 280

Coupling XV. consisting of three distinct parts which join

into one another ; is well explained in the plate . . 329 281
Observations on the first cost' of this coupling, or universal

joint, showing its excellency 330 281

Coupling XVI. a contrivance executed in Buchanan's

time at Manchester 331 282

Observations showing that the principles of this coupling

agree with those of the square coupling .... 332 282

Advantages of this coupling 333 282

Coupling XVII. as used extensively at Glasgow . . 334 282
Observations on the advantages of this kind of coupling . 335 283
Section II. — Of the couplings of upright shafts ... — 283
Square coupling applied to these shafts .... 336 283
Coupling XVIII. is described by the diagram in the plate 337 283

CONTENTS. XXXVll

*

Art. Pa^ie

Coupling XIX.-- This also is well described by the diagram

in the plate 338 284

Obserrations on this kind of coupling, as for light work,
especially flour mills, and connecting the feeder with the
stone-spindle 339 284

Coupling XX. — This also is explained by the diagram in
the plate 340 284

Obsenrations showing this a good and simple mode of
coupling upright shafts 341 284

CHAPTER III.

GsNERAL Observations — 285

The laiger the parts of couplings can be made, so much the
better 342 285

The further the point of stress is from the axis, the couplings

will be more durable ....... 343 285

Ezemplificatioiu>f this in the handspike, capstan bar, or simi-
UirleTer 343 285

Practical illustrations of the correctness of these observations
in point of durability 344 285

In along line of shafts, the couplings, where there is only one
bearing, should be so arranged that the unsupported end
of the shaft should be as far as possible from the part
subject to lateral pressure ; illustrated by diagram . . 345 28 G

Oiling of couplings renders them more durable . . . 346 286

Fly wheel often used in a long line of couplings . . 347 286

Table representing the dimensions, stress, and durability of
ooapHngs, in nine cases of facts ; combining the resistance
in horses' power, the revolutions per minute, the number
of years' work, the side of the square in inches, the length
of the box, and the comparative stress .... 348 287

Obsbbvations. — I. On the durability of these couplings . 349 288

II. CSrcmnstances affecting their durability . . .350 288

III. Standard of power and strength .351 288

lY. Vdod^ affects the durability 352 280

V. daasification of couplings with respect to durability . 353 289
BvpFunmrTABT Obsbbvations. — I. On the durability of

eoiiidings 354 289

IL Thar durability depends mainly on accuracy of work-

nmhip 355 289

nL Jhmnrj of dmvlnlity deduced from construction . 356 289

XXXviii CONTENTS,

An.

ESSAY IV.

On the methoils of disengaging and re-^engaging machinery

while in motion -— 291

Intboduction. — Plan followed in this Essay ... — 291

Methods of disengaging shafts 357 292

Of the vU inertia of matter 358 292

Note. — Dr. Young's definition of this term ... — 292

Inertia simply indicates that matter never changes its state,

unless there be a change in the power or powers acting

upon it 359 293

Note. — Newton's definition of vis inertia .... — 293
lUustration of the strength of machinery by throwing a wheel

into gear 360 294

Division of the subject — engaging and disengaging ma-

chineiy — into two parts 361 294

I. Of methods used when motion is communicated by means

of bands, belts or chains. II . Of methods when motion

is communicated by means of wheel-work . . . 361 294
Method I. — The sliding pulley, Fig. 1, an old contrivance;

description of ; engraving and plate .... 362 294
Observations relating to the application of this contrivance

to cotton carding machines 363 295

Method II. — The bayonet, Fig. 2 ; full description of this

invention 364 295

Note, — Pointing out some of the many ways of making and

appKing this contrivance — 296

Observations showing the superiority of the bayonet to the

sliding pulley 365 296

Method III. — Of the lock pulley; description of this in-
vention, and how it unlocks and disengages the pulley . 366 297
Observations showing that this invention has never been

much adopted 367 297

Method IV. — The fast and loose pulley; description of this

invention 368 297

Observations on the belts used in machinery running over

pulleys 369 298

The fast and loose pulley remarkable for simplicity . • 370 298
Its application in cotton mills is now general . . . 370 298
Method V. — ^Description of this method— or sack tackle 371 298
Observatioiit on the sack-tackle 372 299

CONTENTS. XXXIX

Art.

Sbctior II.— Of the metliods used when motion is oonyeyed

hy means of wheel-work 373 299

Of throwing a wheel into gear 373 299

MiTHOD Vh — ^DisengBging and re-engagiDg wheels by means

ofbridgea 374 299

Obaervationa on this mode of disengaging wheels 375 300
MiTHOD YII. — ^Wheel with sliding clutch, which may be en-
gaged or disengaged at pleasure, described and illustrated 376 300

Obaervationa on its utility 377 301

Mbthob VIIL— The friction clutch .... 378 301
Observations. — ^Applicability of this contrivance to the

largest machinery 379 302

MsTHOD IX. — The friction cones 380 302

Observations. — ^Application of these cones to sack-tackle 381 303

MsTHOD X.^- Wheels acting by friction .... 382 303
Obeervations. — Used with good effect in machinery for

raismgcoal 383 303

Mbthod XI. — Tackle for raising sacks in a brewhouse 384 304
Observations on the ingenuity of this invention . . . 385 304
IfXTHOD XII. — Self disengaging coupling— figure represent-
ing the coupling as disengaged 386 304

il^.^-IUustration of this method founded on Coulomb's ex-
periments 386 304

Observations.— This coupling very useful when turning

lathes are driven by wheel-work 387 305

Jfoie.'-^On the comparative merits of wheels 387 305

ESSAY V.

On mechanism for equalizing the motion of mills, denomi-
nated lifi tenters, engine governors, and water-wheel go-
ynmon — 307

IlTTBODUcnoN. — Showing that this essay relates to machinery
not leas curious in its construction than useful in practice.'

Of the steam-engine governor, and throttle-valve described .

JVoCstd— Descriptive of the throttle- valve ....

SaonON h — The steam-engine governor — its particular cou-
■tnietioii ........

Opention d the governor and throttle-valve

Fopokr dewription of the whole apparatus

▼HmUmm of (he pendulum

—

307

388

308

388

308

389

308

390

309

391

309

392

310

xl CONTENTS.

Art.

Of the lengths of pendulums and oscillations in one minute

of time 393 310

Example showing how to find the lengths of pendulums . 394 310

Section I. — Of the windmill lift-tenter .... 395 311

First construction of lift-tenters for windmiUs . . . 396 311

Second construction of lift^tenters, drawn at Liyerpool . 397 312
Section III. — Of governors applied to water-wheels, and

made on various constructions 398 312

First construction of the water-wheel governor . . . 399 313
Method of lifting a wheel out of gear when a mill is

stopped . . 400 315

Second construction of a water-wheel governor . . .401 315

Third construction of a water-wheel governor . . . 402 315
Ohservation shewing that wheel-work is preferahle to hands

and pulleys 402 316

Fourth construction of the water- wheel governor . . 403 316

Fifth construction of the water-wheel governor . . . 404 317

Appendix on the velocity of water-wheels . . . — 318
Difficulty of finding a law of universal application for giving

different degrees of velocity to water-mills . . . 405 318

Experiments on the Rothesay Mills, hy Buchanan . . 406 319
Methods of ascertaining the proportional quantities of water

used hy the old mill 407 320

Smeaton and Buchanan's experiments compared . . 408 321
Buchanan's experiments are consistent with the experiments

of Smeaton 409 321

Roherton's observations on Buchanan's and Banks's experi-
ments 410 322

Illustration of these remarks 410 323

Roherton's remarks on overshot wheels, and comparative

value of work and water used in performing that work . 411 325
On overshot wheels. — The two principal elements to be

considered in the theory of wheels . . . .412 326
On the proportion of the radius of the wheel to the height

of the fall 412 326

Demonstration of this proportion 412 326

Important practical rule or maxim deduced therefrom 412 327
Method of finding the effective height of the fall and radius

of the wheel 412 327

Of the velocity of the circumference of the wheel to pro-
duce a maximum of dTect 413 328

CONTENTS. xli

Art. Page

Friction eqaal to two thirds the moving power — the velocity
of the drcumference of an overshot wheels in feet per
second, should he 2*67 times tiie square root of the whole

height of the M in feet 413 329

To determine the part of the fall which wiU give the water

the same velocity as the wheel 413 329

Comparison of these results with the experiments of

Smeaton 413 329

On compntiBg the power of overshot water-wheels . • 414 330
Equation for the effective force of the water . . .414 330

Ditto for the mechanical power 414 330

When the wheel is supplied at the summit, the power is

equal to half the weight of water supplied to the wheel . 414 330
Comparative power of overshot and hreast wheels . .414 331
Two points of view under which tiie power of a water-
wheel must he considered 414 331

Method of estimating the horses' power which any water-
wheel may have 415 331

Examples shewing the horses' power in overshot and hreast

wheels 415 332

And also the effective force when the water flows on either
at the summit or the level of the axis .... 416 332

Of the power of hreast wheels 417 332

Smeaton's comparison of the mechanical power of an imder-

shot and overshot wheel 417 332

ESSAY VI.

On changing the velocity of machinery while in motion . — 334
Intboduction. — Division of machinery into mill-work and

smaller machinery. — The mechanism descrihed in this

Essay helongs to the latter class -* 334

SicnON I. — Of turning lathe friction, and of helts of the

same length working on opposite pulleys . . .419 335
Ohaervations on the series of truncated cones, &c., in this

contrivance 420 335

Of alteniate cones, or one cone giving motion to another . 421 336
Ofaaermtion on this piece of mechanism .... 422 336
Allentioii of velocity hy wheels moving one another hy

findion 423 336

Ohiii f alkim on the pecoliar use of these wheels or cones . 424 337

xUi CONTENTS.

An. Pv
Sbotion n. — Moles, well adiqrted for apiiudng all kind* of

«/» 4S5 337

James Crompton, the inyentor of die mule . *S5 337

Williun Kelly of Lanark's patent 425 338

Velocity of spindles called doable speed . . ■ • 4S5 338
Contrivances to show the progress of improTemcnt in this
species of machinejj, and hence the first constnictioii for

donble speed *»6 338

Obscnalions on this conatnction **7 338

Beeond construction for donble speed . . . ■ 438 889

ObsorvtitioiiB on this construction *^9 840

Third construction for douUe speed *30 340

Obsemtioiis on this ooMtruction ^1 . ^1

ESSAY VII.

On the fisming of mill-woik ^ 34i

pRBPACB— relating to the moving parts of machinery . — 342

SxoTiON I.— Pecnliaritiea of framing of mill-work . . 433 342

Causes which subject it to speedy decay .... 433 343
Qualides which miU-work shonld poeseea to make it dnraUe,

alrmgA, sfj^uss, aitd mJuiHf 433 343

Construction should he such that any particnlar part may be

repaired or renewed with the least possble derangement

to the other parts 434 343

Of repkeing dufts 435 343

Fiiotion diminished by the elastic powv id madunety 436 344

Sktiok II. — Of the bearings of shafts .... 437 344
Of steps, bushes, bnaata, pillow hloeka, plumber hloc^ pe-

dsatals 437 844

The subslancee nasd fo pilktwa 438 844

iB^woTtueiits by Mr. Hnrray, of Leeds .... 438 345
Mstkods adopts^ by Oe Shd&eM griadcn in the oso of

tborotiat 438 845

A'stw upon nesal and wooden pillows .... — 345

Of the teims of Mi^w, anj of npngbt Aafb — first mode . 439 346

Seeondmode 440 346

Om«« in wUn^ tbe 1^101 and step do Mt awver w«U . 44t 347

TW <|y.fi>nn«d |«i«t 44S 347

BiMwab's Mod* t«f nuMSf fawit* in a iaid Vr wiaai of a

. 443 347

CONTENTS. Xllii

Art. Page

Breasts and bnshes 444 347

Fnnnels and spindles 445 847

Sktion III. — Of wooden framing — 848

Headstock framing 446 848

Fimming for lying shafts 447 848

Methods of framing the parts, and suspending the shafts from

a ceiling 448 849

Of the framing of upright shafts— of screws and wedges . 449 849
Of the framing of upright and lying shafts, connected by be-

Telled wheels 450 849

Bridge supported by doves 451 849

Respecting the decay of timber, and the means of prevent-
ing that decay 452 849

Sbction IV. — Of cast iron framing 458 850

The resistance of cast iron to compression . . . 458 850

Remarks on the uniform strength of cast iron . . 454 851

Of the strength of cast iron beams 455 851

Illustrations of sections of cast iron beams . . . 456 851

Limit of atrength \» fracture^ oi stiffness \s flexure . 456 851

Of feathered cast iron framing 457 852

Methods of making cast iron framing to imitate wooden

framing 457 852

Of wood and iron bridges for sustaining shafts . . .458 Z5Z

Hollow cylinder applicable in many cases .... 459 ^5Z

Headstock of cast iron 460 Z5Z

Various modes of suspending shafts from ceilings . .461 S5Z

Bleaching machine called squeezers . . . . . 462 853

Description of these squeezers 462 253

ESSAY VIII.

Ctoometrical and practical method for finding the centres of
gravity of mill- wheels; illustrated by examples, in which
two, three, and four wheels compose the system upon one

and the same shaft — 354

Method of finding the centre of gravity of two bodies . 463 354

Geometrical construction of this method — showing that the

centre of gravity is known in terms of the masses . . 468 854

Flncticd BxHen^^MuUipli/ either body by the whde distance be-
iweem Anreemiru: divide the product by the sum of the
1/ Cl# fmHetU witt be the distance from the centre of

Xliv CONTENTS.

Art.

gravity of that body opposite to the one by which the whole

distance is multiplied 464 356

Example to illustrate the rule 464 356

Analytical investigation when the weight of the shaft is in-
cluded 465 356

Practical Rule. — To twice the weight of either hody^ add the
whole weight of the lever or connecting har^ and mtdtiplg
the sum by the central distance ; then divide the product by
twice the mass compounded of the bodies and the bar, and
the quotient wiU be the distance of the centre of gravity
from that body opposite to the one whose double is employed

in the first step of the operation 466 357

Example I. — To iUustrate the rule, and shew the positions
of the wheels relating to the common centre of gravity of

the shaft 466 357

ExAMPLB lid — Bodies of unequal weight at the ends of the

shafts, to find common centre of gravity . . . 466 357

Of three bodies connected by an inflexible bar . . . 467 358
Demonstration of their common centre of gravity . . 467 358
Practical rule derived irom the demonstration . . . 468 358
Example showing how to find the common centre of gravity

of three bodies 468 358

Dr. Jamieson's method of verifying these results in his Me-
chanics for Practical Men 469 359

The cases of utility consistent with this theorem . . 470 360
Example to illustrate the position of an intermediate wheel,
or that the whole weight may be on the middle of the

shaft 470 360

When the distance is known or limited by situation, and the
common centre of gravity must fiedl at the middle of that

system 471 361

Example to illustrate this case, there being 3 wheels of im-
equal weights to be supported by a girder placed at the
common centre of gravity of the system . . . 472 362
Verification of the result now obtained .... 473 363
When the weight of the axle of the wheels is given . . 474 363
Example of 3 wheels of imequal weights on a shaft, and it
is required to find the common centre of gravity for a

support 474 364

Of the centre of gravity of four or more bodies situate in the

same right line • ... • • . • 475 365

Dcmonstntioa of lliis ciue, wliicb is but an cxtcasion of tbe
fonner ......... 475

—MuUiplg llie maffnitttdt or density of tach body by Us

Tttlive diitance from the htginning of the tyttem, and

' divide the turn of ike products by the. turn of lie bodies for

lie diilajiee of the centre of gravity sought . . . 476
EiAMPLK I. — Of four bodies on the some shaft, and it is

required to find their coraraon centre of support . . 476
Geometrical construutioD, to shew the example or similar

examples may be worked mechanically . . . 477

Example II. — Sliewing the exact distance of each of four

nbecls from the common centre of grarity . . . 477

Of the centres of gravity of cones, and of a conic frustnm . 479

Rule. — To the sum of the squares of the rmlii of tite two ends

add their prodari, then multiply the sum liy 4, and reserve the

remit for a dicisor. — To three tivitt tite tquare of the radius

^^ ef the greater emf, add the square of the radius of the less

^^L end, together vith twice the product of the radii, and mul-

^^P liply the sum by the height of the frustum for a dividend.

^^B — Then, divide the dividend by the reserved divisor, and

the piotienl will express the distance betieeen the centre of

magnitude of the lees end, and the centre of gravity of the

frustum 476

Of die centre of gmvity of the Burface of a cylinder . . 478
Of the centre of gravity of a circular arc ... 478

Of tbe centre of gravity of a parabola, Bemiparabolo, and
paiabobc conoid ........ 478

I Table of numbers, sqnares, cubes, square roots, and cabe
^H KWU 479

^1 APPENDIX B.

Remarks on the introduction of the slide principle in tools
and machines employed in tbe production of moctiinery,

by James Nosmytb 480

icwing tbe increased perfection of the workmanship ; ma-
I nnal dexterity could not bavc effected those productions . 481
'egeometrioal figures; \iie line, pUme, circle, cylinder,
aoA sphere 482

xlvi CONTENTS.

Alt. Page

The dexterity of the hand and eye of the workman . 483 394

Mechanical contriyances for holdingy applying^ and dvreding

the motions of a cutting tool 484 395

Accession of power hy the slide rest principle ... — 396
Comparison of this power to that of the steam engme itself . 486 397
Hhistration of the figures in the turning lathe . . . 487 398
niustration of the slide rest principle. — Fig. 1 representing
the system of hand turning hefore the introduction of the

rest 488 398

Tool holted firmly to the rest, which slides along at the com-
mand of the machinist, illustrated hy Fig. 2. . . 489 399
Method of communicating motion by the hand of the work-
man, or by the introduction of the self-acting principle,

explained in Fig. 3 489 399

Application of this operation where neither the hand nor the

eye of the workman can be used 490 401

The mechanical means of operating on the most ddicate or
most ponderous masses of matter by means of the slide
rest, are the results of the late Henry Maudsky's enthu-
siastic devotion to mechanical science .... 491 401
Application of the slide rest to other important processes in

constructiye science 492 402

The planing machine explained : it enables workmen to pro-
duce improved tools 494-496 403, 404

Figure 4 represents the general arrangement of parts exist-
ing in the planing machine 497-501 404-407

To the slide rest we are indebted for the planing machine . 500 406
Also in the screw-cutting machine we have simply a slide

rest This is illustrated by Fig. 5 501 406

Again, in the case of the wheel-cutting machine we have the

slide rest in full existence, as is shewn in Fig. 6. . 502-^04 407-410
Observations respecting the fbnn of tools employed for turn-
ing and planing iron, brass, &c«, together with remarks on
the hardening and tempering of such tools 505 410

Disgimms illustrating the fbius of tools for planing, or

shaving metal, &e. — 411

F^. I. Stfongth, but not acuteness -^ 413

Fig. 8. Acuteness, but not sUrfo^ — 413

F^. 3. Aoutenoss and strc^ngth — 413

Good therefore for pbnittg and laming .... — 414
lUustratien applied to the use of the joiner's plane — 415

CONTENTS. Xlvii

An. Page

Abo in the fonns of drilk 505 416

Explanation of a tool gauge, to ascertain whether any tool

be ground or formed to the proper angle . . . _ 417
This gauge will answer for every kind of planing or turning

tool whatever

Oeneral explanation of the plates 418-469

Ihdbz ..,•.. .... 471

ON

THE TEETH OF WHEELS,

ESSAY I.

PREFACE.

Led from situation, as well as curiosity, to attend ver}^ mi-
nutely to some parts of practical mechanics, one of the ob-
jects which early attracted the notice of the author of
the following short Essay, was the figure of the Teeth
of \^Tieels. He observed, that, in forming these teeth,
workmen followed rules for which they could assign no sa-
tisfectory reason. Nor did he then find in books the in-
formation he wanted : the subject seemed to him to require
a detail and simplification, which no English writer, with
whom he was acquainted, had given it. Afterwards, in-
deed, he found that some French mathematicians had
treated it with much attention. But their works, though
sufficiently clear to those who have studied mathematics,
are too abstract to be of general utility. In the following
Essay, therefore, such an elucidation of the subject has
been attempted, as might render it plain to the operative
mechanic — an object, which will appear the more import-
ant, the more we consider the great variety of useful pur-
poses to which wheel-work is applied.

d

1 PREFACE. QeSSAY I.

De La Hire and Camus are the two French writers,
who have treated most extensively this branch of mechan-
ics.— From the work of the latter, who has written more
accurately, and more fully, the author has borrowed
largely ; nor has he scrupled to take from others, whatever
he found to suit his purpose, and to make the fullest use of
the communications of his friends.

Of the methods followed, it will be sufficient to remark,
that the subject naturally suggested these two general di-
visions— First, the Principles of the Configuration of the
Teeth of Wheels : — Secondly, the application of these to
practice.

The first chapter contains the Principles — the second,
their Application, with certain modifications — 1st, to
Spur GeaVy under which arc comprehended the Wlieel
and Trundle ; the Wfteel and Pinion ; the internal
Pinion^ and the Rack and Pinion. — And, 2dly, to Bevel
Gear.

A third chapter is added, which contains a manner of
forming Spur Wheels^ upon principles somewhat diflFerent
from those considered in the preceding chapter.

In the following pages, no pretensions are made, either
to invention or profound investigation. The writer has
studied perspicuity alone, and will have completely at-
tained his object, if he has only been fortunate enough to
give such a view of the various kinds of teeth, as will en-
able the artist to form some judgment of their respective
merits, and to execute any of them with accuracy and ease.
For this purpose it has been his aim to divest every part
of the subject of obscurity, and to accommodate it to those
who possess not the advantages of a mathematical educa-
tion. But ho is far from saying, that they will not find
some difficulties, particularly in the first chapter ; nor will
they, i)erhai>8, fully understmid the truths it contains, till
they SCO their relation to practice pointed out in the sc-

BSSAT 1.3 PBEFACE. 11

cond. He found, lihat without becoming exceediagly pro-
lix, there was no aToiding the use of some mathematical
terms, but of these he has given definitions, either as
the terms themselyes occur, or at the conclusion of the
Essay*.

* This Pre&ce was written seycnJ years before the translation of Camus
mB pabliahed.

1

ErmrriONsr

1. When two toothed whcek act upon one another, the
greater is called the IV/teel, and the lesser the Pinion.

2. Instead of the pinion, the trundle is sometimes used,
Bach as is here represented. It is likewise known by the
s of lantern and wallower.

3 pinions and trundles are employed for the same
purposes, when the action of two wheels is spoken of, in
general, the trundle is comprehended under the name pinion.

4, The teeth of wheels and of pinions, are comprehended

r the general term. Teeth. Wliere the teeth are of

jhe same piece with the body of the wheel, they are called,

roperly, faef/t ; when they are each of a particular piece,

y aru called cogs. Tiie teeth of pinions are called leaoes,

and those of a trundle staves.

2 GENERAL DEFINITIONS.

V.

5. Whea the action of wheels is spoken of in general,
under the dame teeth, are comprehended teeth, (properly so
called,) cogs, leaves, and staves,

VI.

6. The straight line bf, which joins the centres bf,
of a pinion and wheel, which act together, is called the
line of centres.

VII.

7. When the line of centres bp is divided into two
parts, A B, A F, proportional to the number of the teeth
in the wheel, and in the pinion, these two parts, a b, a p,
are named proportional radii.

It may be proper in this place to show, in what manner
the line of centres is to be divided in the proportion of
the number of teeth in the wheel and pinion ; and for this
purpose, we shall denote the length of the line of centres
by I; the number of teeth in the pinion by p, and the
number of teeth in the wheel by w; then by the definition,

GENERAL DEFINITIONS. S

the line / is to be divided into two parts, having the ratio of
ptow.

Let or = the lesser segment, and y = the greater.
Then we have

py = WX9 and x + y = 1;
M by diAioi aad tr«»p»iti«., we obtaL

^= , and^ = /-ir/

P
and by comparing these values of ^, we get

(p + w)x = p I9 and therefore it is a: =— :£ — .
^ p + w

tin 7

and in like manner it is y = .

RULE

For the proportional radius of the pinion. Multiply the
length of the line of centres by the number of teeth in the
pinion, and divide by the number of teeth in both the
wheel and pinion.

For the proportional radius of the wheel. Multiply the
length of the line of centres by the number of teeth in the
wheel, and divide by the number of teeth in both the wheel
and pinion.

vin.

8. If from the centres b f are described, with the pro-
portional radii, the circles xa, ra; these circles represent
two cylinders, which touch in the point a as if they had
teeth infinitely small, or as if one of them were conducted
by the other by contaction only. These circles I shall call
proportional circles ; or, as they are termed by millwrights,
pitch lines.

IX.

9. The right lines, b k, f q, drawn from the centres of
the pinion and wheel, to the extremities of their respective
teeth, are called real radii.

b2

1 -^

CHAPTER I.

OF THE PRINCIPLES OF THE CONFIGURATION OF THE

TEETH OF WHEELS.

10. In the construction of machines, the proper forma-
tion of the teeth of wheels is an object of much importance.
Though experience may often enable the merely practical
mechanic to approach, in this respect, to some degree of
perfection, yet, being ignorant of principle, his work is
always conducted with uncertainty, and he generally pro-
duces machines expensive in working, and defectiye in re-
gularity, eflFect, and duration.

For when the acting parts of a machine are not truly
formed, it may be so loaded as just to be in equilibrio with
its work in the most favourable situation of its parts, but
when it changes into a less favourable situation, the machine
will stop, or at least, stagger, hobble, or work unequally.

The best figure, therefore, which can be given to the
teeth, is that which shaU cause them always to act equally
and similarly, in situations equally favourable, and which
shaU consequentiy give the machine the property of being
moved uniformly by a power constant and equal ; or, in
other words, ensure an uniformity of pressure and velocity.

Were the teeth of wheels infinitely small, their action
would be regarded as that of cylinders, simply touching,
hanng the property required. The finite and sensible
teeth gi^-en to wheels will, therefore, be of the most advan-
tageous figure, when one wheel conducts another, as if they
simply touched ; or when their pitch lines have in every
part of their revolution etjual velocities.

I

I

I

I

[ CHAP. I.] ON THE TEETH OF WHEELS. 5

That teeth have this property, when formed in a certain
manner, will be evident from the foUowinff proposition and
its connections'.

PKOPOSITION.

1 1. When teeth are of such a form, that a perpendicular
H E 1 1 (Fig. 2. p. 2.) drawn to the tangent to the edge of the
tooth in tho point of contact e, cuts the line of centres at
the termination a, of their proportional radii, their pitcft
lines shall have in corresponding places, equal velocities,
whether the wheel drives the pinion, or the pinion the
wheel ; that is to say, that they will move each other as if
they merely touched t.

The manner of fonning teeth of wlieels here refciTcil to by oar Author,
would ensure an equable communicatiou of power or motion in the imogin-
when the rubbiug parts have no seusibJe friction ; but in no other;
except it he possible to contrive a prat'ticable form for teeth baring the pro-
perty of Teoderiiig the friction uniform during the action of each pair of
loelJi : this has not yet been accomplished. Hence it appears that practical
men have not without reason been doubtful of tho adrantages of the kind
of t«eUi proposed by mathematical i^Titcrs ; for that the iriction of teeth is
sot uoifonD, Dr. Yonng has proved in a letter, which forms a Tolnable port
of thisEeaay, (see Art. 66 — 71.) And we have o practical proof of the
>uie thing b the unequal wear of teeth, (sec Art. 40.) The best means
of K<raidiiig the inequality produced by friction seems to be, to make the
h>Mh "oi smalt and as numerous as is consistent with strength and dura-
bfli*T," (Art. 65.) These limits, with reqiect to strength and durahiiity, I
will endeBTour to establish In the additions to Art. 121, and those following
it. And since, in adopting the principle of short teeth, tlie curved surface
of cad tootti will become so small tljat a circular arc may be employed in-
stead of the proper curve, we slmll, in the additions, point out the mode of
dMcribing ciicnlar arcs to answer this purpose.

+ For tint manner of drawing this perpendicular, see Art. 18.

{ This bnug a Fundamental proportion, it is of importance that it should
'ie •rcQ understood; wesbnll therefore, in this note, attempt a popular illus-

ition of it.

It u deoioiislroble tliat iho line H B (Pig. 2.) has the same proportion to
the line I r which a b Ims to a p. For tunce b b and f i are paroUcI, each of
them being perpendicular to u i ; it follows that the triangles asm and a f i

6 ON THE TEETH OF WHEELS. []CHAP. 1.

We shall now proceed to show, that the epicycloid gives
the property to the teeth of wheels required in the preced-
ing proposition, and shall begin with some definitions re-
specting that curve.

Before we proceed with our Author, it will be an advan-
tage to examine this proposition more particularly.

12. Let AH (Fig. 2.) be the direction of the force of the
wheel to turn the pinion, and b h a line perpendicular to a h,
drawn to the centre of motion b. The eflFect of the force
to turn the pinion will be directly as the distance of its
direction from the centre of motion, or as hb. Also, the
angular velocity generated will be inversely as the distance

of the direction from the same centre, or as — . * Conse-

HB

«

quently, the quantity of motion communicated to the pinion
is as — ; that is, in an invariable ratio ; but by the same

HB

reasoning it may be proved that the force of the wheel at a
is invariable ; and therefore, the pinion will be moved in
the same manner as if it were moved by contact at a, when
HA is perpendicular to the common tangent of the surfaces
in contact at e.

The same may be proved when the pinion drives the
wheel. But this, as well as the more detailed investiga-
tions of Camus (on the Teeth of Wheels, Art. 521.) and
his followers, neglects the eflfect of friction. Let the eflFect
of the friction of the surfaces be represented by :r, when
the pressure and velocity of these surfaces are each equal

are similar, or equiangular ; but the sides about the equal angles of equian-
gular triangles are proportional : therefore, it is

HB : IF :: ab : af.
Now let us suppose h b and i f to be levers, and h i a string, the one lever
pressing from the other, would act upon it with just the same force that the
pinion and wheel do at the point a, where the pitch lines touch ; or, in
other words, as if the circle x acted on the circle r, by means of a string, as
pulleys do on each other by a band.

I

CH-IP. 1.3 ON THE TEETH OF WHEELS,

to Hnily, or 1 ; then the ratio will be

1

H B ( i — J")

; which is invariable only when the friction is invari-
able. But when the teeth are very short, and formed so
that the motion would be uniform were the friction uniform,
it is perhaps the best practical method of forming teeth.

DEFINITIONS.

13. If upon the same immoveable plane are placed two
circles, CNP, calmk, (Fig. 3 and 4.) which touch each
other in the point c, and the former, with a supposed style
or tracer in its circumference at the point c, is made to re-
volve round the circumference of the latter, the style c,
during the revolution, will describe upon the plane calmk
Ibe curve cegdk, which is called an epicycloid. The
fpic^cloid thereibre, is a curve generated by a point in one
circle revolving about another, either on the concavity or
wnTCxity of its circumference, and thus it differs from the
I cycloid, which is generated by the revolution of a

s along a straight line. The cycloid, however, has
me times been assimilated with the epicycloid, by con-
lering the straight line as the circumference of a circle of
I the tUamcter is infinite.

\ 1-k The circle c n p, which, in revolvmg describes the
picycloid, is called the generating circle of the epicycloid,

1 the arc calmk of the immoveable circle, upon which

! generating circle revolves, is called the hase of the

Ofcloid.
\ Epicycloids are distinguished into two sorts, exterior

I interior.

8

ON THE TEETH OF WHEELS. [CHAP. L

Fio. 3.

III.

15. When the generating circle revolves without the
circle of its hase, as in Fig. 3, the epicycloid is called an
exterior epicycloid.

Fio. 4.

And when the generating circle roUs within the circle of
its hase, as in Fig. 4, the epicycloid is called an interior
epicycloid.

COROLLARIES.

I.

16. As the generating circle in revolving from its first
situation, c n p, to different portions, ae f, l gh, &c. ; applies

OS THE TEETH (

9

lluccesfflTely all the parts of its circumference to those of its

is evident the hsse, c a l m K, of the epicycloid h

I the circumference of the generating circle c n p c,

and each such portion, as c a, or c l, &c., of the base, is

equal to each part ea, or gl, of the circumference of the

I generating circle.

Hence a method of drawing the epicycloid, by deacrih-

ing the circles aef, lgh, &c., which have all the same

ndii as the generating circle cnp, and touch the base

I CALHK in anypoints a, l, &:c.; and by making the length

I of the arcs ae, lg, &c., taken from the points of contact

I with the base, equal to the arcs ac, cl, &c*.

Haring thus determined as many points, E, G, &c., as

MY be necessary, the curve c e c d k, which shall pass

ftbrough them and the point c, where the supposed style

i generating circle was supposed to begin its tract,

I shall be an epicycloid t.

II.

I"- \Vhen the generating circle cnp revolves within the
circle of its base, (Fig. 5,) and has for its diameter the
radius Bc of its base, the point c, the place of the style
wing the revolution of the generating circle, will always
mtiQue in the diameter c B k. Hence the epicycloid de-
irihed 6y the style c is a straight line, and a diameter of

* Id pncticc, tlik is most cosily done, and witli suifident accuracy, by
g cneli arc of tbe base, as at A c, into a number of Bmall equal parts,
d bv Betting off tbe same number upon each arc of tie generating circle.

1 1 To diKW tlie epicycloid mecbanically, make the circle of the boee and
M genemtiDg circle of wood, and linving lixod a tracer in the circumference

Ctke gmeniting circle, let the base remain at rest, and the tracer, during

a rolHag nf the genemting circle, will draw an epicycloid. In order to

o ciidcs move with more accuracy, a small piece of tape may have one

F its ends noded to the circumference of the one circle, and the olher cud

B the oiber drde.

10 ON THE TEETH OF WHEELS. [CHAP. I.

the circle of its hose*; and the circumference of the gene-
rating circle c n p being half that of the base, the commence-
ment c and termination k of the epicycloid, must divide
the circumference of the base into two equal parts, and the
diameter ab of the generating circle being half that of kc
of the base, when the generating circle is in the middle of
its progress, the point c must be in the centre of the circle
of the base c (or coincident with b) ; hence we have a point c
at the origin b, in the middle, and k at the end of the
epicycloid, which all lie in c k, the diameter of the base,
and the whole epicycloid may be considered as coinciding
with CK, the diameter of the baset.

Fig. 5.

III.

18. When the generating circle of the epicycloid, as in
Fig. 6, is in any position, a, e, b, touching the circumfer-
ence of its base in any point a, a straight liney drawn from

* Upon this principle a parallel motion has been constructed. It is used
by Messrs. Fenton, Murray, and Wood, in some of their smaller steam
engines.

For a short account of it, see Gregory's Mechanics, vol. ii. p. 265.

t It would carry us too far into mathematics for many readers, were we
strictly to demonstrate, that eyery point of the epicycloid must lie in the
diameter of the base ; what is said, howeyer, will satisfy them of the truth.
The mathematical reader will find a demonstration of this in ^^ Cours de
Mathematique, par Camus," iy. No. 538 ; or En^ish Translation of that
part which treats of the Teeth of Wheels, and from which Buchanan bor-
rowed laigcly in this part of his work.

f

CHAP. I.^ ON THE TEETH OF WHEELS. 11

the point of contact a to the point e, actiuiUy describing
Ae epicycloid^ will be perpendicular to it.

This will be evident by supposing the circles to be poly-
gons, having a great number of sides. For when turning
on any of the summits, the tracer describes a small part of
a circle from that summit as a centre, and will consequently
be perpendicular to it.

Pio, G.

IV,

19. Let us imagine in the same plane three circles, r, x,
T, Fig. 7> which touch in the same point a, and which con-
sequently have their centres, f, b, g, in a straight line, and
are moveable round their centres only.

Suppose a style fixed in the circumference of the circle
T, and that the three circles are made to turn by the move-
ment of one of them : if we make each of the arcs, ah, a c,
equal to a e, then the style placed in e shall have described
on the plane of the circle r, a portion cy, of an exterio r
epicycloid, and on the plane of the circle x, a portion h e
of an interior epicycloid.

12

ON THE TEETH OF WHEELS* [^CHAP. I.

Pig. 7.

The two epicycloids J c e, he, traced in the same time hy
the style e, touch in the point e. For the straight line ae,
drawn from the point a, where the generating circle y
touches its base rc, shall be perpendicular to the two epi-
cycloids, and the straight line h e shall touch the epicycloid
in the point e*.

V.

20. Let us next suppose, that the generating circle y
has for a diameter the radius ab of the circle x, within
which it is placed, and that the three circles, r, x, y, touch
continually in the point a, as in Fig. 8.

The interior epicycloid h e, which touches the exterior
c E, shaU be a straight line directed towards the centre b
of the circle x. Art. 16, and consequently a portion of the
radius b h, which shall always touch the exterior epicycloid
c E in the point e, where it shall be met by the perpendicu-
lar AE.

* Because any triangle wliich can be inscribed in a semicircle, is a right-
angled triangle. For manner of drawing a perpendicular on the end of a
line, see supplementary definitions, Art. 83.

CHAP. I.] ON THE TEETH OF WHEELS.

13

Pig. 8.

Hence it follows, that when the two circles^ r, x, touch
continually, and the one causes the other to turn by con*
tact at the point a, if we imagine a radius b h in the circle
X ; and haying made a c equal to a h, there will be described
by the point c, an exterior epicycloid ce, which has for a
generating circle y, the diameter of which is equal to the
radius bh, this radius bh, during the movement of the
circles r, x, shall always touch the epicycloid in the point £,
where this epicycloid shall be cut by the straight line a e
perpendicular to its curve.

Thus instead of supposing, that one of the two circles
R, X, turns forward the other by the point of contact a, let
it be supposed, that the one is made to push forward the
radius bh, of the circle x, by an epicycloid ce attached to
the circle a, and described by the movement of the circle
T, the diameter of which is equal 4o the radius b h.

One may be able thus reciprocally to make the epicycloid
c £, attached to the circle r, push forward by a radius b h
a circle x ; and by means of the epicycloid c e, and of the

14

' THE TEETH OF WHEBLSi

[chap. I.

radius, bh, the two circles, r, x, may be able to conduct
themselves as if put forward by the point of contact a *.

For suppose the radius, bh, and the epicycloid, ce, to be
teeth oi wheels, x and t ; and the perpendicular ae, from
the touching sur&ces in all situations, cuts the line of
centres at the termination a of their proportional radii.
But we saw. Art. H, that when this was the case, the pro-
portional circles must have equal velocities.

It is principaUjf from this, that we shall deduce the best
Jigure which can be given to the teeth of wheels and pinions,
when one part of the wheel and pinion, or of both, ought to
be a straight line tending to the centre of such wheel or
pinion.

VI.

@1. If in the same plane we have hut two circles, r, t.
Fig. 10, which touch in the point a, and if the movement
of the one communicate itself to the other, hy this point of
contact, any point e of the circumference of the circle t,
describe upon the plane of the moveable circle r, an epicy-
cloid CE.

Fio.

* To be Batiefied of this experimentally,
make aay two drcles of wood, aa in Fig. fi ;
to the ciTcomference of one of them a, fix *
piece of wood b, fonned into an epicycloid,
generated by a circle half the diameter of c
upon A as s base.

On the circle c, draw the line c d, and cnt
out the part bounded by that line and c b.

If you cause one of the circles to move
tke other by the parts b, c d, both circles will
bsTO the same velocity; as may be aacer-
l^ed by putting a mark oppodte-any pomt
in the circumference of each circle before they be^n to move, and anotber
after they stop, and the distance between ^rtiich, measuring by the arcs, wiQ
be found equal.

CHAF. lO

OK THE TEETH OP WHEELS.

Fio. 10.

Suppose this epicycloid attached to the circle r, it (the
epicycloid) shall conduct the circle y, pushiog it round by
the point e of its circumference, in the same manner as the
circle r might conduct the same circle t in commimicating
motion to it by the point of contact a.

And in like manner, the point E of the circumference of
the circle y, turns the circle r, in pushing it by the epicy-
cloid CE, supposed to be attached to b, in the same way
that the circle y would conduct the circle r in communicat-
ing its motion by the point of contact a *.

* The experiment to prove this is umilar
to the fonner, bnt with this diSerencc, that
in the circuinfereiice of one of them, a, is
fixed a fine needle, which is made to act
against a piece of wood, fonned into an
epicycloid, fixed upon the other, b, which
epicycloid is generated by a upon b as a
1»se.

16 ON THE TEETH OF WHEELS. [CHAP. I.

The same mode of proof applies here that did to the
corollary immediately preceding.

This last corollary enables us to determine the best
figure which can be given to the teeth of wheels j when the
pinion shaU be a trundle composed of staves.

We shall likewise determine from it the most advan-
tageous figure which can be given to the teeth of a pinion^
when the wheel shall have staves in place of teeth.

22. In addition to the properties of the epicycloid men-
tioned above, there are several others of a curious and sci-
entific nature, which may perhaps be not improperly intro-
duced in this place, although they may not be immediately
applicable to the construction of the teeth of wheel work.

1. If the generating and quiescent circle have to each
other any commensurable ratio, then is the epicy-
cloid thus generated both rectifiable and quadrable ;
that is, both its length and area are exactly deter-
mmable.

2. If the generating and quiescent circles are incom-
mensurable with each other, then the epicycloid is
unquadrable, but it is still rectifiable ; that is, the
area in this case cannot be foimd in finite terms,
although the length of the curve is exactly assign-
able.

23. To these we may also add the following rules for
finding the lengths of epicycloidal curves, and the areas
which they enclose.

RULE L

As the semidiameter of the quiescent circle, is to the
sum of the diameters of the two circles, so is double the
versed sine of the arc of the generant, which has passed
over any portion of the quiescent circle, to the length of the
epicycloidal arc generated by the point which touched the
quiescent circle or base at the beginning of the motion.

CHAP. I.] ON THE TEETH OF WHEELS. 17

When the whole arc is required, the versed sine becomes the
diameter of the generant The length of any arc of an in-
terior epicycloid is found in a similar manner, only using
the difference of the diameters in the second term of the
proportion instead of the sum.

RULE II.

To find the area of an epicycloid ; it is, as the radius of
the quiescent circle is to three times that radius, plus
twice the radius of the generant, so is the circular segment
AE, to the epicycloidal sector aec. Or, so is the whole
area of the generant, to the whole area of the epicycloid.
This rule applies to both the exterior and interior epicy-
cloid.

A general proposition for the area of all cycloids and
epicycloids is given by Dr. Halley, and is as follows, viz. :
That the area of a cycloid or epicycloid, either primary,
curtate or prolate, is to the area of its generating circle, as
the ram of double the velocity of the centre, and velocity
of the circular motion, to the velocity of the circular mo-
tion.

The same proportion holds good m reference to any parts
generated in those curves, and the analogous segments of
tte generating circle.

CHAPTER IL

OF THE APPLICATION OF THE PRINCIPLES OF THE
CONFIGURATION OF THE TEETH OF WHEELS.

24. Having endeavoured to show, tha4; an epicycloid is
a curve, whereby two circles may conduct themselves as if
put forward by the simple contact of their circumferences,
I shall now attempt a practical explanation of this curve,
in giving the best form to the teeth of wheels.

SECTION I.

OF 8PUB GSABS.

25. By Spur Oeers is understood wheels acting toge-
ther, and in the same plane, with their axes parallel ; under
this head the wheel and trundle come first to be con-
sidered.

OF THB WHBBL AKD TBUNDLB.

26. To determine the figure of the teeth of the wheel,
which depends always upon that of the staves of the trun-
dle, we shall first suppose the staves to be indefinitely small,
and represented (Fig. 12) on the end of the trundle by the
points. A, E, H, &c. : when we have found the figure of the
teeth proper to conduct the indefinitely small staves, (which
are used for demonstration only,) we shall, by means of
that figure, trace the true form which should be given to
the teeth of wheels to conduct trundles with cvlindric staves
of some magnitude. Thus the solution of this case» na-
turally di\-ides itsi^f into twi> parts.

CBAP. 11.3 OS THE TEETH OF WHEELS. 19

Fm. 12.

TO Tina THB nOURB OF THB TEETH WBBK THE STAVES ARE
IHDBFINITBLY SHALL.

37. Draw the proportional circles, c ac and e ak, and
divide each of them into the number of equal parts which
it should have of teeth'.

* Thii opnatioa ia called by millwrights tMinff tff the pitch. By tbe
|Htcli is undentood the dlstBoce between the centres of two condgaoiu
teeth.

so ON THE TEETH OF WHEELS* [CHAP. II.

We have seen*, if the circle cac, which touches the
circle e a e, have attached to its circumference an epicycloid,
c E, described by the point e of the circumference of the
circle e a e rolling upon the circle, c a c, the epicycloid con-
ducts the circle e a e by the point e, as if conducted by
contact at a, and consequently the circumferences of the
two circles shall have the same velocity.

The epicycloid, c e, is then the best figure which can be
given to the teeth of a wheel to conduct a trundle, the
staves of which are indefinitely small, and therefore must
move the stave e, in the direction from a towards e, until
a second stave arrive, and be taken in the line of centres
by a second epicycloid, a b, which shall in like manner con-
duct this stave, a, until the arrival of another stave, e, in
the said line: and thus the other staves of the trundle
shall be conducted by the other epicycloids of the wheel.

Here it may be observed, though perhaps already evi-
dent, that it is the convex side of the epicycloid which
must be used : for though it be useful in some machines,
to make the concave side of a single epicycloid conduct a
point of a single piece moveable on a centre, yet were a
number of teeth so formed, it would be impossible for them
to act on a number of staves, for they would be so hooked
and entangled as not to move forward in the smallest de-
gree.

Were it wished that the wheel should move the trundle
in both directions, it is obvious that each tooth of the wheel
should have its opposite sides, c e, l m, formed into equal
epicycloids.

As we have supposed the staves of the trundle indefi-
nitely small, were the teeth of the wheel also perfect
figures, and djually distanced, there would be no need of
other than indefinitely small spaces between the adjacent

"" See Chip. L Article 21.

[ CHAP. 11.3 ON THE TEETH OF WHEELS.

n

teeth of the wheel ; but as perfect precision is not to be
expected, a space more or leas, such as al, must be left
between them, to enable the wheel, notwithstanding the in-
equalities of the teeth and staves, to move the pinion.

We have hitherto supposed (he teeth uf the wheel con-
ducted by the staves of the trundle, hut it is evident, had
the teeth of the wheel the same figure, when conducted by
the staves, the wheel and trmidle would retain thp property
of moving with the same velocity. It may only be observed,
that the staves of the tnmdle conduct the teeth of the
wheel in approaching the line of centres, while the teeth
of the wheel conduct the staves of the trundle in their pro-
gress from that line*.

[XO nUD THE PIOURBS OV 1
OP TBB TBUtlBLB A

E TEETH OP THE WHHKL, WHEN
f CVt-lNDEHS OP A PIN

I

28. Consider the trundle at first as having infinitely

IsmaD staves, represented by the centres of the staves,
A, E, u, &c., and trace, as above mentioned, the teeth c L p,
A Q.s, &c. of the wlieel, as if it had to conduct a trundle
with infinitely small staves : observing to leave a small
space, such as a l, between all the teeth, in order that they
may act freely.

Describe, with the radius of the staves, upon the plane
of each tooth, as many small arcs as may be convenient,
ba\'iag all their centres in the two epicycloids which form
the teeth.

Trace, by means of these little arcs, two curves, such as
»o, so, parallel to the epicycloids, and then you will have
I inclosed the space, kos, which is the figure all the teeth of
I the wheel ought to have beyond its proportional circle.

• Soe Article 33 of this chupter.

1

t THE TEETH OF WHEELS. (JCHAP. II.

Fia, 13.

For if wc suppose, that the centre e, of a stave, is con-
ductetl by the tooth cpl ; the curve ro, which is parallel
to the epicycloid ci", and which is placed at the distance of
the radius of the stave e. shall always touch the circumfer-
ence of that stave.

Thus the cur%-e ro shall conduct the eylindric stave, as
if the tooth c r i. cundncted the centre of that stave, and
conisequently the twtlh ros, shall he a proper figure to
wnduct the trundle, with eylindric staves.

The eunvil ^larts of the teeth of the wheel, being deter-
mined as above, the spac«s ts >&> &c. should be cut out.

[ CHAP. II.] ON THE TEETH OF WHEELS, 'iS

in order to admit that part of the staves which extends be-
yond the proportional circle of the trundle.

ma^i

TO OESCRIBB THE TEETH C

' A WIIEEI.. FOR A
C1BCUL*B ARCS.

h

^H 29. L«t c D he the line of centres ; e e the pitch line of
^^ the trundle ; and f f that of the wheel ; and suppose the
cpQtre of the stave a to be in the line of centres c D ; then
place one foot of the compasses in the centre of the stave
A, and describe the arc be, which is the form of the tooth.
The part of the teeth of the wheel, within the pitch line,
I may be described with circular arcs as in the figure.

Teeth formed in this manner will not sensibly differ from

Ithose described according to the principles laid down in the

ding articles, when the length of each tooth is not

f greater than is necessarj-. The reader will easily perceive,

that the radius for describing the teeth, is equal to the

pitch diminished by half the diameter of the stave ; and

I aUo that the centres of those arcs will always be in the

I proportional circle, or pitch line of the wheel.

^me authors have imagined that the friction of the
I wheel and trundle might be reduced Ity making the staves

Q4t ON THE TEETH OF WHEELS. [CHAP. II.

revolve ; but it could not be effected so far as to balance
the extra labour of construction, and where the strain
would be considerable, it would become quite impracticable.
(See Emerson's Mechanics, prop. 119, rule 9 ; and Trans-
actions of the Society of Arts, voL xxxv. p. 128.)

Smeaton appears to have been very partial to the wheel
and trundle, when the trundle was executed with cast iron
staves ; these he recommended to be of an oval figure, and
made smooth by grinding them. (Smeaton's Reports, voL
i. p. 316 ; voL ii. p. 391 and 423.)

It may be demonstrated that the least real radius of the
wheel should be equal to the proportional radius added to
half the pitch; when the necessary allowances are made
for wear, (see Art. 43.) and when the staves are of the
same diameter as the thickness of the teeth. But when
the staves are larger than the teeth, as in the figure, a less
real radius is required.

Having considered the case of a wheel and trundle, with
cylindric staves acting together, we are now to explain that
of a wheel and pinion, two sides of the figure of whose
teeth are straight lines directed to its centre.

OF THE WHEEL AND PINION.

To find the figure of the teeth and leaves of a wheel
and pinion^ when thai part of the teeth and leaves^ which
lies within t/ieir respective proportioned circles are straight
lines directed to the centres of these circles.

30. Having set off upon the proportional circles, the
points G, Q, L, and o, o, h, ice, according to the thick-
ness of the teeth and leaves, draw lines from these points,
tending towards the centre of their respective circles, to
serve as the sides of the spaces between the teeth and be-
tween the leaves, the depth of which spaces must be such
as to give room for the action of the curved parts of the
teeth and leaves.

Then describe upon the extremities of the sides of each
tooth, epicycloids, such as qd, ld, with the generating
circle t, the diameter of which is equal to the proportional
radius of the pinion, upon the circumference of the propor-
tional circle of the wheel as a base.

The mode of forming the teeth being thus shown, that of
the leaves will be' plain, v being the generating circle of
their epicycloid, upon the circumference of the circle of the
proportional pinion as a base.

We have seen*, if the radius bh of the proportional
* Ch^. I. Article 20.

S6 ON THE TEETH OF WHEELS. [CHAP. U.

pinion, be pushed by an epicycloid cp, generated by the
circle y, upon the pitch line of the wheel, and projecting
therefrom, the pinion shall turn with the same velocity as
the wheeL

In the same manner it may be proved, that the same
eflFect will be produced, if the epicycloid o m m, attached to
the pinion, be pushed towards the line of centres, by the
radius, l f, of the wheel.

Lastly, the two opposite sides of the teeth, and those of
the leaves, ought to have the same figure, for the ease of
action, and to give the wfieel and pinion the liberty of being
moved in either direction.

From these principles it will be evident, that the figure
here given to the teeth, will make the wheel and pinion
move with perfect regularity.

31. To make that part of a tooth which is within the
pitch line or proportional circle a straight line, as proposed
by the Author, seems to be the most advantageous form,
because it causes least pressure on the axes.

When the teeth are small, and do not begin to act till
they arrive at the line of centres, the teeth of the wheel,
when the wheel drives the pinion, or the leaves of the pinion,
when the pinion drives the wheel, may be described by a
circular arc, of which the radius is equal to the pitch ; aod
of which the centre is in the pitch line of the wheel or
pinion.

This method will always enable a workman to execute
short teeth nearer to the true form than any pattern tooth
will enable him to do. Pattern teeth and compound
curves, are things that may on some occasions be very
useful ; where the teeth are long, and of considerable mag-
nitude in respect to that of the wheel or pinion to which
they belong. But in all the ordinary forms of wheel-work
such operations must consume an immense quantity of
valuable labour to attain even the same degree of accuracy

I CH,*P. 11-3 ON THE TEETH OF WHEELS. 27

that is at once obtained by means of circular arcs. When
a Iiattem tooth is neccssarj', one of its adjustments should
be the centre of the wheel ; and not two points in its cir-
cumference, as projrosed in Imison'a Elements of Science
and Art, (Vol. I. p. 103,) because the latter method at
least doubles the risk of error in adjusting the pattern.

Wlicn part of the action takes place before the teeth
arrive at the line of centres, the method of forming teeth
proposed by our Author, (Art. 41,) seems to he equal, if
not superior, to any other. And its practical application
is shown in Art. 41*2.

pini
poii
dra

. sbaj

REMARKS.

32. As it is the curved part of the teeth of the wheel,
it should push the straight flank hk, of those of the
pinion, in removing from the line of centres, and as the
point E, where the flank is acted upon, is a perpendicular
drawn from a, it shall be always that by which the wheel
sbaQ push, it is clear, that when the extremity- p, of the

licycloid c p, reaches the point e, it shall cease to move
tooth H K ; if the extremity p, arrive at the point e, be-

►rc the flank o n, of the following tooth of the pinion has
reached the line of centres, the curved part, o, m, m, of
this tooth, must be pushed by the straight flank l i, of the
following tooth of the wheel, till the flank o n, reaches that
tine : so that in this case, the wheel conducts the pinion,
at one time before, and, at another, beyond the line of
centres.

But wore it so, that the extremity p, did not reach the
point E, till after the flank, o n, hail arrived at the line of
centres ; it would not be necessary, that the curved parts
of the leaves should be pushed by the flanks of the teeth.
Thus, in this case, the wheel would conduct the pimon, in
pushing its leaves beyond the line of centres only.

3S. It is the general opinion of those who are in the

28 ON THE TEETH OF WHEELS. [cHAP. XI.

practice of constructing wheel work, that teeth ought, if
possible, never to begin to act before they reach the line of
centres, as that mode of action is thought to occasion much
unnecessary friction*. The cause of this great unneces-
sary friction, when the teeth are of wood, appears to be the
following :

Friction depends not only upon the pressiure made on
moving bodies, but on the inequalities of the surface upon
which they move ; and as the surfaces even of the most
highly polished bodies have some inequalities, whenever
two of them are pressed together, the inequalities of the
one must enter the other.

Suppose A and b to be a wheel and pinion, having wooden
teeth, as they would appear through a microscope ; it is
impossible, though there be no other resistance than that
arising from friction, to move them towards the line of
centres, until either the centres, on which the wheels turn,
give way, or some of the small inequalities, c, <f, of the
teeth be broken oflFf .

On the other hand, a very small force will move the
teeth outwards from the line of centres, as the small ine-
qualities, c, d^ and a, 5, may then slide over one another
without being broken ; for the teeth, when so working, are
mutually receding from each other in their point of contact,

* The increase of friction is not the only disadTantage from part of the
action taking place before the teeth arnTe at the line of centres. For when
a machine becomes wonu it causes considerable irr^ularity in its move-
mentSy in consequence of the action, in approaching the line of centres,
tending to spread the axes of the wheels ; while the action, in receding from
that lino, tends to draw these axes together ; and hence occasions more
inrc^ular action, friction^ and wear in the machine, in proportion to the wear
of the parts.

We hare an obvious remedy in making pinions sufficiently large to allow
of the action being wholly on one side of the line of centres.

t See a further iUuslratioQ of thb subject in ike Supplementary Obscrra-
tious* Art* 6$, &e«

HAP. 11.]

ON THE TEETH OP WHEELS.

and the wheels move on their centres with ease ; and where-
as, in the first case, they must have a tendency to force the
utres on which they turn outward, fi'om their true position,
I the second, they have no such tendency*.

, when the teeth are of metal, this unnecessary

friction seems to arise principally after the teeth arc in

some degree worn. (See Fig. 18, in p. 36.) The teeth in

that case have a kind of seat formed at their bottom, and

the curve at the outward extremity is too much inclined to

the radius, and very abrupt. In the action which takes

place before the line of centres, the eliding of the teeth of

; conducting wheel along those of the conducted, has a

ndency to accumulate hardened grease, dust, sand, &c.,

( the bottom, which getting between the abrupt extremity

■ the tooth and the seat at the bottom, become like the

Ebv-stone of an arch, and must require often a considerable

fcrce to bruise the teeth in this situation past the line of

ceutres.

• Tbc true eaiisc of tlic increase of frictioti is shown in Art. €7. Tbe
mnbon againKt the grain of lUa wood will ecoicely have a scDsible eSbct,
ncept when liia tceUi aic new.

iSm TBE TEETH 0& WHEELS. [CHAP. IL

Tbt» it appeanVy that the fridioD of teeth, approaching
the line of oentreB, u modi greater dian in receding from
it But in cases where the pimoo is smalls the action in
approaching to the line of centres, cannot be altogether
prevented. M. Camos, in his ^ Conrs de Mathematique,"
has demonstrated, that a wheel of 50 teeth cannot conduct
a pinion of 7 leaves, without their acting partly before they
arrive in the line of centres. He also proves the same with
regard to 57 teeth and 8 leaves, 64 and 9» 7^ su^d 10*.

34. Since the calculations which Camus has given are

confined to particular cases, and troublesome to apply to

new ones, because his method is indirect ; we will, in these

' additional articles, investigate a rule, which will be found

somewhat more general, and of easier application.

I^t B bo the centre of the pinion, and c the centre of
the wheel ; then we have to ascertain the relation between
the number of teeth on the pinion and the number of
those on the wheel, so that any tooth d of the wheel may
arrive at the line of centres before the preceding tooth of
the wheel quits the tooth a of the pinion.

The last point of contact cannot be beyond the middle
point of the tooth, and therefore the line a c will be in the
middle of it, a being the point of contact. Also ad will be
porpondieular to b a, the point d being in the pitch lines.

Now by the principles of trigonometry we have this pro-

_A» • • BC. sm. c Tfc * •
portion; ab : bc :: sm. c : sm. a = . But sm.

AB

A — sin. c = sin. b; therefore, sin. c (— — 1) = sin. b.

AB

The primitiw radius of a wheel or pinion being proportional
to the umuber of its teeth, it may always be represented by
that number ; and if n be the number of teeth in the wheel.

CHAP. 11.3 ON THE TEETH OF WHEELS. 31

Pio. B.

and n those of the pinion, we shall have b c = n + n. The
line B A, being a side of a right-angled triangle, considering
the radius unity, is found by this proportion; 1 : cos.

B : : n : AB = 71. cos. b. And the angle b is —

qnently, sin. c (— — 1) = sin. b = sin. c ( 360 )

AB " ""

N +W
( t

n cos. -

35. As the sin. c is, in practical cases, sensibly equal to
its corresponding arc, we may use the arc for the sine, which
will render the equation much more , simple. The arc is

a X 34161 3 _ 47124

i which being substituted fbr sin.

c, we have

47124 n (I -cos.—)

„ , .360 360 1 . 2x360.
Or, because sin. — x cos. — =~-9m.

(Gregory's Trigonometry, Chap. iv. Art. 20.)

32 ON THE TEETH OF WHEELS. [CHAP. II.

360^

9'4248 (1— COS.— )

N =

. 720 9-4248
sin.

n n

This equation gives a result nearer to the conditions re-
quired in the actual construction of wheels and pinions
than when the sine is used instead of the arc ; for it makes
an allowance for the wearing away of the teeth. It sup-
poses the teeth of the pinion to he the same size as the
teeth of the wheel, which is the least thickness that can
he given consistent with strength and durahility.

S&. The ahove equation shows, that when a pinion has
less than 10 leaves, it cannot he conducted uniformly hy a
wheel with any numher of teeth whatever, unless they act
partly hefore they arrive at the line of centres ; because
the denominator of the second member of the equation be-
comes negative when n is less than 10.

This is manifest, for when n = 10, the angle in the nu-
merator is 36 degrees, and that in the denominator is 72.
But the natural sine of 72^ as shown by the Trigonome-
trical Tables, is '95106 when estimated in parts of the
radius, and the second term in the denominator when di-
vided by 10, becomes •94248, which being less than the
sine of 72**, the difference between them is a positive quan-
tity, and will continue so for any value of n that is greater
than 10; but for any integral value of n that is less than
10, the second term in the denominator will exceed the
first, and consequently, the difierence must be negative,
and a negative number of teeth in a wheel is obviously
absurd.

But a pinion of 10 leaves may be moved uniformly by a
wheel with 209 teeth, when the whole of the action is
after they arrive at the line of centres ; for in this case

Bap. II.] ON THE TEETH OP WHEELS.

33

n= 10, andbyatableof§^ines,wefindcos.36'' = -80902; aud

8iii.7'2' = -95106. Therefore

9-4-i2-i-8(i --soooa)

= 209 = N.

■95106 -■94(<i48
In like manner it may be calculated that a pinion of 11
leaves may be moved uniformly by a wheel hav-iog not less
I than 28 teeth ; and a pinion of 1 2 leaves when the wheel has
Kit less than 15 teeth; so that the whole of the action
lay be after the teeth arrive at the line of centres.
137. The same equation may easily be extended to the
5 of the wheel and trundle, and as it docs not appear to
have been investigated, we shall here show the result
under the following conditions j 1. The extremity of each
tooth of the wheel is to be one-third of its thickness at the
root. 2. The centre of the stave is to be in the pitch line
when the point of the tooth quits the preceding stave.
When limittid to these circumstances the equation in Art.

I>

540 7-33

Whence it appears that a trundle with less than eight
staves cannot be moved uniformly by a wheel with any
Dumber of teeth whatever.

TTic same is true also of stave-formed teeth (see Art.
41.) when the wheel drives the pinion,

38. From what we have already said, it will be evident,
when the pinion consists of such a number of teeth, as to
be conducted uniformly by the wheel in receding only from
the line of centres, that, except in small numbers, the epi-
cycloid is necessary on the conductors only, whether it be
a wheel or pinion. For instance, in Fig. 16, which repre-
sents two wheels of equal numbers, a is the conducting,
and B the conducted wheel. But it is to be observed.

I

34

ON THE TEETH OF WHEELS. [CHAF. II.

when of two wheels acting on each other, sometimes the
one, and sometimes the other, is the conductor, the teeth
of both should be epicycloidal, as in Fig. 14. p. 25.

When the teeth of the conducted wheel or pinion are
acted upon by those of the conductor, in reccing only,
from the line of centres, it may be remarked, if they were
perfectly made, and of durable materials, it would be lu-
necessary to extend the conducted teeth beyond their pro-
portional circle. But these properties being unattainable,
and as the angles which terminate their sides, would be
apt to cut the conducting teeth, and occasion an irregular
motion, it is proper to form the extremities of the teeth of
the conductor, in the manner represented in the figure by
the dotted lines.

89. Sometimes it may be requisite to have but few teeth
in the pinion. In such cases, in the conducted, whether
wheel or pinion, Buchanan preferred staves to teeth, pro-
perly so called, or to leaves, because a trundle or wheel,
whose staves are cylindric, will be less acted upon in ap-
proaching the line of centres, and consequently have less
friction than a pinion or wheel, the sides of whose teeth
tend to the centre.

Thb will appear by Fig. 17, which represents a stave, a.

HAP. 11.3 ON THE TEETH OF WHEELS.

35

f a trundle, and a leaf, b, of a pinion, turning round on
•&e same centre, a, and a tooth adapted to each, turning on
a common centre, b. The thickness of each of the teeth,
and the proportional circle of both wheels, are the same,
and the proportional circles of the pinions are also equal,
and teeth are each made of the greatest length which the
intersection of the curves will admit, which turns out con-
siderably greater in the tooth adapted to the stave. The
shaded parts represent the tooth adapted to, and acting
upon, the stave ; and the dotted lines represent the tooth
adapted to, and acting upon, the leaf. The teeth, in both
cases, are represented as just at the point where they
would cease to move the leaves or staves uniformly ; and
it appears the stave is conducted considerably further bo-
yond the lino of centres than the leaf; hence the stave will
be less acted upon in approaching the line of centres.

■JO. A trundle has besides another considerable ad>-an-
tage over a pinion ; which is, that it wears much more

36 ON THE TEETH OF WHEELS. [cHAP. II.

Fio. 18.

equally. Every one experienced in wheel work knows,
that when a pinion comes to be considerably worn, the
leaves take somewhat of the form represented at a, which
is evidently the cause of a great deal of unnecessary fric-
tion, and strain in a machine. Whereas no such thing
happens to the trundle*. The trundle has, however, a de-
fect perhaps as bad, if not worse, that of weakness. Its
staves being supported at the ends only, arc not long in
use before they become quite unable to bear any consider-
able strain, and for this reason, it is now in a great
measure disused in machines. It however appears to Mr.
Buchanan, that a wheel might be made, which would com-
bine the advantages of both the pinion and the trundle,
and he accordingly had some wheels made on that idea,
and they appear to answer every expectation.

* This is a mistake, as trundles, in consequence of the surfacee of contact
b^ng Binal], become soon indented by pressure, and wear and cease to turn
round in their sockets.

CHAP. 11.3 ON THE TEETH OF WHEELS.

37

These wheels were made of cast iron. Thcv were each
cast of one solid mass. The upper figure represents the
t'Jge view, and the lower the section of one of them j
irliereby is shown the maimer in which the teeth are sup-
[lorted, like the staves of a trundle at each end, and like
the leaves of a pinion at the roots, but so verj' thin there,
as to run no risk of having the eommon fault of pinions
just now noticed. They were difficult to mould : but
were they to come more into use, he had no doubt in-
genious workmen would soon get over this obstacle*.

41. It is mentioned above, in cases where the pinion had

few teeth, that in the conducted, whether wheel or pinion,

£tavc8 should he preferred ; but it is obvious, that the

I method just described, of making a small trundle of cast

^Hpn, would not apply to a wheel of a great number of

^^Hves. Nor is it in that case so necessary, as the greater

^Tne number of teeth are, the longer they will be in losing

their proper figure. In such cases, therefore, staves,

strictly speaking, should not be used, but teeth made so as

' Bj casting separate plntes, with indentn
D togetber, &c jiinton might he made siillic
1 frequently in crane-work, wii
radng the wheels getting out c

0 fit the teeth, and bolting
:nt1y strung ; auch a method
re it has the iniporiMit wl-

ON THE TEETH OF WHEELS.

[chap.

to produce the same effect— that is, having their acting
parts of the figiure of a stave.

What is meant will be better understood by inspecting
the figure, where the lines show the alteration necessary

on the tooth a, in order to make it produce the effect of a
stave ; which stave is represented by the faint dots. The
dotted lines on d, represent the alteration requisite to
adapt it to the stave, it being necessar}', as formerly proved,
to have it a different epicycloid from what is required to
adapt it to a tooth, whose acting part Is a straight line,
tending to the centre of its proportional circle.

42. Teeth seem to be very well adapted for various
purposes, when formed on the principle recommended in
the preceding article. We therefore will endeavour to show
a simple method of describing such teeth.

CHAP. II.] ON THE TEETH OF WHEELS.

3U

lU

I

It most be observed that the teeth to resemble staves
are [0 be always on the conducted wheel or pinion ; thus
affonlinp the peculiar advantage of the wheel and trundle
iu I'itlicr increasing or diminishing velocity.

^V Let the teeth be divided as usual on the pitch lines, e k,
I PC; and on the conducted wheel c describe circles, as
ihough (here were to be staves. Conceive the centre of
one of these stave teeth to be in the line of centres at a,
;ini| draw the line a b joining the centres of the stave teeth,
riiun the radius a i, from the centre a, will describe the
curved side i c of the tooth of the conductor, and the
cuired part fi a of the conducted wheel. And since this
t is equal to the pitch diminished by half the diame-
■ the circlfi of the stave teeth, and the centres will
■ays be in the pitch lines of the wheels ; all the other
I may be easily described.

. The real radius, when the wheel is the conductor,
■ be vcrj- easily calculated with sufficient accuracy in

40 ON THE TEETH OF WHEELS. [CHAP. II.

this manner : Xet b dhe drawn towards the centre, so that
dcia equal to ^ of the tooth ; and make a d perpendicular

to bd; then we shall have ^/Ai* — AflP = 6rf. But a b
is f of the pitch, when the teeth of the pinion are of the
same thickness as those of the wheel, and a d is sensihly
equal to -5^ of the pitch ; therefore if p = the pitch,

p X >/ r^ — jTT =^bd.:=: '4714 p. And as in practice we

may always regard b d as the difference between the real
and proportional radius ; it being only a very small quan-
tity in excess*, we have this rule :

The real radius of a wheel, of the construction now de-
scribed, should be equal to the proportional radius added
to '47 times the pitch : and '47 times the pitch, is very
little less than half the pitch. But when the teeth of the
pinion arc thicker than those of the wheel, the real radius
may be less than is given by this rule.

The same rule applies to a wheel to drive a trundle.

This approximate method was chosen by Tredgold in
preference to a more accurate one, because the result is
exhibited in those terms which are most directly compar-
able with the proportions founded on practical experience.

When the pinion is the conductor, the real radius should
be the secant of the angle contained by the pitch, when
the proportional radius is considered the radius of that
angle.

Thus, c e is the angle contained by the pitch, and e d is
sensibly equal to the proportional radius ; but by the form
of the teeth r e is perpendicular to e d, consequently r d is
the secant to the arc or angle c e ; and the real radius re-
quired to impel the wheel till the succeeding tooth begins
to act.

* The excess is the difference between the radius and the secant of the
angle containing -J^ of the pitch.

^^tbeir sid
^V It noi

I

BAP. 11.3 OS THE TEETH OF WHEELS. 41

44. Hitherto the conducting teeth have heen considered,
as being always made as long as the epicycloidic tbrm of
tlieir sides would admit. This however is not always ne-

and in some cases may be improper.

It now remains to show, the smallest real radius a wheel
adapted to a trundle can have, without destroying the uni-
formity of the motion.

\VTien the stave e (Fig. 13, p. 22.) shall have been con-
ducted to the situation in which it is represented, the point
S of the following stave, shall be in the line of centres g f.
The stave a, in its turn, may then be conducted by the
following tooth, t y v, and then it shall no longer be abso-
lutely necessary, that the tooth n o s conducted the stave e.
The tooth k o s may therefore be terminated in the point
X, where it should touch the stave e, when the point t of
the following stave shall be in the line of centres, and the
distance x f of this touching point, from the centre of the
wheel, shall be the least real radius which can be given to
the wheel.

To determine the point x, draw from the centre of the
stave E to the point r, the straight line e t, and where this
line meets the circumference of the stave e, you have the
point required*.

45. The smallest real radius which can be given to a
wheel, adapted to the leaf of a pinion, or the teeth of a
vhccl, must evidently be terminated by that point a of its
tooth, which is in contact with the tooth or leaf e, after it
lias conducted it just until the tooth following beirins to
act : thus ab\% the smallest real radius of the wheel c t.

* Sec Art. SO. and 43., where eucL proportions as are applicable in
pi»ctic« are ^ven.
+ Wlicn the wheel isconductor, if we pursue the some mode of coIculBtioD

aa in Aft. 43, making p= the pitch, we shall haver X ^/\ = '553 p.

Tbal i>, wben the teeth of the conductor do not begin to act till they arrive

42 ON THE TEETH OF WHEELS. [CHAP. H.

But, in practice, perfect accuracy is not to be expected ;
and though it were even practicable to have wheels per-
fectly accurate when new, yet the moment they are put in
motion they begin to wear, and deviate from the true
figure of their teeth. It would therefore be attended with
bad consequences, to make the real radius no greater than
what we have here determined it to be, which is the least
which can be given. How much greater it should be, may
be determined by circiunstances*.

46. But it appears to us, that when wheels are made
with their conducting teeth only epicycloidic, and the mo-
tion is steady, there is not much danger of their being too
long ; for the longer the teeth are, the greater number of
them will be in action at the same time, and consequently
the strain will be more general, which will cause them to
retain their true form longer. Besides, though a tooth,
fit>m any accident, should be broken, the wheel will con-
tinue to go very well for a long time ; whereas, had its
teeth been short, the wheel would, by such an accident,
have been rendered useless. We are however aware, that
very long teeth are less able to sustain any sudden stress
upon their extremities t.

But even supposing the teeth, made in the manner above
described, to be no longer than those formed epicycloidic,
upon both the conductor and conducted, yet the former

at the line of centres, the real radius should be equal to "553 times the
pitch added to the proportional radius. But when the pinion is conductor,
the proportional radius is to the real radius, as the radius to the secant of
the arc equal to the pitch.

* Buchanan had been informed, that Mr. Watt drew the figure of teeth
with segments and points, what is below as well as what is above the pitch
lines, and that for small strains and great velocities, he used the pitch line
near the root of the driver, but upon other occasions, he used it just in the
middle, as giving the greatest number of tovA^hing points^ which certainly adds
to the strength of the wheels.

t See the last paragraph of Art 70.

CHAP. U.] ON THE TEBTR OF WHEELS. 43

will luTe less frictioii, and consequently wear longer than
the latter, and if th^ be of the same length, and the same

diickness at the roots, they most he equally strong. Let
Fig. 21 represent wheels having the teeth of both con-

ductor and conducted epicycloidic In Fig. S@, those of
the OHiductor only are epicycloids, and by inspecting the

44 ON THE TEETH OF WHEELS. [CHAP. II.

figure, it will be made obvious, that the teeth of a b, even
when the wheels are new, must act as much upon each
other, in approaching the line of centres, as they do in re-
ceding from it. Whereas the teeth of c d, when new, do
not begin to act until they arrive in the line of centres ;
and c conducts d much further beyond that line than a
does B ; and even when much worn, c d acts but very little
before the line of centres. But it was formerly observed,
that when pinions have few teeth, they must act before
they arrive in the line of centres, and consequently in such
cases, the teeth of both the conductor and conducted,
ought to be epicycloidic. Hence arises one of the dis-
advantages of wheels and pinions having few teeth, a fault
carefully avoided by every good mechanic.

It has been mentioned to me, that the following rule, in
order to determine the length of the teeth of wheels, is
employed by the ingenious Mr. Murray oi Leeds*.

BULB TO DBTBBMINB THB LBNOTB OP THB TBBTH OP WHBBLS.

47. Perpendicular to the line of centres c d, draw the
line A B, a tangent to the pitch lines. Take half the pitch,
that is, half the distance between the centres of two ad-
joining teeth, within a pair of compasses, setting their
points upon the pitch lines £ and f, parallel with the line
of centres c d, draw the line a 6, and where that is cut by
the line a b at c, gives the points of the teeth of wheel and
pinion t.

* This we are infonned was communicated to him by the late Mr. Rapp,
of Manchester, a native of Germany.

t This rule is founded on the properties of involute teeth ; for it may
easily be proved that the line a b will always be divided in the same ratio as
the pitch line divides the distance between the centres of the wheels ; and
that the pitch line always divides the length of the teeth in that ratio, in in-
volute teeth. Hence the observations in the following article of the text,
are to be understood as if made on involute teeth ; the properties of which
our author has not been much acquainted with. See Chap. III. Art. 62.

GBAP. II.3 OX THB TEETH OF WHEELS. 4^

Fkk 23.

OBSERVATIONS.

48. On this rule, Buchanan remarks that it does not
seem to he founded on any satisfactory principle ; were the
pinion, at all times, the conductor, he should not perhaps
differ ^m Mr. Murray, because the action of the teeth
would he, in that case, generally ojier their arrival at the
line of centres.

But in case the wheel were the conductor, the action of
the teeth would generally he almost entirely in approaching
the line of centres.

The evils arising from this mode of action, have already
been clearly proved, (see Art. 37,) it is therefore unneces-
sary here to repeat them.

When the wheel and pinion are nearly of the same

46

Oir TBI TEETH OF WHEELS. £CHAP. IL

diameters, as in Fig. S3, Uie effects are not so obvious as
when the pinion is much smaller than the wheel, as in
Fig. 24.

OF THE INTERNAL PINION.

49. When a pinion is to act internally, as in Fig. 25, it
is evident, that the teeth may be formed on the principles
already laid down, with this difference only, that the epicy-
cloid generated by the proportional circle of the pinion
upon that of the wheel, should bo an interior epicycloid.

The internal pinion may be adopted in many cases with
advantage, as it has less Motion than the external one*.

* A parallel motion upon this pnnciple has recently been erected at the
Buk of England.

CHiP. U.3 ON THE TEETH OF WHEELS. 4?

Vio. 25.

5a To illaatrate this, (See Fig. 26.) let a be the pitch
tine of a wheel, B that of an internal pinion, and c that of
an external pinion.

Suppose the drde a to he moved till the point a arriTes
U b, and that the points c t^ in the circles b c, have both

48 ON THB TEETH OF WHEELS. [CHAP. lU

moyed oyer a space equal to a b. Now it is evident, that
the distance from c to 6 is much less than that fix)m b to d^
and consequently had the circles moved one another hy
means of teeth, a tooth of the interior circle b, in the same
part of a revolution, would have slid over a smaller part of
a tooth of the circle a, than a tooth of the exterior circle c,
and therefore woidd have had less velocity. But other
things heing equal, the less the velocity, the less the fric-
tion ; an interior pinion has consequently less friction than
an exterior one*.

It is upon this principle, that hovelled wheels have less
friction than external spur wheels ; bevelled wheels acting
in a mean situation between external and internal spur
wheels t.

OF THB BACK AND PINION.

51. What is called the rack and pinion, is used for
various purposes in mechanics ; as m jacks for raifiing
great weights, and for the opening and shutting of sluices.

The rack and pinion should be made upon the principles
of spur gears ; with this difference only, that in forming
the teeth, the cycloid is, for reasons obvious from its defi-
nition, used in place of the epicycloid t.

Doctor Johnson gives this definition of the cycloid : " A
geometrical curve, of which the genesis may be conceived
by imagining a nail in the circumference of a wheel : the
line which the head of the nail describes in the air, while
the wheel revolves in a right line, is the cycloid.'*

Thus A B c, is a cycloid generated by the point a, in the
circle d, while it revolves on the right line a c §.

♦ See Art Q5.

t The notion that bevelled wheels have less friction is not correct ; unless
in the case where the teeth are in the concave surface of a cone, as in Fig.

■J

t See Art 71.

§ Velocity makes no difference in friction.

CHAP, n.3 ON THE TEETH OF WHEELS.

Pio. 27.

The subjoined figure represents the teeth of a rack and
pinion, formed in what seems the best mode in cases where
a great weight is attached to the rack.

The leaves of the pinion are made as long as the curve
will admit, in order to prevent them from beginning to act
before they arrive in the Une passing through the centre of
the pinion, perpendicular to the rack. Were they to act
much before they arrived in that line, which may be con-
sidered as the line of centres, and against a very great
weight, they would be apt to jam, and run the risk of their
being broken, or, at least, very much increase the friction*.

5S. The construction above proposed for the rack and
pnioD not being exactly correct, I will here endeavour to
remedy that defect

If a pinion move a rack, and the dotted line a b be the

* See thu Chi^ Art. 33 ; eleo the SupplemenUrj Observetioiia.

fitdt tine of the nek ; and tbe dotted ckde c ■ Ae ^tA
tine of the jmuoa ; then the cnmd ade c d «f Ae loaA
of the pnioa fboold be an iuTiJiile of a tinit. Or, k is

I
that curve which a point in a cord iroakl describe aa the
pinion, were the pinion turned by drawing the cord con-
stantly iu the direction a b ; the cord being supposed to be
wound round the pitch circle of the pinion. Now as b d
is the part of the cord which unwinds from the arc c b, it
is obvious that the pitch lines more with equal velocities;
and the force acting constantly at tbe same distance from
the centre of motion, the force wiU be constant, except
that variable part which is lost in ftiction.

53. In order that the teeth of the rack may be durable,
and not liable to cut the face of the teeth of the pinion,
they should extend beyond the pitch line of the rack;
and the teeth of the pinion should be of sufficient length
for one to move the rack, till the following tooth arrives at
the point b. To determine the length that will fulfil the
latter condition, or what amounts to the same thing, t«

CHIP. I[.] OS THE TEETH OF WHEELS. 51

d thp real radius of the pmion, wc may suppose a line
(imivu from the point r, to the centre e of the pinion ; also
make r b perpendicular to c e, and f d porpendiciUar to
Then, by similar triangles wo have f b : b d : : B E

e real radius — — But u d is equal to the pitch,

FB

d p B to five sixths of the pitch, therefore the real ra-

■ diiis=: . That is, the real radius should be six

5

J Was of the proportional radius.

In ordinan,- cases, the curved surfaces of the teeth of

the pinion may be described from centres in the pitch circle,

with the radius d b. And instead of making the teeth of

the rack square to the pitch line, they may be described

hv (he same radius a o, from points in the pitch line of the

ratk ; and the ends of the teeth as well as the hollows to

receive them, in the pinion may be semicircular. This

Diode of forming the teeth will make them very strong

n'tbout affecting the motion.

Si. liVhen the rack impels the pinion, the curved face

of each of the teeth of the rack, should be a portion of a

cycloid, (as a a. Fig. 27,) and the leaves of the pinion

straight lines radiating from the centre of the pinion ; the

diameter of the generating circle for describing the cy-

cloidal teeth should be half tlie proportional diameter of

the pinion.

SECTION II.

OP BHVBL GEAR.

965. Hitherto our inquirj- has been confined to what is
Bod spur gear, or the action of wheels and pinions whose
s are parallel i we come now to speak of what is called
lel gear, or the action of wheels of which the axes are
to each other. As we formerly regarded the
of gpur gear, with teeth indefinitely small, .ir the

52

ON THE TEETH OF WHEELS. [CHAP. II.

rolling of cylinders upon the surfeice of each other, we may
now regard the action of bevel gear with such teeth, as the
rolling of cones in a similar manner.

In order to illustrate this, let us suppose it is required
to make one wheel move another, the axes of which are not
parallel

Fio. 29.

c

Fio. 30.

Fio. 31.

>B

Fio. 32.

Let A B, A 0, be their axes, and d e, e f, their propor-
tional diameters or pitch lines.

II.]

ON THE TEETH OF WHEELS.

53

To the point a, where the axes intersect, draw a e, a f,
then DAE, and e a v, shall be the outline of two
"Cones*, which rolling the one upon the surface of the
other, will hoth revolve, so that, like two cylinders with
their axes parallelt, all the corresponding points in each,
shall move in every part of their revolution with equal
velocity.

For, suppose any touching point v, the diameters of the
cones at that point shall bear exactly the same proportion
to one another, that their bases do. The same may be
said of every other point on their surfaces, and conse-
quently they shall revolve in the same manner as two cy-
linders having their axes parallel, and the cones may be
considered as bevel wheels with indefinitely small teeth.

But, in practice, we require finite and sensible teeth :
in bevel gear, these are made similar to those of spur gear,
with this difference, that in spur gear, they are parallel ;
but in bevel gear they must, as is evident, diminish in
length and thickness, as they approach the summit of the
cone.

The teeth may be made of any breadth, according to
the strength required, and they are thereby enabled to
overcome a much greater resistance, and work smoother,
than is possible for a common face wheel and trundle,
which, for that reason, are now superseded by bevel gear.
5(J. The epicycloid, which gives the true curve to the
teeth of bevel gear, differs from that used in spur gear, in
being- generated by the rolhng of one cone upon the sur-
face of another, while their summits coincide.

* TIloM cones we shall call llie proportioiin] cones uF tliu wliecl and

f liia Uthc observed, when Uic
V^or of vDc of tlic COIIC8 becomes it
e 30. the ttoiut ji is ill the san

certain inclinations, tlie
II Figure 32 ; in others, i

54 ON THE TEETH OF WHEELS. [CHAF. IL

Thus for example, in the figure, the curve, a b c, is de-
scribed by a supposed style, fixed in the point a, of the
circumference of the base of the cone a d e, while it rolls
upon the cone f c a e.

The style a, being always at the same distance from the
point E, where the summit of the cone is fixed, all the points
of the curve a b c, shall be equidistant from the point e,
and consequently upon the surface of a sphere which shall
have the point e for its centre.

Hence the curve is called a spherical epicycloid.

The circle a g d h, which in rolling describes the sphe-
rical epicycloid, is named the generating circle of that
curve ; and the part a c, of the circumference upon which
it rolls, is called the base of the epicycloid.

When the sphere is given upon which the spherical epi-
cycloid is required to be traced, and we know the size and
position of the rolling cone which should generate this epi-
cycloid, it will be easy, from what has been said relative to
the plane epic}'cloid*, to find as many points of the curve,
as may be neccssai}', and it will be evident, that what is
said on spur gear, respecting the most advantageous figure
of their teeth, is all appUcable to bevel gear ; vrith this

• Sec Chap. I. Art. 17.

■. u.]

ON THE TEETH OF WHEELS.

fi'renec, that the sphcricaJ is substituted for the plane
epicycloid.

lu iirder therefore to avoid tedious repetitions, we shall
immediately proeccd to give some account of what spems
I tho best practical method of laying down the lines nocos-
sary to the right construction of bevel gear.

57. Ha\ing calculated the projKirtional diameters or
pilth lines of the wheel and pinion, draw their axea, a b,
A c, in the proposed direction with respect to each other.

lien the wlicels are in action. Parallel to a b, and at the

iBtance of half the proportional diameter of the wheel,
draw the line n e. In the same manner, draw f d at the
distance of half the proportional diameter of the pinion
from A c. From the point d, where these lines intersect,
draw the line d g, perpendicular to a b, and also the line
11 H, perpendicular to c a. Make g i equal to i d, and k h
equa) to k d. Then u o is what we shall call the priti-

pai diameter, or the dinmeter at the pitch line of the
«!, and D II that of the pinion.
\ Join G a, u a, u a. Then o a d, is the outline of the

»portional cone of the wheel, and n a h, that of the

L Now proceed to draw the teeth of ihe wheel. With the

56

ON THE T^TH OF WHEELS. [CHAP. H.

distance a a, from a, as a centre, sweep a smaU arc> sxich
as G a; at the priDcipal diameter to their extremity, set off
the length of the teeth, from otob, and draw b c tending to a.
The line h c represents the breadth of the teeth, which,
according to circumstances, may be more or less ; only it
is to be observed, that if continued to the point a, the
teeth near that point would be so small as to be of little or
no use.

Describe the arc c e, concentric to i a* ; and from o to

Ji set off part of the required length of the tooth, from the
principal diameter to the root: then dra.vij'g tending to
A, the line/g- becomes the root of the tooth. Parallel to

yg, draw ae, then a,fge, represent the section of the
solid ring of the wheel. The particular direction of the
line a e is no way essential ; all that is necessary is, that
the ring be of sufficient strength for the purpose to which
it is to be applied : but patterns for cast iron wheels are
usually made as represented in the plate.

Fig. 35.

' In practice, it is found easier, tuid sufficiently accurate, to use, instead
or these curtes, stnugbt lines, as near as may be, in the same direction with

caAP. II.J ON THE TEETH OF WHEELS.

Flo. 36.

I

^H Haii-ing thus drami a section of a tooth at g, draw in
^B^ Ae same manner one at d, then d, i, g, l, will be the sec-
tion of the wheel, in which e, h, i, l, a, represent the space
oaupied by the arms. The dimensions of these, and their
particular form, may however be varied according to
fircumstances.

The mode of drawing the section of this pinion will now
be obvious, by inspecting the figure, where it will be ob-
fierred, that the teeth of the pinion are made a little
broader than those of the wheel. This is a practice gene-
rally followed, as the teeth by this means wear more
i-qually than otherwise they would.

5S. For the use of young mechanics, I will, in these ad-
^^Udons, attempt to free the principles of constructing be-
^^blle<l wheels of part of their intricacy, first briefly noticing
^Hte old principles.

The teeth of bevelled wheels, for moving one another
uniformly, may be formed according to different principles.
"hose which have been delivered, are, first, When the
I drives the pinion, the acting faces of the teeth of
3 wheel, should be portions of a spherical epicycloid,
nerated by the revolution of a cone, (as described in Art.

58 ON THE TEETH OF WHEELS. [CRAP. II.

59)) of which the base is half the diameter of the
pinion, and the teeth of the pinion plane surfaces directed
to its centre. (Camus on the Teeth of Wheels, Art 569.
Brewster, Edin. Ency. voL xiii. p. 575.) Second, If the
teeth of the pinion be staves, or formed to act as stay^es,
then, when the wheel drives the pinion, the acting faces
of the teeth of the wheel should be portions of a curve pa-
rallel to a spherical epicycloid, generated by the revolution
of a cone, of which the base is equal to the diameter of
the pinion. (Camus, Art 560. Brewster, Edin. Ency. vol.
xiii. p. 575.) To these a third may be added, that is.
When the pinion drives the wheel, the most advantageous
form for the acting faces of the teeth of the pinion, will
be a spherical involute of a circle, the teeth of the wheel
being plane surfeu^s directed to its axis.

The description of these curves is not a very simple
operation, nor yet adapted for application .in practice ; I
shall therefore propose a new method, which appears to
have escaped the notice of former inquirers, one which is
general ; spur wheels, racks, &c. being particular cases of
its application. See Fig. Ej p. 59.)

59. If A B be the axis of a pinion, and a c the axis of
the wheel, d e the proportional diameter of pinion, and d f
that of the wheel, if c b be made perpendicular to a d ;
then, the proper form for the acting surfaces of the teeth,
may be described upon the surfaces of the cones deb, and
D F c. And since the surfaces of these cones can be spread
out, or developed upon a plane surface, the form of the
teeth proper for communicating equable motion, may be
drawn upon a plane.

60. But the developement of a cone is a sector of a
circle of which the radius is the slant height of the cone,
and the arc equal to the circumference of the base of the
cone. Thus, if the arc d g, be described with the radius
c D i this arc d g will be the developement of part of the

CHAP. II.] ON THE TEETH OF WHEELS,

59

circumference of the proportional circle or pitch line, of
which the diameter is f d, then d c g is part of the deve-
lopement of the cone, and in the same manner the deve-
lopement dbh, of part of the cone dbe, may be de-
scribed. For in practice, a small portion of the develope-
ment is sufficient.

Now, if the sectors that would cover part of the cones
DBE, and DCF, be considered portions of spur wheels,
and the teeth be formed, so that these sectors would move
one another equably, bv the methods described for spur
wheels, (see Art, 29, 31, 4.2, G3,) on sheet copper, or any
other flexible body that could be applied upon the sur-
faces of the cones, the outline of the teeth formed by
these patterns, would be such as are adapted for bevelled
wheels.

61. The breadth of the teeth do, being settled by the

60 ON THE TEETH OF WHEELS. (^CHAP. II.

parallel to c b. Then the form of the interior end of the
teeth may be found by spreading out the cones dcf^ and
dhe. But this is more easily done by making d k parallel
to the axis a c, and from the point k^ where this line cuts
c B, with the radius c Ar, describe the arc kg^ which is part
of the developement of the proportional circle at d. Now
lines drawn from the teeth on the arc d, to the centre c,
will determine the magnitude of the teeth at the arc k ;
and the teeth may be described there accordingly, as is
shown in the figure. Also, when di \b drawn parallel to
A B, we shall have b i equal the radius of the developement
of an arc of the proportional circle of the pinion at d ; and
on this arc the pattern teeth should be described for the
interior ends of the teeth of the pinion, as indicated by the
lines in the figure.

Thin copper will, be very well adapted for the patterns,
and it will be desirable that there should be two teeth
upon each pattern, but more will not be necessary.

The limit of the pitch will be the same as for spur
wheels, (see Art. 138,) and the breadth of the teeth is
also to be regulated by the same rules.

The length of the teeth, the friction of them, and the
peculiar advantages of the diflferent modes of forming them,
may be considered on the developed pitch lines in the same
manner as if they were the pitch lines of spur wheels ; con-
sequently every remark that applies to the one, applies to
the* other. Indeed, the only difficulty in this construction
of the teeth of bevelled wheels, consists in applying the
patterns correctly to the conic surface whereon the ends of
the teeth are to be described ; but it is a difficulty which
is very easily overcome by having proper lines on that sur-
face, to adjust the corresponding lines on the pattern by.

Perhaps it will be of use to make a few small models of
wheels, in order to fully understand the process, and apply
it with certainty of success. For though it is extremely

CHAP, n.3 ON THE TEETH OF WHEELS. 6I

simple to those accustomed to work by developed patterns,
such as carpenters, joiners, and masons*, it may not ap-
pear so, at first, to others.

* See the Art Joinsbt, Supplement to ihe Encyclopaedia Britannica,
{. 11—17; or Nicholsons Carpenter's Guide, p. 18.

6s ON THE TEETH OF WHEELS. [CHAP. HI.

CHAPTER IIL

62. We shall now proceed to describe a mode of forming the
teeth of spur wheels, the first hint of which, we have been
informed, was given by Professor Robison, of Edinburgh.
In order to understand the description and demonstration,
it will be necessary to recollect what was proved in Chap-
ter I. Art. 11, viz.: That if a wheel move uniformly, it is
necessary, in order to move another wheel uniformly, that
the form of the teeth be such as that the perpendicular
from the touching surfaces in all situaiions^ cut the line of
centres, in the same point a, which point divides the line
of centres, so that the one part a b, shall be to the other
A G, as the number of teeth in the one wheel b, is to the
number of teeth in the other g.

This being understood, let b and g be the centres of the
two wheels, and def, ghij the rings upon which the teeth
are placed ; the diameters of which rings are to one an-
other, as their number of teeth. If the thread kd^h^
lapped round the ring ; as it folds up, its extremity Ar, will
describe the curve klm. It is evident, that the thread in
describing the curve, is perpendicular to it ; and the thread,
^, Ijf^ A, is therefore perpendicular to the curve at /, or a.
In the same manner it may be shown, that the point n, of
the thread g n^ will describe a similar (5urve, nop^ which
is perpendicular to the thread gn^ hOj i a, at the points w,
0 and A. If therefore m, a, Z, Ar, be a curve formed by the
evolution of the ring b, and nop^he a curve formed by
the evolution of the ring of the wheel o, a line drawn
through the point of contact a, perpendicular to the touch-
ing surfaces, will touch both rings in the points f and f.

CHAP, in.] ON THE TEETH OF WHEELS.

63

w^K ad the two curres, supposing; thcra teeth, will act as if
^Bllieone pxilled the other by the thread / /.' The line ij".

wtD be the line of action ; that is, the teeth will always
touch each other in a point of this line, and since this line
alwATS passes through the same point of the line of centres,
H G, the actinn will be invariable ; so that if the one move
onifonnly, the other will also move unifonulj", and two
wMghts which balance in anj/ one position, will balance in
aU positions of the teeth.

It is obvious, that though these teeth must work, both
before and after passing the line of centres, that they will
work with equal truth, whether pitched deep or shallow ;
a quality peculiar to them, and of ver\' great importance.

(i3. The following properties of involute teeth arc of
most iniportane« in wheclwork.

I. When I he wheel drives the pinion, the greater part

64 ON THE TEETH OF WHEELS. [CHAP. III.

of the action will take place before the teeth arrive at the
line of centres.

2. When the pinion drives the wheel, they act chiefly
after passing the line of centres; and therefore involute
teeth are most adapted for this case.

3. When the teeth are long, the action is very oblique,
causing much unnecessary stress upon the axes ; particu-
larly when the wheel impels the pinion.

It may be remarked, that when a thread winds ofi^ from
one circle to another, and these circles touch one another
at the circumference, a point in the thread will describe an
epicycloid ; and the same epicycloid would be described by
making the circle from which the thread winds ofi^, the
generating circle, the other being the base.

The length of involute teeth may be determined with
sudfficient accuracy by the rule, Art. 47-

SUPPLEMENTARY OBSERVATIONS.

64. The foregoing Essay was written several years be-
fore the publication of the Supplement to the Encyclopa^a
Britannica. Professor Robison has there (VoL II. page
103, 106) described and recommended the mode which
will be found, Chap. III. of forming teeth of wheels by
involutes of circles. Dr. Brewster, however, in his second
Edition of Ferguson's Lectures, VoL II. page 227, ob-
serves, that this principle is not new ; De la Hire having
long ago considered the involute of a circle, as the last of
the exterior epicycloids ; which it may be proved to be, if
we consider the generating straight line as a curve of in-
finite radius*.

Professor Robison sayst, that ^* this form of teeth ad-

♦ See Art 71.

t Encyclop»dia Britannica, Volume XX. page 104. See also Rees's
Cydopcedia, Art. Clock Movement

Bap. 111.] ON THE TEETH OF WHEELS. t)5

mits of several teeth to be acting at the same time, (twice
the number that can be admitted in M. De la Hire's me-
thod.) This, by dividing the pressure among several teeth,
diminishes its quantity on any one of them, and therefore
diminishes the dents or impressions which they unavoid-
ably make on each other. It is not altogether free from
sliding and friction, but the whole of it can hardly be
said to be sensible. The whole slide of a tooth, three
inches long, belonging to a wheel of ten feet diameter,
does Dot amount to one sixtieth of an inch, a quantity al-
together insignificant'.

In the same article, this highly respectable philosopher
was mistaken, in supposing, with other eminent authors,
that the mutunl action of the teeth, (when formed into epi-
cycloids, by the method of M. Camus,) is absolutely witli-
ouijriction, and in saying, " That one tooth only applies
iUetfto the other, and rolls on it, but does not slide or
KUB an it in the smallest degree, TTiis makes them, last
long, or rather does not allmo them to wear" A very
slight examination of the figures given in various parts of
the preceding Essay, will, I hope, show, that the point of
contact must slide from the pitch line of the conducting
h outwards. Dr. Young, in his Natural Philosophy,
•Vol, II. page 183, says, that ** a form [of teeth] without
JHction, is perfectly impracltcable, although, fjr a single
tootht possible."

65. In the first volume of the same work, he makes the
foUofring judicious observations on our present subject :

" It baa been supposed by some of the best authors that
the epicycloida! tooth has also the advantage of completely
aToiding friction ; this is however by jio means true, and
it is even impracticable to invent any form for the teeth of
a wheel, which will enable them to act on other teeth
ithout friction.

• Sm Art. 70.

cont
Lloot;
fVo\.

66 ON THE TEETH OF WHEELS. []CHAP. III.

^^ In order to diminish it as much as possible, the teeth
must be as small and as numerous as is consistent with
strength and durability ; for the effect of fidction always
increases with the distance of the point of contact from the
line joining the centres of the wheels.

" In calculating the quantity of the friction, the velo-
city with which the parts slide over each other has gene-
rally been taken for its measure : this is a slight inaccuracy
of conception, for, as we have already seen, the actual re-
sistance is not at all increased by increasing the relative
velocity ; but the effect of that resistance, in retarding the
motion of the wheels, may be shown, from the general
laws of mechanics, to be proportional to the relative velo-
city thus ascertained. When it is possible to make one
wheel act on teeth fixed in the concave surface of another,
the friction may be thus diminished in the proportion of
the difference of the diameters to their sum.

^* If the face of the teeth, where they are in contact, is
too much inclined to the radius, their mutual fricticm is
not much affected, but a great pressure on their axes is
produced ; and this occasions a strain on the machinery,
as well as an increase of the friction on the axes.'^ *

The concluding part of these observations appears to me
peculiarly applicable to the figure of teeth described in our
last chapter ; for in wearing, they will be more liable than
many other forms, to have the face of the teethf where tihey
are in contact^ too much inclined to the radius.

REMARKS ON THE FRICTION OF WHEEL WORK, AND ON
THE FORMS BEST SUITED FOR TEETH,

IN A LBTTBB FBOM DB. TOUNO.

'66. ** I have been considering your observations on
the difiereuce of the friction, accordingly as the teeth

"* Youngs LccUiKs, VoL I. p. 176.

. in.]

ON THE TEETH OF WHEELS.

fi7

»

y

r

bmch before or after the line of centres ; at first I was
disposed to doubt of the fact ; but, upon more mature
examination, I found that, like many other practical oh-

tservations, they went beyond the scope of the doctrines of
theoretical writers. 1 cannot however perfectly agree with
you as to the explanation of the fact ; but I will state to
you briefly my opinion on the subject, not having leisure
at present to enter into a more ample discussion.

" The magnitude of the friction has usually been esti-
mated by the relative velocity of the surfaces concerned ;
a mode of calculation, which, as I have observed in ray
lecture on machinery, is only so far correct, as it shows the
comparative effect of a given friction in retarding the ma-
chine. But in fact the primitive friction itself is liable to
variation, according to the obliquity of the surfaces ; for
since the friction is nearly proportional to the mutual pros-
sure, it will he greater or less, as the direction of these
Borfaccs is more or less inclined to the radii, the force of
rotation being supposed to he given : and, what Is of still
more immediate importance to the resolution of the diffi-
culty in question, the direction of the force, by which the
one wheel acts on the other, is not to be considered as per.

.fendicuiar to the surface of the teeth, but as oblique to it,

%eing 8o situated as to oppose the joint residt of the direct
resiatance and the friction j that is, as being inclined to the
Burfaee in a certain constant angle, which a late anonj-mous
writer has called the angle of repose, and which is equal

|to the inclination of a plane, on which one of the sub.

hitances concerned would begin to slide on the other by its
grantatiim.

67. " Let the tooth a impel the tooth b with the given
fiirce A c, perpendicular to the common surface of the
; make cad equal to the angle of repose, then the
mu^t act in the direction a d, and making c d parallel
the radius a e, ad, wilt he the actual pressure : then

68 ON THE TEETH OF WHEELS. [CHAP* IH.

Fio. 38.

drawing d f parallel to the radius a 6, a f will be the effec*
tive force in the direction a c, and f c will be the loss by
friction. Again, if b impel a, the angle of repose must lie
on the other side of a c, and c h must be parallel to a g,
and H I to AE, and the friction in this case will be i c,
which is obviously less than f c.

" Hence we may easily calculate the magnitude of the
resistance f c, or i c, produced by friction, calling the force

A c unity ; for c d becomes -^ , and f c = c d. -I =

S. ADC S. CFD

5. CAD S. CDF S. CAD S. GAE I j • xi.

' and m the same

S. ADC S. CFD S. (CAE -f CAD) S. GAC

_ ^.CAH 5. CHI ^.CAD ^.GAE tj .1.

manner ic = . =— ^ n . . Both

^.AHC S.CIH 5. (G AC + CAD J ^.EAC

these quantities vary ultimately as the angle formed by the
radii, and vanish when the point of contact is in the line
of the centres ; and in this case the common theory agrees
with this calculation* When c a e is always a right angle,
as in the epicycloidal tooth commonly reconmiended, the

CHAP. Itl.3 ON THE TEETH OF WHEELS.

69

friction F c varies, in the different positions of the teeth, as
-1— : — , or as cdt. o a c ; that is, if .\ k be made constant,

(. GAC
as K L.

68. *' Since therefore it is demonstrable, that the friction
or pressure is always greater in approaching the line of the
centres, than at an equal distance beyond it, it must ob-
viously be desirable that the contact should be rather after
Ihan before the passage of the teeth over that line, although
it is better that it should be at a small distance before,
than at a much greater distance beyond it. Hence the
impelling teeth ought to be of such a form, as to accelerate
the motion of the impelled a little before, and a little more
after, the passage of the line of the centres, and then to
retard it again, so that the next tooth may succeed to a
similar operation.

69. ** A wheel acting on a trundle, with cylindrical
staves, has in this respect an advantage over two wheels
with teeth, since the curve, fitted for impelling the trundle,
\a adapted only to act on it beyond the line of the centres.
This curve may however be formed more easily, and at the
same time more advantageously, than by the method which
has hitherto been recommended : for if we employ an epi-
cycloid described by the rolling of a circle, which would
just touch the internal surface of all the staves of the
trundle, on the circumference of the wheel, the trundle
irill at first be accelerated a very little, and will then bo
allowed to fall back from each tooth to the succeeding one,
soon after its passage over the line of the centres. The same
form will also answer verj' well, when the trundle is to im-
pel the wheel, although this mode of action produces a

_ greater friction than the former.

^K 70> *' A similar advantage may be obtained in teeth of
H|Dy other form, by finishing them in such a manner as to
^hrqject a very little beyond the regular outline, at the point

70 ON THE TEETH OF WHEELS* [CHAP. UU

which is intended to come into contact a little beycmd the
line of the centres. Such a corrected outline may he do-
scrihed at once, if it he required. If the tooth is to he
formed into an involute of a circle, having fitted a thread
or fine wire to the circumference of the wheel, find the
point of contact at the instant when the end of the wire is
describing the part of the tooth which is to act at, or a
little before, the line of the centres ; cut off fix)m the wheel,
beyond this point, an arc equal to the distance of the
centres of two adjoining teeth, and fix a pin in the tangent
at the same point, that is, in the continuation of the part
of the wire which is unrolled, at such a distance as just to
streteh the part which is left loose by the removal of the
arc : the pin thus fixed, and the remainder of the circle,
will serve as bases for continuing the evolution of the wire,
and the description of the tooth. The same position of the
wire will show the outline of a basis proper for describing
by means of a circle rolled on it, the curve which must be
substituted for the form of any epicycloidal tooth, which
might have been described by causing the same circle to
roll on the simple circumference of the wheel as a basis ;
the curved part of the tooth beginning, in this case, at the
point of contact first mentioned.

" If it be objected, that in such an arrangement, the
equability of the motion would be lost, and a shake would
be created ; it may be answered, that the inequality would
be utterly imperceptible in practice. But I do not know,
that the form, thus determined, would have any material
advantage over teeth made as short as possible, or so cut
away as not to act before the passage of the line of centres,
which may easily be done in all cases, nearly in the same
way as you have shown with respect to epicycloidal teeth*

" The advantage of dividing the pressure among several
teeth ought not to be purchased at the expense of an in-
crease of friction, since the property of greater durability

n

CHAP. Ul.] ON THE TEETH OF WHEELS. 7 1

may be obtained, in an equal degree, by simply making the
wheels thicker, without materially adding to the friction :
uuJ in fact, although the momentary pressure on each
tooth may be leadened by dividing it, yet its duration is in-
creased in the same proportion.

71 . "I must beg leave to observe, that the form proper
for the teeth of a pinion, acting on a rack, is the involute
of a circle, and not a cycloid. The cycloid would be a
proper form for the teeth of the rack, if they were intended
to impel the pinion.

" It has been remarked that the form of the involute of
a circle is not immediately deducible from the general
principle of La Hire ■, and the remark is strictly true, since
the curves, formed, according to that principle, fi-om two
contiguous circles as bases, could not act on each other
without a further separation of the centres, which would
render the demonstration inadequate. But I have ob-
sen-ed in the Additions to my second volume, p. x. the
principle may be extended to any other curves, as well .
as circles and straight lines : and if we employ an equian-
gular spiral, instead of a straight line, we shall have the
involutee, exactly as they are recommended for practice."

SUPPLEMENTARY DEFINITIONS.

72. An angle is the inclination of two Unes to one an-
other which meeting do not lie in one line.

73. A triangle is a figure contained by three straight

72

ON THE TEETH OF WHEELS. |^CHAP. lU.

74. A circle is a plane figure contained by one line,
whicli is called the circumference, and is such, that all
straight lines drawn from a certain point within the figure
to the circumference, are equal to one another, and this
point is called the centre of the circle.

75. The radius of a circle, is a straight line drawn from
the centre to the circumference. The word radii is used,
when more than one such line is spoken of.

76. The diameter of a circle, is a straight line drawn
through the centre, and terminated both ways by the cir-
cumference.

77- The arc of a circle is any part of its circumference.

GBAF« nu} ON THE TEETH OF WHEELS. 73

78. A chord of an arc, is a straight Ime joming the two
extremities of the arc

79« A tangent of a circle is a straight line, which pasf
through a point in the circumference without cutting it

80. A polygon is a figure, having more than four sides.
The term is seldom applied to figures that have less than
fiye sides.

81. Parallel straight lines are such as are in the same
plane, and which, heing continued ever so far either way,
never meet

83. The word perpendicular is the same with square, as
used by workmen.

In order to draw from a given point, a, in a given line
B c, another line perpendicular to it.

Take a e, equal to a f, and from the points f and £, with
any radius greater than a b, make the intersection d ;
draw D A, which will be perpendicular to b c.

74

OK THE TEETH OF WHEELS* [cHAP. HI.

y<ru

\

p ~ E

83. If it be required, from the end of a given straight
line A B, to raise a perpendicular,

Take the point c nearer to a than b ; about the centre
c, with the radius c a, describe the circle, e a d ; through
the points e and c, draw the Une £ d, and join a d, which
will be perpendicular to A B ; and bad, is called a right angle.

E V

7^ A

84. To let fall from a given point a, a perpendicular
upon a given straight line b c.

About the given point, describe a circle cutting b c in e
and F ; from the points e and f, make the secticm n, an4
draw the line a d from a towards n, and a d is the pw-
pendicular required.

B-

^

7f

MAP. III.]] OV THE TEETH OF WHEELS. 7^

85. A cylinder is a body having two flat surfaces, and
one circular. For instance, a roller is a cylinder.

cone is a solid body, of which the base is a circle,

and which ends in a point.

m

K

^V 87- ^'^elocity is a term equivalent to speed.

88. Mr. Smeaton thus defines the term power : " The
word power, as used in practical mechanics, I apprehend
to signify the exertion of strength, gravitation, impulse, or
pressure, compounded with motion, to be capable of pro-
ducing an eflect ; and that no eflect is properly mechanical,
but what requires such a kind of power to produce it."*

89- A proposition is a sentence in which any thing is
affirmed.

90. A corollary is an inference or deduction.

91. The extract from Smeaton's works in Art. 88, con-
veys very little information respecting power ; and yet it is
necessary that everj- mechanic should have correct ideas
on this subject, which will be a sufficient reason for intro-
dudug a further explanation here.

iti>B'5 Minecllancous Papers, p. 30. See An. 91.

76 ON THE TEETH OF WHEELS. [CHAP. HI.

92* Power is the general term for that which causes
motion or rest For hodies in nature are in a state of rest,
only, when the opposing powers acting upon them are in
equilibrium.

But this general term, power, is divided into several
particular ones according to the circumstances under which
it acts.

93. When power is, or can be, balanced at rest, it seems
to be most proper to call it force ; but, to distinguish more
precisely the circumstances of its action, it is necessary to
employ the simple terms, weighty pressure^ and stress, and
also the compound terms, force of attraction, force of gra-
vity, cohesive force, centripetal force, centrifugal force, and
others of a like nature.

94. But when a body is in motion, its power, at any in-
stant, or at any point in its path, is usually termed fna-
mentum, or moving fprce, or quantity of motion. It is this
species of power which Sir Isaac Newton makes the object
of his second definition. (Mathematical Principles of Natu-
ral Philosophy, Book I.) Some writers propose to use the
term energy instead of momentum, (See Edin. Rev. voL
xii. p. ISO,) but there does not appear to be sufficient rea-
son for adopting it, the other having been in a consider-
able degree restricted to this species of power*.

Here we take the liberty of remarking, that neither
the measure of momentum nor that of any other kind of
power, has any relation whatever to time ; for momentum
simply expresses the quantity of power in a moving body at
a particular instant, without reference to the rate of accu-
mulation, or to the efiect it would produce ; and similar
remarks apply to other species of power.

95. It is further necessary, both for practical and scien-

* The tenn energy has also been applied to the product of the mass of
the body into the square of its velocity. See Dr. Young's Nat. Phil. VoL
II. Art. 347.

HAP. HI.3 ON THE TEETH OF WHEELS. 77

tlBc purposes, to have a term to designate that power which
is equivalent to momentum, wlien the velocity is uniform.
Smealon employed the term rnevlianical power for this piu--
po9e ; and since this term is sanctioned by the language of
all writers on the first principles of meclianics ; and the
simple machines, by means of which such power is modified
to produce the desired effect, have always been called the
mechanical powers*. We think it will be found desirable

use the term mechanical power in preference to any
«ther that has been proposed. The term impetus is ob-
jectionable, because it indicates a degree of violence in the
action of power, which does not agree with what takes
place in the most common applications of mechanical powei-.
And it is questionable, whether its proposer did not intend
it to be a measure of effect.

We must now attempt to inform the reader, more par-
ticularly, of the circumstances to which these different mo-
difications of power apply, and in so doing, we shall have
occasion to place a most interesting department of mechani-
cal science in a different light from what it has been regarded
by my predecessors.

JK>. Force is immediately comparable with the weight of
a quiescent body. Its intensity, direction, and equilibrium,
are the proper objects of that part of mechanics called sta-
tics, or hydrostatics, and aerostatics, when the body exert-
ing force is fluid.

97- Alomentuin, or the force of a moving body, is pro-
portional to the quantity of matter in the body, multiplied
by its velocit)' at that instant when the comparison is made.
Its rate of increase and decrease, its direction, and equili-
briara, are the proper objects of those parts of mechanics
called dvnamics and bydrodjiiamics. In fact, statics is

* See Dr. Jamieson's Mecbanics for Pmctical Men, oomprisiDg Treatises
00 the CoiD[Hieition and Hesolution of ForceE, the Centre of Gravity, and
the Mcchiuiic»I Powen.

78 ON THE TEETH OF WHEELS. [CHAP. III.

only that particular case of dynamics when the velocity is
nothing. In like manner we simplify an important part of
the science of mechanics hy separating all prohlems in which
the velocity is miiform ; hecause in that case the length of
the line the body moves over, is proportional to the velo-
city.

98. Mechanical power then is, a particular name for the
momentum of a body in uniform motion ; in that case, it is
proportional, to the quantity of matter in motion, multiplied
by the length of the line through which it acts ; conse-
quently, in aU problenis where the motions are uniform,
(and there can be no difficulty in distinguishing such pro-
blems,) this measure of power may be employed, and its
motion, equilibrium, and direction, determined accordingly.
Every person conversant with the management of such pro-
blems must be aware of the advantage of this mode of in-
vestigation ; it applies to the motion of water-wheels, of
wind-miUs, of rivers, the resistance of fluids, &c. &c, and
in general to the motion of machines. It has often been
partially applied in considering the equilibrium of mechani-
cal powers, (see Wood^s Mechanics, prop. xxx. and xxxi.,)
but we are not aware of its having been previously pointed
out as a general principle, with the object of forming a dis-
tinct branch of mechanics. Some writers have confounded
measure of power with measure of effect so far as to sup*
pose, that mechanical power is identical with the quantity
of matter multiplied into the square of its velocity; we
hope the true nature of mechanical power is here so de«
fined as to prevent a recurrence of a like mistake.

It is much to be regretted that power has not been made
the basis of all mechanical science, in the place of motion;
for motion is merely an affection or mode of matter acted
upon by unbalanced force. For, in the practical applicsr
tion of mechanics, it would prevent error ; and in the theory
of mechanics, that interesting phenomenon» the aocumuhh

CHAP. III.] ON THE TEETH OF WHEELS. 79

Hon of power ^ must have been forced upon the attention of
philosophers.

99* Buchanan had long employed himself in making
a collection of facts respecting wheels actually in use in
millwork, and, by arranging them agreeably to his own
views, draw such useful practical inferences as might bene-
fit workmen generally. All the facts he had been able to
collect and arrange will be found in the following Chapter.

Time and other circumstances did not allow him to enter
more minutely into the subject. Yet these hints, even in
their present state, may lead to a fuller investigation, and
they will not be altogether without some advantage, espe-
cially as nothing exactly of the same kind has hitherto been
published in this country.

With respect to the elementary propositions which guide
U8 in this inquiry into the proportional strength of the
teeth of wheels, Buchanan did not enter into their demon-
strations. To the artisan, unacquainted with mathematics,
they would be unintelligible ; and the mathematician can
either demonstrate them himself, or have recourse to those
elementary writings where the demonstrations may be found :
of these last, as being more generally accessible, we refer
to " Emerson's Mechanics," quarto edition ; and to the
volumes of the "Encyclopaedia Metropolitana" comprising
machinery, and edited by Professor Barlow, of Woolwich ;
also Dr. Robinson's " Mechanical Philosophy," as edited
by Brewster, and the excellent paper of Mr. Willis, on the
** Teeth of Wheels," published originally in the second
volume of the Transactions of the Institution of Civil En-
gineers, London, 1838, and added as an Appendix to this
work of Robertson Buchanan.

CHAPTER IV.

A PRACTICAL INQUIRY RESPECTING THE STRENGTH AND
DURABILITY OF THE TEETH OF WHEELS USED IN MILL-
WORK.

100. Having treated of the forms of the teeth of wheels,
we come now to consider their proportional strength with
relation to the resistance they have to overcome.

We are aware that, owing to a great variety of circmn-
stances, this subject is involved in much difficulty, and that
it is no easy task to form any general rule with regard to
the pitches and breadths of the teeth of wheels. We do
not pretend to more than a mere approximation towards
general rules ; yet, were this judiciously done, we are of
opinion, that it might be useful to the millwright, who has
not had leisure or opportunity for scientific inquiries. A
rule, though not absolutely perfect, is better in all cases,
than to have no guide whatever.

And it is too evident to require proof, that it is essential
to the beauty and utility of any machine, that the strength
and bulk of its several parts be duly proportioned to the
stress or wear to which the parts may be subject.

Some general observations on the wheel work of mills,
will serve greatly to simplify our inquiries on the subject.

GENERAL OBSERVATIONS ON THE WHEEL WORK OP MILLS.

101. Mistaken attempts at economy have often prompted
the use of wheels of too small diameter. This is an evil
which ought carefully to be avoided. Knowing the pres-

CUAP. IV.] ON THE TEETH OF WHEELS.

stire on the teeth, we cannot with propriety reduce the
(iiameter of a wheel below a certain measure.
Suppose, for instance, a water wheel of 20 horses' power,
I moving at the pitch line with a velocity of 3^ feet per second.
' It IB known, that a pinion of 4 feet diameter, might work
uito it, without impropriety ; hut we also know, that it
Hould be exceedingly improper to substitute a pinion of
only one foot diameter, although the pressure and velocity
at the pitch lines in both cases would he, in- a certain sense,
the same. In the case of the small pinion, however, a
much greater stress would be thrown on the journeys (or
journals) of the shaft. Not, indeed, on account of torsion
or twist, but on account of transverse strain, arising, as
well from greater direct pressure, as from the tendency
which the oblique action of the teeth, particularly when
somewhat worn, would have to produce great friction, and
to force the pinion from the wheel, and make it hear harder
on the journals. The small pinion is also evidently liable
to wear much faster, on account of the more frequent re-
currence of the friction of each particular tooth.

That these observations are not without foundation, is
knairn to millwrights of experience. They have found a
great saving of power, by altering corn mills, for example,
from the old plan of using only one wheel and pinion, (or
trundie,) to the method of bringing up the motion, by
means of more wheels and pinions, and of larger diameters
and finer pitches.

The increase of power has often by these means been
nearlv doubled, while the tear and wear has been much
lussened ; although it is evident, the machinery, thus al-
tered, was more complex.

The due consideration of the proper communication of
ihe original power, is of great importance for the construc-
Uoo of mills on the best principles. It may easily be

82 ON THE TEETH OF WHEELS. f CHAP. IV.

seen, that in many cases, a very great portion of the ori-
ginal power is expended, before any force is actually ap-
plied to the work intended to be performed.

Notwithstanding the modem improvements in this de-
partment, there is still much to be done. In the usual
modes of constructing mills, due attention is seldom given
to scientific principles. It is certain, however, that were
these principles better attended to, much power, that is
unnecessarily expended, would be saved. In general, this
might be in a great measure obtained, by bringing on the
desired motions in a gradual manner, beginning with the
first very slow, and gradually bringing up the desired mo-
tions, by wheels and pinions of larger diameters. This is
a subject which should be well considered before we can de-
termine, in any particular case, what ought to be the pitch
of the wheels. In the case above alluded to, where the
supposition is a pinion of 4 feet diameter, or of 1 foot dia-
meter ; it is obvious, that the same pitch for both would
not be prudent. That for the small pinion, ought to be
much less than that which might be allowed in the case of
the larger pinion. It is also equally ob\dous, that the
breadth of the teeth, in the case of the small pinion, ought
to be much greater than that in the case of the larger
pinion.

102. It is evident, however, that although great ad-
vantage may often be derived from a fine pitch, that there
is a limit in this respect, as also with regard to the breadth.
We shall endeavour to find some trace of this limit in what
follows ; and that we may the better do this, we shall call in
the aid of propositions, which are true with respect to pieces
of timber, or metal, subjected to ordinary cases of pressure.
It is allowed, that they cannot here, in strictness, be de-
monstratedj as applicable to wheelwork. Yet they wiU, for
want of better light, serve at least to prevent any material

niAP. rv.] ON THE TEETH OF WHEELS. S.S

[)ractical error with regard to the strength of the teeth of
ffhech. For it is to be remembered, that we are not so

I much here in search of truths of curious or profound mathe-
matical speculation, as of that kind of evidence of which
the subject admits, and which may l)e sufficiently satisfac-
tory for any practical purpose.

It most, however, be understood, that we suppose the
diameters made sufficiently great to prevent the evils which
lie have already noticed, and in the annexed table, (Art.
120,} are some examples in actual use, which have been
foimd in practice sufficiently durable. We would particu-
larly recommend attention to those of Boidton and Watt,
whose most extensive practice, as well as scientific know-
ledge, renders their work a model well worthy the atten-
tion of millwrights.

As cast-iron pinions are now generally used, and as the
teeth of the pinion are most subject to wear, I think we
are safe, in the present inquiry, in considering them all as
cast-iron.

TTie laws to which we have alluded in this investigation
are these : —

I

^^B lOS. The strength of any piece of timber, or metal,
^^g^Qse section is a rectangle, is in direct proportion to the
' hraadth, and as the square of the depth.

Let B D be any beam, placed horizontally, and fixed at
ihc cod Bc, and let afg be the perpendicular section in
which the fracture is supposed to take place. Divide the
depth A P into an infinite number of equal parts at n, b, c, rf,
, whose aj^^egate is n =af, and through each of those

G Q

PRINCIPLES OF PROPORTIONING THE STRENGTH OF
TEETH OF WHEELS.

PROPOSITION I

S4>

ON THE TEETH OP WHEELS. [^CHAP. IT.

divisioDS suppose sbraight lines to be drawn parallel to fg
the upper side of the beam. Then let any force be applied
at p in the direction dp to break the beam at af. Now,

since the strength of the timber is nothing but the force by
which the particles cohere together, the breaking of the
timber is nothing but overcoming this force and separating
the parts from one another.

Let the force of cohesion of any one of the parts be de-
noted by unity, and imagine dAO, qa6, qac, &c., to be so
many bent levers whose fulcrum is at a ; we have then
to inquire what will be the sum of all the forces applied at
Q the extremity of the levers, to break the beam at a.

Now by the property of the lever, the power applied at q
to equal or overcome the resistances at a, a, b, c, &c., will be

0 Aa \b AC o iAF r ,1 T.'i-

— , — , — , — , &c., to — J or because the cohesive force

AQ AQ AQ AQ AQ

of any one of the filaments in the section is represented by

unity, it wUl be

0 1 2 3 0 . n
— , — , — , — , &c., to — *

AQ AQ AQ AQ AQ

Consequently, the effect of all the forces applied at q to
break the beam ; that is, the whole strength of the beam
will be, as

-L (0 + 1 + fi

AQ

J + . . . . + ra), or as

AQ

Therefore, since aq is given, the strength of the beam is

IV.]

ON THE TEETH OF WHEELS.

as the square of the depth, or as af* = n*. Now, it is ob-
vious, that if the breadth fg be increased iii any proportion,
the strength of the parts must be increased in the same
proportion. So that the absolute lateral strength of the
beam, will be
^L as FG X af'.

104. Hence may be inferred, that the strength of teeth
of wheels, moving at the same velocity, and under the same
circumstances, is directly in proportion to their breadth,
and as the square of their thickness. Thus, for example,
if we double the breadth, we only double the strength ;
but if we double the thickness, in other words, double the
pitch, keeping the original breadth, we increase the strength
four times.

For although when wheels are working accurately, the
stnun is at the same time divided over several teeth, yet as
a very small inaccuracy, or even the interposition of any
small body, such as a chip of wood or stone, throws the
whole stress upon a single tooth in practice ; therefore, and
in order to simplify this case, we may consider the strength
of a single tooth, as resisting the pressure of the whole
work.

But as the length of teeth commonly varies with the pitch,
this circumstance must be taken into account, and the most
ample view we can take of it seems to be, that of having
the strain of each tooth, thrown all to the outward extre-
mitv ; we have then the following proposition to guide this
part of our inquiry.

105. Ifanifj'vrce be applied laierallj/ to a lever or beam,
e stress upon any place, is directly as the Jbrce and its
iejrom that place.

86 ON THE TEETH OF WHEELS. [CHAP. IV.

For suppose paf to be a bent lever; it is evident
that the greater the power applied at p, the greater is the

force exerted at f to separate the particles of the beam in
that place. Also the greater the distance a p, the greater
effect has any power applied at p to overcome the cohesion
of the wood at f ; and therefore, the whole stress depends
upon both.

PROPOSITION III.

106. The pitch being the same^ the stress is inversebf
as the velocity.

This is obvious, for the teeth of wheels, and the wheels
themselves, which act with greater force, must be propor-
tionally stronger ; and in any combination of wheels and
axles, the strength must diminish gradually from the weight
to the power, so that at every part it may be reciprocally
as the velocity of that part.

For example — ^if the pitch lines of one pair of wheels be
moving at the rate of 6 feet in a second, and another
pair of wheels, in every other respect under the same cir-
cumstances, be moving at the rate of 3 feet in a second, the
stress on the latter is double of that on the former.

107. This proposition is true only in the wheels of the
same machine. To render it universal, the first movers of
all machines must be reduced to the same standard ; which
may be done as follows, where the horse power is supposed
to be the standard.

If p be the power of the first mover, in anv machine.

CHIP. IV.] ON THE TEETH OF WHEELS. 8?

and V lis velocity ; also, let w be the velocity of any part
(HI which it is necessary to determine the stress. Then,

as (' : v:: p : stress = - — . (Wood's Mechanics, Prop.

I'
XWl.)

\W taking the same value of the horse power as is
used throughout this Work; that is 200 lbs. moved
at the rate of 3| feet per second, and using h to re-

present the number of horses, we have

2tX)H X 3|

Bv using i- = the stress, we get rid of the fractions,

nd have a sufficiently accurate measure of the force.
Hence, if the power of a machine be equal to any number
k of horses, the stress at any pitch line of which the velo-

j is V feet per second will be = i—

['

In a hke manner, the stress at the surface of any journal
r shaft may be found.
But, in those cases where the same first mover gives
lotion to different trains of machinery, the stress, at any
part of any one of them, should be measured by the greatest
uumber of horses' power necessarj' to perform the work as-
Tied to that train. That is, if ii be the number of

i that could perform the work, then - — — = the stress

V

\ anv point in the train moving with the velocity r.
' We shall confine our attention for the present to wheels
iBving cast-iron teeth ; and in order to take experience as
r guide, several examples in the annexed tables, actually
i use, are selected.

I The pitch, velocity, and strain, are all stated ; the strain
I measured bv the horsed power, at which the resistance
\ valued. Horses' power is a term now in general use, lo

88 ON THE TEETH OF WHEELS. [CHAP. IV.

express the force required, in order to drive any kind of
mill, and it may be proper here to give some further ac-
count of it

horses' power.

108. Although horses are not all of one strength, yet
there is a certain force now generally agreed upon among
those who construct steam engines, which force is denomi-
nated a horses power ^ and hence, steam engines are dis-
tinguished, in size, by the niunber of horses* power to which
they are said to be equaL

The measure of a mechanical effect equal to a horse's
power, has been much disputed: this I believe to be a
matter of little consequence, if the measure be generally
understood, since there is no such thing as bringing this
into any real measure. Some horses will work double of
others, and horses of one country will work more than those
of another. Desagulier's measure is, that a horse will
walk at the rate of 9\ miles per hour, against a resistance
of 200 lbs. * and which gives, as a number for comparisons,
44,000 ; that is, the raising of 1 lb. 44,000 feet in a minute;
or, what amounts to the same, the raising of 44,000 lbs. 1
foot in a minute.

Emerson's measure is the same as Desaguliers's, (see
Emerson's Mechanics, p. 178,) and Mr. Smeaton's result
is 22916 lbs. under the same circumstances!.

James Watt found, from repeated experiments, that
33,000 lbs. 1 foot per minute, was the average value of a
horse's power; but his engines were calculated to work
equal to 44,000 lbs. 1 foot per minute.

* When working 8 hours a day, (Desaguliere's Course of Experimental
Philosophy, Vol. I. p. 241,) 2^ miles per hour is equal to 220 feet per
minute, or 3J feet per second.

t Reports, Vol. I. p. 229. But Desaguliers gives the immediate power
of a horse, Smeaton the effect of that power applied to raise water : hence
the friction of the machinery should he added to Smeaton's horse's power.

nt.ii'. IV.3

ON THE TEETH OF ■

89

But, that he allows only ^3,000 in his calculations, ap-
' pliej to mills, considering the difference as being lost in
the friction of the engine itself.

109. It is common in practice, to reckon, that it requires
one horse*s power to drive 100 spindles with preparation
'if cotton water twist.

110. One thousand spindles with preparation cotton
mule yam.

HI. Sevcnty.fivc spindles with preparation flax yam.
We beg leave here to make the following extract, on the
subject of animal force, from Dr. Young's Natural Philo-
sophy, Vol, II. p. 1(j5.

11 '2. "In order to compare the different estimates of
llif force of moving powers, it will be convenient to take a
unit which may be considered as the mean effect of the
labour of an active man, working to the greatest possible
advantage, and without impediment j this will be found,
upon a moderate estimation, sufficient to raise 10 pounds
in feet in a second, for 10 hours in a day ; or to raise 100
[>ounds, which is the weight of 1'2 wine gallons of water, 1
ftMit in a second, or 36,000 feet in a day, or 3,000,000
pounds, or 4-32,00{.) gallons, 1 foot in a day ; this we may
call a force of 1, continued 3G,000."

113. Immediate force of men and horses, without de-
■^ duction for friction.

"A tnan of orJinary strength can turn a winch,

nnlb a force of 30 pouiidB. and with a velocity of

3| feet in l" for 10 hours a day."— Deaaguliers

"Two men working at a windlass, ivith

handle? at right angles, can raise 70 pounds more

c««ilr than one can raise 30."— Desaguliera

" For a "hort time, a man may exert a force
8f> [KiundB, with a fly, when the mution is

!Uv ijuieV." — Desaguliers

*• A honw can draw, with a forceof 200 pounds,

, ndlni an lioiir for 6 lionrs in the day

M Willi s force of 2*0, only 6 hours."— De-

I

90

ON THE TEETH OF WHEELS. [CHAP. IV*

114. Performance of men and horses by machines.

^' A man can raise, by a good common pump,
a hogshead of water 10 feet high in a minute,
for a whole day." — Desaguliers

^' By means of pumps, a horse can raise 250
hogsheads of water 10 feet high in an hour." —
Smeaton's Reports*

Force.

Conti.
nuance.

Day*!
Woii.

•875
'864,

• • •
8h.

•875

1 15. The power of men and horses to move machines,
has very frequently been made a subject of investigation by
writers on mechanics ; but yet it appears possible to con-
sider it in a different manner, which will furnish results
from principles somewhat more strictly practical than those
which have hitherto been made the basis of calculation.

116. It is almost always a necessary condition, that the
moving power of a machine be sensibly uniform, and con-
sequently the power which moves it, should be uniform, or
of that kind termed mechanical power, see Art. 98.

117- Let PD be the mechanical power of a horse, which
can be continued during a whole day, and also day after
day, without exhausting its strength. Then, since the ex-
ertion of the muscles will be constant, the product pd will
be a constant quantity, for the degree of exertion will not
be altered by altering either d, the distance passed over in
a second, or the force p. But, let d be the distance when
the force p is the least possible corresponding to the man-
ner of action the machine requires, or in other words, when
the power to move the machine to produce useful effect
would be nothing. And let p dhe the mechanical power,
when the distance passed over in a second is d. Then,
PB^pdf and, rf(jo — p) = the mechanical power exerted
on the machine, which is to be the greatest possible. But

p = ^, therefore jo (rf — — ) = a maximum.

D D

» Vol. I. p. 229.

HAP- t>'.3

ON THE TEETH OF WHEELS.

Now it may be shown by the principles of maxima, &c.,
(hat this quantity is a maximum when rf = ^ d. For

(P
putdng the expression p (d — —) into fluxions, and equat-
ing the fluxion with zero, or 0, we get

•idd

0;

I

^und consequently, by transposition it becomes

^K ^^ = 1, or d = k D.

^^H||HKfbre, a man or a horse acts with the greatest ad-
^^Bp^ on a machine when he moves with half the velocity
■ Iw rmild continue at, were the effective resistance of the

machine nothing.
That portion of the mechanical power which is efficient

in impelling' the machine will be ^ pd. For since rf = -J d,

p= — = ^/), or2p=^; hence (f(^ — p) =^ D (2p — p)

= ^ PD.

The force p, and the distance moved through d, will
each vary in the same man or horse according to the man-
ner of spphing the force ; but their product will be nearly
a constant quantity. In any case, one of these quantities
may be determined from experience, and in many instances
both of them, and that one may always be supplied by cal-
culation which cannot be found by experience.

118. When the effective force is nothing, it must not be

leretooc] that a roan or a horse is acting against a force

lal to his own weight, in any case whatever; for as has

Wn observed by Dr. Young, (Vol. 1. p. 132, and 212,) in

walking, the resistance overcome is not exactly comparable

weight, and the same may be remarked on other

of exerting force.

"Wben a man ascends vertically, his velocity is reduced to

>ut one half of his horizontal velocity, indicating that he

Hqnt

92 ON THE TEETH OF WHEELS. ^I^^^^* ^*

acts against a double resistance; therefore when a man
ascending a ladder, carries a load, the maximum effect will
take place when his ascending velocity is about one fourth
of the velocity he can walk horizontally without a load.

A man of ordinary strength will not be able to walk,
unloaded, at a quicker rate than 3| miles an hour, if this
exertion is to be continued for 10 hours every day. In-
deed, those who examine the subject with a view to a £edr
average, will find this to be about the extreme velocity that
can be continued, without injury, for any considerable
time.

According, therefore, to our investigation, a man ought
to move with half this velocity to produce a maximum
effect ; that is, at the rate of If mile an hour, which is
about 2^ feet per second.

But this supposes the whole load to be the useful effect,
whereas part of it must consist of the apparatus employed
to carry it, or the friction of an intermediate machine, or
other circumstances of a like nature. About one fifth of
the velocity may be considered equivalent, at an average,
to the force lost in ftiction, &c., in all cases; in many it will
exceed one fifth. Hence the maximum of useful effect will
take place when the velocity is 2 feet per second, or about
11 furlongs an hour, continued for 10 hours each day.

Smeaton is said to have made numerous comparisons,
from which he concluded that the mechanical power of a
man is equivalent to 3750 lbs. moving at the velocity of
one foot per minute * ; and taking this average to be near
the true one, as I have reason to conclude it is, we have

."^'Ji^ 31 -25 lbs. Therefore, we make the average me-
2x60 ^

chanical power of a man 31*25 lbs. moving at the velocity
of 2 feet per second, when the useful effect is the greatest
possible ; or half a cubic foot of water raised two feet per

♦ Art. Water, Rees's Cyclopaedia.

lAP. IV.]

ON THE TEETH OF WHEELS.

93

second ; a very convenient expression for hydrodjmamical
inquiries.

tif a man ascend a vertical ladder, according to a pre-
ling remark, (p. 91,) the velocity which corresponds to
• maximum of useful effect will he 1 foot per second, and
the load double that which he carries horizontally ; conse-
quently the average of useful effect is 6'2'5 lbs. raised one
foot per second.

Bricklayers' labourers in London ascend ladders with a
load of about 80 lbs. besides the hod ; sometimes at the
rate of one toot per second, but more frequently about 9
inches per second.

Ascending stairs is more fatiguing to the muscles of the
legs than ascending a ladder ; and therefore the useful
effect is less, till a person has become accustomed to this
kind of labour. And it is also to be observed that the
space moved over is increased, unnecessarily, except where
the horizontal distance is part of the path over which the
load is to be moved.

We ought not to be surprised at the opposite conclusions
here obtained, from those of other theoretical inquirers,
when the data they have proceeded from are considered.
For their data have been the extremes of force and velo-
cit}', without any knowledge of the true laws which con-
nect theni-

1 19. The force of a horse is, at an average, about equal
to tliat of six men, according to various estimates ; and the
rate of travelling about the same, perhaps rather less than
that of a man, when his exertion is continued for 8 hours j
quently the velocity corresponding to the maximum

set, will be about '2^ feet per second. Whence, the
• mechanical power of a horse may be estimated at

7i lbs. moving with a velocity of 2^ feet per second, or

ubic feet of water raised 2^ feet per second. The day's
L being 8 hours.

94 ON THE TEETH OF WHEELS. [CHAP. IV.

This estimate of the power of a horse is equal to 28, 125 lbs.
raised one foot per minute ; nearly a mean between Watfs
and Smeaton's. But since our author has employed
that of Desaguliers, it was not very easy to make a change
in this work ; and still more objectionable to employ two
measures of different values.

French writers use as a dynamical unit a given measure
of water raised through a given space ; and the Americans
use a similar measure. In my opinion, it is preferable to
make a horse's power the dynamical unit ; because, a prac-
tical man has a more correct idea of the quantity of me-
chanical power expressed by this unit, than he can have of
any one less frequently under his observation ; besides, it
is a power often employed to move machines ; and there-
fore is a familiar measure of comparison.

• 3 ON THE TEETH OF WHEELS. 9^ ^^H

TABLB OP PITCHES 01' WHEELS IN ACTUAL USB IN MILLWOHK.

■

=i

s

u.

Wheel.

Pinion.

1

1

1

^1

i"i

«-.

,

1

I

Is
Is

i
f

y

y

1

•s

1

1

1

1

1

ll

ill

1

li

pi

sis

■

=

£

p.

r.

a

0

£

^H

fe^'

10

^

51

"■. 1°'

'\'.°'

55

■

feed', B

30

3

101

W

"9J24

"3

3is"47

3'41

3-as

4-489

riml. C

15

3

6

■204

4j

16 3

44

20

3 "e

4-

3-8

5-06

rheel', D

oi

a

4

207

16 51

50

3 11!

7-27

3-

7-27

=ill. E

I

21

01

3

a Oi

22

12-9

1 51

400

-949

12-65

■31, F

1

2i

4J

fli

3

6 01

IflJ

1313

45-0

■949

U-2S

1

j^R

M

3i

6

96

IB

8 0

42

4332

3 6

2-5

7-95

6-625

■

BTh'

46

3

8

132

17i

54

50

1-7

11-

8-2

■

KPl

93

3

G

lie

10

8 10

1-87

8-78

5-47

■

■VTk

U

3

5

64

25

5 1

29

55

2 4

3-57

6-65

7-91

HPK'L

QO

2\

5

90

18

A II

38

42-63

2 7

2-5

6-57

4-64

SSTm

10

2i

d|

77

25

* n

40

48-5

2 41

5-75

6-2

11-88

Btio-. N

6

2|

51

60

28

3 7

27

1 71

8-75

5-25

15-31

Enrf. O

4

21

4:

48

32

•2 10

25

6111

1 6

11-87

4'S

18-99

liito. P

2

2

41

62

3 6

2

375

8 8

t-t

10

li

6

77

25

a 10

40

48'5

1 in

6-

5-

10

12

a

66

44

2 8

48

60-5

1 9

25

5-99

4-95

■

?/Slf ™"i.. ^^l "^i " -" fpu-a fttar .» a«ro- to. ,h. ..™u> .. it i. .«rin, much

• »«oribiw'h«itliilheuioeintu' """ " i""" emin.it

«ia« ID thl. MiinB, whirh hu b»ii l« y«n u •ork, U ibe wioi of binilih in th« ipui-«h»
ter duMt ln£ivi bHffl 6 ItHdM ot mote, » Oic» will nol ImI h^ « long « Uie IwveI— liMi, uid

M hu mndiD imn . u>d hn Docd' WDriilpg fur lhi» y«T> pul.

AlMlMHiramxiuibcKiolfiit

^^H

»t*!**TIOK (IF THE TABLE OF WHBEI.S IN ACTUAL USB IN MILLWORK.

1

!he wheels arc all reduced to what may be called one ^^|

nniDatioD. ^^H

Irst — By proportioning their breadths all to what they ^^|

lid be to have the same stTeng;th, if the resistances ^^|

B equal to tl>e work of a steam engine of ten horses* ^^H

^^1

(condly — By supposing their pitch lines all brought to ^^H

Mne velocity of three feet per second, and proportion- ^^M

96

ON THE TEETH OF WHEELS. [CHAP. IV.

ing their breadths accordingly. I have chosen this parti-
cular velocity of 3 feet per second, because it is the velocity
very common for overshot water wheels.

Such cases as appear to have worn too rapidly, are
marked, which may tend to discover the limit in point of
breadth.

Column I contains the horses' power.
2 . . . . pitch in inches.

3 . . • . breadth of teeth in inches.

4 . . • • number of teeth of wheeL

5 . . . . revolutions of wheel per minute.
6 . . . . diameter of wheel.

7 • . . • number of teeth of pinion.

8 . . . • revolution of pinion per minute.

9 • • • « diameter of pinion.

10 ... . breadth proportionate to 10 horses'

power, and at the present ve-
locity.

11. . . . present velocity per second in feet

12 . . . . breadth in inches proportionate

to 10 horses* power, at 3 feet
per second. That is, all the
cases reduced to the same de-
nomination.

OBSERVATIONS ON THE TABLE OF WHEELS IN ACTUAL USE IN MILL WORK.

121. Having reduced the examples in the table, in the
manner already described, to one denomination, the results
approach nearer, considering all circumstances, than could
have been expected.

1st. In two of the cases, viz. b and d, it appears, that
the wheels were rather too narrow for their work. These,
however, have been working about sixteen years, and may

I CirAP. IV.3 ON THE TEETH OF WHEELS.

97

[ jet continue for a long time. — u is ■t'4'89 inches in breadth,
I then reduced to 10 liorses' power, at 3 feet per second. —
D is 7'^ inches in breadth. The pitch of both is three
I inches.

Three inches being a pitch in ver\' general use for the
first motion of mills, could the proper breadth for this pitch
be ascertained, it would servo as a verj" useful standard.

Rules, to be of practical use, must be easy of remem-
brance, as well as easy of application.

Let us, therefore, assume a simple standard, and try
liow far it will bear the test of esperience ; for, as Du
Buat justly observes, " It is an excellent method, in the re-
search of obscure difficult truths, to suppose a theory pre-
existent, founded upon the most probable principles, from
which may be determined, the choice of some direct ex-
periments, proper to evince the fallacy or accuracy of the
principles proposed."

The actual cases in the table may be considered as
satisfactory experiments. Let us, therefore, try the fol-
lowing simple rule, and compare some of its results with
those cases.

Rule I. for pitch of three inches, when the velocity is

tree feet per second, at the pitch line.
Make the teeth as many inches broad as the number of
rsesi' power which it has to resist.

For example, for nine horses' power, make the teeth
ne inches broad.

|ld& Taking this example then as a point from which

1 set off, and supposing the same breadth of nine inches

istanl, we shall, in the first following table, state various

tches, and (by Proposition I.) shall first square these

iches, to find the number of horses' power, equal to the

igth and durability of the pitch, when the teeth are all

; length. The strength, thus found, will be inserted

98 ON THE TEETH OF WHEEL8. [CHAP. IV.

■

But, as the lengths generally vary as the pitches, taking
the same point (three inches pitch, nine inches hroad)
from which to set off, we shall diminish the value of the
strength, as we ascend, and increase as we descend, agree-
ahly to Proposition II. The results will be found in
column z.

But as in this investigation durability is of equal import-
ance with strength, perhaps the true proportion may be
somewhere between the results in the columns y and z.

DESCRIPTION OF THE SIX FOLLOWING TABLES OF PITCHES.

In all the following tables, the column w contains the
pitch in inches.

Column X contains the breadth of the teeth also in
inches.

Column Y is formed upon the supposition, that the teeth
of all the pitches were of the same length, and contains the
strength and durability of the teeth valued in horses^
power.

Column z contains the strength and durability of the
teeth, also valued in horses' power, upon the supposition,
that the lengths of the teeth are in the same proportion to
one another, as the pitches ; and that the strength (by
Prop. II.) is inversely as the length. Having taken a
three inch pitch as our standard, the two last colunms, Y
and z, exactly coincide for that pitch ; the column z, de-
creasing upwards from that point, and increasing down-
wards in an inverse ratio of the lengths.

It is evident, that the tables upon these principles al-
ready laid down, might be greatly extended. But it is
hoped, that these will be sufficient for our present purpose.

CHAP. IV.3 ON THE TBETH OF WHEELS.

99

I. TABLB OF PITCHES

123. The velocity of the pitch line being three feet per
second, and the breadth of the teeth nine inches.

w

Y

Z

Pitch in
inches.

Value of

Value of

stren^ in

strength in

hones* power.

horses* power.

4

16-

12-

34

12-25

10-5

3

9-

9-

2i

6-65

7-5

2

4-

6-

H

2-25

4-5

1

1-

3-

Supposing again, that the pitches were the same as in the
first table, and that the breadths were made, in each par*
ticular case, just double the pitch, then the horses' power
would vary as in the following table, which is calculated
by taking the above table, and by direct proportion, finding
the horses' power equal to each particular breadth, when the
breadth is just double the pitch.

II. TABLE OF PITCHES

124. The velocity being three feet per second, and the
breadth of the teeth double each pitch.

w

X

Y

Z

Pitdiin
inches.

Twice the
pitch in
breadth.

Value of

strength in

horses* power.

Value of

strength in

horses* power.

4
3
2
1

8
7
6
5
4
3
2

14-22
9-53
6-

3-47

1-77

•75

•22

10-66

8-17
6-

4-16
2-65
1-5
'66

H 2

100

ON THE TEETH OF WHEELS. [CHAP. IV.

The strength being directly as the breadth, (by Prop. I.)
it is easy from this to find, by the Rule of Three direct,
the horses' power equal to any given breadth of these
pitches.

The stress being inversely as the velocity, (by Prop. II.)
the horses' power equal to any of these pitches, and breadths,
may be easily found for any other velocity.

But, perhaps, it will be of advantage to take a different
view of the subject, by fixing upon a case in the table of
wheels in actual use, and proportioning breadths and
pitches from it for various velocities, resistances, &c.

For this purpose we shall select the case h, erected by
Boulton and Watt, being a pitch of three inches, the teeth
8 inches broad, moving with a velocity of 11 feet per
second, and having a resistance valued at the power of 46
horses.

Then by following a similar procedure, with the two
foregoing tables, we have the Tables III. and IV. propor-
tionate to the case h at a velocity of 1 1 feet per second.

III. — TABLE OP PITCHES

125. Proportionate to h in the Table of Wheels. The
breadth of teeth (8 inches) and velocity (eleven feet per
second) being constant.

w

Y

Z

Pitrh in
inches.

Hones* power.

Value of

strength in

horses* power.

4

3i
3

2i
2

1

81-77
6261
46-
31-94
20-44
11-5
511

61-33

53-66

46-

38-33

30-66

23-

15-33

CHAP, iy.3 ON THE TEETH OF WHEELS.

101

IV. — TABLE OP PITCHES

126. Proportionate to h, the breadth being in this case
double the pitch. The velocity being constant, (eleven feet
per second,) but the breadths double.

w

X

Y

Z

Pitch in
inches.

Breadthfl

double

the pitch in

inches.

Hones' power.

Value of

strength in

horses* power.

4

H
3

2i
2

IJ
1

8

7
6
5
4
3
2

81-77
54-78
34-5
19-95
10-22
4-31
1-28

61-33
46-95
34-5
23*94
15-33
8-62
3-84

But it may be satisfactory, in order to compare with the
tables, first and second, to reduce the velocity to three
feet per second.

The two following tables, therefore, are calculated ac-
cordingly at that velocity.

v. — TABLE OP PITCHES

127. Proportionate to h, at a velocity of three feet per
second ; the breadth being constantly eight inches.

w

Y

Z .

1^'x 1 •

Value of

Pitch m
inches.

Horses* power.

strenia^h in
horses' power.

4

22-30

16-72

H

17-07

14-63

3

12-54

12-54

2i

8-71

10-45

2

5-57

8-36

li

3-13

6-26

1

1-39

4-17

102

ON THE T£fiTH OF ¥mE£L8.

[chap.

VI. — ^TABLB OP PITCHES

128. Proportionate to h, at a velocity of tliree feet; tb.e
breadths being double each pitch.

w

X

Y

Z

Pitch in
incfaei.

Brcsutn of

teeth
in inches.

Hones* power.

Value of

alrength in

hones* povrer.

4

3

2i

2

1

8

7
6
5
4
3
2

22-30
14-93
9-4
5-44
2-78
117
0-34

16-72
12-79
9-4
6-53

4-17
2-34
1-02

From a comparison of the second table with the sixth,
it appears, that the rule we have annexed, is at least safe
in point of strength ; for by that rule, a three inch pitchy
sis inches broadj is equal to a strain of six horses. Whereas
in the sixth table, the same pitch and velocity, at six inches
breadth, has strength valued at nine and nearly a half
horses* power.

But when we consider how much more liable, from sand,
&c., teeth attached to water wheels are to wear, than those
which are properly greased and free from sand, the results
correspond as nearly as could be expected.

129* Rule II. So that, taking h as a standard, we may
conclude, that, for a pitch of three inches^ with a velocity
of three feet per second, every inch of breadth may be
valued at one and a half horses* power.

The first rule (Art. 121.) I think therefore may safely be
followed for teeth attached to water wheels, and the above
conclusion for wheels in all situations where they are pro-
perly greased and free from sand.

The conclusions here drawn, wiU, I think, give teeth

CHAP. IV.] ON THE TEETH OF WHEELS. 103

fufficiently durable and strong for the work whi(.'h they

may have to perform. This first conclusion gives, perhaps,

loo great a result ; how much may with prudence be de-

I ducted from the results of either, those of experience will

rietemiine. But to the young millwright, I would advise,

I of the two extremes, rather to err in makiug his work too

slrong. Durability ought not for a moment to bo out of

light in the arrangements of wheelwork ; there are many

parts of machines subjected to greater stress, but not liable

to wear; whereas the t«eth of wheels, the moment they

begin to act, begin to change from their original form, and

to become progressively less strong.

130. In millwork, at present, the breadth of the teeth,
as commonly executed by the best masters, seems to be
from about twice to thrice the pitch.

It is, perhaps, not easy to determine what proiwrtion is
on the whole the most advantageous for the breadth of

Eth. A fine pitch, on the one hand, gives a smooth mo-
1, and the teeth will rub less on each other ; but an in-
aso of breadth increases in some degree the friction*.
'the durability, as well as the strength of teeth, is per-
haps nearly in direct proportion to their breadth.

tlSl. After sending the foregoing " Jnqiiirif respecting
\e Strength of the Teeth itf JVheels " to the press, Mr.
ohn Robcrton, engineer, perused a manuscript copy of it,
and was so obliging as to communicate to me the substance
^of what follows. His rule, it will be readily perceived, is
Hbnnded on the principles laid down in the "Inquiry;"
Hnt it is more simple, and perhaps more accurate than the
mode of approximation which occurred to me. It is how-
I mt satisfactory to find, that the table formed on his rule,
^'hich from his experience he is of opinion cannot be far
the truth,) very nearly coincides in its results with
pies fifth and sixth. It may be observed, that he founds
* See Art. C7 and 68.

104 ON THE TEETH OF WHEELS. [CHAP. IV.

his calculations upon the thickness of the teeih^ wluch in
all cases he supposes a little less than half the pitch, which
proportion is very common in practice. When hoth wheel
and pinion, however, are of cast iron, it is evident, that,
heing more liahle to wear, the teeth of the pinion ought to
he thicker than those of the wheel.

CONSTRUCTION OP THE FOLLOWING TABLB.

132. The thickness of the teethj in each of the lines, is
varied one-tenth of an inch. The breadth of the teeth is
always four times as much as their thickness. The strength
of the teeth is ascertained by multiplying the square of
their thickness into their breadth^ taken in inches and
tenths, &c. The pitch is found by multiplying the thick-
ness of the teeth fry 2*1. The number that represents the
strength of the teeth, will also represent the number of
horses* power, at a velocity of about four feet per second.
Thus in the table where the pitch isS'\5 inches^ the thick-
ness of the teeth 1*5 inches^ and the breadth 6* inches^ the
strength is valued at 13^ horsed power ^ with a velocity of
four feet per second at the pitch line.

rCH-tP. IV.] ON THE TEETH OF WHEELS.

k 3ABLB OP PITCHES OP WHEELS,

13S. With the breadth and thickness of the teeth, and
Ijhe correBponding number of horses' power, moying at the
* pitch line at the rate of three feet, of four feet, of six feet,
and of eight feet per second.

f

Strength of

iTiict-

leeOi, or

HorWB-

Honun'

Pbchb

nsBof

BrettJih

Hones' poser
at three feet
pCTBccond.

poneral

power at

iulHl.

teetbm
incha.

ininSes.

J^foTfra'^r

I^^Di

wcond.

' 3'fl9

1-il

7-fl

27-43

20-57

4114

54-85

. n$

1-8

7-2

23

32

17-49

34-98

46-64

1 a-37

1-7

6-8

19

65

14-73

29-4G

39-28

1 3'36

le

e-4

16

38

12-28

2i-,5G

32-74

1 3-13

1-5

e-

13

5

10-12

20-24

26-98

2-8*

1-4

5-e

10

97

8-22

16-44

21-92

' 2-73

13

5-2

8

78

6-58

13-16

17-54

J-52

1-2

+■8

fi

91

5-18

10-36

13-81

S-31

1-1

*1

5

32

3-99

7-98

10-64

21

!-0

4-

4

0

3-0

B'O

8-0

i-8a

■iJ

3-6

S

91

2-18

4-36

5-81

I-fi8

■s

3-2

2

04

1-53

3-06

3-08

1-47

-7

2-8

1

37

1.027

2-04

2-72

\u

-6

2'4

86

■64

1-38

1-84

1-05

■5

2-

■5

•375

-75

r

James Carmichael, millwright, (of Dundee.^ made the

following- remarks to Buchanan on the strength, &c., of

wheelwork.

ilS-t. " Sir — It is a corroboration of the truth of the

iles of pitches, that Mr. Roberton's table coincides very

ly with columns marked y in jour tables ; but he

to have overlooked the propriety of taking the

:h of the teeth into bis calculations. I am, therefore,

gtill of opinion, that the true value is in the columns

aiarked z in your tables.

Admitting the truth of the fundamental propositions.

106

ON THE TEETH OF WII&&L8. j^CfiAP; IT^

and, from a comparison of the tables of pitches, I would
propose the following rule, which is on the same principle
as that of columns z in your tables, for calculating the
proportionate strength of the teeth of wheels.

" Rule. — Multiply the breadth of the teeth by the square
of the thickness, and divide the product by the length.
The quotient will be the proportionate strength in horses*
power, with a velocity of 2*27 feet per second.

'* By that rule I have calculated the following table ;
and, for the sake of comparison, I have taken three cases
from Mr. Roberton's table, and three from your Tables 3d
and 5th.

(b

EXPLANATION OF THE TABLE.

" Column 2 contains the thickness of the teeth. The
pitch is found by multiplying the thickness by 2'1 * ; and
the length is found by multiplying the thickness by l*2t.

" Column 5 contains the proportionate strength, and
also the number of horses* power (proportionate to the case
H, see p. 100) which the teeth are equal to, with a velocity
of 2*27 feet per second.

1

2

3

4

5

6

7

8

Strength of
teeth, or

number of

horses*

power, at

2-27 feet

Pitch in
inches.

Thick-
ness of
teeth in
inches.

Breadth

of teeth

in inches.

Length

of teeth

in inches.

Horses*

power at

three feet

per second.

Horses*

power at

SIX feet per

second.

Horses*

power at

eleven feet

per second.

per second.

3-9

1-9

7-6

2-28

11-73

15-46

30-92

56-84

2-9

1-4.

56

1-68

6-53

- 8-63

17-26

31-64

2-1

1-

4-

1-2

3-33

4-4

8-8

16-1

4-

1-904

8-

2-285

12-698

16-78

33-56

61-52

3-

1-428

8-

1-714

9-523

12-58

25-16

46-

IJ

•714

8-

•857

4-752

6-27

12-54

23-02

* That is in order to make the space between the teeth a little wider
than the thickness of a tooth. See Art. 132,
t Respecting the length of teeth, see Art. 145.

cn.ip. IV.]

ON THE TEETH OF WHEELS.

107

" REMARKS.

"1st. The last three cases in the table are taken from
the Tables 3d and 5tb, and the results coincide so well
with the columns marked z, that I presume the rule is
just.

" 2dly. The first three cases are from Roberton's table ;
ihe second case is very near the same as in his table -, but
the first is considerably less, and the third considerably
more. Hence I infer that Roberton has taken his data
from a pitch about three inches.

" 3dly. If any two wheels have the length and thickness
of their teeth in the same proportion to their respective
pitches, the breadth of the teeth and the velocity being
the same, i/ie strength will be dirextli/ as the pitches. The
truth of this is deduced from the columns marked z in
Tables 1st, 3d, and 5th."

135. Table of pitches of wheelwork, with the brendtk
and thickness of the teeth, and the corresponding strength
io horses* power, calculated by Carmichael's rule.

TtWt-

Breadth

Lennh
oftoHh
LDiDches

Sireneth of

Hor-e,'

Hoi««'

Hor««'

!>iidib

teeth in

uftedh
iaiDchis.

tW fe^l

power al
iii feet

power at
clcvpti fret

inchea.

p«f second.

[ler lecond.

3-90

!•»

7-0

2-28

12-03

15-90

31-80

58-30

■^■78

1-8

7-2

2-16

10-80

U-27

28-5*

52-32

3-57

1-7

6-8

20*

9-63

12-72

. 25'54

46-68

3-36

1-6

6-4

1-92

8-53

11-27

32-54

41-32

315

1-5

6-0

1-8U

7-50

9-91

19-82

3G-33

2»*

1-4

5-e

l-(58

e-53

8-G3

17-26

31-64

2-73

1-3

52

l-.')6

5-63

7-44

14-88

27-28

S-52

1-2

4.-8

1-+4

4-80

G-34

12-C8

23-24

S-31

1-1

+■4

1-32

+■03

5-32

1 0-84

1954

S-10

l-«

40

1-20

3-33

4-40

6-81

ie-15

1-89

0-3

3-6

1-08

2-70

3-57

7-1*

13-09

< l-M

(1-8

3-2

0-96

2-13

2-81

5-62

10-33

,3-»"

0-7

2-8

0-8*

l-«3

215

4-30

7-88

1-86

0-«

8-4

0-72

1-20

1-.S9

3-18

3-83

: 1-05

0-5

2-0

O'liO

0-83

110

220

4-03

108

ON THE TEETH OF' WHEELS. [CHAP. IV.

136. It is not perhaps quite so difficult, as our author
imagined, to determine, from first principles, the strength
proper for teeth of wheels, and such a method must always
be preferred to empirical rules. We sh|dl here show how to
apply those principles which will give the reader an oppor-
tunity of comparing the two methods.

In the first place let us consider under what circum-
stances the strain on a tooth will be the greatest possible-
Let A B c D be the side of a tooth, then it will be evident,
that the strain will be greatest, when the stress is thrown

upon one comer of the tooth, as at c, whether it be from
irregular action or from any substance getting between the
teeth.

In such a case, it may be shown by the rules of mcurima
and minimaj that e c being equal to c b, the strain will be
greatest in the line e b ; and, in the case of fracture, it
would take place according to that line.

Since the thickness of a tooth is not regular, we shall
have a result, sufficiently near for this purpose, if we
express the relation between the stress and strain by

^ =-^ ^ ^^ ^ ^ — . (Essay on Cast Iron. Art

V 8(fc) ^ ^

81, and Art. 107 of this Essay,) which, wheny =
15,300 lbs. on a square inch, reduces to — ZJi=rf* =

V

the square of the thickness of the tooth in inches. But, a
tooth should be capable of resisting this stress, when it is con-
siderably worn by friction ; and an allowance fully equal to

CHAP. IV.] ON THE TEETH OF WHEELS. 109

that which ought to take place before renewing the wearing

parts of the machine, will be one third of the thickness of a

tooth. Now to allow of this degree of wear in the tooth, and

that it shoidd remain equal to the stress, it may be easily

shown that the tooth should be capable of resisting 2^ times

. , ^ , A ■■''5G H „ S H ,

(he power at the first ; therefore — — = d" : or 7 ^/ - = "■

^\Tiere h is the number of horses which are equal to the
poirer of the first mover ; v the velocity of the pitch line
of the wheel in feet per second, and d the thickness of a
looth in inches.

Tliis investigation furnishes an easy practical rule for
the thickness of teeth ; and, consequently, for the pitch of
wheels and pinions ; we shall give it in words at length,
nith an example, and then proceed to determine a rule for
the breadth of teeth.

137. Find the number of horses which are equivalent
to the power of the first mover of the train of machinery,
and divide that number by the velocity, in feet per second,
of the pitch line of the pinion or wheel ; extract the square
root of the quotient, and three fourths of this root will be
the least thickness of the tooth for the wheel or pinion, in
laches.

1 LEAST QUA.VTril- OF PITCH FOR A WHEEL OB P

^Hl38. If the thickness of the teeth of the pinion be in-
^tonded to be the same as those of the wheel, multiply the

:)ili.-kncse above determined by 2*1, the product will be the

Tiiich required.
^HBut wc may observe, that if a pinion makes three turns.

110 ON THE TEETH OF WHEELS. [CHAP. IV.

for example, while the wheel makes one, the teeth of the
pinion will be worn three times the quantity of those of the
wheeL Hence, to provide against such excess of wear, if
the pinion makes n revolutions, while the wheel makes

one, the pitch should be d inches, and the thickness

3

of the teeth of the pinion d inches, when d is the

thickness of the teeth of the wheeL

EXAMPLE.

139. Let the force of the first mover be equivalent to
ten horses, and the velocity of the pitch line three feet per
second. Dividing 10 by 3, we have 33- ; and the square
root of 3^ (by the Table of Powers, Art 479.) is 1-83
nearly; and f x 1*83 = 1*45 inches for the thickness of
the teeth of the wheel.

Again, suppose the pinion to turn twice while the wheel

turns once, then n = 2, and x 1*45 = 3*5 inches

3

the pitch.
And X 1 '45 = 1 "93 inches, the thickness of the

teeth of the pinion.

THICKNESS OP WOODEN TEETH.

140. The kind of wood employed for teeth, is usually
about one fourth of the strength of cast iron, and since
the thickness of the teeth should vary inversely as the
square root of the power of the material, the square root
of \ being i, wooden teeth should be twice the thickness
of cast iron teeth. The pitch of course will be greater in
the same proportion.

CHAP. IV.] ON THE TEETH OF WHEELS.

TO DSTERMINE THE BRBADTH OF CAST IRON TEETH.

UI. That case where a beam is fixed at one end, and
thi- load acts at the other, applies to the teeth of wheels,

in their general state of action, and the stress is i -.

fArt. 107.) Ilcnce, when / = the length, and b =

the breadth of a tooth, T12iL' = 212 b d\ (Essar/ on

Out Iron, Art. 116.) And to allow one third of the

thickness of the tooth for wear, the equation becomes
.'I j( -j-.i^) jj I

— = 212 b d*. But we have already seen, that

^=•££^1 therefore ?ijlli^^^= -556 b x 212, or

H!=b.

This calculation informs us what breadth is essential
tor strength ; that is, the breadth should never be less
ihan 1-2 multiplied by the length of the tooth; but, it
may be proved that the durability is nearly in direct pro-
portion to the breadth, and inversely as the pressure.
Some of the maxims of our author, as far as regards the
breadths of teeth, agree well with the theory of durability j
but the conclusions respecting strength and the limits of
pitch are not so much to be relied upon ; indeed the facts
drawn from practical construction are not so well adapted
for tlie latter object.

1 4-^2. If we suppose that teeth sis inches in breadth are
idfficienfly durable for a power equivalent to 10 horses,

en the pitch line moves at the rate of 3 feet per second j
the Table of Wheels in p. 95, Art. 120. seems to indi-
that this supposition is near the truth ; therefore

'"' (■ " / 3 X 6 X H 1-8 H TT, . • I.,- 1 ..1.

- ; o : : - : o = = ^. Ihat is, multiply the

■j F 10 w "

HpBC^ po

power hy X"8, and divide by the velocity of the

112 ON THE TEETH OF WHEELS. [CHAP. IV.

pitch line in feet per second, the quotient will be the
breadth in inches.

EXAMPLE.

Taking the case h in the table Art. 120, we have

= — =7'53 inches: Messrs. Boulton and Watt

V 11

in this case made the breadth 8 inches.

T ,, , ., 1 1'8h 1*8 X 14

In the case k, by the same makers, = —pm =

•^ V 6-65

3'8 inches. The breadth actually employed was 5 inches.

Hence it appears, that, in a considerable range of power

and velocity, our formula gives results below those actually

employed, but the breadth assigned by the table calculated

by Mr. Roberton, Art. 133, is always vastly below ours in

the greater powers ; indeed it is manifestly erroneous in

the breadths, for where the moving power is doubled, the

breadth is increased only one third. In the thickness of

teeth, it nearly agrees with our rule.

BREADTH OP WOODEN TEETH.

143. For wooden teeth we may take, as the basis of a
practical rule, the case n in the table, Art. 120 ; which, ex-
pressed in the nearest whole number, is — = 6 in inches.

V

EXAMPLE.

In the case o ; h =4 horses, and v =4*8 feet per second ;

therefore, — = =4*17 inches the breadth : the actual

i; 4-8 '

breadth used was 4f inches.

,v.]

ON THE TEETH OK WHEELS.

113

OP THE GIBENOTH OF aiAVE9 FOR TBUNDLBS.

144. TTiis is a subject our author has not touched upon,
but we consider it necessary to examine the strength of
stares, because trundles seem capable of improvement, and
thi'v have some advantages which toothed pinions have not.

If tlie length of a stave in feet ho I, its diameter in

inches «/, and the stress upon it ^ ; which is supposed

ti) act at the weakest part of the stave ; that is, in the
middle of its length ; then, by the rule for the strength of

cjlmders, (Sssai/ on Cast Iron', Art. 129,) =rf^

Now, if it be made to resist 3^ times the power when first

made, in order to allow for wear : we shall have *^ '

500 V

=d', or ( ) =d; or with sufficient accuracy, it is

^L EXAMPLE.

^r Let the power of the first mover be equal to 10 horses,
the velocitv of the pitch line 3 feet per second, and the

rth of the stave -{i of a foot ; then 2 i^) ' = a (l^ilA' ) *
' ^ V ' ^ 3 '

ix 1*442 = 2"884' inches the diameter of the stave.

p45. We take the liberty of inserting the following tablo

I a respectable periodical publication, as we purpose to

K its application, knowing it maybe of use to millwrights f-

' TredgoM'ii work, whicL we occasionally quote.

t This very useful table was printed in a small pamjihlet, price one

I 1803, but IE nt present out of print. Wben tbe nuniber of

1 10, look for tbe nulius of double tbe number of tcetb,

Pit win be the mdius required. Wbeu the number of teeth ex-

^look for the radius of half tbe number of teeth, the double of

Kieh win Ik tie otie required.

114

ON THE TEETH OF WHEELS. [CHAP.ITb

TABLE OP THE RADII OP WHEELS, PROM TEN TO THRU BUlHttlD IIRI,

THE PITCH* BEING TWO INCHES.
BY B. DONKIN, ESQ., CIVIL ENGINEER, LONMIT.

No. of

Radius in

No. of

RadiuB in

No. of

Radius in

No. of

Rib.

teeth.

inches.

teeth.

inches.

teeth.

inches.

teeth.

•^^^L^

10

3-236

47

14-972

84

26-741

121

88-5«

11

3-549

48

15-290

85

27063

122

88-881

12

3-864

49

15-608

86

27-381

123

39-151

13

4-179

50

15-926

87

27-699

124

39-47

U

4-494

51

16-244

88

28-017

125

89^

15

4*810

52

16-562

89

28-336

126

40-11

16

5-126

53

16-880

90

28-654

127

40-4S

17

5-442

54

17-198

91

28-972

128

40-74

18

5-759

55

17-517

92

29-290

129

41-06

19

6-076

56

17-835

93

29-608

130

41-38

20

6-392

57

18-153

94

29-927

131

4f7a

21

6-710

58

18-471

95

30-245

132

42-OS

22

7-027

59

18-789

96

30-563

133

42-33

23

7-344

60

19-107

97

30-881

134

42-63

24

7-661

61

19-425

98

31-200

135

42-97

25

7-979

62

19-744

99

31-518

136

43-28

2(J

8-296

63

20-062

100

31-836

137

43-61

27

8-614

64

20-380

101

32-155

138

48-93

28

8-931

65

20-698

102

32-473

139

44-24

29

9-249

66

21-016

103

32-791

140

44-56

30

9-567

67

21-335

104

33-109

141

44-88

31

9-885

68

21-653

105

33-427

142

45-26

32

10-202

69

21-971

106

33-746

143

45-5S

33

10-520

70

22-289

107

34-064

144

45-84

34

10-838

71

22-607

108

34-382

145

46-15

35

11-156

72

22-926

109

34-700

146

46-47

36

11-474

73

23-244

110

35-018

147

46-79

37

11-792

74

23-562

111

35-337

148

47-11

38

12-110

75

23-880

112

35-655

149

47-43

39

12-428

76

24-198

113

35-974

150

47-75

40

12-746

1 t

24-517

114

36-292

151

48-06

41

1 3-064

78

24-835

115

36-61 1

152

48-38

42

13-382

79

25-153

116

36-929

153

48-70

43

13-700

80

25-471

117

37-247

154

49-08

44

14-018

81

25-790

118

37-565

155

49-34

45

14-336 1

82

26-108

119

37-883

156

49-66

46

14-654

83

26-426

120

38-202

157

49-97

* By the pitch is understood the distance between the oentrM of H
contiguous teeth ; and by the radius is understood the distance betwaea t
centre of the wheel and the centre of each tooth. For any other pill
say, as two inches is to the radius in the table, so is the gi^en pitdi fee i
radius required.

pp. IV.] ON THE TF,ETU OF WHEKLS. 115 ^^^|

B bdkub

No. of

lUdiluiD

No. of

lUdius in

No. of

Htulimin

^^H

■ iach«>.

teeth.

iMLh.

inches.

teeth.

inchau

1

lig 1 ao-ane

194

61-755

230

73-214

266

84-673

I!3 oO-ei5

195

62-073

231

73-532

267

84-901

m 50933

196

62-392

232

73-850

268

83-300

^^^^1

1 51-25)

197

62-710

233

74-168

260

83-627

^^^^H

i 51-569

198

63-028

234

74-487

270

85-946

^^^^H

'( 51-888

100

63-346

235

74-805

271

86-264

^^^^H

H 52-20e

200

63-665

236

75-123

272

86-582

^^^^1

ffi 52-524

201

63-983

237

75-441

273

86-900

^^^^1

«B 52-8*3

202

64-301

238

75-700

274

87-219

^^^H

fi7 53-1 Gl

203

64-620

230

76-078

275

87-537

■^^^B

S8 fi3-i79

204

64-938

240

76-397

276

87-855

^^^^M

W 53-798

205

65-236

241

76-715

277

88-174

^^^^M

W 5VII8

206

63-574

242

77-1133

278

88-492

1 5t-434

207

63-893

243

77-351

279

88-810

^^^^M

5*--52

208

6B-211

244

77-670

280

89-120

^^^^M

55-071

209

66-529

245

77-988

281

80-447

^^^^M

; 55-380

210

66-848

246

78-306

282

80-763

^^^^M

55-7fl7

211

67-166

247

78-623

283

90-084

^^^^M

56-026-

212

67-484

248

78-943

284

80-402

^^^H

5B344

213

67-803

249

79-261

285

90-720

^^^H

■- 58-662

214

68-121

230

70-580

286

91-038

^^^^1

P 56-980

213

68-439

251

70-898

287

91-357

^^^H

57-299

216

68-757

252

80-216

288

91-675

^^^H

57-617

217

69-075

253

80-534

289

91-993

^^^^1

57-935

218

69-394

234

80-853

290

92-312

^^^^M

58-253

219

69-712

253

81-171

291

92-630

^^^^M

58-572

220

70-031

256

81-489

292

92-948

^^^^M

58-890

221

70-349

257

81-808

293

93-267

^^^^M

5S-2O0

222

70-667

258

82-126

294

03-585

^^^^M

59-327

223

70-985

259

82-444

295

93-903

59-845

224

71-304

260

82-763

296

04-222

^^^H

eo-163

225

71-622

261

83-081

297

94-540

^^^H

00-482

22G

71-941

262

83-399

298

94-858

^^^H

90-800

227

72-258

263

83-717

299

95-177

^^^H

6M18

22S

72-577

264

84-036

300

95-495

^^^H

6I-43S

229

72-805

265

84-354

^^M

OF AO&ANGmC THE NUMBERS OF WHEEL-WORK. ^^^H

, In a machine, the velocity of the impelled pniat ^^^|

be to that of the working point in the ratio which ^^H

pted to the maximuni effect of the moving power on ^^^|

le port, and the best working effect on the other part. ^^^|

ther arrangement of the relative motions of the parts ^^H

. M

116 ON THE TEETH OF WHEELS. [cHAP. IV.

of a machine must clearly be attended with a loss of power,
or the work will not be done properly. But when the best
working velocity is known, and also that which enables the
first mover to produce the greatest effect ; the proper ar-
rangement 6f the numbers of the teeth of the wheels and
pinions is a very simple operation. The subject has been
treated of for particular machines, by several writers ; but
since it has been chiefly under a somewhat erroneous view
of the real nature of the maximum effect of machines, it
will be perhaps of use to give a general formula, and a few
particular examples, to save the trouble of reference, and
render the work somewhat more complete*.

It will be an advantage to advertise the young mechanic
of one or two essential particulars, before proceeding to the
principal object.

147. In the first place, when the wheels drive the
pinions, the number of teeth in any one pinion should not
be less than 8 ; but rather let there be 11 or 12 if it can
be done conveniently. And in the particular form of teeth
described in Art. 30, the number of teeth in a pinion
should not be less than 10 ; but it would be better to have
13 or 14. (See Art. 34 to 37.)

148. Secondly, when the pinions drive the wheels, the
number of teeth on a pinion may be less ; but it will not
in anv case be desirable to have fewer than 6 teeth on a
pinion ; and give the preference to 8 or 9, where it can be
done with convenience.

149. Thirdly, the number of teeth in a wheel should
be prime to the number of teeth in its pinion ; that is, the

* The methods of adjusting the numbers of wheel- work, so that the con-
temporary revolutions may be always in a given ratio, is a distinct branch
of this subject, chiefly useful in clock and watch- work, planetary machines,
and the like ; and since our plan does not include the construction of such
machines, the reader, desirous of such information, may consult Camus on
the Teeth of Wheels.

CHAP. IV.3 ON THE TEETH OF AVHEELS. 11?

number representing the teeth in the wheel should not be
divisible by the number of teeth in the pinion without a
remainder. And as the numbers of pinions will in general
be first settled, it will be an advantage to take a prime
number for each pinion, as 7> 11> 13, 175 19, 23, &c., be-
cause such numbers are seldomer factors than others.
But when it happens that a prime number can be directly
fixed upon for the wheel, any whole number which ap-
proaches near to the required ratio will answer for the
pinion; as minute accuracy is not required. A prime
number for the wheel, or one which is not divisible by the
number of the pinion, is esteemed the best, because the
same teeth will not always come together, and the wear
will be more uniform.

150. Foiurthly, if it be desired that a given increase or
decrease of velocity should be communicated with the least
quantity of wheel- work, it has been shown that the number
of teeth on each pinion should be to the number on its
wheel, 08 1 : 3-59. (Dr. Young's Nat. Phil. Vol. II. Art.
S66.) But, on account of the space required for several
wheels, and the expense of them, it will often be necessary
to have 5 or 6 times the number of teeth on the wheel that
there is on the pinion. The ratio of 1 : 6 should however
not be exceeded, unless there be some other important
reason for a higher ratio.

151. Of calculating the Numbers for Wheel-Work.
Let n be the number of revolutions per minute for the first
axis^ to which the moving power gives motion ; and n the
number of revolutions per minute of the last axis, where

the resistance or working point is. Then, n : n : : 1 : — »

which is the ratio the velocity is to be increased or di-
mimshed.

If this ratio shpuld not exceed 1:6a single wheel and

118 ON THE TEETH OF WHEELS. [CHAP. IV^*

pinion will be sufficient ; but when it exceeds that ratioy
more will be necessary.

When each of the pinions has the same number of teedx,
and each wheel the same number ; then, the ratio of th.^
number on a pinion, will be to the number on a wheel

as 1 : ^; oras 1 : (^V' ., ^

The number of pinions being a.

But it is sometimes necessary to adapt the trains of ma-
chinery to produce diflferent velocities at the working points,
and it is on this account often desirable to vary the size of

the wheels. Then, the ratio 1 : - must be decomposed into

factors suitable to the nature of the work to be done ; if
those factors be any numbers a, b, c, &c. the ratio will be

1 : a X 6 X c, &c. : — (2.)

The first mover should be as near as possible to the re-
sistance. But, when it is absolutely necessary to perform
operations at a considerable distance from the first mover,
the velocity of the communicating shafts should be brought
up, as near as possible, to the first mover, to that which is
most advantageous for the difierent species of work.

But the velocities of the parts of machines are often
given in feet per second ; let v be the velocity in feet per
second ; then 60 1; is the feet described in a minute. Also
let d X 3*1416 be the circumference which moves with the

velocity V ; then g,^^^^^ = —^ — being the revoluticms

which the axis makes in a minute.

When the first mover acts with a velocity v at the dis-

, . -. , . ^ 10-09t7
tance ^ a from the axis; then — j — = n ; and the ratio

CIMP. IV.] ON THE TEETH OF WHEELS. 119

6f the teeth on tlie pinions should be to those on the wheels

(3.)

^9f)-09tt
And, when the velocity v of the working point is also given,

19-09 V

■ N ; and the ratio of the teeth on the

nitiionsshouldbeto those on the wheels as 1 : ( — \z (4.)

If the velocity is to be decreased, then it will be as the
fheeU are to the pinions, instead of the pinions to the
wheels. The use of these proportions will be best illus-
irated by examples.

I5'i Example I. Let it be required to calculate the
numbers for a com mill moved by an overshot water-wheel.
This case comes under Proportion (l). where o, the
number of pinions, will never exceed 2, and not often more
than 1. And in order that the grain may not be too much
heated in grinding, the velocity of the circumference of the
miUstonc, should not he greater than 23 feet per second,
hence v = 23 ; and d will be the diameter of the millstone
in feet.

153. If we suppose the wheel and its concomitants to
offer no resistance to the impulse of the water, it is manifest
that the velocity of the circumference will then be the
wme as that with which the water strikes it, or that which
.ue to the whole height of the fall, and will therefore be
iS9ed by V = -J '■Zgh; where h denotes the height of
fall in feet ; « the velocity in feet per second, and g =
feet, the velocity generated by gravity.
But it is not consistent with the laws of nature that a
machine can he put in motion without offering some re-
liance to the moving power ; for, in the first place, the

120 ON THE TEETH OF WHEELS. [CHAP. IV.

friction of the parts has to be oyercome, and it must be a
machine of a very simple construction indeed, if the ac-
complishment of this alone does not expend one-half of the
force applied : but in the generality of combinations, it
will be found to balance nearly two-thirds of the power ap-
plied ; we will therefore be pretty near the truth by as-
suming the friction as equivalent to two-thirds of the
moving power.

In the next place, there is the resistance to be overcome
at the working point, and this is equivalent to the quan-
tity of work to be performed, which must therefore expend
the remaining third of the moving power. ^ It thence ap-
pears, that, in the case of an overshot water-wheel, the
height of the fall which produces the velocity of the cir-
cumference, must be divided into two parts ; one part to over-
come the friction, and the otiier to produce the useful
effect

Let e be that part of the fall which is competent to over-
come the friction of the loaded machine only; or that
which corresponds to the velocity when the useful effect is
nothing ; and let x be that part of the fall producing the
velocity corresponding to the maximum of useful effect
It is therefore manifest, that the effective force of the water
on the wheel, when the work done is the greatest possible,
will always be proportional to A — a: ; and when the work
done is nothings the effective force will always be proper-
tional to h — e. Now, the difference of these two quan-
tities drawn into the velocity must be a maximum when
the greatest effect is obtained ; hence we have v (A — or —
A -h ^) = ?^ (^ — a:) a maximum. But, by the laws of fall-
ing bodies, we have v= >/ Q,gx\ let this be substituted for

V in the above expression, and it becomes ^/ Q,gx (e — x)y
a maximum ; or by dropping the constant factor 2 g*,
it is e X ^ or — dL maximum. Let this expression be

CHAP.IY.J ON THE TEETH OF WHEELS. 121

thrown into fluxions and equated with zero or nothing,
and it becomes

^ € <jb X ■"" <u «ir X — v/»

Gmsequentlj, by transposition and reduction, we get

Bat we have stated above that two-thirds of the moving
power is expended in overcoming the friction ; hence we
ha?e« = A — f-A = JA; therefore, ^=iA, and the velo-
city corresponding to or is

t?= >/64ixjA = 2-673 ^h.

We have next to determine the diameter of the wheel
in relation to the heigl;it of the fall when the effect is a
maximum ; and for this purpose, let ^ be equal to that por-
tion of the circumference which is loaded with water, and x
equal the arc comprehended between the point of impact
and the horizontal radius ; then, by the principles of men-
suration, the magnitude of the solid which represents the

effective force, is ^ i {jLH V where h is the section of

the stream supplying the buckets. But this, by the ques-
tion, is to be a maximum. Let it therefore be thrown into
fluxions, and put equal to zero, and we get

from which by transposition we get

and reducing the quadratic by the rules of algebra, it be-
comes

■ar=^(l-^/f).
But 1 - >/i= 1 - 70711 =-29289 ; hence we get

ar=-29289^; and^ =

•29289

122 ON THE TEETH OF WHEELS. [CHAP. IV.

It is, however, obvious, that the difference between ihe
two arcs <f> and a:, must be equal to a quadrant ; that is,

,gQggQ-ar=90^ and this gives x^SJ"" 1&.

Let r denote the radius of the wheel, estimated from
the centre to the remote point of the bucket ; then will
r (1 4- sin. 37* 16') express the eflGective height of the ML
But the natural sine of 37** 16', is '60553 ; hence we have
1*60553 r for the effective height; and if the absolute
height of the fall be equivalent to nine-eighths of the
effective height, we shall have A=l*8062 r ; consequently,
r — *554f h and rf=2 r=l*108 h. Having thus determined
the values of v and d^ let them be substituted in the ap-
propriate Proportion number (4) preceding, and we get

1 : ^ — ^ — , when there is only one pinion j

but when there are two pinions, it becomes

asl:{?:^}*.

In practice, the height of the fall and the diameter of
the millstone will always be known ; in the present in-
stance let D = 5 feet, and A = 16 feet ; then we have

>/ 16 = 4, and therefore it is

^ . 9*54 X 4

X ■ f

5
or as 1 : 7*632. Now it will be better, in this case, to have
two pinions, since the ratio is greater than 1:6; there-
fore as 1 : 7*632^ or as 1 : 2*763, so is the teeth in each
pinion to the teeth in its wheeL And making the prime
number 11 the number of teeth of each pinion, we shall
have 1 1 X 2*763 = 30, the nearest whole number for the
teeth of each wheel. The thickness of the teeth being
found by the rules for that purpose, (see Art 137,) the

CHAP. IV.] ON THE TEETH OF WHEELS. I'iS

radius of the wheels and pinions will be found by the table
of radii, (Art. 14.5.)

;ijain, let the fall be 4'8-l- feet, the diameter of the mill-
stone being 5 feet as before ; then, .j k = 'i'Q, and

1: — becomes as 1 : 4'1976. Here one pinion or trun-

D

file will be best ; and making the teeth on the pinion 1 1
those on the wheel will be 11 x 4'-1976 = 4i6 in the nearest
wliolc number.

The young millwright will find it useful to calculate a
table by these rules, which might be extended to under-
shot wheels, and exhibit at one view the whole construction
mills.

154-. Example II. Let it be required to arrange the
lumbers for a machine for raising water, where the pumps
are to make N strokes per minute, the velocity of the
moving power being v feet per second, and the diameter of
the circle described by the power d feet.

This case is an example of the use of Proportion (3), or

1 : (^ — 1„ — J-. Now let the moving power be a horse,

where v is 2^ feet per second, (Art. 119,) and d the dia-
meter of his track, 30 feet ; the pump to make 20 strokes

em

r

minute, or n=20, then the ratio is I : ■., t

19-09 x^i^

1 : 1'2*6 nearly. Therefore if the pinion have 13 teeth,
wheel should have 13x 12-6 = 164 teeth in the nearest
whole numbers.

But we should prefer making two piniona, and then the

ratio will be 1 : v liJ*6 or as 1 : 3-549 nearly, and each

pinion having 1 1 teeth, the wheels should each have

H] X 3-549 = 39 teeth.

^HfThese examples will perhaps be sufficient to explain the
^^bdc of calculation ; and when once it is understood, it
^^Bl vur)- easily be applied to other cases ; and will be

134 ON THE TEETH OF WHEELS. [CHAP. IV.

found somewhat more convenient than the methods usually
followed.

PRACTICAL OBSERVATIONS WITH REGARD TO THE MAKING OF PATTERNS

FOR CAST IRON WHEELS.

155. Having determined the pitch of the wheel strong
enough for the purpose to which it is to be applied, the
thickness of the tooth serves to regulate the proportionate
strength of the other parts.

A very respectable millwright informs me, that he has
for a considerable time adopted the following rule for de-
termining the length of the teeth of wheels, the practical
efficacy of which he has found quite satisfactory.

Rule. — Make the length of the teeth equal to the pitchy
deducting freedom^ (by the freedom is meant the distance
at the top of one tooth, and the root of another measured
at the line of centres,) in other words, the distance from
root to root of the teeth, at the line of teeth when the
wheels are in action, exactly equal to the pitch.

For example — ^he makes the teeth of two inches' pitch, 1
inch and ii in length, which is allowing iV of freedom.

Another respectable millwright, who has had much ex.
perience, particularly in miUs moved by horses, has for a
considerable time past made the teeth of his wheels in
length only one half of the pitch, and works them as deef
as possible without the point touching the bottoms. Be-
fore he fell on this expedient, he found the teeth exceed-
ingly liable to be broken fix)m any sudden motion of the
horses*.

Indeed, ujwn reflection, it will be found there is no oc-
casion for more freedom, than that the point of the tooth
of the one wheel shall just clear the ring of the other ;

* lUcspccting the length of icclb, sscc Art. 2D, 42, luid 47.

MAP. IV,] ON THE TEETH OF WHEELS. 195

more than this must only serve to weaken the teeth. The
Diixle of gearing, however, ahove alhided to, is more neces-
sary in horse mills than where the moving power is steady
lad regular.

HattoQ (on Clock-work) recommends making the dis-
lance of the pitch line | of what we call the thickness of
the tooth. Thus, suppose the rule applied to a two inch
pitch and that the tooth and space were exactly equal, then
the tooth would project f of an inch beyond the pitch line,
(nil its root would be as far within the pitch line, as to re-
eeive freely the tooth intended to act on it: suppose it also
J, then the tooth would be 1 J inch long, besides the free-
dom, which, making as above, iV, the tooth would be in
»11 lU inch long.

156. But it is to be remarked, that the millwright, in
naking his pattern for a cast iron wheel, has to attend to
I drcumstance arising from the nature of that material.
The pattern must not only be of such a form as to be suf-
ficiently strong, calculating by the bulk of the parts, but
|lso proportioned, so that when the fluid metal is poured
B the mould, it may cool in every part nearly at the same
bie.
HTien due attention is not paid to this circumstance, as
e metal is cooling, if it contract faster in one part than
I another, it will be apt to break somewhere, just as a
haking glass is broken by suddenly cooling or heating in
ly particular part of it. In all patterns for cast iron,
kout J of an inch to the foot, should he allowed for the
Btraction of the metal in cooling.

Attention must also be paid to taper the several parts,
that they may rise freely without injuring the mould,

1 the founder is drawing them out of the sand. A
lie observation of the operations of a common foundry,
1 better instruct on this part of the subject than many
We niav obser\'C, however, that about Vs of an

126

ON THE TEETH OF WHEELS.

[chap.

inch, in a depth of 6 inches, is commonly a
taper.

157* Attending to those circumstances, weoffer the folio
ing proportions as having heen found to answer in practi

Make the thickness of the ring a e equal to the thie
ness of the tooth a c near its root. When the ring*
made thinner than the root of the tooth, the ring common,
gives way to a strain, which would not hreak the tooth.

Make the arm, at the part where it proceeds from
ring, of the same hreadth and thickness as the ring ;

18

'7

Pio. 2.

Fig. 3.

Fio. 4.

at the junction «» ^ let it he so formed as to take off any
acute angle which would he apt to break off in sand.

CHAP. IV.2 ^^ THE TEETH OF WHEELS. 127

The arms should become larger as they approach the
centre of the wheel, (see Emerson, Prop. 119, Rule 8,)
and the eye e, should be sufficiently strong to resist the
driving of the wedges, by means of which it is to be fixed
on the shaft. This cannot be brought easily to calculation.

On the other hand, care must be taken not to make the
eye so thick as to endanger unequal cooling.

It should be somewhat broader than the breadth of the
teeth, in order that it may be the firmer on the shaft : this
breadth must be greater in proportion as the wheel is large.

When the ring a e is about an inch thick, it is common
to make the eye about an inch and a quarter thickness, and
about one-fifth broader than the ring, when the wheel is
about four feet diameter.

Small wheels have generally but four arms, but it being
improper to have a great space of the ring unsupported, the
number of arms should be increased in large wheels.

In order to strengthen the arms with little increase of
metal, it is not unusual to make them feathered, which is
done by adding a thin plate to the metal at right angles
to the arm, as represented by figure third. Fig. 4 is a
section of Fig. 3, at a b.

The same rules apply to bevelled wheels ; of the prac-
tical mode of laying down the working drawings of which
we have already spoken. But it is proper to observe, that
the eye of a bevelled wheel should be placed more on that
side which is furthest from the centre of the ideal cone of
which the wheel forms a part.

158. When wheels are beyond a certain size, it becomes
necessary to have patterns sometimes made for them, cast
in parts, which are afterwards united by means of bolts.

To prevent the bad eflects of unequal contraction, the
arms may be forked or curved, as in the second figure ; the
forked or curved parts are commonly of the same radius
as tihe wheel, and spring from the half length of the arms.

128 ON THE TEETH OF WHEELS. [CHAP. !>?"•

MATBBIALS OF PATTERNS.

1 59. The patterns should be made of well-seasoned wooci
The most proper is clean mahogany*, but that being no^^
very expensive, white deal is most commonly used. BeeaXl
is very often used for the teeth, and being a close graine*^
wood, it may be made very smooth.

It is almost superfluous to say, that the workmanship of
wheel patterns should be such as to produce great accuracy^
and a smooth surface, the former being essential to the
good movement of the wheels, and the latter to make the
patterns produce a good and clean impression in the sand

It is a conmion practice, in many places, to make teeth
very large in the pattern, and after fixing the wheels on
their shafts, to chip and file the teeth to the proper size ;
but we doubt whether this practice be really advantageous ;
for besides the great time which it occupies thus to dresa
the iron teeth, and the consequent expense, there is the
loss of the outer skin (if we may use the expression) of the
cast iron, which is by far its most smooth and durable part.
In cotton mills, therefore, this absurd method is now bat
seldom practised t.

* The common chestnut tree is equal to any wood that can bo used ; and
its dimensions adapt it equally well for moulds with Honduras mahogany.

t Messrs. Peel, Williams, and Co., have, after great time, trouble, and
expense, made and arranged a very great number of patterns of wheels, so
as to suit almost every case that can in practice occur. They have published
a complete list of them, which they intend inserting also in the '^ Repertory
of Arts." In my opinion, what they have done is a material national be-
nefit ; their expense, I am informed, for patterns, has not been less than
four thousand pounds. There is, however, every reason to think, that it
will be an excellent thing ultimately for themselves, as well as of great
practical utility to the public. — Buchanan.

CHAPTER V.

» THE USE OP CHARTS, AND SOME FURTHER EXPLANATION
[ 0? THE CONSTRUCTION OF THE TABLES OF PITCHES OF
[ VHEEL-WORK.

. When quantities of any kind, such as time, space,
My, &c., expressed in numbers, are mentioned, it often
s a painful exertion of the mind to recollect and
e them. Hence the utility of bringing them into one
r in tables. But there is another motlc of comparing
iDtities not so generally practised, though, in many
I much more easy and satisfactory to the mind. I
I to charts, in which, instead of using figures, as in
tafiles, the quantities are geometrically represented. This
■ done by dinding the sides of a square or rectangle into
' i|ual parts, and drawing parallel lines at right angles from
tiL' divisions. The quantities are pointed off at certain
itersections of these scales.

llli. WTicn the quantities increase or decrease in arith-
in'tical proportion, as 1, 2, 3, 4, &c., that proportion will
!k' represented by a straight line, which will pass through
thfse points.
\(>'2. But supposing the quantities to increase in geome-
ial proportion, as 1, 4, 9, l6, &c., the line passing
'rdugh the points of intersection will form a curve.
These two cases will be best explained by examples.
K).'?. F^rst, Suppose the value of any thing to increase as
'eight, — the scale on the one side of the square will
I represent the value, and that on another the weight.
at AC, I'ig. I. Plate I., represent weight, (say ounces.)
AD value, (say shillings.) Now, suppose we mark tho

130 ON THE TEETH OF WHEELS. [CHAP. Y.

price of four ounces, it is done by placing a dot opposite to
four, on the line of value, and opposite to four on the line
of weight, at the intersections of the perpendiculars from
these points, which intersection is marked by d on the figuie.
In the same manner, we may mark the value of 1 2, 3, 5,
6 ounces. These points are marked a b c e fy and the
straight line a b passes through them all, and shews the
regular progress of the proportion.

It is of no consequence whether the divisions on the line
A c be greater or less than those on a d, provided the lines
be divided into equal parts.

164. Second y Suppose an accelerating motion, such as
that of a falling body, is to be laid down on a chart, — ^this
motion increases as the squares of the times ; that is, the
body falls a certain distance in the first second of time, four
times that distance in the next second, and nine times in
the third second, &c.

These points are accordingly marked in Figure ^ by a
opposite to 1 on both scales, by b opposite 2 on the scale
of time, A D and 4 on that of motion a c ; by c, opposite S,
on a d, and 9 on a c, &c., the line a b passing through these
points forms a curve.

165. When the proportion of any kind is regular, the
curve has a regular easy sweep ; if otherwise, the curve will
undulate, or have irregular windings. Hence, it is a good
mode of proving many kinds of tables, to lay down the
quantities thus geometrically ; for if there be any material
error, when the proportion ought to be regular, an elbow
will appear in the line a b.

166. Much calculation, too, may often be saved, for when
a few of the principal points at some distance from each
other are obtained in the curve, the rest of it may be easily
found, by drawing the curve between them with a slip of
thin wood, or any other such means of producing an easy
curve.

CHAP. V.3 ON THE TEETH OF WHEELS.

131

167. Charts, on similar principles, are used for many
purposes ; for example, there are biographical charts, show-
ing the periods when eminent men appeared, and the rela-
tive length of their lives. They are also used for represent-
ing revenue of any kind, which generally forms an undu-
lating line, as does also the charts of the heights of the ba-
rometer, or the temperature indicated by the thermometer.
The heights of mountains, the tides — ^in short, they may be
considered as merely scales of equal parts, and, of course,
are applicable to all subjects capable of being represented
by numbers.

168. In order to give a more distinct comparative view
of the tables of pitches, in the ^^ Essay on the Teeth of
Wheels,*' we shall lay their contents down in one chart, but
previously collect all these tables, and give some further
explanation of their mode of construction.

TABLES OP PITCHES OF WHEEL-WORK.
(See Chap. IV., Art. 123—129.)

TABLE I*.

Velocity of the pitch line being 3 feet per second, and
breadth of teeth 9 inches.

w

X

Y

Z

Htehin
inches.

Breedth

of teeth in

inches.

Value of

strength

in hontcs*

power.

Value of

strength

in honics'

power.

4

8i
3

2i
2

1

9
9
9
9
9
9
9

16-
12-25

9-

6-25

4-

2-25

1-

12-
10-5

9-

7-5

6-

4-5

3-

* Tables I. and 11. are for teeth attached to water-wheels, where liable
to be worn by aand and water.

k2

las

ON THE TEETH OF WHEELS.

[CH

TABLl U.

The velocity being 3 feet per second, and the bi
of the teeth double each pitch.

w

X

Y

Z

Ktcfain

Twice the
Dieedth.

Value of

Itrength

in hones'

power.

Value of

ttrmgth

in homt*

power.

4

3*
3

2

1*

1

8
7
6
5
4
3
2

14-22
9-53
6-

3-47

1-77

•75

•22

10-66
8-17
6-

4-16
2-65
1-5
'66

TABLB in*.

The velocity being 1 1 feet per second, and the br
of teeth 8 inches.

w

X

Y

Z

Pitch in
inches.

BreMlth

of teeth in

inches.

Value of

strength

in horses*

power.

Value of

strength

in horses*

power.

4

3i
3

H

2
1

00 00 00 00 00 00 00

81-77
62-61
46-
31*94
20-44
11-5
5*11

61-33

53-66

46-

38-33

30-66

23*

15-33

• TiWce III^ IV^ V^ and VI. are for teeth properly greased,
from sand.

UP. v.]

ON THE TEETH OF WHEELS.

133

TABLE IV.

The velocity being 1 1 feet per second, and the breadth
)able each pitch.

w

X

Y

Z

Pitrhin
inches.

Twice the
pitch in
breadth.

Value of

strength

in hones*

power.

Value.of

strength

in hones*

power.

4

3i

3

H
2

H
1

8
7
6
5
4
3
2

81-77
54-78
34-5
19-95
10*22
4-31
1-28

61-33
46-95
34-5
23-94
15-33
8-62
3-84

TABLE V.

The velocity being 3 feet per second, and breadth of
Beth 8 inches.

w

X

Y

Z

Pitch in
inches.

Breadth

of teeth in

inches.

Value of

strength

in horses*

power.

Value of

strength

in horses'

power.

4

3

H
2

1

8
8
8
8
8
8
8

22-30
1707
12-54

8-71
5-57
3-13
1-39

16-72

14-63

12-54

10-45

8-36

6-26

417

134

ON THE TEETH OF WHEELS. [CHAP. Y.

TABLB Vr.

The velocity being 3 feet per second, and breadth double
each pitch.

w

X

Y

Z

PHchin
inches.

Twice the
pitch in
breadth*

Value of

strength

in hones'

power*

Value of

strengui

in hones'

power.

4

3

H
2

1

8
7
6
5
4
3
2

22-30
14-93
9-4
5*44
2-78
117
0-34

16-72
12-79
9-4
6-53
4-17
2-34
1-02

BEFEBENCB TO TABLB I*, ART* 123, IN THB E8SAT ON XHB

TBBTH OF WHBBL8*

169. Column X is omitted in Tables I., III., and V.,
being only a repetition of the same breadth for all the
pitches of each table, but as being perhaps plainer, they
are inserted here.

The numbers in column y are found by squaring the
pitch in column w. — (See Proposition I. p. 83.)

EXAMPLE.

The square of 4, (the pitch in inches) >= 16, the value
of strength in horses* power. — (See the first line of table.)

The column z is found by inverse proportion. — (See
Prop. II. p. 85.) taking three inches, (the standard pitch,)
always as the first term, the pitch column, w, as the
second, and the horses' power, in column y, as the third
term.

CHAP, v.] ON THE TEETH OF WHEELS. 135

EXAMPLE.

hi In. HonaT power. Honw* power.

3:4:: 16 : 12 (See first line of table.)
3:1:: 1 : 3 (See last line of table.)

&BPBRSNCB TO TABLE II.

170. The numbers in column y are found here by direct
proportion from Table I., nine inches (the breadth in
TaMe I.) being always the first term of the proportion ;
the horses' power in y, Table I. the second, and the
breadth in x. Table II. the third term.

EXAMPLE.

iB-Bomr power. In. R

9 : 16: : 8 : 14-22 (See Table II. line first). Column
2 is found, as in Table I. by inverse proportion, 3 inches
(the standard pitch) being always the first term. Thus,

^ Ib. Honei^ power. Honei* power.

3:4:: 14-22 : 10-66.— (See first line of table.)

RBPBRBNCB TO TABLB III.

171. The numbers in column y are found by direct pro-
portion, taking 9 (the square of the standard pitch of
three inches) as the first term, and the square of the pitch
in w as the second term.

EXAMPLE.

As 9 (the square of 3 inch pitch)

Is to 16 (the square of 4 inch pitch).

So is 46 horses' power (the value in column y of 3 inch

pitch)
To 81*77.— (See first line of table.)

136 ON THE TEETH OF WHEELS. [CHAP. Y.

Column z is found by inverse proportion, as in former
tables.

EXAMPLE.

In. In. Honcft* power. Hones* power.

3 : 4 : : 8177 : 61-33 (See Istlme of table.)

BEFKBBNCB TO TABLE IV.

172. The numbers in column y are found here in a
manner similar to Table II. by direct proportion.

EXAMPLE.

In. In. Hortc** power. Horaet* power.

8:7:: 62-61 : 54-78.— (See 2d line of table.)
The numbers in column z are found, as in the former
tables, by inverse proportion,

In. In. Hortct* power. Hones* power.

3 : 4 : : 8177 : 61-33.— (See 1st line of table.)

KEFEBENCE TO TABLE V.

173. The numbers in column y of this table are found
by direct proportion from column y of Table IV., 1 1 feet
(velocity per second) being always the first term, and 3
feet (velocity) the second term.

Ft. Ft. Hones' power. Horaes' power

Thus, 11 : 3 : : 81-77 : 22-30.— (See Istlme of table.)
Column z is found by inverse proportion, as in all the
former tables :

In. In. Horses' power. Horses* power.

Thus, 3:4:: 22-30 : 16'72.— (See Istlme of table.)

RBFBBBNCB TO TABLB VI.

174. The numbers in column y and z in this taUe^ are

I: CHAP, v.] ON THE TEETH OF WHEELS, 137

I fbnoed from Table V. in the same mamicr as those columns
I in Table II, are formed from Table I. The only differ-
lence in Table VI. from Table V. is that which arises
■from the difference of the breadth of the teeth.

EXPLANATION OF THE CHABT.

175. The scale on the line a b represents tlic pitch in
[inches.

The scale on the line a c the horses' power ; a single

iinple will probably be sufficient to illustrate the use of

[fbe chart.

Suppose the pitch to be 3^ inches, let it be required to

1 the horses' power to which that pitch is equal when

Kmg at 3 feet per second, in situations where properly

P greased and free from sand — observe, where the line from

I file pitch 3^ intersects the curve z of Table VI. perpen-

iitular to the point of intersection, on the line ac, will he

Had 12'79 on the scale or the horses' power to which 3^

ch, when the teeth are 7 inches broad, is equal, after

Jting allowance for the length of the teeth.

OBSERVATIONS.

176. Ist, It will he observed, that the curves y and z
intersect each other, for all the tables on the pitch line
marked 3 inches, because that is the standard. (See Ist
Essay, p. 96, 97-)

177. '2d, The curves, continued from 1 inch pitch down-
irard, unite in the points marked 0, being the commence-
ment of the scale of pitches, and this part of the curve

i the horses' power equal to any fraction of an inch,

L178. 3d, It has been already observed, p, 98, that durability

Ewell as strength should be considered in this investiga-

Tbe true proportion, therefore, may be somewhere

len the curves y and z, but nearer to z than y. For

138 ON THE TEETH OF WHEELS. [cHAP. ▼.

although long teeth will be more easily brok^i than short
ones, yet while they do not break, the strain being gene-
rally diffused over a greater number of teeth, they will
wear longer*.

179. The pitch is laid down on a b, real measure, so
that if the pitch should happen to be fractional, it may be
taken by a pair of compasses and applied to the chart,
which will at once indicate the power to which it may bet
equal at certain velocities.

Among the writers who have turned their attention to
the forms of the Teeth of Wheels, Professor Willis, (^i
Cambridge, stands pre-eminent ; and we are greatly io..
debted to that gentleman for his liberality in permitting
us to insert the following appendix, his Essay on the Teeth of
Wheels, and which appeared originally in the second volume
of the Transactions of the Institution of Civil Engineer^
and for the additions he has since made to that paper.

• See Art. 70.

APPENDIX A.

OH THE TEETH OF IFHEELS. BY R. WILLIS, M.A., F.H.S.,
H.J1.1NST.C.E., JACKSONIAN PROFESSOll OF NATURAL PHI-
LOSOPHY IN THE UNIVEESITY OF CAMBRIDGE.

180. The investigation of the proper curves to be given to

'k' teeth of wheels, has been a favourite occupation with

matliematicians of the highest eminence, and the geometry

uf ihe subject may be considered to be very nearly com-

pk'te.

Its application to the requirements of modem construc-

D appeared to me to be susceptible of improvement, and

therefore ventured to lay before the Institution of Civil

gineers the following suggestions, in which I en-

lToure<l to point out forms possessing properties more

I than those hitherto adopted, as well as some prac-

d methods of tracing readily the outlines of the teeth.

SECTION I.
ON THE CURVES ADAPTED TO PRACTICE.

■There are an infinite number of forms which will answer

I conditions of enabling the teeth of one wheel to com<

plicate equable motion to those of another, for it can be

n that, under certain limitations, if any form of tooth

Segiren, another may be determined which will work cor-

140 ON THE TEETH OF WHEELS. [^APPEND. ^^

rectly with it*. A simple instrument which famishes a
practical solution of this problem, wiU probably carry mo:«re
conviction to the minds of practical men than the demtk-^.
strations which have been given by the writers referred -to
below. Let a pair of boards he prepared, having th^r
edges, AB, CD, Fig. 1, formed truly circular. Attach to

one of them by any simple clamp the shape of the given
tooth E, cut out in pasteboard, and to the other a piece of
stiff paper secured by means of drawing pins ; the shape
E must be raised slightly above the surface of its board, so
as to allow the paper which is appended to the other to
slide under it, as is shewn in the figure. Make the cir-
cular edges of the two hoards roU together, and in each
successive position draw the outline of the shape e upon
the paper below it. The result of all these intersectjng
lines will be a bounding curve, which from the very mode
of its description wiU touch the shape e at some point of
its edge in every one of the successive positions. But as
these positions were all obtained by making one circular
edge roll upon the other, so it is clear, that if the new
curve be cut out and made to touch e, the motion produced
1^ the mere contact of these two curves wUl he exactly the

* Vide De la Hire, Traite des Epicydoidcs. Young's Natural Philo-
sophy, Vol. I. page 176- Ait;, Cambridge Philosophical TnuiBactioits,
Vol. II. page 277.

APPEND. A.] ON THE TEETH OF WHEELS. 141

same as that caused by the rolling of the circular edges,
and therefore perfectly uniform.

Many forms of e, tried in this manner, will prove un-
tractable, for some of the successive portions of its edge
may cover up and obliterate parts of the curve that have
been previously drawn. These are forms that fall under
the limitations alluded to, but it is unnecessary here to in-
vestigate the general reasons for this effect, as the propo-
sition in question is well known and recognized by mathe-
maticians, although not so well understood by practical
men.

From among the infinity of curves that may be offered,
the epicycloids and involutes have been universally pre-
ferred, on account of the facility with which they can be
mechanically described, and perhaps because they admit of
ready and independent demonstrations of their possessing
the properties required. But the practice has hitherto
been confined to that class of epicycloids which work cor-
rectly with straight lines or circles. Teeth formed upon
these principles possess this inconvenience : a wheel of a
given pitch and number of teeth, say 40, if it be made to
work correctly with a wheel of 50 teeth of the same
pitch, will not work correctly with a wheel of 100 teeth of
the same pitch. This is obvious, for the diameter of the
describing circle by which the epicycloid is formed must
be made equal to the radius of the pitch circle of the
wheel with which the teeth are to work, and will therefore
be twice as large in the second case as in the first.

In the old style of mill-work, in which the teeth of
wheels always consisted of wooden cogs, this property
offered no very serious impediment, although, as we shall
seep it introduced some complication of method ; but in the
modem practice of making cast iron wheels, the objection
18 a very serious one. A founder must make a new pat-
tern of a wheel of 40 teeth for every combination that it

14^ ON THE TEETH OF WHEELS* [APPEND.

may be required to make of this wheel with others, an^^
the same for a wheel of any other number. Besides, i ^
often happens in machinery, that one wheel is required tc^^
drive two or more whose number of teeth are different, aodL
in this case the teeth cannot be correctly formed at all on
the common principles ; and again, the perfiscticm of ma^
dunery is impaired from the temptation to employ in one
combination patterns that have been formed for some other
combination very nearly the same ; for example, to make a
wheel of 40 teeth that has been formed to work with one
of 80, serve for a required combination of 40 with 85.

It is essential, therefore, that the teeth of wheels should,
if possible, be so formed as to allow a given wheel to woiIl
correctly with any other wheel of the same pitch. Now it
has long been known that involute teeth have this very pro-
perty, but the objections to these teeth on the score of the
obliquity of their action have operated fatally against their
introduction*. I shall now, therefore, explain a method
of imparting to epicycloidal teeth this properly, and thai
without making them deviate very much frt>m the general
form that has been established by practice.

To effect this, it is merely necessary to employ a propo*
sition well known and stated by almost every writer on the
subject, namely. If there be two pitch circles touching
each other, then an epicycloidal tooth formed by causing
a given describing circle to roll on the exterior circumfer-
ence of the one, will work correctly with an interior epi-
cycloid, formed by causing the same describing circle to
roll on the interior circumference of the other t.

This proposition having been already demonstrated, it
is unnecessary for me to dwell upon it longer than to re-
mark, that our author, like all the other writers on the
subject, has passed from it, to recommend for practice that

* Vide Hawkins's Notes to Camus, page 161.
t Vide Art, 19.

trEND. A.^ ON THE TEETH OF WHEELS. 143

particular case of it in which the describing circle being
made equal in diameter to the radius of the pitch line, the
interior epicycloid becomes a radial straight Hoe, the in-
couTOTiences of which practice I have shewn*.

The following corollary is, I believe, new, and consti-
tutes the basis of the system I propose to explain.

Conilliirt/. If for a set of wheels of the same pitch, a
constant describing circle he taken, and employed to trace
those portions of the teeth which project beyond each pitch
line by rolling on the exterior circumference, and those
«iuch lie within it by roiling on its interior circumference :
then any two wheq^ of this set will work correctly togc-
ther.

For, in the first place, it is well known and can be
shewn from general principles, that the portion of tooth
jrithin the pitch line of a driving wheel, works only with
the portion that lies bei/oiid the pitch line of its follower,
and that its action is confined to the approach of the point
of contact to the line of centres. After the point of con-
t of the teeth has passed that line, then the case is re-
, and the portion of the driving tooth which lies be-
i the pitch line is in contact only with some part of the
lower*8 tooth which lies within its pitch line.
Mow as a constant describing circle is used fur the whole
it is clear that the proposition will apply to any pair
of wheeU both before and after the teeth have passed the
line of centres, for in each case we have an exterior epicy-
cloid working with an interior epicycloid, and both have
been drawn by the same describing circle, that is, by the
ctaistant circle of the set.

To carry this scheme into practice, it only remains to
■ttlc! the proper diameter to be given to this constant dc-

' Vide Brewsler'fl Ferguson, Voi. II. p. 223. CamuG, p. 27, or 25 new

144 ON THE TEETH OF WHEELS. [APPEND, j^

scribing circle, which may be done by considering tli,
effect this diameter has upon the form of the tooth.

Let Bc Tif Fig. S, be a pitch circle whose centre is c,
then upon this system the flank of the tooth, or that por-
tion which lies within the pitch circle, will be an arc of an

Fig. 2.

interior epicycloid (or hypocycloid) mfn or mn. Now if
the describing circle be of half the diameter of the pitch
line, the flank will become a straight line coinciding with
the radius on. If the describing circle be of less than
half the diameter of the pitch line, the flank mn will be
concave, and the base of the tooth will spread ; but if the
describing circle be of more than half the diameter, the
flank mfn will be convex, and the base of the tooth lessen
inwards, a form manifestly unpractical and useless. Hence
the describing circle must not be greater than half the dia-
meter of the pitch line.

On the other hand, if the diameter be too small, the
base of the tooth will spread inconveniently, and the curv-
ature of the exterior epicycloids be injuriously increased,
therefore, on these grounds, it should be made as large as
it can consistently with the limitation just stated, so that
we finally obtain this rule for finding the diameter of the
constant describing circle for a set of wheels.

IPPKVD. A.] OS THE TEETH OF WHEELS. H5

Make it equal to the radios of the least pitch circle of
the stt.
And as pinions should never have less than 12 or 14
!th, it would be well to establish one of these numbers
that least pitch circle.

The proposition and corollary being perfectly general,
apply to racks, which must be considered as very large
Is, and also to annular or internal wheels. Accord-
ly, if the constant describing circle be employed in
ring their teeth, they will work correctly with any wheel
the set.

It rill be seen that this system is more easy of practice
the workman than the old one. Every epicycloid re-
es two circular or rather segmental templets, which are
illy cut out of thin board. One of these, which may
termed the pitch templet, has its edge formed into an
of the pitch line of the wheel ; the other, which re-
lents the describing circle, and may be called the de-
bing templet, has its circular edge formed accordingly,
I tracing point is fixed upon the circumference of the
ST, and the workman having previously described an
of the pitch circle of the wheel upon his drawing
d, fixes the pitch templet, so that its edge may coincide
this arc, and then causing the describing templet to
upoQ the pitch templet, he traces the arc of the re-
epicycloid,

bw on the old system, a set of wheels requires as many
ilets as there are pitch circles in the set, and also as
f describing templets, but on the system just explained,
one describing templet is needed. As, however, the
s of the l«eth within the pitch circles become curves
i)f straight lines, it is necessarj' to have concave
ilets a<lapted to the pitch circles, upon whose edges the
ibiiig templet may be made to roll for the purpose of
the proper interior epicycloid. The best way is

D

14G ON THE TEETH OF WHEELS.

to make each pitch templet with two edges, one cootox and
the other coDcaTe, as in the figure, and to write the dia-
meter upon each of them.

ON A POBM OP INCKBASBD STBBITaTH.

In a laige class of machinen-, the wheels constantly
move in the same direction, and whenever this is the case,
it is possible to increase the strength of the teeth in a veiy
great d^jee, by an alteration of the common form repre-
sented in Figure 3.

Let AB, CD, be the acting iaces of the teeth of a pair of
wheels, of which mn, ks, are parts of the pitch lines-
Now, according to the ordinary practice, the backs of the
teeth would be formed exactly in the same maimer as the
acting faces, as shewn by the dotted tines, and this enables
the teeth to work backwards or forwards at pleasure if re-
quired. If, however, the back is never required to ac^ the
strength of the tooth will be nearly doubled by "■■^"g it
of the form BegK, that is, by taking off the portian tm*,

'0

ON THE TEETH OF WHEELS.

147

and filling up the nook eng. Teeth so formed will clear
eich other quite as well as those formed in the usual man-
iier, with the advantage of a root of nearly double extent,
snd as the acting faces remain of the usual form, they will
fork together just as the ordinary teeth do. Strictly
speaking, the back b eg should be an arc of an involute so
proportioned as to work correctly with the corresponding
b.tck (if the tooth of the other wheel. For then, as the
W'ks of the teeth would drive each other truly, they are
Bure to clear each other ; and besides, if the machinery be
made accidentally to run backwards, the teeth will still
work, although with a considerable divergent pressure upon
lb axes'. It will be quite near enough, however, to make
the back an arc of a circle described through three points
Beg', the first of which, b, should be taken a little way
from the point of the tooth in order to blunt it slightly j
the second, e, on the pitch circle is set off in the usual
inimner, so that le may be about i^ths of the pitch j and
the third point, g, may be found by dividing ak into five
I, and taking gK equal to one of them. The space
s required to enable the point of the corresponding
b to clear itself.
3 form resembles the saw-shaped teeth which have
employed occasionally by mechanists, for example,
»rding to Mr. Reid, in his Horology, p. 100,) Lepine,
, had in some of his watches the teeth and pinion-
s of a saw-teeth form, hut I am not aware that the
ntage of this shape has ever been systematically shewn,
J principle of its formation given.

A divergent pre^ure will do no hsnn, ttecanse the kmd of mo-
J to which I propose to adapt this form, never drives backwards,

e workiog presiure i% upon it, but only during some preyioiia od-
mts, when the only pressure to be overcome is that produced hy

T by the friction of the parta of the engine upon each otiier.
L 2

148 ON THE TEETH OF WHEELS. [APPEND. A.

SECTION 11.

ON A PRACTICAL APPROXIMATION TO THE TRUE FORM

BY ARCS OF CIRCLES.

Although the practice in the best workshops is to de-
scribe the shape of a tooth carefully with templets in the
manner just described, yet this is not done for every tooth
in the wheel or pattern ; on the contrary, having traced
the shape of a single tooth in this manner, the workman
next finds with his compasses, by trial, a centre and small
radius by which an arc of a circle can be described that
will coincide as nearly as he can manage to make it with
the templet-traced epicycloid. Then having struck up(m
the face of the rough cogs a circle concentric with the
pitch circle, and whose distance from it is equal to that of
the centre of his arc, he adjusts his compasses to the small
radius, and always keeping one point in the circle just de-
scribed, he steps with the other to each cog in succession,
they having been previously divided into equal parts cor-
responding to the pitch and breadth of the teeth. Upon
each cog he describes two arcs, one to the right and the
other to the left, which serve him as guides in shaping and
finishing the acting faces.

The portion of curve employed in a tooth is so short,
that a circular arc would be quite sufficiently accurate, if
its centre and radius were determined more correctly than
by this coarse mode of triaL This consideration induced
me to investigate the method I am about to describe, in
which the examination of the nature and properties of the
curves made use of for teeth is entirely dispensed with. I
have deduced a simple construction by which a pair of
centres may at once be assigned for a given pair of wheels,
from whence, if arcs of circles be struck and employed for

APPENO. A.'] ON THE TEETH UF WHEF.tS.

149

^M ttie working faces of teeth, they will answer the purpose of

^H enabling these wheels to work correctly together*.

^^ I shall first explain the methods that arise from their

tofijtruction for the use of practical men, and then add the

theon- upon which they are founded.
The working face of each tooth may be formed of one arc

of a circle, or of two. In Fig. 8, Plate 20, each face is formed

» of a single arc op, and the resulting tooth has considerable
snali^ to the involute tooth, and hke it has the fault of
acting with rather too much obliquity j but the mode of
describing it, on the other hand, is exceedingly simple, and
any two wheels whose teeth are thus formed, will work cor-
rectly together. For small teeth, I am inclined to think
tliis method would answer the purpose very well. It only
remains to show how the centre p and radius i't of the arc
oTp is to be determined.

Let A T be the radius of the pitch circle of the proposed

Hvlieel. Upon at describe a semicircle tpa, and from t

Hlbt off t p equal to one quarter of the radius ; then will i>

^De the centre from which, if an arc op be described through

T, the required side of the tooth will be obtained.

Or, construct a bevil in brass, of which the angle at t
&hall be exactly equal to 75° Sty, and graduate the side tp

* Eut(ir, in his secooJ paper on the teeth of wheels, (N. C. Pet. XI.

_'ii9,) li&s with his usual ftbility investigated the proper ciutck, by examin-

■ I'j the rchttion between their radii of curvature at every point. This me-

^iiod hoB uMurally fondui^ted him to results of a similar nature to those

li I have given in tlie following pages, and he Ruggests that a smaJl arc

(die circle of curvature would suffice in practice for the forms of teetb.

II given some geometrical constmctionB for this purpose, and has then

jn finally to recommend the involute oa the best cuire, this paper

a fact, the first in which that curve ts pointed out as possessing the

d pmperties. To Euler, then, belongs the merit of first suggesting

p sabiititudon of an arc of the circle of cun-atiire for the real curve, a

I wkicb bos been, as for as I know, neglected by every succeeding

"writer. This may perhaps be utlribiitcd to the abstruse manner in whicb

b<> hu treated the subject.

150 ON THE TEETH OF WHEELS. [APPEND. A.

into a scale of quarter inches, as in the figure. Apply the
plain side of this bevil to the radius at of the proposed
wheel, and its point t to the pitch circle ; read off the length
of the radius a t in inches upon the reduced scale t p, and
the point p so indicated will he the centre of the tooth as
before. Thus in the figure, at is four inches, and p is
found at 4 upon the scale.

When the side of the tooth is formed of two arcs of cirdeB,
the forms shown in Figs. 9 and 10 are obtained : these re»
present the same teeth in different relative positions. In
Fig. 9 the tooth abc is approaching the line of centres ab,
and in Fig. 10 the same tooth abc is retiring from it. The
portion of tooth ab which lies within the pitch circle, is
described from a centre p, Fig. 9 ; and the portion be which
lies beyond the pitch circle is described from a centre p, fig.
10. The resulting form is a very strong one, possessing
the property that any two wheels of a set will work to-
gether. Any practical man may convince himself of the
degree of accuracy with which this is effected, by describing
according to this method, on a large scale, (say six inches
pitch,) a pinion of twelve or fourteen teeth, and a few teeth
both of a wheel of fifty and of a rack. These teeth may be
cut out of thin board, and it will be found that any two ci
them will work correctly together with a degree of pre-
cision amply sufficient for practice. To facilitate the descrip-
tion of teeth as much as possible, I have thrown the system
into the form of an instrument, which I have termed an
Odontagraph, and which any one may make for themselves
out of a sheet of card paper, by observing the following in-
structions*.

FED^ Fig. 11. represents this instrument on a scale oi
one quarter of the originaL The angle d <f is exactly J5%
and the side k^f is graduated into a scale of half inches,

* Those who arc not disposed to take the trouble, may obtain it
plcte of Mos^n^. Holtzupfcl of Charing Cross.

APPEND, A.] ON THE TEETH OF WHEELS. 151

each half inch being divided into ten parts. The half

inches are numbered from zero at /, both ways towards the

extremity of the scale, 0, 10, 20, up to about 200

upwards, and 40 downwards.

Upon the plain surface of the card are placed the tables

which follow:

TABLES SHEWING THE PLACE OF THE CENTBES

UPON THE SCALE.

1 CENTRES FOR TEETH WITHIN THE PITCH CIRCLE.

' PnCB IN MCHEa AND tlL^IS.

i

I

i

5

1

H

n

1}

2

n

2i

3

H

13 I32

48

6*

80

96

129

160

193

225

257

289

321

386

450

U |l7

26

35

13

52

69

87

104

121

139

156

173

208

242

\5 .12

18

25

31

37

49

62

7*

86

9S

123

148

173

16

10

15

20

25

30

40

50

59

69

79

89

i)9

1^1

IT

8

13

17

21

25

34.

42

SO

59

67

75

84

101

117

IS

7

n

15

19

22

30

37

45

52

59

67

74

89

104

19

10

13

17

20

27

35

40

47

54

6(1

67

80

94

20

e

9

12

16

18

25

31

37

43

49

56

62

74

80

22

5

8

11

U

16

22

33

39

43

49

54

65

76

a*

7

10

12

15

20

25

30

35

40

45

49

59

69

26

...

9

11

14

18

23

27

32

37

41

46

55

64

28

»

6

13

22

26

30

35

40

43

52

60

30

...

8

in

12

17

21

25

29

33

37

41

49

58

33

...

9

11

16

19

23

26

30

34

38

45

.13

to

...

5

7

15

18

21

25

28

32

35

42

49

eti

=

e

8

9

13

15

19

22

25

28

31

37

43

80 ...

4

7

12

17

20

23

26

29

35

41

100

"s

11

14
13

16

19

22
21

25
24

28
27

34
32

39
38

m

5

wt. . 2

e

7

10

12

15

17

20

22

25

30

34

1 CENTRES FOR TEETH OUTSIDE THE PITCH CIRCLE.

' f,t™ in iscbm akd p*mT«.

Number
■■fTwrt.

i

8

1

i

i

1

U

H

'?

2

2J

2!

3

3i

18

7

2

2

3

4

5

e

~

9

10

11

12

15

17

15

3

7

A

10

11

12

14

17

19

V

4

'5

B

8

9

11

12

14

15

18

21

^tm/t

3

'4

7

9

10

12

14

16

18

21

25

1

6

8
9

10
n

11
12
13

14

13

14
15

16

15
16
17
18
19

17
18
19

20
21

19
20
21
22
23

23

25
26

27

26
29
30
31
33

's

7

B

«

t;

*

i'ft

12

!5

17

20

22

25

90

34

i

L

L9S

OS THK

One example viHe^iiiai
meiit* Let it be reqoiRd id
fer a wheel of 29 ceech of -5

Describe an arc t/ of tfae jwj^iici
offnpoD it T/, equal to tfaepDciL
radial lines bt, b/. For the
applj the slant edge of the
placing its extrenntr / oo the pitek

of a tao&

Gorde, and ■!
me; drnr
pifim GRv
^e rsSal Bat nt,
IB. the apnSi

In the table, headed Centre* J^ teedk ntibn Ae fUk
circle^ look down the oofamm of -5 isA pbcft^ and oppoHte
to 30 teeth, wfaicb is the nearest namber n> dac rehired,
win be foond the mnnber 40- The pant r, indkatsdci
the drawing board br the pmitWw of this number on die
scale of equal parts marked, 5ca/e o^cmlrer ftfteeA rittts
^'/cA circle^ is the centre reqaired, from which the arc e/
most be drawn with a radios re.

The centre for the arc de^ which lies oatdde the pitdi
circle, is formed in a manner predselT similar, br iqiplyiDg
the slant edge of the scale to the radial line bt. The
number ^21 obtained from the table of Centres fior teeth
etrtside the pitch circle will indicate the positicHi of this
centre upon the ^Scale of centres Jbr teeth outside the pitch
circle^ namely at r.

The radius of the wheel may be found, by help of the
following table and rule. Multiply the number correspond-
ing to the given pitch in this table by the number of teeth
required, the product will be the radius of the pitch circle in
inches and decimals. Thus,forawheelof^ teeth of 3 inches
pitch, multiply '4774' by 29» and the radius is 13*84 inches.

Pitch.

Facton.

! PHch.

Facton.

3J

•5570 '

■

•1989

3

•4774

1

•1591

' 91

•3979

1

•1193

2:

•3581

! 1

•0994

2

•3183

1

•0795

If

•2785

1

•0597

l|

•2387

1

•0398

VPEND. A.] ON THE TEETH OF WHESI

The curve def, Fig. 11, is also true for an annular wfieel
t the same number of teeth, ^becoming', of course, the
point of the tooth, and d its root. For a Rack, the pitch
T t will be a straight line, and b /, b t bo drawn perpen-
dicular to it, at a distance ironi each other equal to the
jitch. The numbers for pitches not inserted in the table,
wy bo obtained from the column of some other pitch, by
liirect proportion. Thus for 4 inch pitch, by doubling the
imb^-s in the column of the 2 inch piteh, for 1^ by
fmhling ^\, and so on ; or if the difference be small, the
nlamn belonging to the nearest piteh may bo employed,
nthout a serious error ; or more accurately a number may
il taken half way between those given in the two nearest
jBtumns.

I No tabular numbers are given for twelve teeth, for with-
B the piteh circle such teeth are bounded by radial lines.
I But without using the Odontagraph, the geometrical
istruction shown in Figs. 9 and 10 may be employed,
s must be of course drawn to the real size of the wheels
D question.

Let A B be the centres of a pair of wheels, r the point of
oontingcncc of their piteh circles; through t draw ktk,
making on angle of 15" with the line of centres, and hi-
sected in T; also draw pt perpendicular to ktk, tk may
be of any length less than the least radius of the piteh
circles- There are thus obtained two points k, one near to
the right hand centre b, and the other to the left hand
centre s. The first is thus employed in Fig. 10, to obtain
the arcs be, ef. Join hk and produce it to q; join ak in-
tersecting QT in p. Set off T » equal to half the pitch, and
with centre f and radius m describe the arc be outside
the pitch circle of the left hand wheel, and with centre <i
and radius qm describe the are e/" within the pitch circle of
! opposite wheel, whose centre of motion is b ; then will
! arcs work truly together. In like manner Fig. 9.
L and produce it to meet the line tqp in p ; join i

154 ON THE TEETH OF WHEELS. [APPEND. i%.

intersecting the same line in q. Set off Tm equal to haJf
the pitch, and with centre p and radius Ftn describe thAt
portion of the tooth ab which lies within the pitch cirdle
&T) and with centre q and radius Qm describe the tootli
ed^ which lies beyond the pitch circle eT.

These rules must be observed in both pases, namely, thAt
the half pitch Tn, Tm is always set off on the opposite
side of the line of centres to the centres pq of the teeth ;
and that the centres of the concave flanks within the pitcl
circle are obtained by joining the centre of the pitch circle
with that Ky which lies nearest to it, and producing the
line to meet tqp : but that the centres of the convex teeti
beyond the pitch circle are obtamed by joining the centre
of the pitch circle with that k, which is most remote from it

Any two wheels in which the length of kt is the same,
will work truly together.

ON TBBTH WORKING WITH TRUNDLES OB RADIAL PLANKS.

The particular applications of the general construction
which I have given, apply only to complete sets of wheels
working together, and it may be as well to shew its use in
obtaining teeth adapted to work with trundles or pin
wheels, as well as teeth in which the flank is a radial line
as in the common form. The diagram of Fig. 12, Plate
20, must be drawn of the full size for any required wheel,
A and B are the centres as usual, fg and hk arcs of the
pitch circles. Upon the radius of the trundle at describe
a semicircle, upon which set oflF from t, tp equal to the
pitch. Draw p t q and let fall a perpendicular b q upon it
from B, intersecting it in q. If the point p be taken for
the centre of the stave or pin, an arc mn described from
Q and touching the stave in m, will be the side of the tooth
required.

If the flanks of the teeth are to be radial lines, then the
portions lying without the pitch circle may be arcs of cir-

.IPPEND. A.] OS THE TEETH OF WHEELS. 155

ties found thus. (Fig. 14.) a and b are the centres ; fg,
ilk, arcs of the pitch circles as before. Upon at describe
an entire circle, and upon its circumference from a and t
set off A J, T7n equal to each other and to about three
quarters of the pitch ; join bz and through mi draw nnq.
intersoctiog bz in a; then an arc described from centre Q
ad struck through m, will be the curved face of the tooth
for BT, and this will work with the radial flank of the
loolh of A T. To find the curved face of the latter tooth
Biakc a similar diagram, in which at and bt exchange
places.

ON CUTTERS.

The Odontagraph is also applicable to the obtaining a
larrect form for the cutters used in shaping the teeth of
tintal wheels. The form of the cutter is that of the space
between two teeth, and in order to shew the nature of the
fhange of form required for different t^eth as well as the
general form itself, I have in Figure 13, Plate 20, drawn
«ith accuracy, and on a large scale, the teeth proper to the
fiFo extreme cases of a pinion of 12 on the one hand, and
3 rack on the other, and have applied these two together,
50 that the central line of the spaces shall coincide, and
I ibiu bring the shapes of the cutters into direct compari-

Bow between these two lie all the forms that are re-

1 for any number of teeth from 12 to a rack, or the

»t possible wheel ; but in making a set of cutters, for

pitches especially, it is by no means necessary to

ke one for every number, as the forms for numbers that

■ close together are so nearly ahke that the errors of

irkmanship would entirely destroy the difference.

■ The \-ariation of form however is much less among high

Bobers than in low ones. For example, the difference of

between a cutter for 150 teeth, and one for 300, is

k greater than that between cutters for 16 and 17 teeth.

156

ON THE TEETH OF WHEELS. ^APPEND, A.

This being the case, it appeared worth while to investi-
gate some rule by which the necessary cutters could be de-
termined for a set of wheels, so as to incur the least pos-
sible chance of error. To this eflFect I have calculated, by
a method sufficiently accurate for the purpose, the follow-
ing series of what may be termed equidistant values (tf
cutters ; that is, a table of cutters so arranged, that the
same difference of form exists between any two consecutive
numbers.

TABLE OF EQUIDISTANT VALUES FOR CUTTEB8.

No. of
Teeth.

Raek.

300

IfiO

100

76

as

uu

ISx

11

This will be a guide in the selection of the wheel to
which each cutter shall be accurately adapted after it has
been determined how many are necessary in a set For
example, if a single cutter were thought sufficient for a set of
very small wheels, it had better be accurately adapted to teeth
of 25, for that value is intermediate between the two ex-
tremes. If three cutters are to suffice for the whole set*
then 76, 25, and 15 must be selected, of which the cutter
76 may be used for all teeth from a rack to 38, the cutter
25 from 38 to 19» and the cutter 15 from 19 to IS, and so

on.

It appears from the figure that the greatest diffisrence (tf
form is at the apex of the tooth, (that is, at the base of the
cutter,) and amounts to *25 inch in 2 inch pitch ; from this
the difference may be ascertained for any smaller pitdi,
and as many cutters interposed as the workman's notion of
his own powers of accuracy may induce him to think ne-
cessary.

Thus if the hundredth of an inch be his limit of aoca-
racy in forming cutters, and he is making a art far Jvlf

APPESD. A.] ON THE TEETH OF WHEELS. 15? ^^^H

inch pitch, where the difference of form is ^ x -25 or -06 ^^^|
nearly, then half a dozen cutters will be sufficient, and ^^H
these must bo made as nearly as possible to suit the wheels ^^H
nf IJO, ^0, 30, 21, 16, 13. ^H
1 The following table contains a selection of numbers for ^^^|
Afferent case:;, which may save trouble. ^^^|

TABLB OP CUTTSBS. ^^^^H

Witrf

»

W

1

1

7»

M

It

1

1

lOU

M

*>

14

_

1

1

.«

H>

so

II

dm

1

1

mo

«7

to

»■

IS

1

ICI

M

■n

M

33

^S|S|

1

u

lm

m

43

a 14

13

1

u

30O

iw

100

70

«to

36

SO

3D' IB

4a

Mia

SiljS

»•

^^

**

1=0

Hw

IS,

uai

4 30

^'r

.;,,»|

^lien the numbers have been selected, the Odontagraph ^^H
nay be employed to draw the figure of the cutter corre- ^^H
'ponding to each wheel, either oa the same scale as the pro- ^^H
posed cutter, or on a much larger scale, which may be ^^^^
^Aerwards reduced proportioQally. ^^^|

SECTION in. ^^H

THEOBY OF THE PRECEDING CONSTRUCTIONS. ^^H

We must first examine the nature of the motion which ^^|
1 i* proclucefi by the pressure of one circular ai-e upon another ^^H
■^n disposed so as to work in the manner of teeth. ^^|
^■lut AB, Fig. 4., he two centres of motion, A-mtj a piece ^^H
^^>nwl into a circular arc described from a centre p, and ^^B
H^*ble of revolvuig round a ; oyip in like manner a cir- 1
^■bIut arc described from q, and capable of revolving round 1
^K now if the arc Um/i be made to press against OMp, so 1
^H^eoraiuuaicatc rotation to it round b, the line pq, join- M

158

ON THE TEETH OF WHEELS. [aPPEND^

Fio. 4.

ing the centres of the arc will necessarily always pas^
through the point of contact m» and will be of a constant
length equal to the sum of the radii, so that in fact the
motion will be exactly the same, if for the circular arcs a
link p Q be substituted, which length is equal to the sum of
the radii pm, qm, and which is jointed to the revolving
pieces at p and q, the places of the centres.

This also shews that a change of the actual lengths of
the radii pm, qm, will not aflPect the motion, so long as the
distance of the centres is constant, for that whether the
circular arcs had been struck through m or m', or even
through a point m^' beyond the centre q, the system would
still have been equivalent to the link pq, jointed to the
arms ap, bq.

It is only necessary then to examine the motion of this
simple system of rods, and then to explain how it may be
employed in forming the teeth of wheels.

Let the rod a p. Fig. 5, be moved into a new position
A/), its extremity will carry with it the end of the link pq,
and communicate through it a motion to the arm b q, by
which it will be driven into the new position b q ; and it is
necessary to know the relative value of this motion to that
of AP, which produced it

Now this relation is continually changing, but its value

APPEND, A.3 ON JTHB TEETH OF WHEELS.

159

Fig. 5.

at any instant may be thus determined. The rod p q during
its motion may be considered as always turning round some
centre or other in space, although the relative position of
that centre to it is continually shifting. Produce the arms
AP, BQ in the requisite directions to meet in k, then will
this point k be the momentary centre. For as the ex-
tremity p moves round the centre a, the direction of its
motion at starting from p must be perpendicular to ap,
therefore the momentary centre will lie somewhere in a p
produced. In like manner the initial motion of the other
extremity q must be perpendicular to b q, and the moment-
ary centre must also lie somewhere in the direction of b q :
therefore it must be in the intersection k of the two lines
AP and bq produced. But since the rod pq turns on the
momentary centre k, the direct motion of p and q are to
each other at any given instant as their radial distances
firom K, that is, as pk to qk, which is true, whether we
eonaider them as the extremities of the rod p q or of the
ndii AF, bq; also the angular motions of the latter will
lie firand by dividing these direct motions by their re-
qieclive radii i therefore we have,

160 ON THE TEETH OF WHEELS. [aPPENDV A,

Angular moti<m of p round a: angular motion o£' q

J PK QK

round b : : — : — .

AP BQ

Draw KL, AM, BK, perpendicular to pq. Then we

have

PK : AP :: kl : am by similar triangles kpl ; apm

bq:qk::bn:kl bqn;klq

at:bt::am:bn atm;tbn

and compounding these three proportions we obtain

PK QK

— : -1- :: BT : AT

AP BQ

that is to say, the angular motion of the arms are to eacb
other at any moment inversely as the segments into which
the direction of the link divides the line joining the centres
of motion, or line of centreSj as it is usually termed. K
now it happens that when the link p q moves into its new
position py, very near to the first, this second position in-
tersects the first in a point l above (or below) the line of
centres, as in the figure ; then the ratio of the segments
A T, B T will be altered into that of a <, b ty consequently the
ratio of the angular motion will be an increasing or de-
creasing ratio, as the case may be. But if the point l
coincide with the line of centres, this ratio will for the mo-
ment remain constant.

Now a little consideration will show that the point of in-
tersection between two successive positions pq, pq of the
link must be at the place where the perpendicular from k
falls upon it. For as k is the momentary centre of motion
of this link, the extremity l of the perpendicular will
begin to move in a line at right angles with it, and conse-
quently will remain in the direction of the first position pq
when the link has passed into the second p 9, that is to say,
it will be the point of intersection of the two positions ;
when, therefore, the rods are in such a position that the
perpendicular from k meets the link p q in the line of cen-
tres, the ratio of the angular motions of a p and b q is con-

PI'EMD. A.] OH THE TEETH OF WHEALS.

Slant: and if in this state of the systpm the pniuts p and u
bo employed (as in Fig. 4) as centres from whence short
arcs are drawn througli any common point w, and applied
ii& leeth, these arcs will manifestly drive each other cor-
rfctly when in the exa«t relative position described, and
Teiy nearly so when removed to a short distance on each
side of it, which is the thing required*.

Now these relative positions of p and q may be deter-
mined by a simple construction founded upon the necessary
i*incidence of l with x.

Let A and b. Fig. G, be the given centres of motion, a u
'lie line of centres divided in t, so that the segments at, ar
shall have the ratio of the required motions ; or in other
fnrds, let T be the point of contact of the pitch lines.
[)raw ptq, making any angle with ab, and through t
ilraw TK perpendicular to it. Upon ptq assume a point p
as a centre, from whence the circular arc or tooth belong-
ing to A is to be drawn. Then, to find the corresponding
ccDlre for ii, join ap, and produce it to meet tk in k, join
Kb and produce it to meet ptq in q. Then will q be the
point required, which will appear by comparing this dia-
smm with Fig. 5.

If the point p had been taken at p', so that the angle
at't were less than a right angle, then the line p'a would
have intersected t k in a point k' on the other side of t,
and this would have thrown q to n' nearer to t.

Again, V might have been assumed on the other side of
i B as at p", but then the driving arc struck through m
Would have been concave. It is not worth while to ex-
amine all the cases that arise from the different relative
positions of the points ; I shall merely show those that are
able to practice.

to rcniork, tlittt u more direct an<I simple de-
migbt have been given Ly employing in-
desirous of avoiding ill a practical paper.

162

ON THE TEETH OF WHEELS. [APPEND. A.

Fig. 6.

The side of the tooth may be formed either of a single
arc or of two. As the are is only an approximation, and
is, strictly speaking, only exact at one point of the action,
it will be better to adopt a figure composed of two arcs of
circles, as we obtain two exact points ; but in that case one
arc should be concave and the other convex, in order to
facilitate their jimction and produce a wider base ; and
thus a figure is formed, as we have seen, very near to that
usually adopted, the convex arc being of course given to
that part which lies outside the pitch circle, and the con-
cave to that which extends within it.

The angle atp is arbitrary, and its value may therefore
be determined from other conditions than those already
stated. If, however, it be made a right angle, it is clear
that the points p and q vanish by coinciding with x j and
if it be made a little less than a right angle, the points p

'•]

THIC TRETH OF WHEELS.

Ui3

d H are thrown so near to t that the radii by which the
Kuxs are struck become too short, and the points of the
teeth too much rounded off.

On the other hand, if the angle a t P is made too acute, the
action of the teeth upon each other at the moment of passing
the lino of centres and elsewhere becomes very oblique,
and an injurious pressure is thereby thrown upon their
axes. By various trials I have fixed upon 75° as the value
of the angle which appears to avoid these two extremes,
and have accordingly employed it in the construction of
the Odontagraph.

Again, the position of the point vi, through which the
sns are to be struck, is also arbitrary, and must be deter-
mined by considering which point of the action we wish to
make the correct point. If the teeth consist of a single
flrc each, the correct point may be fixed at the moment of
passing the line of centres, and therefore the arcs must be
llnick through the point t ; but if the side of the tooth
lie formed of two arcs joined, one lying within, the other
ItftHid the pitch line, then the action of one of them will
b confined to the approach of the point of contact of the
pth to the line of centres, and the action of the other to
I recess from that line, and m must be assumed upon
a principle that the correct pouit of each arc shall
^ nearly in the middle of its action, the mode of doing
ich will appear presently.

10 DEBCBIBE T

I aiNQLH ABC.

If the side of the tooth consist of a single arc, the sys-
tem may be made exceedingly simple, for as the distance
■'!" the point k from t is arbitrary, when the points p or a
■ •.n: not given, suppose it to be taken at an infinite distance,
1 (Fig. ()) ATK and qbk will become parallel to tk,

164

ON THE TEEtH OF WHEELS. [aPP£ND» A.

and perpendicular to ptq, which shews that if lines be
drawn from a and b (Fig. 7) perpendicular to ptq, the
points p and q will be centres, whence if arcs be drawn
through some common point m, or rather in this case t,
these arcs will drive each other correctly.

Fig. 7.

If the angle p t a remain constant for a set of wheels of
this kind, any two of them will work truly together, pro-
vided the arcs be struck through the point t ; for let the
wheel, whose radius is at, be removed, and another
whose radius is a' t be substituted, a'p' drawn perpen-
dicular to T p p' will give the point p' as the centre belong-
ing to the arc oy, and it is clear that this new arc o j will
work as well with kn sls the former one, and also that if
the radius a't had been substituted for bt instead of for
at, by placing it and its corresponding line a'p' in the
situation indicated by the dotted lines b'q', that still the
conditions of the construction would have been satisfied,
and those two wheels worked truly together, and the same
may Ih> shown of any other pair of radii. But if the arcs
wore struck through a point wi, not coinciding with t, then
tho wluH^ls would fall into two groups, in one of which, as
at, a't, tho arcs are struck through a point m on the op-
]H^ito side of tho line of centres to the centre points p, p',
and in tho other, as b t, b't, they are struck through a

APPEND. A,] ON THE TEETH OF WHEELS. l65

point m on the same side of the line of centres as the cen-
tre points qq'. Any wheel out of one of these groups will
work correctly with any wheel taken from the other.

But suppose that a pair of wheels out of one of these
groups he put together, for example, out of that in which
the point m and the centre point of the arc, lie on opposite
sides of the line of centres, and let a t and b t he the radii
of the wheels in question. Now the relative positions of
the points p and q will still he true, but the arcs will no
longer be struck through a common point, one of them
being through w, the other through mf at the same dis-
tance on the opposite side of x, and therefore they will not
work truly together. The arcs of the entire set must
therefore be struck through x, and then any two wheels of
the set will work.

The distance xp is equal to ax x cos axp, and if axp
be fixed at 75*" 3(y, which is a convenient value, then xp =

— , whence the value is very easily found for any given ra-
4

dius, for in this case the value depends upon the rac^us
alone and not on the pitch or number of teeth, as in the
next example. The practical mode of setting out the
teeth has been already explained.

On this system, however, the tooth has but one true
point, that is to say, it is only strictly exact at the moment
of passing the line of centres, and I therefore greatly pre-
fer the construction about to be described, in which the
side of the tooth is made up of two arcs united, and con-
sequently has two points of accuracy. The tooth just de-
scribed has considerable analogy to the involute, and like
it has the fault of acting with too great a degree of obli-
quity. The teeth next to be described are of nearly the
same form as that which has been so long in use, and have,
as well as those of Fig. 8, the property of allowing any
pair of wheels in a set to work together.

166 ON THE TEETH OF WHEELS. f APPEND. A.

TO DESCRIBE TEETH CONSISTING OF TWO ARCS OP CIBCLBS.

Figures 9 and 10, Plate SO, represent a pair of so con-
stituted teeth in contact, fig. 9 shewing their action before
they reach the line of centres, and Fig. 10 after they have
passed that line ; each tooth is formed of two arcs of cir-
cles, a J, bcy de^ ef^ of which the concave ones, a&, e/J are
situated within the pitch circles, and the convex ones, ic,
dcy extend beyond these circles; therefore, from well known
principles, the concave arc a h will drive the convex arc it^
until the point of contact reaches the line of centres, and then
the convex arc h c will begin to drive the concave arc ef.

There are two points in the action of these teeth at whidi
perfect accuracy is attained ; one of them is when the teeth
are in the position of Fig. 99 during the mutual action of
ab and de^ and the other when they are in the position of
Fig. 10, during the action of be and ef; and the arcs are
so set out that these points of the action shall take place,
the one nearly in the middle of the arc of motion before the
line of centres is reached, and the other somewhere about
the middle of the arc of motion that is traversed from the
line of centres until the teeth quit contact.

The construction of these teeth in a set is as follows.
AB, Figures 9 and 10, is the general direction of the Kne
of centres; qpt, as before, is a line making a constant
angle of 75"* with the line of centres ; k t k perpendicular
to QPT and having its two points k set off at equal distances
on each side of t, these points and the lines being inva-
riable for the entire set.

The centres for the convex arcs are found by joining the
centre of each wheel (a. Fig, 10 ; b. Fig. 9) with that
point K which lies on the opposite side of the line qpt.
Thus in Fig. 9, Q is the centre of the convex arc de, found
by joining bk, and in Fig. 10, p is the centre of the convex
arc bCf found by joining ak.

APPEND. A.] ON THE TEETH OF WHEELS. l67

The centres for the concave arcs are found by joining
the centre of each wheel with the k which lies between
it and the line qpt ; thus in Fig. 9* p is the centre
of the concave arc ahj found by joining ak, and pro-
ducing it to meet pqt, and in Fig. 10, q is the centre of
the concave arc efi found by joining bk, and producing it
to meet tpq. Moreover, the whole of these concave and
convex arcs are struck through a point lying beyond t at a
constant distance, Tn, or Tm, which for simplicity's sake I
have assumed equal to half the pitch ; finding that this
will place the correct points of the action at a sufficient dis-
tance o each side of the line of centres.

The consequences of this arrangement will be, that any
pair of teeth so described will, when put together, answer
the conditions of the construction already demonstrated.

(Fig. 6.)

Ist (Fig. 90 Before reaching the line of centres we
have a concave arc ah driving a convex one de, of which
the first has been struck from a centre p, derived from its
nearest k, and the second from a centre q, derived from its
farthest k, consequently both derived from the same k ;
also the arcs have both been struck through a point m, at
the same distance beyond t, and therefore will work truly
together.

Sd. (Fig. 10.) After passing the line of centres, a
coiiYex arc be drives a concave arc efj which in like man-
ner are seen to have been derived from the same k, and to
have been struck through a common point /i, so that al-
though the position of all these points is reversed, the arcs
will, in this case, work truly together.

The same will manifestly be true for every pair of wheels
in the set, for the distances tk and Tm, or tti, are con-
stant for the whole.

168 ON THE TEETH OF WHEELS. [ APPEND. A.

CONSTRUCTION OF THE ODONTAORAPH.

To enable a workman to find these points p and q at
once in every case, I have contrived the instrument ah^y
described, (vide page 150,) which I have termed an Odon-
tagraph, and have represented in Fig. 11, Plate 20, with
the arrangements for describing the tooth fed of Figures
9 and 10. These three drawings being all made to the
same scale will explain each other by comparison.

The instrument, as already mentioned, consists of a kind
of bevil formed of a sheet of card paper, four times the
lineal size of the drawing ef^d, the angle Btkia 75% and
the side A:^f is occupied by a scale of equal parts numbered
from t both ways. An example will show how this instra-
ment is connected with the previous demonstration.

Let the example be a wheel of 26 teeth 4 inch pitch.
Describe an arc ter of the required pitch circle, and set
off upon it ^T equal to the pitch and bisected in ^, draw ra-
dial lines Bty B T. To describe the arc ef within the pitch
circle, apply the slant edge d < of the scale to the upper
radial line b<, as in the figure. In the table headed " Cen-
tres for teeth within the pitch circle," look down the
column of 2 inch pitch, and opposite to 26 teeth will be
found the number 37, which being doubled gives 74- The
point indicated on the drawing board by the position oT
this number at q on the scale of equal parts <f, which,
is marked Scale of centres for teeth within pitch circle^.
in the actual instrument, is the centre required, fronm
which the BTcfe must be drawn with a radius qe.

Now a comparison of this figure with Fig. 10, will sho\iir
that thus far the relative positions, inclinations and dis-
tances have been indicated by the instrument for the point
q (the Q of Fig. 10.) The line b<. Fig. 11, is the same as
BT in Fig. 10.

The centre for the arc erf, which lies outside the pitch

>•]

ON THE TEETH OF WHEELS.

169

circle, is found in a manner precisely similar, by applying
the slant edge of the scale to the lower radial line bt,
placing the instrument in the position indicated by the
(lotted lines. The table of centres for t«eth outside the
foleh circle does not contain SG in its column of Number
of Teeth, therefore the nearest number must be taken,
which in this case is 30, and the number 28 = ^ x 14
in the column of 2 inch pitch, will indicate the position of
the centre q upon the scale t A of centres for teeth, outside
Hupitch circle, this scale being so titled in the actual instru-
Here, again, a comparison of Fig. 11 with Fig. 9.
ill show that this new operation has given the true relative
wdon of the jmint q to the radial line bt and arc de.
] will now explain in a few words the mode of calcidating
le numbers in the table, by way of enabling other persons
I liter any of the conditions. A formula tor tliese num-
a may be obtained as follows. (Vide Fig. 6, 'page l62.)
wn A draw am perpendicular to ttp', then from the

lailar triangles amp, ptk we obtain kt = ^ ~

Let KT =c, at = r, pt =
D.R. sin 0

(1)

R, COS 0 — d'

^'Now the point p being in this case obtained from the k

the opposite side of t to a, this formula belongs to that

rt of the tooth which lies beyond the pitch circle, accord-

[to the principles already laid down. If tk" be taken

lal to T K on the line k t produced, and a point p" ob-

Kd by joining ak", and producing the line to meet

'T, then p" will belong to the part of the tooth within

pitch circle, and the similar triangles amp", p"tk",

igive us for this case the formula

d'h sin fl , ^ , , „

C= ;; i (2); where d=tp'.

B cos tf + D ^ -^

pw the value of c, which represents the equal lines

170 ON THE TEETH OF WHEELS. [aPPEND« A.

KT, or K^^T, may be determined for the whole set, by con-
siderations similar to those already employed in settling
the diameter of the constant describing circle in the first
section of this paper. If the radius at of a wheel be as-
sumed of such a length that a k^' fall perpendicularly upon
k''t, then will the line ak'^p'' become parallel to ptp'',
and consequently the point ^" will go off to infinityi and
the arc which should be struck from it to form the flank (tf
the tooth will become a right line perpendicular to ptp".

If the radius at be taken still smaller with respect to
k^'t, it will be seen (by taking k't larger than at) that in
such a case the point p, will make its appearance on the qp*
posite side of t *, but this makes the flank of the tooth
convex, and drawing inwards so as to be less at the base
than at the pitch line, which is an impracticable form. To
avoid this, and at the same time to make k^^t as large as
possible consistently with this limitation, assume k^^t equal
to R^ sin 6 J where r^ is the least radius of the set. This
value corresponds to the case in which a yl" is perpendico-
lar to k'^t, and necessarily excludes the impracticable
forms ; for since the least radius of the set now corresponds
to that peculiar example in which ak'^p'^ is parallel to ptp",
every other value of at being larger, will throw the points
v^' on the opposite side of t to m, which is the thing re-
quired to produce the concave flank. These observations
apply only to that value of kt which lies nearest the centre
A, and therefore to the flank or portion of tooth within the
pitch circle. As to the opposite value of t k, which cor-
responds to the portion of tooth beyond the pitch circle,
and which it must be remembered is equal to tk'^ it is
clear from the figure that whatever value be given to it, its
point p will always lie between t and m, and the arc of
tooth be convex, supposing it to be struck, as it must boi
through a point near to t.

* Our fonnula then becomes c z: — ^ .

d' — B cos 6

The ^-alue selected for k"t (namely r' sin 6) will tbere-
isaitKT. Substitute now this value for c in the for-

B (1) and ('2), and after arranging the terms we obtain

following values of d and d'.

'•]

ON THE TEETH OF WHEELS.

171

h'b cos S

. . (3) andD'=-

• C-t)

low D and u' (that is tp and tp") are the distances of
centre points of the arcs measured from t, and it will
leen by comparing the diagrams with the description of
Odontagraph, that the numbers in the columns of each
ii are the values of d and d', corresponding to the num-

of teeth in each wheel given in the first column, or,

1 is the same thing, to the values of the radii n and it',
find these numbers for a given pitch, substitute in (3)
(♦) the particular values of r' and B, and by help of
ble of logarithms, the values of u and d' belonging to

my values of r as may be thought necessary, may be
puted, and thus the column of numbers obtained for
; pitch. Tlioae of the other pitches may be derived
\ the first by common proportion. In this way I formed
table, assuming 12 tor the least number of teeth, and
for the value of 8, and employing a scale of half inches
tenths in which to express the values of d in the near-
irbole numbers, because I foimd that a unit of the twen-
b of an inch was sufficiently small to avoid practical error.
t i» unnecessarj' to have numbers corresponding to
y wheel, for the error produced by taking those which
Dg to the nearest as directed, is so small as to be un-
wiable in practice. I have calculated the amount and
re of these errors by way of obtaining a principle for the
ber and arrangement of the wheels selected. It is
Bcessarj- to go at length into these calculations, which

, &om very simple considerations, but I will briefly
I the results.

e difference of form between the tooth of one wheel and
Other ia due to two causes, (1) the difference of curva-

1J2 ON THE TEETH OF WHEELS. FaPFEND. A.

ture, which is provided for in the Odontagraph by placing the
compasses at the different points of the scale of equal parts,
(2) the variation of the angle /bt, (Fig. 11,) which is met
by placing the instrument upon the two radii in succession.

The first cause is the onlv one with which these calcukr
tions are concerned. Now in three inch pitch the great-
est difference of form produced by mere curvature in the
portion of tooth which lies beyond the pitch circle, is only
•Qt inch between the extreme cases of a pinion of twelve
and a rack, and in the acting part of the arc within the
pitch circle is *1 inch, so that as all the other forms lie be-
tween these, it is clear that if we select only four or five
examples for the outer side of the tooth and ten or twelve
for the inner side, that we can never incur an error of more
than the to oth of an inch in three inch pitch by always
taking the nearest number in the manner directed, and a
proportionably smaller error in smaller pitches. But to
ensure this, the selected numbers should be so taken, that
their respective forms shall lie between the extremes at
equal distances. Now it appears that the variation of finrm
is much greater among the teeth of small numbers than
among the larger ones, and that in fact the numbers in the
two following series are so arranged that the curves cor-
responding to them possess this required property.

For the outer side of the tooth, 12, 14, 17, 21, 26^ 34^
47, 73, 148, Rack.

For the inner side, 12, 13, 14, 15, 16, 17, 19, 22, 26,
33, 46, 87, Rack.

Now these numbers, although strictly correct, would be
verj' inconvenient and uncouth in practice if employed for
a table like that in question, where convenience manifegtly
requires that the numbers, if not consecutive, should always
proceed either by twos or fives, or by whole tens, and so on.
They arc only given as guides . in the selection, and bj
comparing them with the actual table, their liae in flie
formation of the first column will be evident

ESSAY II.

ON THE SHAFTS OF MILLS.

CHAPTER I.

IHl. To make these Essays useful to operative mecha-

Inica; to save engineers and managers of manufactories
ibe trouble of much explanation in giving directions to
fcremen and others, who are to carry their ideas into effect ;
lo give workmen some notion of the principles on which
their work should be conducted ; and to construct machi-
nerj upon true principles, which is ultimately the most
economical plan of proceeding ; we introduce the following
extract from a respectable periodical publication*, as it
appears appUcable to the subjects of these papers, and may
induce the reader's taste for entering upon a new and un-
trodden path.

" A country in which manufactures are extensively
fslablished, and conducted with spirit, as in Britain, be-
'■omes by degrees a country of machinerj-. For invon-
ti'ins to diminish the quantity of human labour employed,
"ill be more ingenious in construction, more powerful in
opi-ration, and of more general use, in proportion to the
iitessity of furnishing a greater quantity of commodities
^' moderate and equable prices. The bodily exertions of

• Kt-lectic RiivJew, Dec. 180C, Art. XII.

174 ON THE SHAFTS OF MILLS. [ CHAP. I.

workmen, in whatever branch of labour, have their limits,
and excessive efforts, if unduly prolonged, irremediably
destroy the health and vigour of those who pursue thenL
But machines may be continued in activity day and night,
week after week, and month after month ; having in them-
selves no life which suffers a sensible consumption, no prin-
ciple of activity whose energy requires a pause to effect its
recovery or renovation.

" We have seen the manufactures of our own country
solicit the aid of every hand that could be spared from its
agriculture, and seek in distant lands for labourers of
every age to supply the mill or to throw the shuttle. We
have seen ingenuity exerted to its utmost, to contrive and
to construct those machines which these labourers were to
superintend and assist. We remember the time when
these constructions were the dread and the hatred of the
manufacturers, but we believe the most ignorant workman
of the present day acknowledges their utility, and would
with difficulty be induced to relinquish that very imple-
ment which his father or grandfather would have gladly
committed to the flames.

" Considering then the importance of machines to
shorten labour, and the number of persons who are mter-
ested in them, as proprietors, as inventors, or as con-
structors, it is wonderful that so little has hitherto been com-
municated on this subject by the medium of the press.

" The process towards perfection in complicated machi-
nery is perhaps too generally the reverse of what might be
expected. When practice has shewn the importance of a
machine, science takes it up, investigates its principles,
analyses its movements, and connects them by the assist-
ance of mathematical precision.

^* Mathematicians are seldom inventors, and workmen
are rarely men of science, yet the mutual assistance of
study and practice is necessary, to perfect the subject which

■0

ON THE SHAFTS OF MILLS,

175

each is intent on improving." It was Buchanan's aim,
then, to come between these two classes, and to make them
better acquainted, and more useful to each other. How
far he has succeeded in the attempt, must be left to the de-
termination of time.

Lin common with all writers on similar subjects, he cx-
prienced considerable difficulty in finding precise technical
mirds to express the different parts of mill-work. Those
wiiich are used by millwrights in different districts being
Teiy different from each other. It is hoped, however, that
the explanations which we have given of the terms, will
niake them sufficiently clear.

LWith regard to this particular Essay on the Shafts of

I, the subject is treated in a manner similar to that

wed in the Inquiry into the Strength and Durability

of (he Teeth of Wheels. For the reasons there given,

Buchanan did not here enter into the demonstrations of

the elementar)' propositions which serve to guide the in-

qniry. He was at pains, however, to collect and arrange

etB respecting gudgeons and journals in actual use, upon

)kh to ground calculations. This method he considered

being much more certain than rearing calculations

I insulated experiments made on the cohesive strength

.materiais. These, however, are of great value, and

tome cAses he has endeavoured to apply them. The

ious Tables given in the course of the Essay, will be

peat use to the millwright in finding without trouble

lizes of gudgeons and journals for any case that may

r in practice, and the principles on which they are

fd are laid down in so plain a manner that he will

iy understand and apply them. These Tables may be

Indered as certain great lines drawn to guide the mill-

[ht in his work, and even allowing they may not be ab-

tely true, he may find from experience how near they

^ be approached with safety.

176 ON THE SHAFTS OF MILLS. [CHAP. I.

18& To proportion the diameters of axles to the stress
they have to hear, is in mill-work of great practical import-
ance. On the one hand, if the shafts he made too weak,
it is evident they must soon give way ; and on the oth^
hand, if made too strong, they occasion not only unnecessary
expense in the construction of the machinery, hut, what is in
most cases still worse, a waste of power from unnecessary
friction. It is therefore desirable, that the millwright
should have some rules to guide him in this very important
part of his business ; a part which has hitherto in most
cases been conducted entirely at random. This Essay gives
such a practical view of the subject as shall enable the mill-
wright to proceed with greater certainty.

183. Until of late years, most of the shafts used in mill-
work were constructed of timber. The use of cast iron in
this and other parts of mill- work, however, has now become
almost universal. For this improvement we are perhaps
indebted to those who are engaged in the cotton manufiic-
ture. After Arkwright's invention, it became a great ob-
ject with them to save time in the erection of machineiy,
and to render it as durable as possible ; for every stoppage
was attended with great loss, by throwing idle the numbers
of people necessary in cotton mills.

Besides the expense attending the repair, what had per-
haps still more weight with them was, that the profits at
that period on cotton spinning, were almost unparalleled in
any other branch of manufacture.

Another circumstance which tended very much to the
advancement of mill-work, arose from James Watt's im-
provement of the steam engine, which enabled cotton-spin-
ners and other manufacturers who required power to work
their machinery to carry on their business in towns. Hence
power and people might, without trouble, be concentrated
on the most eligible spot, and the great expense and disad-
vantages avoided which are attendant on colonizing the

CHAP. 1-3 ON THE SHAFTS OF MILLS. 177

remote situatioiis in which powerful fells of water are com-
monly found. The questions of health and morals belong
to the l^islator, not to the civil engineer.

The introduction of cast iron then may be considered as
a kind of new era in the history of mills, without the use of
iMch it would not have been possible, with the same num-
ber of operative mechanics, to have constructed one tenth
part of the machinery which has of late years been erected
in Great Britain.

CHAPTER 11.

SECTION I.

GENERAL DESCRIPTION OP 8HAPT8.

184. The axles used in mill-work are commonly den<
minated, when of a large size, sfu^ ; those which
smaller, are usually called spindles. Thus, for exampl
we say the shaft of a water-wheel ; the spindle which
ries the millstone of a com-milL

185. When shafts lie in a horizontal direction, they are
called It/ing or horizontal shafts ; when vertical, they axe
termed upright or vertical shafts.

186. Shafts are usually made of wood or of iron. Large
wooden shafts are generally made either of solid oak, or are
built of fir-logs. The scarcity of large oak occasioned the
built shafts of fir to come into more general use. The
latest improvement made on wooden shafts, was that of
having what are called cross-tailed gudgeons * . Before that
improvement, it was attended with very great trouble and
expense to keep the gudgeons from becoming loose in the
shafts. Indeed, it was found impracticable to keep them
fast for any considerable time.

187. Fig- 1> Plate XL represents a wooden shaft, with
the gudgeons in use previously to the last improvement.
They are called laid-in gudgeons, a b c is the gudgeon
somewhat in the form of the letter T. One of the tails, c,
was let into a mortice, and the rest of the gudgeon sunk into

* The gudgeon is the arbour or spindle on which the shaft turns.

CHAP. II.] ON THi*-.^^ .^S OF MILLS. 179

its place in the centre of the shaft. In order to accomplish
this, it was necessary to cut out the part, from b to d,
Fig. 1, No. 2. After the gudgeon was laid in its place, the
vacant part was filled up by the piece of wood d e, Fig. 1,
No. 1. The shaft was then hooped, and the end of it
driven full of wedges, in order to fasten the gudgeon. This
gudgeon is shown in perspective. Fig. 1, No. 3.

188. Fig. 2. represents a wooden shaft, with cross- tailed
gudgeons. This kind of gudgeon is made of cast iron, and
being thin in the cross-tails, let in from the end of the shaft,
it leaves the wood much more entire than the laicUin gud-
geotij while its cross-arms take a much firmer hold. After
it is let in, the hoops are driven on the end of the shaft,
when warm, and lay firm hold of the ends of the cross-tails.
The wood is then wedged up, which makes the gudgeons
perfectly fast. Fig. 3. is a perspective view of a cross-tailed
gudgeon, and Fig. 4. its profile. It is cast with the round
part undermost ; for which reason the pattern must have a
taper, to make it rise out of the sand. This taper has, in
the cross-tails, another very important use, that of giving
the gudgeon the advantage of dove-tailed joints with the
timber of the shaft when it is wedged up. Instead of
wrought iron hoops, cross-tailed gudgeons sometimes have
a hoop of cast iron cast along with the tails, as represented
by Fig. 5.

189* When it is considered that the direction of the
stress which tends to loosen the gudgeon in a wooden shaft
is continually changing, and that such action is exerted
upon wood, a material that is so very easily permanently
compressed, it will not be wonderful that it should have
been found difficult to render them firm and lasting. The
last method, viz. that where the hoop is cast along with
the cross-tails, seems to be far preferable to the other ; but
perhaps it seldom happens that the hoop part is of sufficient
length. It may be proved, that when the diameter of the

180 ON THE SHAFTS OF MILLS. [CHAF. XL

shaft is sufficient for the stram upon it, the length of tb6
hoop should he equal to the square of the diameter pf
the shaft divided hy the length of the shaft; otherwise
there will be no certainty of the gudgeon remauung
permanently fixed*

190. An improved method of fixing gudgeons is do-
scribed in the Transactions of the Society of Arts, &c.f
Vol. xxxi. p. 223 ; it consists in casting the gudgeon with
cross-arms, which fit into proper notches in an octagonal
box of cast iron that has been previously fixed upon the
end of the shaft. The arms of the gudgeon are retained
in their places by screw-bolts.

In Plate [IV. A,] Fig. 1, 2, and 3, a a represents the
end of the wooden shaft, which is supposed to be made of
an octagonal form, b b is the cast iron box accurately fit-
ted on the end of the shaft, and wedged tight. The end of
the box has a projecting flanch a a, with four notches to re-
ceive the cross-arms bbjdd of the gudgeon c. These cross-
arms are firmly fixed to the box by four screw-bolts, which
pass through the flanch, and the ends of the cross-arms.
The section. Fig. 3, shews the box b b on the end of the
shaft, with the gudgeon c, and its cross-arms separated ; to
explain a further precaution which is necessary for strength.
This precaution consists in the cross-arms having projec-
tions, e Cy which enter the end of the box, and keep the gud-
geon true to its centre, and prevent any lateral strain on
the bolts. When the gudgeon of a wheel is fitted accord-
ing to this method, it can be easily removed when it is so
far worn that a new one is necessary ; and the new one may
be inserted without injury to the end of the shaft.

This improved method was invented by Robert Hughes.

It is obvious that the length of the box should be regu-
lated by the rule stated in Art. 189*. The real advantage
gained by this mode of fixing seems to be, that of retaining

* The rule suppotjes the shaft to be proportioned to the stress upou it

ICH.1!'. It.] ON THE SHAFTS OF MILLS. 181

f the end of the wooden shaft more perfect, with the means
\(d renewing the gudgeon, without injurj* to the shaft,
lyi. Cast iron shafts are sometimes made hollow cylin-
ders, and sometimes they are made solid, and of various
figures. It is demonstrable, that a hollow cylinder is much
stronger, with the same quantity of matter, than it woidd
hfl if made into a solid of the same length. This law is
YCij observable in the beautiful economy of nature ; for in-
slance, the stalks of plants, the quills of birds, the bones
(^animals. But, in the works of art, numberless obstacles
■to perfection continually occur. In this particular case the
■ttpcnse of making small shafts hollow, would be very great j
(1 another objection is, the difficulty of making such cast-
s perfect. .Shafts of a small diameter are, therefore,
monly made solid.
, Fig. 6, Plate II. represents a cast iron cylindrical
It consists of three parts, the body, a u c d, and the

0 gudgeons, a e c, and B i' D c. The gudgeons are turned
d carefully fitted into the ends of the body, which is bored

1 turned to receive them. They are then fixed with
sorew-bolts, which pass through the flanches, as may be seen
!j_v the figure •-

This kind of shaft will obviously be variously constructed,
according to circumstances. That represented in the
figure was made for a cast iron water-wheel. The use of the
small projections h, h, &c., is to prevent the eye of the arms
from shifting roxmd on the shaft. When cylindrical shafts
are not used, what are called feathered shafts are often
I adopted.

^Kl93. Fig 7, Plate III. represents this construction of a
^Hiift. It probably took its name from its resemblance to
The feathered part of an arrow. It may be here remarked,

In this constroction, the resUtoncc to twisting depends entirely on the
difEcultiet appear to be encoimtereil jn costing without cor-
indiiig wlnuitngee, cilhcr in strength or beauty.

^rik, and eomc d
^^bmidiiig wlnui

18S ON THE SHAFTS OP HILLS. [CHAF. H.

that shafts of this species, as often constructed, are by do
means calculated to withstand the twist brought upon them
by the strain of the machinery. From the breadth of tlie
feathers, their strength to withstand lateral pressure, is, no
doubt, considerable -, but wanting substance between the
feathers, they are liable to continual tremor. Where fea-
thers are applied to shafts it is preferable to keep the body
of the shaft fully as strong as the gudgeon, or journal* ^ and
apply the feathers merely to prevent bending in the mid-
dle, as Fig. 7) No. 4. But the simple square. Fig. 8, is
more easily made, and has been found in practice, at least
as advantageous as any other form that has been tried for
solid shafts t.

Having given this general account of shafts, we come
next to consider the causes from which the stress on them
arises.

SECTION 11.

OF THB KINDS OF STRESS TO WHICH SHAFTS ARE SUBJECT.

194. There are two kinds of stress to which shafts are
liable : first, lateral stress^ by which they may be broken
across: secondly, stress arising from torsion^ by which
they may be wrenched or twisted.

All horizontal shafts are liable to the first kind of
stress, viz. lateral stress ; and some have no other strain

* Journals^ or journeys^ are gudgeons subject to torsion.

t It is easily proved, that the best form for a revolving shaft is a cylinder,
and tliat in any other form the flexure will be irregular, and consequently
produce irregular wear on the gudgeons and brasses ; but when a shaft is
to be adapted for placing wheels on any part of its length, a square section
is convenient ; in all other cases, the section ought to be circular with pro-
jections, as at H, H, Fig. 6, Plate II. By making four of these projections
continue throughout the length upon a cylindrical shaft, all the advantage
and convenience of a square one would be obtained, with very little irregular
flexure. The projections should not be greater than is necessary to ^x the
wheels firmly on the shaft.

ICHAP. 11.] ON THE SHAFTS OF MILLS.

I whatever ; as, for instance, a water-wheel shaft, where the
motion is conuuunicated from teeth, on the shrouding. See
iPlatelll. Fig. 9.

In Fig. 10, the stress on the upright shaft arises from
Itorsion only ; excepting what may proceed from the inac-
Jturacy of the teeth of the wheels, which, if great, will
■iKcasioD a considerable lateral thrust.

A vertical shaft, which gives or receives motion by
I Deans of a pulley, has thereby a lateral pressure brought
iponiL

In Fig. 11 and 1'2 the stress is compounded of lateral
weaure, arising from the weight of the wheels a and a,
Ind that of the shaft itself, and of the torsion or twist pro-
fdnced hotween the wheels a and b.

The following remarks of John Hoberton, engineer,
I relate to the subject of this Essay, and contain much
I useful matter which will be acceptable to the reader ; to
[whom it must be demonstrable, " that by a judicious ar-
I Tangement of wheels and pinions, in many cases much of
I lie stress and friction may be avoided. This is a doctrine
Impractical mechanics of very great importance, when we
Icooeider, that in many cases almost the whole of the im*
I felling power is expended in overcoming the friction of
|tlie machinery.

195. *' Let A, Fig. 1, Plate IV., be an overshot water-
Ifheel. Let the line a b represent the line of direction of
e centre of gravity of the water in the buckets. On the
■trcmity of this wheel let there be a toothed wheel acting
Bto the pinion b. It is obvious that, independently of the
weight of the wheel, the whole weight of the water will be
supported by the axis c, and the teeth of the wheel at d ;
and the weight which each will sustain, will be in the ratio
of ce to ed. That is, by the principles of the lever, the
weight on the gudgeon will be as the distance e d, while
that on the teeth will be as ce\ or if the radius of the

184 ON THE SHAFTS OF MILLS. [ CHAP. II.

toothed wheel were ce^ the teeth would sustain the whole
weight of the water, leaviog no weight on the gudgeon Init
that of the wheeL Again, let the wheel b be removed to c,
it is evident that the gudgeon c, would have to sustain the
weight of the water, and a great deal more. That is,
the weight on the gudgeon would be increased as e({to
ec. The true relation of the stress in the two cases, is

edi — ^ ^^ . So that it is evident, the stress and friction
dc-^-ed

of the gudgeon miist depend, in a great measure, on the on

of the toothed wheel attached to the water- wheels and to the

situation of the pinion b.

196. '^ Again, let there be a wheel at a. Fig. % fixed od
the end or middle of a shaft working into the wheel or
pinion b, of any size. — ^Let the teeth move in the direcdon
ah. It is evident, that the gudgeon or shaft will tend to
move in the contrary direction, that is, in the direction ci^
and with the very same force that the teeth act upon eadi
other, as action and reaction are equal and in contrary &
rections, and for the same reason, the gudgeon of the wheel
b, will tend to move in the direction e i, with the very same
force, that is, the same force as the action of the teeth on
each other. Indeed, in any single pair of wheels, of what-
ever form or construction, the tendency to break or bend
the shaft, or cause friction, is the same as the action of the
teeth on each other.

" Now, if the above wheels were made of a double size, it
is evident that the acting power on them would be only
one half, and consequently, one half of the strain to break
the shaft or cause friction *.

197. " In the case of an intervening wheel, the force or

* The object of this remark is, apparently, to show the superiority of
large wheels ; but it is clear that the acting power would be the same with
the same first mover ; and if the resistance be diminished on one shaft, an-
other must be added to give the proposed velocity to the working point.

CHAP. 11.3 ON THE SHAFTS OF MILLS. 185

tendency to break the shaft depends on the situation of such
intervening wheeL Thus, if it be placed in a direct line
betwixt the centres of the conducted and conducting wheels,
as at A, Fig. 3, the shaft or gudgeon will tend to move in
the line ah or ha^ (according to the direction of the con-
ductor,) with double the force of the action of the teeth.

'« If the axis of the intervening wheel form a right angle
with the axis of the other two wheels, the force to break the
shaft will be to that of a pair of single wheels, or the ac-
tion of the teeth on them, as the diagonal of a square is to
one of its sides. That is, the direction of the force, and its
intensity, will be represented by the diagonal, it being evi-
dent, from the well-known laws of mechanics, that by the
action of the wheel b, Fig. 4, No. 1, the centre of the in-
tervening wheel A, would tend to move in the line a c, and
by its action in the wheel e, it would tend to move in the line
a d. Let a d and a c represent the forces in these directions,
complete the square or parallelogram, the diagonal of which
will both represent the force and its direction. See also
Kg. 4, No. 2, and No. 3.

198. " On the other hand, if a wheel be placed betwixt
two others, as a. Fig. 3, where a is the conductor, the teeth
of which act with equal force on each of the wheels b and c,
it is evident that the strain is wholly taken off the shaft, the
forces being equal and opposite to each other * ; and in Fig. 5,
where a is supposed to be the conductor, and b and c the
conducted wheels on which the teeth on each bear equally,
— it is plain from what has been already stated, that the
direction of the forces will h^da and a c, and letting a d and
a c represent the direction and intensity of these forces, and

* The abaft ought not, however, in any case, to be entirely freed from
piMHin ID this manner ; because its motion will not be so steady and re-
giikr M when there is some considerable pressure on the gudgeons. And
dura is nothing more iigurious in machinery than a hobbling, unsteady
tliii pomt leqmras the engineer s most careful attention.

186 ON THE SHAFTS OF MILLS. []CHAF. It.

completing the parallelogram, we have the diagonal n ft to
represent the compound direction and intensity of the force.

199- " There is another point that is worthy of attention :
that isy the place on the shaft where the wheels are fixed.
If a wheel is put on at the end of a shaft to drive any other
or others, it is clear, that the whole or nearly the whole of
the stress will be at the end of the shaft or joumaL If the
wheel be placed in the middle of the shaft, the strain to
break it will be greatest at that part, but the force will be
resisted equally by each journal. Indeed, on whatever part
of the shaft a wheel is placed, at that very place is the
greatest (cross) strain on the shaft, and the force on each
journal will be in the inverse ratio of the distance of the
wheel from the ends of the shaft.

" In Fig. 6, let a represent a shaft, either upright or lying.
If a single wheel, as b, fixed upon it, drive two pinions, as
c D, directly opposite to each other, the shaft or journal will
not be afiected thereby *, but if two wheels are placed go
the shaft, driving each a pinion, as in Fig. 7^ both journals
will be afiected, and the greatest strain to break the shaft
will be at the arms of the wheel, or close to them, and be-
twixt the arms and journal. In this case, the middle point
of the shaft, as at a, being in a state of contrary pressure,
and therefore no strain on that part, it may be considered
as a lever on each side of a ; this being the fixed point,
and the greatest strain on the shaft being at the wheels.
The strain on the journal will be inversely as the distance
of the wheels from the middle point a, it being underBtood
that the wheels, pinions, and resistances are all the same.

200. ^^ Again, if two wheels are placed on a shaft, and
two pinions, both on the same side of the shaft, the journals
must support the strain or force of the action of the teeth
of both wheels, or indeed whatever number of wheels is on
a shaft ; and working into pinions on one side, the jounal

* See the note to Art. I9S.

■■]

ON THE SHAFTS OF MILLS.

187

Aat shaft must support a pressure equal to the whole of
r action into the teeth of the wheels or pinions into
ch they are connected ; and the strain on the shaft to
ik it, depends on the situation of the wheels, as they

be placed on the shaft: it being imderstood, as for-
ly, that directly at the place where the wheels or pinion
[ed on the shaft, that it must support a pressure equal
le pressure of the teeth of the wheel in its eorrespond-
wheel or pinion. And according as these pressures
bine, or act in a contrary direction to each other, so
; the strain on the journals, or tendency to break or
: the shaft be.

Hence the importance of placing wheels and pinions,
at their action on each other may be in contrary direc-
{, or so as to avoid, as much as possible, the strain on
shaft or journals.

>l. " In lying shafts, where circumstances may require,

ly be advisable to have the main or heaviest shaft on
lift of the wheels, as this will take off a considerable

of their weight or friction on the journals. The other
I will naturally be on the fall of the wheel, (which is
mod in this case much lighter than on the main shaft,)
will be prevented from jolting upwards, from both its

weight, and a pressure equal to that on the teeth of

wheels, being supported by the journals. It being
lent that whatever the wheels are, that the same pres-
6 that is on the teeth of the wheels will be equallv the

on both shafts, the one tending to increase the weight
Bie shaft on the journals, and the other to diminish it."
102. Roberton, in the course of his business, made seve-
obscrvations on the foregoing subjects. Late in his ca-

a case of this nature occurred, of considerable import-
at the flour-mill erected near the Slitt Mills of Pa-

'. " The water-wheel, 16 feet diameter, makes about

* On llie Itfiiiks iif the Civdc.

188 ON THE SHAFTS OF MILLS. [CHAP. II.

10 or 11 revolutions per minute, driving in general two
pair of stones ; the pit wheel about 6^ feet diameter. The
consequence was, that the machinery, firaming, &c., were
not of sufficient strength to bear the force applied by the
pit wheels, (though they were very strong, and well exe-
cuted for ordinary cases,) the shafit, framing, &c'., were in a
high state of tremor, the machinery working in a rough
and straining manner, and the wooden teeth (5 inches
broad, pitch about 4^) could not stand for any length of
time. The mill was altered, by enlarging the pinion, to
let the wheel run at 14 or 15 turns per minute ; and the
pit wheel enlarged about one foot diameter ; afterwards the
mill wrought very well.

'^ In conunon corn-mills, and many others, there is a great
deal of the impelling power lost by the smallness of the
pinions, &c., which causes a great friction on the shaft or
spindle, as well as by the friction of the teeth. Were both
wheel and pinion increased in diameter, much advantage
would arise, not only in saving of power, but in the tear
and wear of machinery. It is on this account that a com-
mill, constructed in the double way, other circumstances
being the same, performs much more work than in the
single way.

203. " In short, more things of this nature take place in
machinery, than the most of operative mechanics, or even
philosophers, are aware of ; and there can be no doubt, that,
with a knowledge of the affecting causes, a vast deal might
be done in saving power. For example, in a cotton*mill, the
main shaft generally makes from 40 to 50 revolutions per
minute. Let the weight of the shafts and machinery, the
size of the journals, and the effect of the wheels on these
shafts, and friction of journals, be taken into the account,
in the one case, and let the main shaft be supposed to be
reduced to half of the former velocity, that is, from 20 to
25 turns per minute, and let the strength of the shaft.

CHAP. II.] ON THE SHAFTS OF MILLS. 189

together with a due proportion of wheels, he augmented,
so as to preserve the same firmness in every part of the
machinery as in the former, and that the ultimate part of
the machinery may he hrought to the same speed as
formerly, it will he found on investigation, that in the last
case there will be a considerable saving in the first, or im-
pelling power.

" It must not be lost sight of, that, by reducing the
velocity of the main shaft, a judicious increase of the
diameters of the wheels thereon is absolutely necessary.
Indeed, without strict attention to matters of that kind, the
effect of any alteration that may be proposed or made is
very precarious. A due regard to the proper diameter of
wheels, according to the work they have to perform, is a
matter of very great importance ; and it will be found, on
general inquiry, that in most cases of machinery, it would
be prudent to have the wheels and pinions of large dia-
meter ; and, as we said before, by increasing their size,
the force, strain, and friction on the shafts and journals
are diminished in the same ratio.

" Suppose, in a mill, that a range of Ijring shafts, of 80 or
100 feet long, together with wheels, &c., fixed on them,
weighed yOOOlbs ; the journals 4 inches diameter, making
46 turns per minute ; the surface of the journal would at
that rate move at about 50 feet per minute : and supposing
that the friction was equal to one third of the weight, we
should have 7000 ^ 3 = 2333 x 50 = 11 6650 -r 44000 =
2'66 ; that is, nearly 2f horses' power expended in over-
coming the friction of these shafts*.

" Again, were these shafts and journals extended to 5

* Thai is, Taluing the horses' power at 44000 lbs. 1 foot per minute.
Sm page 88, ^ Horses Power" Art 108.

The quantity of frietion is much overrated, but the loss of power would
be very considerable on the lowest estimate ; and, therefore, the proper situ-
ation for the fint mover in a system of machmery is of some importance.

O

190 ON THE SHAFTS OF MILLS. [ CHAP. 11,

inches diameter, and their numher of turns reduced to one
half of the former ; that is, to 23 turns per minute, the
surface of the journal would move at the rate of 31 feet
per minute, and the weight of the wheel and shafts in-
creased one hal^ or say, to 10000 lbs., we should have
10000-=-3=333Sx31 = 103323-?-44000=2-348, that i8»
nearly Q^ horses' power ; so that in this last case there
would be a saving of something more than three tenths of
a horse's power, and the machinery would be, in every re-
spect, improved.

204. " In a horse-gin, where a pinion is driven by a
toothed wheel on the gin, the friction, or strain on the
journals, depends on the situation of the horse-beam; at
any moment of time when the lever or hors^-beam is above
or below the pinion, the friction on the shaft will be the
least, and when in the opposite direction, the friction and
force on the shaft will be the greatest, and the general
friction will be as the size of the main wheel ; that is, the
smaller the main wheel is, the friction and force on the
shaft will be the greater, and the larger this wheel is, the
friction, &c., will be the less ; it being understood that the
friction or strain on the gudgeon, or axle, on account of
the reaction of the teeth of the wheel is meant^ and not
the strain on the shaft, to twist it, nor any other friction or
strain whatever.

205. " In a water-wheel turning machinery, in many
cases, it is most advisable to have the toothed wheel of the
same diameter. In this case, whatever power or force is
applied to the wheel, the same must be resisted by the
teeth of the wheel ; and it also follows, of course, thai
whatever is the size of the wheel, or pinion which is driven
by the main wheel, that the very same strain is on the
shaft ; that is, the same as on the teeth ; and the shaft
must be sufficiently strong to withstand the pressure. But
another circumstance occurs, that by increasing the size of

CHAP. II.] ON THE SHAFTS OF MILLS. 191

the pinion, there must necessarily be increase of the dia-
meter of the shaft to withstand the twist. The shaft by no
means, however, keeps pace with the augmentation of the
wheels, but is only as the cube-root ; for instance, a water-
wheel of 16 feet diameter would work very ill into a pinion
of 12 inches, supposing its shaft or journal to be 3 inches
diameter. Again, let the pinion be increased to 2 feet
diameter, a shaft of 3f inches will be sufficient to with-
stand the twist, the friction on the journals will be much
less, and the strength to resist the strain will also be much
increased. General rules cannot be laid down with accuracy
in these things. It is the particular circumstances of the
case that will guide a machinist in the construction of any
piece of machinery ; and if he be not well acquainted with
these, it cannot be expected that his schemes will be well
arranged.

** The above observations are no doubt at variance with
the opinions of those who are continually holding up the
simplification of machinery. For instance, Fenwick's
* Essay on the Simplification of Machinery.' * In many
cases it is prudent to make machinery of a more complex
nature than it is sometimes constructed ; and in order that
it may be easier driven, that the tear and wear may be
lessened, as well as the ultimate expense and trouble of
attending it.'' t

* The miLirimg of Fen\«ick consider friction only, and arc so far correct ;
but of the wear and tear, and friction of teeth, he has taken no account.
His 6th maxim, that ^^ Small wheels are equally as generative as large
wheeliB, if the same ratio of size be preserved," is true only within certain
limite, (see Art 33,) but these limits being assigned, his other maxims hold
tin the stress on the moving parts is less from a greater number of small
wheels, than from fewer large ones.

t John Robbbton, to whom the world is largely indebted, not merely
for the fer^cMiig dever and scientific remarks, but for much of modem im-
praTfOWDt, WM, in his day, (the beginning of the present centory,) one of
thesMNl diitingmahed millwri^ts or engineers in Olasgow.

o2

192 ON THE SHAFTS OF MILLS. [cHAP. II.

206. In the essay to which Roberton in the forcing
observations alludes, Mr. Fenwick infers, (p. 64s) that
'* the most perfect machine is that which operates with the
fewest moving parts.^* But Fenwick seems to have beai
misled here by a desire to generalize. Simplicity is, no
doubt, a most desirable quality in a machine, provided it
can be obtained without making a sacrifice of power or of
durability. A sledge has fewer moving parts, and in that
sense is more simple than a steam carriage, yet no one
with truth could say that a sledge is more perfect*.

Nor, perhaps, does the simplicitif of a machine consist
strictly in having^/few; moving parts. If the parts of a ma-
chine be few, they are perhaps more easily taken in by the
eye at one view, which may make them more easily compre-
hended by the mind, and in that sense be more simple. But
in machinery, the kind of simplicity at which we ought to
aim, has more regard to the manner of action than to the
number of the moving parts. Thus, for example, when a
weight is to be raised, if one wheel worked by a screw he
employed, the machine consists of fewer part^ than two
wheels and two pinions, applied to the same purpose. But
in this last case, the manner of action is really more simple;
for the action and resistance are directly opposed in the
same line ; whereas, in the case of the screw, the action is
oblique, and experience shows that it has much more fric-
tion, and much less durability.

What is here said of manner of action is applicahle in
comparing machines, consisting each of trains of wheels
and pinions ; for the most durable, by longest maintaining
the true figure of the teeth, will ultimately be the most
simple in the manner of its action. It is evident, that
when the teeth become much worn, that the manner of

* The Essay by Fenwick here alluded to, is wholly confined to the <^
plicatum of wheel- work ; consequently the remarks in this and the follow-
ing paragraph are not applicable to his Essay.

....]

ON THE SHAFTS OF MILLS.

I'J^

becomes proportionably more oblique and less

lie. Respecting the durability of the wheel-work of

we may refer the reader to Art. UX).

It can be of very little use to give rules for the dia-

of gudgeons, or for the strength of shafts, unless

be accompanied with some method of estimating the

ng force. Our author has not touched upon this

^ of his subject ; and therefore we have inserted Ro-

berton's remarks, with the ^iew that the information which

convey, respecting the stress on shafts and gudgeons,

be studied in their proper order ; we shall add some

ional inquiries to these articles,

'. Let AB, Plate [IV. A.] Fig. 4., represent a shaft,

one wheel at d, and another at c ; these wheels being

any size whatever. If a power act upon the wheel d at

thi! point F, and the resistance be at w, the stress arising

from these forces will cause a pressure on both the gudgeons ;

the line wp being drawn, cutting the axis at some

E ; the stress upon the shaft and gudgeons will be

same as if a force equal to the power and resistance

together, were applied at the point e. Hence it is clear.

It when the wheels differ considerably in size, the

IpoD next the smaller wheel will have to sustain the

iter part of the stress.

08. But if the resistance were at w, the power and re-
ttice in this case being at the same side of the shaft,
pressure will be downward on one gudgeon and up-
i upon the other. For the descending power p is re-
sd by the support of the gudgeon at a, and by the re-
at w ; but the power not falling between these
it will tend to raise the gudgeon b.
KJ. Again, if the resistance be at the point c on the
or under side of the wheel c, the pressure at the
u will be wholly in a lateral direction ; conse-

194 ON THE SHAFTS OF MILLS. [^CHAP. H.

quently, when the impelling power moves with the wfaed,
the stress on the gudgeons will vary considerahly both iB
intensity and direction.

If the plan of the shaft and wheels he drawn to a scale,
it will be easy to compute the pressures in these difimnt
cases, and to compare them ; and perhaps a young machinist
will feel some pleasure in such comparisons, where he
would have been fearful of engaging with a mass of algebra.

Case 1. The power and resistance being at opposite
sides of the shaft. Draw the line a b in the middle of the
shaft, and also draw the line wp. Then to find the stresB
upon the gudgeon b, we shall have a b : a e : : power added

to the resistance : stressonthe gudgeon b; or ^ ^^

AB

the stress upon the gudgeon b. Also ab : be :: power
added to the resistance : stress on the gudgeon a =
be X (p + w)

■ ■ •

ab

Case 2. The power and resistance being at the same
side of the shaft. Draw the lines Ate and pb which cut
one another at f. Then, the line pb mav be considered a
lever with its fulcrum at f ; and to find the stress neces-
sary to keep the gudgeon b down, we have bf : pf :: power

at p : stress at b = = the stress on the gudgeon B.

B F

This stress will obviously be opposed to the weight of the
shaft and wheels. Also, af : wf :: the resistance at w :

stress on the gudgeon at a, = . This stress will be

AF

to add to the stress from the weight of the shaft and wheels.

Here it may be remarked, that it is desirable that the
greater pressure on any gudgeon should always, when con-
venient, be in the direction of gravity, to prevent the un-
pleasant jolts which take place when the machine is pat

ON THE SHAFTS OF MILLS.

So motion, when the pressure is upwards upon any of
igadgeons.

210. But the preceding cases suppose that the power at
and the weight or resistance at w, are parallel, they are,
wever, oft^n ohlique in respect to one another. Let k,
p. 5, (i, and 7. on a plan of such wheels, be the axis ;
I p be that point in the eircnmference of one of them
Bpe the power acts in the direction d p ; and let w be
i point where the resistance acts in the direction dw.
ben DE will be the direction of the stress upon the axis,
d if D c be made proportional to the resistance ; or u A
opcH'tional to the power ; and if the parallelogram Dcab
>C4nnpleted, «d will be proportion^ to the whole stress
on the axis, which is obviously greatest in Fig. 5, and
It in Fig. f). Now, make cd perpendicular to de, tlien
i is the pressure on the axis caused by the resistance at
) circumference of the lesser wheel ; and ad will be the
on the axis from the power at the circumference
the large wheel. But it must be remarked, that when
8 direction of the stress on the axis falls between the di-
tion <rf the power and that of the resistance, as in Figs.
Hid 7 ; the whole stress will he either in the direction
1) as in Kg. 5, or ED as in Fig. 7i and proportional to
I. Whereas, if the directions of the power and rosist-
Ke be both on the same side of the axis, as in Fig. 0,
I »-ill be the pressure on the axis at the place of the small
leel w, in the direction e d, tending to raise the axis ;
Q rfa the pressure on the axis at the large wheel p, in
B direction de ; or opposite to the pressure at the wheel
i while iiD is the difference between these pressures,

the whole tendency of the axis to rise.
The figures are all drawn to the same size, and the
T and resistance being represented by equal lines in
figurt', the ad-vantage or disadvantage of any particu-

196 ON THE SHAFTS OF MILLS. [CHAP. ^

lar construction will be seen by inspection ; and the rea^
may easily multiply examples, by drawing more figures.

The relation and intensity of the pressures might hsLr
been shown in a more elegant manner by the arithmetic
sines, but perhaps not so satisfactorily to the minds of mo
of my readers. And when the rule and compasses are :
the hand, the relations are more easily, and with less ris
of mistake, ascertained by drawing the wheels and axis a
shown in the figures ; as it merely requires a little know,
ledge of the composition and resolution of forces.

Thus, make d c, from a scale of equal parts, equal to
the number of cwts. in the resistance at w, and make ac
parallel to the power pd, and cd perpendicular to de.
Then d d, measured from the same scale, is the stress at
the centre of the wheel w in cwts. and ad the stress in
cwts. at the centre of the wheel p.

The stress from the resistance and power being deter-
mined, that which is caused by the weight of the shafts
and wheels themselves will be easily calculated. See Art
245 and the following articles.

The case of lateral stress being the most simple, we
shall, in the first place, examine it, confining, for the pre-
sent, our attention to the gudgeons only. The bodies of
shafts will be afterwards considered.

CHAPTER III.

It TEg STRENGTH (

> ODDOBONS WHBRB THE STBESS IS PRODUCED BIT
LATERAL PRESSURB DNLV.

The gudgeons having all the weight on the shafit
mpport, ought to be made sufficiently strong for that
while, to avoid unnecessarj' friction, they should
ifflade as small in diameter as possible, conEistently with

lent strength and durability.
■When we are able to determine the diameters of the
gudgeons, or journals, this serves as a foundation for the
Jroportions of the other parts of the shafts.

ilthough wTought iron will bear a greater weight than
iron, yet cast iron being not only cheaper, but much
» easily formed into convenient shapes, gudgeons
Bow most commonly made of that material. We shall,
refore, in the first place, confine our attention to the
Hbeters of gudgeons made of cast iron. Here it may
^per to state the following proposition :
ill Prop. I — Solid cifUnders of the same letigtk have
lateral strength as (lie cube of their diameters ', Jbr
ffieral, the lateral strength of any pieces of iron or
whose sections are similar Jigures, are as the cubes
fe timilar sides of t/ie sections.
That is, if a gudgeon of two inches be sufficient to sup-

Soe Bmemou'ti 4to edition, prop. 67, cor. 2. Gregory 'a Mochanics,
iuL I7U. cor. 3.

198 ON THE SHAFTS OF MILLS. [CHAP. lU.

port a certain weight, a gudgeon of four inches will sap-
port eight times as much.

From this law, it is evident, that were all gudgeons
made of iron of the very same quality, knowing the
strength sufficient in any one case, it would be easy to cal-
culate what it should be in any other case.

But as there is a great variety, in point of strength, in
different kinds of iron ; Welch cast iron, for instance,
being stronger, as some think, by one fourth, than that
made in Scotland ; it is prudent to calculate upon the
weakest. Mr. Banks observes % that, '^Iron is much
more uniform in its strength, than wood ; yet it appears
that there is some difference in different kinds of ore, cr
iron-stone ; there is also a difference from the same for*
nace, perhaps owing to the degree of heat which it has
when poured into the mould.'*

213. The strength of a gudgeon is limited by the strain
it will bear without permanent derangement of its struc-
ture, for the length is always so small in regard to the diar
meter that the flexure will be, in all practical cases, insen*
sible.

The calculated stress should include every kind of force
acting on the axis or shaft ; and the diameter should he
determined, so that the gudgeon would be capable of re-
sisting the whole stress if it were thrown upon the extreme
point of its bearing, (see Art. 217- )

When w is the utmost amoxmt of the stress in cwts., and
/ the length of the gudgeon in inches, from the shoulder t^
the extreme point of bearing, it is shewn (Essay on Cas

Iron, Art. 138.) that Cli^ZJiJ^* == rf; the diameter o

5

the gudgeon in inches. Or, 0*42 ( w /) * = rf.

214. But an allowance should be made for wear, which
will be nearly directly as the stress, and inversely as the

* Banks s Power of Machines, p. 94.

ON THE SHAFTS OF MILLS. 19!)

len^h of the gudgeon's bearing ; consequently the length
of tiie gudgeon should be greater in the same ratio as the
stress is greater. And till some more certain principles
of jjrojKjrtioning the gudgeons of a machine so as to be of
equal duration shall be found, we may allow one fifth of
tbe diameter as a provision against wear where no gritty
substance is likely to affect it, and one third in all cases
irhere the gudgeons are exposed to gritty matters.

RcLE — In the former case, the rule will become 0*5
(«/)! =d. That is, multiply the stress in ewts. by the
length of the gudgeon in inches, and the cube root of the
product being multiplied by 0-6, will give the diameter of
ik gudgeon in inches.

WTicn a gudgeon is likely to wear much from the nature
i the machine or its particular situation, multiply the
•. root of the product by O'fi, instead of 0-5. Gud-
s of water-wheels may be included in the class which
« exposed to considerable wearj

I K the stress on one gudgeon be equal to the weight of
|e wheel, and the wheel be at the middle point, that part
■ the stress which is produced by the action of the moving
Heer and resistance, must be considered equal to half the
ight of the wheel ; for only half the weight will bear on
K of the gudgeons in this case. But it often happens
it the wheel is considerably nearer to one bearing than
B other, and in such cases, the rule of the author woidd
B likely to mislead.

[ Since our author has given it as a general principle, that
K diameter of a gudgeon should be equal to the cube
f^»t of the weight supported in cwts., it will be desirable
"> compare our rule witli that principle ; first assuming
Ml the weight ia actually equal to the stress upon the
IRlgeon. Now, in tbat case, the rules will be the same
D 0-5/1 = 1, or / = 2 1 = 1"20. And in the second rule,

200 ON THE SHAFTS OF MILLS. [CHAP. UI.

when /= (— ) =1*185 inches. Therefore whenever the
•6

length of the gudgeon exceeds ahout 1 inch and ^, the rule
gives the diameter too smalL Again, if we take the actual
stress to be only half the weight of the wheel in cwts.,
which is clearly in all cases less than the real stress, the
above numbers should be multiplied by the cube root of %
which will show that the author's rule becomes in defect
again when the length exceeds 1*587 inches.

We shall proceed in this inquiry, on principles siinilar
to those which have been followed in the Inquiry respect-
ing the Strength and Durability of the Teeth of WhedSf
namely, by taking a number of cases from mill- work in ac-
tual use, and drawing inferences from them. This metbod
of making inferences, by collecting and arranging facts re-
specting miU-work, being much safer and more usefiilf
than founding calculations upon experiments, often made
on a small scale, and under circumstances very different
from those which occur in practice with machinery. We
begin with considering the gudgeons of water-wheels.

SECTION II.

OP GUDGEONS OP WATER-WHBELS.

215. In the following table, water-wheels of various
weights are collected, and the diameters of the gudgeons
in actual use, stated. The weights of the cast iron wheels
were found, from the weight of the castings, &c., of which
they are composed. The wooden wheels are estimated,
making allowance for the wood becoming heavier by being
soaked in water. We are aware, however, that besides the
mere weight of the wheel, other circumstances should be

. .„.]

ON THE SHAFTS Of MILLS.

201

&en into account', such as the weight of water in the
buckets, and the pressure brought on the gudgeon by the
resistance of the work, &c. But we shall follow the ge-
neral result of those cases only, in which the gudgeons
havo been found sufficiently strong, and we apprehend
that those extraneous causes on the one hand, will not
affect our rules more than the different qualities of cast

BwiU the strength of gudgeons on the other handt.

fhe water-wheels in the table were all in the middle of

ir respective shafts. Both gudgeons of each wheel had,

lefore, equal stress.

216. Description of the first Table of Gudgeons of

'dtr-telieels.

Column 1. contains letters to distinguish the wheels.

2. shews the material of which the wheel is
made.

3. the diameter in feet,
■i. the width in feet.

5. the kind of wheel.

6. the diameter of the gudgeon in inches.

* See Encyclopfedia Biitannica, ulicle Rotation.

* These remarks of Bobcrtaon Buclianan render it necessary to say a
■ule9 founded on empirical principles, and particularly when

ire not minutely detuled for those cases on which the
I m founded.

It rule far the strength of gudgeons in the text, supposes the stress to
llnjti proportional to the weight of the wheel, but thia supposition
triy ever corresponds «nth the truth ; consequently, in the cited proc-
t, if the stress was not the greatest possible, in regard to the weight
;lb wheel, and the meta! of an inferior quahty, Uiis rule may lead us
Iwrious errors. But the practical eases ore not described, and therefore
>t judge, from any thing in the teat, of the safety of the rule.
B mlc ia to be formed by any process, every cause of stress should
imiiilered, and where siinphcity is desirable, the stress should be rcpre-
d by a quantity which is certain to equal it, even in an extreme case.
(Uaot err greatly, if the error be always on the dde of strength.

302 ON THB SHAFTS OF HILLS. f {»AP. m.

Column 7. the weight of £ome of the wheels, in tou
and cwtB.

8. the weight of some of the wheels, in cwti.
and qrs.

9. contains the cube root of the weight
The use of this column is to compare the several gud-
geons with the law contained in IVop. I. For, were tH
the gudgeons duly proportioned to the weight they hafe to
sustain, they would be to one another as the cube roots (tf
their weights.

TABLE I. GUDGEONS OF WATER-WHEELS.

1

2

3

i

5

G

7

8

9

y

i

'tl

wrighiof

Wright of

Wheel

^

3

i

whHilllll

fheeli tn

node of

Elod.

lontind

cwu.u>d

Df wdshl

II

1

cint.

*""■

Incou.

A

Wood

24

12

Ovcrshol

7

23

14

474

7-796974

B

Cart iron

16

6

ditto

7

12

Q40

3

6-214464

C-

Caitiron

16

8

ditto

6i

16

io

930

6-91IM23Th?MJ(~

D

Caalimn

1

wheel *nd

[^

4i

ditto

24

480

7 -829736 ouu*.

buckets

f

1

E

Wood

16

fl

ditto

6

10

211

5-95334 1|

F

Wood

121

7

ditto

6

5

14

114

4-8488081

G

Wood

3-2

11

ditto

10

H

Hi

10

ditto

7

Wood

«

ditto

8

K

Rr

i»

lO

ditm

10

• Tbe holhn ihuft oT C

OBIBRVAIIONS I

I THB PtBST TABLB OF QUDOEOHS.

217. Particular care should be taken that the axis of
the gudgeon be exactly in a line with the axis of the shaft
which it supports, otherwise the motion will be imequal,
and at one part of the revolution the stress will be thrown

CHAP. III.] ON THE SHAFTS OF MILLS. 203

to the point of the gudgeon; this would endanger its
breaking, more particularly if very long, though otherwise
sufficiently strong*. In the case h, we have an instance
of a gudgeon breaking, from being made too long. In
practice it is a good method to turn the gudgeons of a
wooden shaft, after they arc fixed in their places, a second
time, in order to render them quite true.

218. From comparing the diameters of the gudgeons
(column 6) with the cube root of the weight in cwts. of the
wheel, (col. 9,) it will be found that they approach one
another. In other words, the • cube root of the weight in
cwts. is nearly equal to the diameter in inches. In the
case c, the gudgeon broke only from being a bad casting,
although it is smaller in proportion to its weight, than
most of the other cases. Since it was renewed of the same
size, it has continued to support its work. When, there-
fore, we can ascertain the weight of a water-wheel, we
have a very simple rule for finding the diameter which the
gudgeon ought to have.

RULE I.

219. The cube root of the weight of a water-wheel^ in
hundredweights^ is nearly equal to the diameter in inches
of a cast iron gudgeon sufficiently strong to support such
wheelf.

We say nearly^ it being evidently most prudent to make
the gudgeon a little more rather than less in diameter, and
to make aUowance for wearing.

* The possibility of such a cause of failure should be guarded against in
proportioning the strength of a gudgeon. See Art. 213.

t In water-wheels, the stress is not proportional to the weight of the
wlieeL See Art 212 — ^215 inclusive, where more correct principles are
iiifiettigited. Alao tee the cantions in note to Art. 215.

S04 . ON THE SHAFTS OF MILLS. [CHAP. 10.

EXAMPLE.

Suppose a water-wheel to weigh 12 tons, 0 cwt, 3 qn.
what ought to he the diameter of a cast-iron gudgeon, suf-
ficiently strong to support the wheel ?

12 tons = 240 cwt. 3 qrs.

The cuhe root of 24075 = 6*221 Answ. That is, the
diameter of the gudgeon should not he less than 6*^ dia-
meter. It ought to be rather more, to allow for wearing,
&c. See B, in the first table of gudgeons.

220. As the weights of wooden water-wheels cannot be
accurately known, without a good deal of calculation, it is
desirable to have some more ready method for practical
purposes.

The weights of overshot, or bucket water-wheels, will
be to one another nearly as their circumferences, or dia-
meters and breadth. — ^We say nearly^ because the arms will
make large wheels heavy in rather a greater proportion.
Hence the following rule is formed on the direct proportion
of the sole and buckets, adding one-half of the diameter
increased in the duplicate ratio or square of the diameter.

RULE IL

For fcooden water-wheels^ multiply the diameter in fid
by the width also in feet, to which add the square of half
of the diameter. The cube root of the sum will be nearly
equal to the diameter of the gudgeon in inches.

EXAMPLE.

Suppose a wiioden water-wheel 1 2 feet diameter and 7
feet wide, (^see e in Table !!• of Gudgeons.)

CHAP. III.J ON THE SHAFTS OF MILLS. ^).5

12 X 7 = 84
The square of 6 « 36

ISO the cube root of which = 4'9S24.24;
that is the gudgeon should not be less than about 5 inches
diameter.

IZPLAHATIOH OP TABLE II. OF WATBB-WHSBL8.

2S1. All the columns, except No. 10, are the same
as in Table X,* The colunm 10 shews the result by
Rule II.

TABLE II GUDGEONS OF WATER-WHEELB.

i

t

DlDin

Wherf

s

i

KIpd.

w^hi or

1

lUnuirlu.

A Wood

24

~ii

Overahol

7

23

14

474

7-796974

7-559525

B Out iron

16

16

ditto

7

12

2

24^

6-231678

C C«t iron

16

8

ditto

»l

16

10

330

6-910423

DOW iron

32

4i

ditto

9

SO

400

7-368063

nfaeeluid

buclieU

K Wood

12

7

ditto

C

5

14

114

4-848808

4-932424

F Wood

S&

11

diao

10

9-471647

G Wood

21

10

ditto

7

6-839903

H Wood

28

6

ditto

8

7-140037

I Wood

16

9

ditto

6

io

211

s-a^i

5-924991

K Wood

18

10

ditto

10

..

6-390676

r IBON ODDOEONS FOB VABIom FUBFOaSS.

2S2. Taking it for granted that the cube root of the
weight of a water-wheel, in hundredweights, is nearly
equal to the diameter in inches of a cast iron gudgeon,
sgfficiently strong to support such wheel, the following

* In then table* Uh wd^t of the wheel b U diffeicot. We mippoae the
Int taUe to 1w the oomct <ma, bat the difference ia iacondderable.

206

ON THE SHAFTS OF MILLS* [CHAP. in.

table of the diameters of gudgeons, and the weights which
they may be supposed to sustain, is formed. It may be of
use for finding the diameters of journals in all cases of
stress arising from lateral pressure, such as grindstone,
intermediate spindles, &c. where the pressure can be ascer-
tained, as well as water-wheels*.

EXPLANATION OF THB TABLE OF CAST *IBON 6UDOB0N&

223. Column 1 contains the diameter in inches, firom 1
to 11 inches.

Column 2 contains the cube of that diameter, or the
hundredweights which the gudgeon may sustain.

N.B. We have already remarked, (Art. 219f)thatit
would be most prudent to make the gudgeon a little more
in diameter than the cube root of the hundredweights.

TABLE OF CAST IRON GUDGEONS.

TX» A •—

Cube of dBameter, or

Diameter in
inches.

Cube of diameter, or

Diameter m

cwts. wnioh toe gua*

cwtB. which the f^nd-

m WKXi^mM^Cm

geon may austun.

m^m,\^mm^^9^

geon may sustam.

1-

1-

6-25

244-140625

1-25

1-953125

e-b

274-625

1-5

3-375

6-75

307-546875

1-75

5-359375

7-

343-

2-

8-

7-25

381-078125

2-25

11-400625

7-5

421-875

2-5

15-625

7-75

465-483375

2-75

20-796875

8-

512-

3-

27-

8-25

561-515625

3-25

34-328125

8-5

614125

3-5

42-875

8-75

669-921875

3-75

52-734375

9-

729-

4-

64-

9-25

781-453125

4-25

76-765625

9-5

875-375

4*5 •

91-125

9-75

926-859375

^•75

107-171875

10-

1000-

5-

125-

10-25

1076-890625

5-25

144-703125

10-5

1157-625

5*5

166-375

10-75

1242-296875

5-75

190-109375

11-

1452-

6-

216-

* See Art. 212.

CHAP* IIL] on the shafts OF MILLS. 207

USE OP THE TABLE.

EXAMPLE L

324. Suppose a cast iron gudgeon of 5f inches diameter,
what weight of a water-wheel would it he capable of sus-
taining?

In the first column find 5*^5. Opposite to which will
be found 190* 109375 hundredweights, which is rather
more than 9i tons, which is the answer. But in practice/
it would perhaps be prudent not to load this gudgeon with
more than nine tons.

EXAMPLE IL

Suppose a grindstone weighing \5\ hundredweights,
required the size of a gudgeon sufficient to sustain this
weight Look in the second column for the nearest weight
to that given, which will be found to be 15*625, opposite
to which, in the first column, is 9,\ inches, the diameter of
the gudgeon required.

But in practice the spindles of grindstones are com-
manly of wrought iron, which metal we shall presently
consider as applied to gudgeons.

SECTION IV.

or MALLBABLE OB WBOUOHT IBON OUDOBOirS.

225. Professor Robison states*, that the cohesive force
of a square inch of cast iron is from 40,000 to 60,000 lbs.,
wionglit iron, from 60,000 to 90,000 lbs.

In the year 1795, Buchanan had occasion to substitute
cast iron gudgeons for those of wrought iron, and made

* KnejdopndiB Britannica, article Strength of Materials, 40.

p2

208 ON THE SHAFTS OF MILLS. []CHAP. HL

some experiments on those metals, firom which he drew the
following inference : that gudgeons qf the same size^ of
cast and of wrought iron^ in practice^ are capable^ at a
medium^ qf sustaining weights without flexure^ in the pro-
portion qf9 to 14*.

Taking it for granted that this proportion is near the
truth, we may find the diameter which any wrought iron
gudgeon ought to have when its lateral pressure is given,
in the following manner :

226. 1. Find the diameter which a cast iron gudgeon
should have to sustain the given pressure, then say, as 14
is to the cube of the diameter of the cast iron gudgeon, so
is 9 to the cube of the diameter of the wrought iron
gudgeon.

2. The root of this last number gives the diameter re-
quired of the wrought iron gudgeon.

* According to Tredgold's experiments, the stiffiiess of good cut iron b
to that of good English malleable iron as 1 is to 1*3 nearly. (Eanjon
Cast Iron, Art. Iron.) But the stifihess of malleable iron is much increued
by hammering, &c.; and in his trials some pains were taken to obtain the
resistance unhammered, which most probably causes the difference.

It is further necessary to observe, that in calculating the Btrength of
gudgeons, the resistance to permanent alteration is the proper mcasoiei be-
cause they are too short to admit of sensible flexure ; and the strain whidi
produces permanent alteration in malleable iron is only 1*12 times that pro-
ducing a like alteration in cast iron.

Hence the diameter of a malleable iron gudgeon should be 0*963 times
that of a cast iron one to bear the same stress. For if a be the diametCT of
the cast iron gudgeon, and b the diameter of the \iTOught iron one ; then
a^ X 1 = ^^ X 1*12 when their strengths are equal; consequently

n2 =*'' °' (TT2)* = *' ^"* (T^* = »•««»'» = *•

The proportion given by our author is about 0*863 azz.by bat his propor-
tion is made from the relative stiflfhess, while the rule applies to the
strength, and therefore it is not correct

CBAP.in.] ON THE SHAFTS OF MILLS. 209

EXAMPLE.

Suppose the lateral pressure to be 125 hundredweights,
the cube root of which is 5, the diameter in inches of the
east iron gudgeon : then say.

As 14 : 125:: 9 : 80-357.
The cube root of which is 4-30887*.

Upon this principle the following table is calculated to
Aaw the proportionate diameters of cast iron and wrought
iran gudgeons.

nPLARATION OF THE TIBLB OF CAST AND WROUGHT IRON OUDOE0N8.

Columns 1 and 2 are the same as those in the table of
cast iron gudgeons.

Column 3 contains numbers in the proportion of 9 to 14
lesg than those of column 2.

Column 4 contains the cube root of column 3, or the
diameters of wrought iron gudgeons, having the same
ttrength as those of cast iron in column 1.

* To find the proportion by the preceding note, mnldply the diameter of
e cut iron gudgeon by 0*963. Thus 5 inches X 0*963 = 4*815 inches,
e diameter for a wrought iron one.

ON THE ■HAVTS OF MILU.

TABLE OF CAST AND WBODGBT IBON G17DOBON8.

1

2

3

4

Dismeterof

Cube of diuneter of
casl iron gudKeoDS, or
the cwte. wEid. the

Cube or iSaineteT

DiamMeiof

ntnughliron

nf

of wrought iroD

Qri,-

gudgeon* may Bustain.

[-III.

)■

r

■6428571

-863

1-25

1-953125

1-2555803

1-063340

1-5

3-375

2-1696427

1-259921

1-75

5-359875

3-4453125

1-514825

2-

8-

51428571

1-709976

S-SB

11-400625

7-3289732

!-»12983*

is

15-625

10-0446428

2154435

S-75

20-798875

13-3694196

2-361335

3-

27-

17-3571428

2-571282

3-25

34-328125

22-0670803

2-802039

3-5

42-875

27-5625

3-018294t

3-75

52-734375

33-9006696

3-239618

*■

64-

41-1428571

3-448217

4-25

76-785625

49-3493303

a-esgsot

4-5

91-125

58-5803571

3-881938

4-75

107-171875

88-896

4-10I5«6

6-

125-

80-357

4'808870

S-26

144-763125

93023

4-530655

5-»

iee-375

106-955

4-747459

5-7S

190109375

122-213

4-959675

e-

216-

138-857

5-180101

e-25

244.-1 40625

156-948

5-394690

e-5

274625

176-545

5-609376

e-75

307-546875

197-709

5-828476

7-

343-

220-500

6-041877

7-26

381-078125

244-979

6-257324

7-i

421-875

271-205

6-471274

7-75

465-484375

299-240

6-686882

8-

512-

329143

6-903436

8-25

561-515625

360-975

7-120367

S-5

614-125

394-795

7-337234

8-75

ee9-921875

430-664

7-553688

9-

729-

468-643

7-769462

8-85

791-453125

508-791

7-984344

9-5

875-375

562-741

8-257263

8-75

926-859375

593-837

8-415541

10-

1000-

642-857

8-631103

10-25

1076-890625

692-287

8-845085

10-5

1157625

744-187

9061309

10-75

1242-396873

798-619

9-279308

II-

1452-

933-428

9-771484

* Bnw)ier aj«, " A gudgwn 2 inches disnieter, wiO tatoin 9S39 Dm. noi
BirnMhT's Edition of FereuiMHi^ Leaim*. Vol I. p. 157.

f The wRiugfaliniD ^ndtlonc tpindle ui«db}- Snuthon, iahk esperimenKon
■u31iurheii]iaii]eler,indcuTiedanDDe«>e^iiDg3TU0Il«. Sae FUL Hi^ VoL X

I

CHAP.m.] ON THE SHAFTS OF MILLS. Sll

USE OP THE TABLE,

EXAMPLE.

^. To find the diameter of a wrought iron gudgeon of
the same strength with one of cast iron of 3 inches diameter.
Look in the 1st column for S, and on the same line in the
4th column will be found 2*571 S82, that is, a little more
than ijj inches, the diameter required of the wrought iron
gudgecm.

The numbers in the Srd column, being the cube of those
in the 4th, another use may be made of this part of the
table. For, supposing the 4th column to represent cast
iron gudgeons, then the Srd column will represent the hun-
dredweights which cast iron gudgeons of those diameters
should sustain.

Before proceeding to consider the bodies of shafts sub-
ject to lateral stress^ we shall inquire into the strength of
joumak of shafts subject to torsion.

CHAPTER IV.

X L

7 JIC2XALS. WBXS THB STRESS ARISES FBOH

a TwrasnjKk, is Aoxnox to lateral stress^

Strength and Durability of

if Wsfesefe^ we luiTe used what is called the hmsei

h Jt^sasfsre iar the strain. We refer the reader

-D ^nac iPB? lan? diere said (Art 108 — 114) in explana-

imL IT :3ac vm vhieh ve shall use here, in measuring the

jriutfOi: CB shafts bv torsion or twisting.

c^ torsioii, as well as that of lateral pressure,
prnpMtionate strength is as the cubes of the

It 3CIT be proper here to remark, that what we had to
g journals, relates to those of cast iron, for
iron will bear more lateral stress, as we
^Art. *25,) yet it is a fact, perhaps not gene-
nZy known, that wrought iron will not resist torsion equal
)£^ cast iron^.

In seme cases a journal has not only torsion to resisf,
bet abo to carry a very heavy fly wheel, and it is prudent

* When ft shaft has a support between the points where the poirer aofi
resuSftDce are applied, the part of the shaft which revolves on this sopport
}< oued a Journal, i

+ See Gr^ry's Mechanics, Vol. I. Aft. 191.

This reference is to a statement that the strength is i awpcrtw^

bot it is not demonstrated.
t The author seems to be under a miatelt^ k

CHAP. IV.3 ON THE SHAFTS OF MILLS. 213

in such cases not merely to make an allowance for the
weight properly balanced, but also for any inaccuracy which
may occasion swagging, which greatly adds to the stress :
but others have hardly any other resistance but what arises
fitmi torsion.

It is further observable that the value for 10 horses in
the smaller engines is much less than in the larger. For
this difference what we have said respecting the weight of
the fly in a great measure accounts : and not only is the
heavier fly to be considered, but also the greater danger of
accidents from a large fly than from one that is of a
smaller diameter.

SECTION 11.

OF PBOPOBTIONINO JOURNALS TO THE STRESS WHICH THB7 HIVB TO

SUSTAIN.

229. The stress any journal has to sustain being as the
horses' power to which the resistance is equal directly ^ and
the number of revolutions which the shaft makes inversely^
it follows: That a resistance for example of 32 horses'
power on a journal making 50 revolutions per minute, has
the very same stress with a resistance of 16 horses' power
on another journal making 25 revolutions per minute.

dS divided by 50 is equal to 16 divided by 25, each of
which gives a quotient of 0*64«. Therefore in all cases
when the horses' power divided by the revolutions per
minute produces the same quotient the stress is the same.

Thus a resistance equal to 50 horses' power making 50
revolutions per minute, produces the very same stress as 10
horses' power making 10 revolutions per minute.

Having therefore fixed on any journal which has been
found sufficiently strong, we may make any other to have
Ae same strength in proportion to the resistance which it
has to overcome in the following manner.

214 ON THE SHAFTS OF BflLIA |^CBAF. IT.

230. Rule. — If it so happen that the hone^ pom,
and the revolutions per minute he the same mumberm Fcr
instance, 50 horses' power making 50 revolutioiiBy 5O-r50
= 1 ; then the cube of the diameter of the jofunal will h
a multiplier, by which to find the cube of the diameter of
the required journal.

But in case the horses' power and the revolatiooi par
minute are different numbers^ then you must suppose them
both the same, and calculate (as in Ex. II.) what» in that
case, would be the proportionate diameter of the joumaL—
The cube of this diameter will be a multiplier, the saine u
mentioned above.

Having found the multiplier, to find the diameter of the
required journal.

Divide the horses' power by the revolutions per minute.

Multiply the quotient by the multiplier^ the cube root of
the product will give the diameter of the journal re-
quired*.

EXAMPLE I.

To find the multiplier from a journal 7^ inches diameter,
where there is an engine of 50 horses' power turning a
shaft, at the rate of 50 revolutions per minute.

Divide the power by the revolutions, that is 50 divided
by 50 is equal to 1 ; the diameter of the journal is 7i
inches ; the cube of this is 4^, which multiplied by 1
produces 420.

In this case it happens that the horses' power, and the
revolutions per minute, are the same numbeTf therefore we
with little trouble find the multiplier.

EXAMPLE n.
PVom a journal of 4 inches diameter, where the horses'

* See Art 233.

CHAP. IV.3 ON THE SHAFTS OP MILLS. 215

is 1 ii, and the revolutioDs per minute 4>8 ; to find
c multiplier.

Now let us suppose both numbers the same, that ie, 12
irses* power, and 12 revolutions.

Here it is evident that there will be four times the
brought on the journal. Its actual diameter was 4

Then the cube of 4 is 64,

C-t multiplied by 4 is 256 inches,

The cube root of which is 6.35,
riiieli is the diameter which the journal ought to have, to
be in proportion to the velocity.

The cube of 6'35 is 26,
iliich is the multiplier required.

11 is to be observed, that in the latter case a much
nailer steam engine is employed than in the former, and,
lerefore, for reasons already given, (Art. 228,) has less

i brought upon the joumaL This accounts for the
ultiplier being less.

We shall now give an example of the application of a inul-
Her; let us take that found in the ease Example I., viz.
80, and see what size of the journal it would give in the
Be Example II., which is an engine of 12 horses' power
id journal making 48 revolutions per minute.

12 divided by 48, equal to -2.5, then multiplied by 420,
B a quotient of 105, that is, the strength of the journals
it be as 420 to 105; but the cube root of 420 is 7^,

dthe cube root of 105 is 4f, which points out that the
TOnials 74 and 4| arc proportioned to their respective
In like manner the following table is calculated;
Hie multiplier being 420.

fSl. Description of the Table of' Journals, proportionate
to D, havin-g 420 ns a multiplier.
Column 1 contains letters to distinguish the cases in

216

ON THE SHAFTS OF MILLS. [CHAP. 1?.

which D is the some as Example L (Art. 230,) andE,
Example II. of the same Art.

Column 2 the horses' power.

Column 3 the revolutions of the journal per minute.

Column 4 the product of the horses' power divided by
the revolutions of the shaft.

Column 5 contams the proportionate strain on each
journal, represented in whole numbers, which are found by
multiplying the product in column 4, by 420, as a molti-
plier.

Column 6 diameters of journals, as really executed in
several steam engines.

TABLE OF JOURNALS

Proportionate to d, having 420 as a multiplier.

1

A
B
D
E
F

2

3

4

5

6

Niimhers of
horses* power.

Revolutions

of journal per

minute.

Product of
power divided
DV the rev. of

the journal.

Proportionate
strain on
journal.

Diameters of

journals from

observation.

32
32
50
12
9

58
19
50
48
55

0-55

1-67

1-0

0-25

0-16

231
701
420
105
67

9;

4

4

OBSERVATIONS.

232. We have ab-eady observed, (Art. 228,) that, be-
sides torsion, the journals of fly-wheel shafts have consider-
able lateral and other stress, arising from the weight and
swagging of their fly wheels, and therefore they ought to
be made stronger than shafts, in other situations. The
multiplier 420, therefore, which we have used in the table.

IV.] ON THE SHAFTS OF MILLS. 217

eld give diameters too great, for some other parts of
ihinerj'. A journal, for instance, subject to torsion,
mediately connected with a water-wheel, has, from the
[ht of the wheel and other causes, considerable lateral
» ; but not 80 much as that of a steam engine. The
Bnal may, therefore, be considerably smaller than would
required for a steam engine fly-wheel shaft subject to
same degree of torsion*.

Again, a secondary shaft driven from a steam engine, a
T-wheel, or horse-gin, by means of wheels, has in gene-
very little lateral stress, compared with the two cases
stated ; and may therefore have a journal smaller than
er, when the degree of torsion is the same.
SS3. For these reasons the three following multipliers
U probably approach near the truth ; that is, for journals
steam engine fly-wheel shafts (where the power is mo-

ite) 400

Journals in immediate connexion with water-wheels t,

other heavy work 300

Journals for the ordinar>' kind of internal mill-work 100

Suppo8e B, in the table, (Art. 231,) 1-67 multiplied by
0, is 668, the cube root of which is 8'74l6 inches, dia-

of journal.

SS*, When the resistance of a journal is equal to the
'ittbg stress, the strain not being sufficient to produce
nnanent derangement in the material, the cube of the
meter of the journal will be equal to 3*78 times the

It wems B better method to use a mode of c&Iculadon which includes
dectof lateToI streBs; see Art. 333.

The reader will jtletme to obaerre, that, when Bnchamui uses the word
ol here, he suppows it subject to torsion. Where there is latenl
KK only, and no tornon, he invariably uses the word gudgeon.

218 ON THE SHAFTS OF MILLS. [CHAP. IT.

number of horses* power divided by the number of revo-
lutions in a second*.

But some allowance must be made for wear, and if this
be made so that the journal shall have sufficient strength
when it is worn down one-sixth of its diameter, the nonher
8*78 should be made 6*01 ; or with sufficient accuracy &

If the number of revolutions in a minute be employed
instead of those in a second, the constant multiplier, 6|
must be multiplied by 60 ; and therefore the constant mul-
tiplier will become 360 ; and a less number ought not in
any case to be employed, because there will always be some
lateral stress in addition to the twisting stress. The re-
sistance of a journal, or its diameter as regards the twist-
ing strain, may be always calculated by the following rules.

Rule. — If n be the number of revolutions in a minute,

N.d?
and d the diameter of the journal in inches, then Qkf;=thfi

number of horses' power the journal is sufficient to resist.
Rule. — If n be the number of revolutions per minute,
and H the number of horses' power moving the train of

machinery, then 7*12 x (— ) = rf, the diameter of the jour-
nal in inches.

EXAMPLE.

Let it be required to find the diameter of a journal for

case B in the table of journals, Art. 231. ; then we have h

equal 32 horses, and the number of revolutions n equal 19;

32
therefore Yq = 1*68421. The cube root of 1-68421 is

found to be 1-19; and 7*12 x 1-19 = 8-4728 inches, the
diameter of the journal, or nearly 8^ inches.

* The reaaon of this rule will be given in treating of the resistance d
shafU.

CHAP. IV.] ON THE SHAFTS OF MILLS. 219

235. But the effect of lateral stress ought always to be
considered, and we have found the strength of a gudgeon
to be 0"6(w/)i = rf; where the stress is wholly lateral,
(Art 212.) Or, 0-216 w/ = flP. And the strength of a
journal, where the stress is altogether twisting, to be

= flP ; therefore, since journals bear both kinds of

stress, we have as a general

Rule. — (0'2l6 w/ -f ) = rf, the diameter in

inches. Where w is the lateral stress upon the journal in
cwts., / the length of the journal in inches, h the number
of horses' power moving the train of machinery, and n the
number of revolutions of the journal per minute.

SECTION III.

236. When the diameter of a journal and its revolutions
per minute, are given, in order to find the horsed power to
which it is equal ; we must invert the preceding operation,
and convert the multiplier into a divisor.

Rule. — Cube the diameter of the journal, divide the
cube by the divisor. The quotient multiplied by the re-
volutions per minute, gives the horses' power, to which
the journal is equal .

EXAMPLE I.— F IN THE TABLE.

Suppose the journal of a steam engine to be 4 inches
diameter, making 55 revolutions per minute, then we shall
use 400 as a divisor.

The cube of 4 is 64, divided by 400 equal to 0*1 6, then
this quotient (0*16) multiplied by 55, gives 8*8 the horses'
power, to which the journal is equal

9S0 ON THE SHAFTS OF MILLS. [CHAP. T

EXAMPLE n.

Suppose again, the same size of a journal, connecti
with heavy machinery, then we must use 200 as a diviso:

The cube of 4 = 64 -r- 200 = -32 x 55 = lyG horsi
power.

EXAMPLE m.

We shall take the same journal for internal work, of tl
ordinary kind, and use 100 as a divisor.

The cube of 4 = 64 -r 100 = -64 x 55 = 35-2 horsa
power.

CHAPTER V.

SECTION I.

ON THE BODIES OF SHAFTS.

237. From what is stated in the preceding part of this
Essay, the millwright will he enahled to approach suffi-
ciently near to the truth, for all practical purposes, in pro-
portioning gudgeons and journals to the stress which they
have to sustain. Taking this for granted, it may he pro-
per for us next to consider what relates to the bodies of
shafts, or those parts which lie hetween the gudgeons or
journals. In this part of our inquiry, we may derive
assistance from the principles which have heen applied hy
writers on mechanics, to the stress of timher and other
materials*. The generality of writers on mechanics, how-
ever, as Dr. Young justly observes, (vol. i. page 136,)
have confined their attention to strength (resistance to
fracture) alone, although there be other very important
properties, which required their consideration. The most
usual as well as the most important effect, produced by the
application of force is flexure : (p. 138, ibid.) stiffness
therefore, as well as strength^ ought to be considered in
determining the form, as well as the quantity of materials,
for any mechanical purpose, more particularly that of a
Bhaffc in mill- work, which in theory may be considered as
an inflednhle straight line. The practical reader should
attend to the distinction between stiffness and strength.

* Few operatiTe mechanics hare a distinct notion of the difference be-
tween ilnngtk and tHfiteis.

Q2fi ON THE SHAFTS OF MILLS. [CHAP.T.

Stiffiiess is that property which resists ^jrt^r^ or ben^ng.

Strength that which resists fracture or hreaking.

The consideration of their limits, may make this plainer.

The limit of stiffness \& flexure ; the limit of strength is
Jracture.

The stiffness of a heam follows laws very different from
those which determine its strength ; these laws we shall
presently consider, and endeavour to shew their application
to practice, with regard to some cases of shafts. But
although those laws may throw considerahle light on the
suhject, yet it must he confessed, that there are many cases
in practice, in which it is very difficult, if not impossible,
to apply them ; for it is very often difficult to estimate
what may he the amount of the lateral pressure on shafts,
arising not only from their own weight and that of the
wheels, upon them, hut also from the thrust, proceeding
from the action of the toothed wheels, and other extnu
neous causes. In cases of this nature, where calculation
fails, much must he done, hy what Smeaton calls feeling*^
which will direct the experienced millwright to make a due
allowance for whatever accidental strain may be Ukelv to
occur.

We shall now proceed to state and apply some of the
laws, respecting stiffness and strength, with regard to
force, applied transversely.

OF LATERAL STIFFNESS AND LATERAL STRENGTH.

238. The " stiffness of any substance, is measured by the
force required to cause it to recede, through a given small
space, in the direction of the force," (Young's Nat. Phil-
vol. i. p. 139.) Its transverse strength is measured by the
pressure required to produce its fracture, or, in other
words, to break it.

* Smeaton 8 Account of Eddjstone Lighthouse, p. 136.

OX THE SHAFTS Of MILLS.

TROPOSITION 11.

Any beams of' equal length have tfieir lateral
'M, [/o bear a load at any point in t/ie length,'] as
breadth and cube of the depths (Young's Nat. Phil,
vol i. p. 139, or ii. art. 333,) and have their lateral
ttretigth, atf the breadth and square of Die depth*, (Gre-
vol. i. art. 169, cor. 1. Emerson, prop. 67.)
IU3, if a square beam measure twice as mucli, on the
le, as another of equal length, it would be sixteen
times as stiff. In other words, it will sustain sixteen times
le weight, without bending.

But, if a square beam be twice aa much on the side as
ither, both being the same length, it will be only eight
B stronger.

!ence we see, that when beams or shafts are of equal
ths, their stiffness, by any increase of thickness, in-
Bes in a higher proportion than their strength.

EXAMPLE 1.

f a beam or shaft be four inches square throughout,

another five inches, both of equal lengths ; what is

■ comparative stiffness f

The cube of % is 6+,

(Mix 4 =256,

The cube of 5, is 1^25,

125 X. 5 =625,

That is, the shaft of live inches is nearly two and a half
stiffer than that of four; in other words, it would
[lire nearly two and a half times the weight to Iwud it.

Ihat h, as tlie cube of the «de of a square beam, sjid in general tlic

Kof any beams wbose sections are similar, as the cube of ibc
diametere of tbc seclioiis.

224 ON THE SHAFTS OF MILLS. [CHAP. V.

EXAMPLE II.

If a beam or shaft be four inches square throughout, and
another five inches, both of equal lengths ; what is their
comparative strength ?
The cube of 4, is 64,
The cube of 5, is 125.
That is, the five inches shaft is nearly twice as strong as
that of four inches ; in other words, it would require nearly
double the weight to break it.

PROPOSITION III.

240. Any beams of different lengths have their stiffness
[to bear a load at any point in the length^ directly as the
breadth and the cube of the depth, and inversely as the
cube of the length, (Young^s Nat. Phil. ii. art. 333,) and
have their strength directly a^ the breadth, and as the
square of the depth, and inversely as the length*, (Young,
vol. ii. art. 335.)

Thus, if a beam be twice as long as another, of the
same breadth and depth, it will have only one eighth of
the stiffness, while it will have one half of the strength.
Hence the stiffness of shafts or beams by any increase of

• This is not strictly true in practice, for " some experiments appear to
shew, that the strength is diminished, in a proportion somewhat greater
than that in which the length is increased." (Young's Nat. Phil. vol. ii. p*
147.)

The variation is caused by the increase of strain which takes place wbc^
the flexure is considerable, (see Elementary Principles of Carpentry, Art
18,) and some decrease of cohesive power when the natural arrangement
of the particles of a body is disturbed more than in a certain degree; but
these causes are insensible in a practical point of view, because we can
never allow the stress to produce so much flexure, nor die strain to be 00
near to fracture, as to make it necessary to allow for such circumstances.

CHAP, v.] ON THE SHAFTS OF MILLS. 225

their length, decrease in a much higher proportion than
that of their strength.

EXAMPLE I.

Suppose a beam or shaft, four feet long and four inches
square throughout, and another eight feet long and seven
inches square ; what is their comparative stifihess ?
The cube of 4 feet, is 64,
The cube of 8 feet, is 512,
512 divided by 64, is equal to 8, that is, when we double
the length, we decrease the stifihess eight times.

The cube of 4 inches is 64, which multiplied by 4 is
equal to 256, a number representing the stifihess of the
four inch shaft.

The cube of 7 inches is 343, multiplied by 7 is equal to
£401, divided by 8 is equal to 300*1, then as 256 is to
300*1, so is the stifihess of the shaft of four inches to that
of a shaft of seven inches.

EXAMPLE n.

Suppose a beam or shaft,^ four feet long and three inches
square, and another eight feet long and seven inches
square, what is their comparative strength ?

The cube of 4 is 64, which represents the strength of
the four inch shaft.

The cube of 7 is 343 ; but the shaft being of double
length, we must halve this sum, to find the number repre-
senting its strength, viz. 343-^2 = 171*5 divided by 100,
that is, as sixty four is to a hundred and seventy-one and
a hal^ 80 is the strength of the short shaft to that of the
long one. Thus the shaft of seven inches, eight feet long,
has nearly two and six tenths times the strength, of the
firar inch four feet long.

2^ ON THE SHAFTS OF MILLS. [CHAP. V.

PROPOSITION IV.

241. Supposing a tubcy indefinitely thin^ to be expanded
into a similar tube of a greater diameter^ but of equd
lengthsy the quantity of matter remaining the same^ tiie
STIFFNESS will be increa^edy in the ratio * of the square of
the diameter^ and the strength in the ratio of the dia-
meter f.

Thus, if the one tube be double the diameter of the
other, it will have four times its stiffness^ but only double
the strength.

Hence, hollow cylinders of equal lengths, by any in-
.crease of diameter, increase in stiffness^ in a much higher
proportion than in strength.

EXAMPLE I.

Suppose two thin narrow cylindrical cast iron shafts, of
equal lengths and weights, the one of one foot diameter,
and the other three feet diameter, required their compara-
tive stiffness ?

The square of 1 is 1,

The square of 3 is 9,
that is, the shaft of three feet diameter, is nine times stiffer
than that of one foot.

* Ratio, that is, proportion.

t This proposition is taken from Dr. Young's Nat. Phil. vol. ii. wl. 339,
where it is followed by this essential limitation. " When a beam of finite
thickness is made hollow, retaining the same quantity of matter, the
strength is increased in a ratio somewhat greater than that of the diameter,
because the tension of the internal fibres at the instant of breaking is m-
creased." Dr. Young has given the correct rule for estimating the strengtb
and stiffness of a hollow cyHnder, at p. 84, (Nat. Phil. vol. ii.) " The
strength of a tube may be found by deducting from the strength of the
whole cylinder that of the part removed, reduced in the ratio of tlie dia-
meters." And observes, that " the strength is in this case in the sune
ratio as the stiffness."

CHAP, v.] ON THE SHAFTS OF MILLS. 227

EXAMPLE II.

Suppose the same shafts, as in example first, required
their comparative strength ?

Diameter one foot,

Diameter three feet,
that is, the three feet shaft is just three times stronger
than that of one foot diameter.

242. In these examples, we have supposed the weight of
the shafts equal, that is, the area of their ends to be equal,
but the strength of any of them would be increased in
proportion to their weight, or the areas of their ends and
diameters, conjointly. (Gregory, vol. i. art. 172, cor. 3.)

Thus, suppose two shafts of equal length and diameter,
the one double the weight of the other, it will be double
the strength*.

243. Professor Robison justly observes, " that this
property of hollow tubes is accompanied also with greater
stifihess, and the superiority in strength and stifiness is so
much the greater, as the surrounding shell is thinner in
proportion to its diameter. Here we see the admirable
wisdom of the Author of nature in forming the bones of
animal limbs hollow. The bones of the arms and legs
have to perform the office of levers, and are thus opposed
to very great transverse strains. By this form they be-
come incomparably stronger and stiffer, and give more
room for the insertion of muscles, while they are lighter
and therefore more agile ; and the same wisdom has made
use of this hollow for other valuable purposes of the ani-
mal economy. In like manner, the quills in the wings of
birds acquire by their thinness the very great strength
which is necessary, while they are so light as to give suffi-
cient buoyancy to the animal, in the rare medium in which

* Sec note to Art. 236.

SS8 ON THE SHAFTS OF MILLS. [cHAP. V.

it must live and fly about The stalks of many plants,
such as all the grasses, and many reeds, are in like man-
ner hollow, and thus possess an extraordinary strength.''
(Ency. Brit, article Strength.)

• Long before this eminent philosopher, the celebrated Ga-
lileo made similar observations, and goes on to say that **if
a wheat straw, which supports an ear that is heavier than
the' whole stalk, were made of the same quantity of mat-
ter but solid, it would bend or break with far greater ease
than it now does. And with the same reason art has ob-
served and experience confirmed, that a hollow cane or
tube of wood or metal, is much stronger and more finn
than if, while it continued of the same weight and length,
it were solid, as it would then of consequence be not so
thick.

It may be proper now to consider the effects called
stressj which are produced on beams or shafts lying hori-
zontally by weights or pressures brought on various parts
of them.

SECTION II.

OP LATERAL STRESS.

244. The stress or strain * are terms used to express
the force which is excited in any body tending to break
it The meaning of the term stress may perhaps be more
clearly understood by contrasting it with the term
strength.

Strength^ as we have already observed, is the property
which resists fracture.

Stress is that which has the tendency to produce Jrac-
ture ; and lateral stress is that particular application of it,
which has the tendency to break a body across.

* Strain is the effect of stress : it is the derangement from the uatnral
state which is caused hv stress.

ON THE SHAFTS OF MILLS.

^9

PROPOSITION V.

77(« stress on a beam arising from one weight
upon it, is proportional to the rectangle of the parts
the beamy and is greatest when the load is laid on the

of the beam. (Ency. Brit. art. Roof, § 19.)
What is meant by the expression rectangle of the parts,
is the product of parts multiplied into each other. Thus,
fcr example ; if a beam be ten feet long, and the weight
Dg two feet from one end, the parts are 2 and 8, which
UtipUed together, would be equal to KJ; but supposing
iveight were hung in the middle, the parts are 5 and 5,
ich multiplied together would produce 2.5.
06. The ends of beams having the whole weight to sup-
rt, the end which is nearest the weight has to support the
utest proportion of it, in the inverse proportion of the
(liatance of the weight from the end. This will be easily
understood from the properties of the lever. For, suppose
As beam instead of being supported by two props or walls,
in Fig. 9. No. 1, to be hung from each end by a rope,
b Fig. 9, No. % it is plain that the beam would receive
Bame support, and suffer the same stress, as if Iving on
pe or walls ; now suppose the weights a and b, to ba-
le the weight w, then a and b, taken together, must be
ll to w, but A must be greater than b in proportion as
1 near to it.

St?" Hence when any beams or shafts are loaded exactly
he middle, each of the ends of the beams or gudgeons
the shafts has half the weight to support, and when the
ight is nearer one end, the end or gudgeon to which it
tearestt has the stress in the inverse proportion of the
In this last case, therefore, the one gudgeon
^t be smaller than the other.
(S. " We may (Uways consider the weight which is

2S0 ON THE SHAFTS OF MILLS. [CHAP. V.

uniformly diffused over any part of a beam as united in
the middle of that part, and if the load is not uniformly
diffiised, we may suppose it united at its centre of gravity."
(Ency. Brit article Roof, § 20.)*

249. It is evidently of importance that a beam or shaft
should in every part be able to resist the strain excited in
that 'psLTt. ** It should therefore be equally strong, because
the piece will nevertheless break where it is not stronger
throughout, and it is useless to make it stronger (relatively
to its strain) in any part, or it will nevertheless equally
foil in the part that is too weak." (Ibid.)

250. From what we have said respecting lateral stress,
it is evident than when a beam lying between two props is
loaded at some intermediate part, that part has to sastain
more stress than the rest. In order to resist this strain,
therefore, and to render the beam equally strong through-
out, it should have its section enlarged at the place of
greatest stress, and hence shafts subject to lateral stress
should swell in the middle, and it will be found that whffl
each section is made proportional to the stress it has to
sustain, that the sides of the shaft will form curves.

251. When the transverse sections of a beam are all
similar, such as circles, squares, or polygons, and the
weight is laid on one place, in order to make it equally
strong throughout its length, the curve of the sides of the
beam becomes what mathematicians call a cubical parabohu
(Ency. Brit. Strength of Materials, 87.) But when the
weight is uniformly diffused all over the beam, the sides of

* When the weight is uniformly diflfused, the stress is the greatest at the
middle of the length, and is equal to half the weight collected in the middk^
But the flexure in the middle produced hy a weight which is uniformlj
diffused, is the same as when five eighths of the load is collected in the
middle of the length. (See art. Carpentry, Supplement to Encydopsdia
Brit. 1817, Prop. F., or Barlow's Essay on the Strength of Timher, p.
117.)

CHAP, y.3 ON THE SHAFTS OF MILLS. 231

the beam must be a different parabolic curve called a
semicubical parabola*.

We come now to examine some of the laws respecting
twisting or torsion.

SECTION in.

OF TOBSION.

PROPOSITION VI.

252. In general the strength of a cylinder or solid axle
hjf which it resists being wrenched asunder hy twisting is
as the cube of its diameter. (Ency. Brit art. Strength of
Materials, 123.)

Thus, if a solid cylinder be double the diameter of
another, it would require eight times the force to wrench
it asunder.

OF HOLLOW AXLES.

253. Hollow axles are stronger to resist twisting than
solid ones containing the same quantity of matter. For if
a hole be bored out of an axle of half its diameter, this re-
duces its weight one-fourth^ (because circles are to one
another as the squares of their diameters,) but the strength
of solid cylinders being as the cubes of their diameters, the
part taken out by boring had only the eighth part of the
strength of the whole cylinder, and therefore when taken
out would reduce the strength of the whole one eighth.

Thus, let the external diameter of the hollow axle be
fiye inches, and that of the hollow part of it four inches,
then the diameter of another cylinder made solid, having
ih6 same quantity of metal with the tube, is three inches.

* ^ Tbe poiabola is a conic section, arising from a cone being cat by a
phnn panUel to one of its sides, or parallel to a plane that touches one
wim of the eone.*

292 ox THE SHAFTS OF MILLS. [CHAP.T.

For 5 multiplied by 5 is equal to 25
4 multiplied by 4 is equal to 16

Difference 9
The square root of 9 is 3. The strength of the solid
cylinder of fire inches diameter may be expressed by the
cube of 5, or 123. Of this the internal part, four inches
diameter, exerts 64, that is, the cube of 4 ; therefore the
strength of the tube is 64, subtracted from 125, is equal to
61, but the strength of the solid axle of the same quantity
of matter, and three inches diameter is expressed b? the
cube of 3 or 27, which is not half of that of the tube.
(Ency. Brit art. Strength of Materials, 124.)*

254. The superiority of strength of hollow tubes over
solid cylinders is much greater in resisting torsion thin
transTcrse or lateral stress. We have seen above, that the
strength to resist torsion of the tube was to that of the
cvlinder as sixtv-one is to twentv-seven ; but Professor
Robison estimates, that their strength to resist transverse
strain is onlv as sixtv-one is to thirtv-two and a half nearly
— and if we calculate according to Dr. Gregor}^*s corollary,
Vol. I. page 109, (see Art. 237 of this Essay,) the result
will be still more in favour of strength to resist torsion;
for bv the last mode of calculation the tube would be to the
cylinder only as fort}-five is to thirty-six ; but the Pro-
fessor's mode of calculation, though less simple, is probably
more accurate than that above alluded tot.

* These calculations are founded on the erroneous supposition that the

tension is equal in every part of the section ; and consequently they are

widely distant from the truth. (See note to Art 254.)

t It has been shewn that the resistance of a cvlinder to torsion k

124'8</* =: R w ; where w is the stress in Ihs. and r the leverage in feet it

acts with ; d being the diameter of the cylinder in inches. (Essay on Cast

Iron, Art. 227.) And when the straining force is considered to act at the

d
surface of the shaft ^ = 1 2 r ; and therefore in this case 20*8 d' zz w.

w. v.]

ON Tiir; sirAiTs of mills.

2SS

Profiwsor Robison mentions (in Enc;\ Brit. art.

;ngth of Materials, § 128) that "when the matter of
B axle is of the most simple texture, such as that of
itak, we do not conceive that the length of the axle has
f influence on the fracture. It is otherwise if it be a
reus texture, like timber ; the fibres are bent before
teking, being twisted into spirals like a corkscrew. The

ph of the axle has somewhat of the influence of a lever
this case, and it is easier wrenched asunder it" long. —
BCordingly we have found it so ; but we have not been

e to reduce this influence to calculation*.

855. All shafts are exposed to lateral stress and twist-

, but it will commonly happen that one of these forces

'. vastly exceed the other ; and consequently, we need

r ad^t the shaft to the resistance of the greater power;

' dte same reaaoning il may be proved tliat, in a hollow cylinder, where
the exterior diameter, and jid the diameter of the hollow part,
!tr' (I — «') = RW ; wlien the force w acU with the leverage R in
hat when the force is applied at the surface of the shaft, 20'8 n'

when the strain is at the surface of the shaft, the reaiatance of a
qrlinder is to that of a hollow one as rf' : »' (1 — »*); and taking the
I Bumple, wtiich our author has quoted from Professor Robison, we
I rf = 3, I) = 5, and n D =: 4, or n = -8, the ratio is 9 : 25
- t09e,) or as 9 : 14-76, when the stress is reduced to the surface of
liafl or cylinder.

U the ratio is (/' : d' ( I — n'] when the lever^e is constant ; that is,
T3-8, instead of 27 to 61. 1 1 must however bo observed, that these
• obt^n only in tiie particular case for which they are here calculated,
fluexample where the leverage is constant the general ratio is (I — ti-}J
- n') ; : strength of a solid cylinder : that of a tube containing the
quantity of matter. And it may be very easily proved that the lateral

in the same rabo.
In wood the lateral adhesion of the fibres being much inferior to their
t cabenon, it b much ivcakcr to rcRist torsion ; the fibres sliditig one
I uotiier very considerably before the rupture takes place. But this
I die lenph no senmble inHuence, when the strain is kept within proper

234 ON THE SHAFTS OF MILLS. [cHAP. V.

but in the first place we must be able to ascertain when
the one or the other must be calculated for.

The resistance of a shaft to a lateral stress must ob-
viously be measured by its stiffiiess to resist flexure, be-
cause it would otherwise play too much on its brasses,
couplings, &c, and occasion irr^ular action in the maclm

From a comparison of shafts in use, it appears, that
about the yi^th part of an inch, for each foot in length, is
the quantity of flexure that may be allowed without sen-
sibly affecting the regularity of its motion.

In a cast iron shaft, supposing it to be a solid cylinder,
if w be the stress when referred to the middle of the length
of the shaft ; / ^ the length in feet, and d = the diameter

in inches : the deflexion in the middle beinsr of an

^ 100

inch, then ^r^ = rf*. (Essay on Cast Iron, Art 218.) or

250 rf*
w = -^.

740h
256. Now if be the twisting power collected at the

surface of the shaft, where v is the velocity of that surface
in feet per second, and h the greatest number of horses*
power that is necessary to work the train of machinery to
which the shaft belongs, (see note to Art, 106,) we have

740 H
124-8cP =^-^ — , (Essay on Cast Iron, Art. 227,) hut the

velocity of the surface of the shaft is equal to its cir-
cumference in feet multiplied by the nimiber of revolutions
in a second. And, if this number of revolutions be n,

then V = 9 ^'^d the equation between the stress and

^^

^ 3-78 H
strain, reduces to cr = — - — •

N

•v.]

ON THE SHAFTS Ol- MILLS,

therefore appearB, that when ( ) = ( o^-j ) t"^

in &OTn torsion will be equal to that from lateral pres-

., 147'2 /H J , , , , ,,

whence if .)- x ( — 1 be less than w, the diameter

the shaft must he determined by the rule for lateral
, if it be greater than w, calculate the diameter by
rule for torsion.

'or the advantage of those who are not much versed in
inlation, it may be of use to remark, that in common
t when the stress in lbs, multiplied by the square of
length in feet, is less than 3,000 ; the twisting strain
I be the greater; and the reverse.
1 a series of lyingshafts, as in Fig. 1, Plate V. or Fig.
Plate VIII. it is necessary to make the journals equal
the twisting strain with the addition of tlie necessary
iwance for wear ; hence it will often be necessary to
1st the shafts to both strains : but in all instances where

lateral stress is less than — „ =*(-)' ^^^ bodies of the
i need not be greater than the journals.
fa other cases, as in Fig. 1'2, Plate III. the principal

Iting strain will be in the part of the shaft between the

(da A and b, while the twisting strain on the gudgeons

I be equal only to the friction.

Having thus stated these laws, as far as seems to be

Ksgar}' for the purpose of our present inquiry, it may be
Dper next to endeavour to apply them more particularly
pactiee, with regard to the proportion of shafts.

SECTION IV.

257. It was already observed, (Art. 211.) that the gud^
or journals having to support the whole stress of the

236 ON THE SHAFTS OF MILLS. [CHAP.V.

shafts to which they belong, their diameters being deter-
mined, serve to guide in determining the proportions of the
whole shaft. They are subject to wear, which the body
seldom is. They ought, therefore, to be sufficiently large
to allow for that wear. It frequently happens, that a shaft
has no lateral pressure excepting that which arises from
its own weight, for instance in a line of coupled horizontal
shafts conveying motion to a distance. In the case of
vertical shafts also there is often little or no lateral stress.
In such cases, when solid cast iron shafts are of moderate
lengths, it is found from experience, that making them
square of the same size throughout *, between the joumalsi
and the measuring a little more on the side than their
diameters, gives them sufficient stiffiiess. Even where there
is considerable lateral stress when the shafts are but short,
making them in this manner is found to give sufficient
stifihess. This square form, in many cases, affords great
convenience for hanging or fixing wheels, pulleys, &C.,
upon them.

258. The gudgeons of water-wheels are often so near
the wheel, that the stress is, in a great measure, taken off
the shaft. Hence some water-wheels are made without
shafts, the gudgeons being fixed to the arms at each side
of the wheel. The sole of the wheel, in these cases, may
be considered as a large hollow axlef,

* See note to Art. 193.

t For a wheel of considerable breadth it will in many instances be an ad-
vantage to employ a comparatively small axis, and to render the wheel firm
by arms and braces. But if the axis be dispensed with, the same degree of
firmness will be gained only by a greater quantity of matter.

The author seems to have had in view the ingenious method of constract-
ing a water-wheel executed by Mr. Bums, at Cartside; (Dr. BrenW*
edition of Ferguson's Lectures, Vol. II. p. 55 ;) but the addition of an ftxw
would be of great use if it were only to prevent the racking strain of the
gudgeons upon the cross arms.

iP. v.]

ON THE SHAFTS OF .VHLLS.

237

59- 1st. Let us now suppose a square cast iron shaft re-
to be made eight feet long', with gudgeons four
les diameter, having considerable lateral stress iu the
die — were the shaft of equal size throughout, it would
lently be weakest in the middle. Its section, in that

I, ought therefore to be considered. We have already
1 (Art. 257) that square shafts, when short, are suffl-
tly stiff, if made as large as the gudgeons ; this shaft

therefore be four inches square at the end. Supposing
section in the middle enlarged to five inches, swelling
I a regular curve (Art. 187) from each end (by Prop.

then

The cube of 4 is fri ;
The cube of 5 is 1'25 ;

is, nearly double. — Therefore, the strength of shafts

; inversely as their length, (Prop. III.,) the section, as
R increased, would make the eight feet shaft nearly as
ng as a shaft subject to the same stress, only four feet
[ and four inches in the middle.

iCt us next see what would be the stiffness of this shaft,
thus swelled in the middle to five inches. Then (hy
fposition II.) the cube of 4 is 64, which, multiplied by
1 equal to ^G. The cube of 5 is 125, multiplied hy 5,
equal to 625 ; therefore, the stiffness being inversely as
cube of the lengths —

The cube root of 256 is 6-3.
The cube root of 625 is 8*5.

refore, the section as thus increased, would make the
It foet shaft about as stiff as one of six feet, subject to
same stress, and four inches throughout.
Bo. Sod. Let us next suppose the point of greatest lateral
sore to be two feet from one end.

238 ON THE SHAFTS OP MILLS. [CHAP. Y.

From the properties of the leyer, the gudgeon next the
point of greatest pressure has three-fourths of the whole to
sustain.

Now the cube of the diameter of the gudgeon is 64; miil«
tiplied by 2, is equal to 128, which represents the strength
of both journals taken together.

3.4ths of 128 is equal to 96.

The cube root of 96 is 4*578, the diameter which the
largest gudgeon ought to have.
l-4th of 128 is 32.

The cube root of 32 is equal to 3*174s the diametor
which the smallest gudgeon ought to have, or that farthest
from the point of pressure.

Haying thus obtained the diameters of the two gudgeons,
each end of the shaft may be made square, equal to its
respective gudgeon. (Art* 257.)

The stress on the shaft being as the rectangle of die
parts, (Prop. V.); in this case it is less than the former, as
twelve is to sixteen — ^for four feet multiplied by four fieet is
equal to 16, which expresses the stress when in the middle
— 6 feet multiplied by two feet is equal to 12, which ex-
presses the stress when two feet from one end.

This shaft would therefore be as strong as the fonner,
if its greatest section were reduced to 4*54.

For the cube of 5 is 125, as 16 : 12 :: 125 : 9375, the
cube of which is 4'54.

It would be as stiff as the former, if its greatest section
were reduced to about 4*65 :

For the cube of 5 is equal to 125; multiplied by 5 is
equal to 625, which expresses the stiffiiess of the former
shaft, as 16 : 12 :: 625 : 468*75, which expresses the
required stiffiiess; and the cube of 4*65 is equal to
100*5444625, which, multiplied by 4*654, is equal to
467*934, which is nearly equal to the required sti&ess.

t.v.]

ON THE SHAFTS OF MILLS.

•iSii

examining shafts in this way, the millwright, aecord-

;to the nature of the case, will be enahled to judge how

they should swell at the place oi greatest sfi-ess.

. As far as regards the strength these examinations

apply ; but not to stiffness ; because the rule for the

itifiness supposes the shaft to be everywhere of the same

section. And it may be here remarked that a cylinder is

iiiffer than any figure which can be inscribed within it ;

hence there is not that advantage in diminishing a shaft

towards the points of support which many people have

led.

ipposing the reader to have considered the principles

estimating the pressure on a shaft (Art. ^7) with

sufficient attention to be able to ascertain the greatest stress

upon one with as much precision as is necessary in these

inlculations ; we shall here give rules for computing the

diameters of different forms of cast iron shafts to resist

leral stress.

CVLINDBICAL SHAPT8 OP CAST IRON.

fthe stress be in the middle, and equal to w cwts. the
ire in the middle not to exceed as many hundredths of
nch as the shaft is feet in length j then by (Art. 255)
')i X H = d the diameter in inches. That is, the fourth
of half the stress in cwts. multiplied by the square
of the length in feet is equal to the diameter in
les.

62. If a cylindrical shaft has no other lateral stress to
lin than its own weight, then by making the proper
titutions, the rule becomes V'OOy/* = d, the diameter
iches.

iliat is, multiply the cube of the length in feet by '007 ;
the square root of this product is the diameter in

PS.

r2

240

ON THE SHAFTS OF MILLS. [^CHAP. ¥

The rule now stated may be easily applied to any case
with the advantage of both accuracy and simplidtyi be-
cause it enables us in every instance to include the effect
of the weight of the shaft itself ; when it is computed in &
tabular form. Let the stress, supposed to be at the mid-
dle, be fi times the weight of the shaft, then ^Ol^Pn^i
in inches.

263. Table of Shc^ of Cast Iron to resist kUerai
Pressure.

Length
in feet

Diameter
in inches.

Diameter
in inches.

Diameter
in inches.

Diameter
in inches.

Diameter
in inches.

2

•237

•31

•44

•54

•62

4

•67

•88

124

1-5

1-76

6

1-23

1-61

228

2^79

322

8

1-9

2-48

3^51

430

4-96

10

2-65

3^47

4-9

6-0

6-93

12

3-48

4-55

6-44

7-89

910

14

4^38

3-74

8-12

9-94

1V4S

16

5-36

7-01

9-92

1215

1402

Own weight
only.

Stress equal

to its own

weight, or

11=1.

Stress dou-
ble its own
weight, or
ii«2.

Stress three
times its

own weight,
or n=8.

Stress four

times its

own weight,

or fiB4.

HOLLOW CYLINDRICAL SHAFTS OP CAST IBON.

264. To compute the diameter of a shaft when it is to
be a hollow cylinder, it is necessary to assume the ratio
between the diameter of the hollow part and that of the
exterior of the shaft, in order to avoid a complicated rule.
Let D be the exterior diameter, and nd the interior one,

then w bfeinff in cwts. the rule will be (— ^ -) =Dthe

^ ^2(1 -N*/

diameter in inches. But if the shaft supports n times its

own weight, then V 1- = d, when the necessary sub-

CHAP, v.] OV THE SHAFTS OF MILLS. 241

stitutions are made. If however some eonvenient number
be fixed upon for n, the rule may be still further simplified.
And I thick one that will be well adapted for practice is
to make the hollow part six-tenths of the exterior dia-
meter. According to this proportion, the rule is V "009 1'n
= D. That is, the cube of the length in feet, multiplied
oy "009, and also by the number of times the weight of
the shaft is contained in the stress, then the square root of
this product is the diameter in inches.

The weight of the shaft in lbs. will be very nearly equal
to the square of the exterior diameter in inchee, multiplied
by l'(j times the length in feet.

^5. Table of Hollow Shajis of Cast Iron to resist
■al Stress.

diamelor

Exterior I Interii
inches, in inchei.

12-8
15-9
19-4

\e we^it of
the abfi.

[t will be necessary to refer to the tables of resistance to
sion, previous to hxing on the diameter of a shaft.

inON SUAPTS TO BESISI LITBRIL STBEM.

B66. If the diameter of the shaft be calculated for cast
], and this diameter be multiplied by "935, the product will
the diameter of a wrought iron shaft of equal stifihess.

242 ON THE SHAFTS OF MILLS. [CHAP. Y.

For in order that shafts of these metals may be equally
stiff, the stiflhess of wrought being 1*3, when that of cast
iron is unity, (see Art. l62, note,) c* x 1 = t(;* x I'Sor
c X '9S5 = w ; where c is the diameter of the cast iron
shaft, and w that of the wrought iron one.

WOODEN SHAFTS.

S67. Suppose a cast iron shaft five inches square at the
point of greatest lateral stress — Required the size it should
be when made of oak to have the same strength ?
The cube of 5 is 125.

Cast iron is four times stronger than oak, (see Banks on
Powers of Machines, &c., p. 94,) therefore 125 x 4 =
500.

The cube root of 500 is 7*93 inches.

268. Shafts should be compared by their stiffiiess, or
their resistance to torsion; but here the question is the
comparative stifl&iess. Now the stifihess of good oak is

to that of cast iron as : 1 (Essay on Cast Iron, Art.

11-2 ^ ^

Oak.) Therefore c*xl=o*x when the shafts

"^ 11-2

are of equal stiffness ; hence 1*83 c = 0 ; where c is the

diameter of the cast iron shaft, and 0 that of the oak one.

And consequently an oak shaft should be 1'8S times the

diameter of a cast iron one to be equally stiff.

And the stiffhess of square shafts being as the fourth

powers of the sides of the shafts, the side of an oak shaft

should be 1*83 times that of an iron one to be of equal

stiffness.

269. Required the size of a fir shaft to have the same
strength as one of cast iron ?

The cube of 5 is 125.
Cast iron — 5'5 times stronger than fir, (Banks,) there-

i^

r.3 ON THE SHAFTS OF MILLS. 243

135 X 5'5 = 687*5 i the cube root of 687-5 is equal

to 8-82 Am.

270- The Btiffhess of red or yellow fir is to that of cast

B — : 1 (Essay on Cast Iron, Art, Fir.) Therefore

c being the diameter of a cast iron shaft, and J" that of a

fir one to resist the same stress, c* x 1 = /* x — -, or c x

1 =^ that is l*7l6 c =J'. Whence it appears that a

shaft of yellow fir should be 1-716 times the diameter of a
cast iron one, to resist the same stress.

Shafts that are square should be in the same ratio.

SJl. In the same manner we might examine various

^kinds of cylindrical shafts, but what has been already

^Bud will suffice. In order to shew, however, that considcr-

^HUe allowance should be made for accidental stress, we

may mention the following fact which lately occurred.

272. The hollow shaft of a water-wheel, in cousequence
^^d one of the gudgeons getting loose, broke quite through
^^kar one end, although it was I6 inches diameter, and two
^Bches thick in the shell. (See Art. 2l6, 1st table of gud-
^^Bons.) The gudgeons of this shaft were not secured by
^^■nches.

273. Hollow cylindrical shafts are often made of equal
size throughout, in order to get large flanches, the better
to secure the gudgeon.

(374. We saw (Art. 254) that the length of a cast iron
aH has no influence on its resistance to torsion, and we
have already said all that seems to be necessary respecting
them. The case is, however, different with regard to
wooden shafts J but we arc yet in wani of facts to reduce

BSIFTS SUBJECT TO TOBSION.

244 ox THE SHAFTS OF MILLS. [CHtf.T.

the influence wliich their length may have^ to calciditnL
As it may give some idea of this influence, we may stile
one fact which came under Buchanan's ohserratioD.

A shaft about 15 feet long, made of fir, had cast ira
journals 2iV diameter ; one of the journals broke firom fur
stress, after working about 16 years ; for a connderabb
time before it broke, the resistance was equal to 7 haati
power, making 11^ revolutions per minute. It gradulfy
wore until it broke ; when that happened, the shaft seeinel
strained to the utmost, so that it might be reckoned m
just equal in strength to the gudgeon ; the shaft was 9}
inches square.

The cube of 9| is 926,

926 divided by 55 (Art 269,) is 16-8 ;

Cube root of 16*8 is 2*6. That is, the fir shaft wodd
be equal to resist the same lateral stress as a square ctst
iron shaft 2*6 on the side.

The cross-tailed gudgeons of wooden shafts commoolj
require them to be made sufficiently large to withstand the
stress which is brought upon them ; often indeed, formerly,
they were much weakened by mortices cut through them
for inserting the arms of wheels ; this practice is now,
however, almost entirely abandoned.

275. The power of a shaft to resist torsion, has been cal-
culated in comparing the resistance to torsion with that to
lateral stress, in Art. 255, therefore it only remains to ap-
ply the equation.

OP CYLINDRICAL SHAFTS OP CAST IRON TO RESIST TORSION.

We have found the equation expressing the relation be-
tween the stress and strain to be — '- = d\ the diar

N

meter in inches. But it will be accurate enough for our
purpose to increase the constant multiplier to 4, in order

•v.]

ON THE SHAFTS OF MILLS.

24s5

aider the computation easier ; with this change, when
■volutions per minute, we have as a practical rule

= d', or(^^^")' = d. That is, divide 240

^ N '

he number of horses' power by the number of revo-
fotions per minute, and the cube root of the quotient will
be the diameter of the shaft in inches.

But the reader must remember that when, in this or any

her case, he represents the power of the first mover by a

' rtain number of horses, he must be certain to make an

ample allowance for any temporary increase of the action

of the moving power.

The following table is calculated by the above rule, and
it will be found useful to compare with the table, Art.
'JtiO.

1^76. Table of Ciflindncal Shafts of Cast Iron to resist
Torsion.

Diuneto

BEVOLOTtO

Nfl or THE

SHAITB w

A «,NUT1.

<.rtl>.lbja
iMbc

Srw.

10re».

SOrar.

SDrer.

«re,.

Wrer.

■

Hor«.-

H,^,

Hon«-

Bona-

Honci-

Honu-

L

pcncr.

pOBir.

1 '

0-17

0-33

0-66

0-99

1-33

166

1 '

056

113

2-25

3-37

4-5

5-62

1 i

^■33

266

5-33

7-99

1066

13-33

I ^

2-6

5-2

10-4

156

20-8

26-0

1 '

i-5

9-0

1600

27-0

3G0

45-0

I ''

7-15

14-3

28-G

+2'9

37-2

71-5

1 ■

10-66

21-33

42-GG

640

85-0

106-6

■ 10

20-63

41-66

83-33

125-0

166-0

20S-3

■12

3600

72-00

144-0

21G-0

288-0

360-0

■n

63-83

127-66

253-33

383-0

510-0

638-3

|.»

85-33

170-66

341-33

5120

C82-0

853-3

7. The same table will serve for hollow cylindrical
% to resist torsion, if the diameter be multiplied by
, and the diameter of the hollow part be six-tenths of

^iS ON THE SHAFTS OF MILLS. [cfiAF. V.

the exterior diameter ; for in that case ( -) = 1'05.

(See Art. 254, note.)

Indits. Indiei. hidiM. Lidies. Ii»dic&
Shafts of solid cylinders ; diameters 8- 10- 12* 14* 16*

HoDow shafts of 1 exterior diameter 8*4 10*5 12*6 14*7 16*8

equal strength; /interior diameter 5* 6*3 7*5 6*8 10*

278. This table applies to vertical shafts, but in hori-
zontal ones there is an additional stress if it be cmly from
their own weight, and much more should there be wheels
on the shaft. Where the lateral stress is small, it may be
allowed for by adding something to the diameter shown bjr
the table, or it may be calculated by the rule at the end at
this article.

Example. — Suppose a vertical shaft is to make SO revo-
lutions per minute, the power of the first mover heing
equal to 18 horses. Look in the column of horses' power
under 20 revolutions ; and opposite 18, the diameter will
be found in the first column to be 6 inches, for cast iron.

279. If the shaft is to be of wrought iron, then mul-
tiply the diameter found by the rule or the table, by 0*963.
(Art. 225, note.) Thus, in the above example, 6 x 0*963
= 5*778 inches, the diameter of a wrought iron shaft to
make 20 revolutions per minute, when the power of the
first mover is equal to 18 horses.

280. If the shaft be of oak, then the power of oak being

when that of cast iron is 1 *, (Art. 268,) and the re-

11*2

sistance to torsion being as the cube of the diameter, we
have (11*2)* = 2*238; and multiply the diameter found
by the rule or table for cast iron shafts by 2*238, and it
will be the diameter for an oak shaft. Thus in the pre-
ceding example, 6 x 2*238 = 13*428 inches for the dia-

* The relative stiffness is used instead of the relative strength, to reduce
the quantity of torsion in wooden shafts.

and

fct

CBAP. V.J ON THE SHAFTS OF MILLS. 247

meter of an oak shaft to make 20 revolutions per minute,
the first mover being equal to 18 horses.

281. When fir is to be used for a shaft, its diameter

should be ( — ) times that of a cast iron one for the

same purpose; but ( — ) = 2'06 nearly ; therefore it

should be 2'06 times the diameter of the cast iron one.

Example. — Let the moving force be equal to 7 horses,
and the number of turns per minute 11^, (see the case

died in Art. 274.,) then by the rule '-^^|^ = 146-09,

and the cube root of 146-09 is 5-267 nearly ; or practically
'3 inches should be the diameter of the shaft were it of
iron. And 2-06 x 5-3 = 10-918 inches, or nearly 11
inches for the diameter of a fir shaft. It seems that a
shaft of 9^ inches square, of fir, was found equal to the
strain ; and one 1 1 inches diameter is at least it stronger.
We have made these calculations directly from the theory
of equal cohesion, but it is so well known a fact that the
lateral cohesion of fir is vastlv inferior to the direct cohe-
sion, that in the rule for fir shafts to resist torsion, an in-
crease of diameter should be allowed by considering the
number of horses' power about ^ more than it is intended
to be ; at least, till experiment shall have given the pre-
cise effect of lateral cohesion in decreasing the force of
shafts to resist torsion.

282. If a shaft have to sustain both lateral stress and
ion, then the sum of the straining forces must be taken ;

and hence bv Art. 261, and 274, we have 1 +__ = rf'.

N 2rf

But in this equation it is difficult to calculate the value of

the diameter, as it is what algebraists call an equation of

the fourth degree. This difficulty may however be easily

■voided by considering 1 d to be 2 only, for then the error

L 28
^^kn-sii

248 ON THE SHAFTS OF MILLS. [CHAP. V.

will always be in excess, except when d is less than unity ;
and it is much better to be in excess than defect Conse-
quently we have as a practical rule ( + — ) =rf,the

diameter of the shaft in inches, when of cast iron. Where
/ is the length in feet between the bearings, h the number
of horses which are equal the power of the first moyer, n
the number of revolutions to be made by the shaft in a
minute, and w the lateral stress in cwts.

Example. — Suppose that a cylindrical shaft of cast iron
is to make 34 revolutions per minute, the power of the first
mover being 6qual to 3 horses, the length of the shaft 8
feet, and the lateral stress 3 cwts., when reduced to the

middle point, then 0^2L^^§Ji^y ^ (gl-lg + 96)*

= (117-18)* = 4-893 inches.

283. With regard to the making of patterns of cast iron
shafts, the reader is referred to what has been said in the
first Essay, relative to the making of patterns for cast iron
wheels, which is, in a great measure, applicable to those of
shafts. Nor is there any thing on this subject to be added
here ; except to remind the millwright that he make the
allowance for contraction of metal, of one-eighth of an inch
to the foot in the pattern.

284. The following table contains the dimensions of
shafts subject to torsion, and to considerable lateral pres-
sure, as they were executed by a respectable millwright
It will serve to shew the sizes of the parts as found in
practice sufficiently strong, and may be foimd useful to
compare with those which would be produced, calculating
on the principles laid down in this Essay.

Column 6th, therefore, shews the diameters which these
journals ought to have, were 400 used as the multiplier.
(See Art Q33.)

CHAP, v.] ON THE SHAFTS OP UILLS.

TABLE OF SHAFTS.

Nuno.

1

2

3

4

5

6

Hemarics.

l

.5

1

■s

!

1

i.

Hi!

lying shaft.

Malleable iron
lyixigBl^ft.

20
18
16
U
12
10
B
8
5
4
3
2
1
6
5
4
3
2
1

20
22
22
24
25
25
27
28
30
32
34
46
40
28
30
32
3i
36
40

6
3|

sj

5

5

*}

l

3

3

2

2

3

3i

2

2

■i

1

11-
11-

106
10-

9-6

9-

9-

8-6

8'6

8-

8-

8-

8-

8-6

8-

8'

8-

8-

7-6

7

3
3i

7,368
6,889
6,621
6,153
5,768
5,428
4,904
4,414
4,061
3,484
1,203
2,802
2,154

Feathered
shafts.

Square
s4ftB.

APPENDIX.

COHESIVE STRENGTH OF DIFFERENT METALS.

285. " We shall tak^ for the measure of cohesion the num-
ber of pounds avoirdupois which are just sufBlcient to tear
asunder a rod or bundle of one inch square. From this it
will be easy to compute the strength corresponding to any
other dimension.

" Gold cast

Silver cast

f20,«
(24,1

" 1st, Metals.

lbs.

20,000

,000

(40,000

(43,000

Japan 19,500

Barbary 22,000

Hungary 31,000

Anglesea 34,000

, Sweden 37,000

Ironcast i^^^OO

(59,000

Copper cast <

APPEND.]

OK THE SHAFTS OF MILLS*

351

Iron bar

Steel bar

Tin cast

lbs.

' Ordinary .... 68,000

Stirian 75,000

Best Swedish and Russian 84,000

Horse nails .... 71>000

j Soft 120,000

( Razor temper . . . 150,000

Malacca 3,100

Banca 3,600

Block 3,800

English block . . . 5,200

grain . . . 6,500

860

Lead cast

Regulus of Antimony 1,000

Zinc 2,600

Bismuth 2,900

" The only author who has put it in our power to judge
of the propriety of his experiments is Muschenbroek. He
has described his method of trial minutely, and it seems
unexceptionable. The woods were all formed into slips fit
for his apparatus, and part of the slip was cut away to a
parallelopiped of i-th of an inch square, and therefore ^th
of a square inch in section. The absolute strengths of a

square inch were as follow.

n>8.

lbs.

Locust tree

, . 20,100

Pomegranate

. 9,750

Jnjeb . . .

. . 18,500

Lemon . . .

. . 9,250

BeeeluOak .

. . 17,300

Tamarind . ,

. . 8,750

Orange . . .

, . 15,500

Fir . . . ,

. . 8,330

Alder . . .

. . 13,900

Walnut . .

. . 8,130

Ehn ....

. . 13,200

Pitch Pine . ,

. . 7,640

Mulberry . .

. . 12,500

Quince . . ,

. . 6,750

WiDow . . .

. . 12,500

Cypress . . .

. . 6,000

Ash ... .

. . 12,000

Poplar . . .

. . 5,500

■a . . .

. . 11,800

Cedar . . .

. . 4^880

tr . . .

. . 10^000

252 ON THE SHAFTS OF MILLS. [aPFEND.

" Muschenbroek has given a very minute detail of
experiments on the ash and the wahiut, stating the weights
which were required to tear asunder slips taken from the
four sides of the tree, and on each side in a regular pro-
gression from the centre to the circumference. The num.
hers of this table corresponding to these two timbers may
therefore be considered as the average of more than 50
trials made of each, and he says that all the others were
made with the same care. We cannot therefore see any
reason for not confiding in the results ; yet they are con-
siderably higher than those given by some other writers.
Pitot says, on the authority of his own experiments,
and of those of Parent, that 60 pounds will just tear
asunder a square line of sound oak, and that it will bear
50 with safety. This gives 8640 for the utmost strength
of a square inch, which is much inferior to Muschenbroek's
valuation.

" We may add to these —

cwt.

Ivory 16,280

Bone 5,250

Horn 8,750

Whalebone 7,500

Tooth of sea calf 4,075

" The reader will surely observe that these numbers
express something more than the utmost cohesion, for the
weights are such as will very quickly, that is, in a minute
or two, tear the rods asunder. It may be said in general
that two thirds of these weights will sensibly impair the
strength after a considerable while ; and that one half is
the utmost that can remain suspended at them, without
risk, for ever ; and it is this last allotment that the en-
gineer should reckon upon in his constructions. There is
however a considerable difference in this respect Woods
of a very straight fibre, such as fir, will be less impaired

APPEND.] ON THE SHAFTS OF MILLS. 053

by any load which is not sufficient to break them inune-
diately.

** According to Emerson, the load which may be safely
suspended to an inch square is as follows :

lbs.

Iron 76,400

Brass 35,600

Hempen Rope 19)600

Ivory 15,700

Oak, box, yew, plum-tree .... 7f850

Elm, ash, beech 6,070

Wakut, plum 5,360

Red fir, holly, elder, plane, crab . . 5,000

Cherry, hazle 4,760

Alder, asp, birch, willows .... 4,290

Lead 430

Freestone 91

" He gives us a practical rule, that a cylinder whose
diameter is d inches, loaded to one fourth of its absolute
stfength, will carry as follows :

cwt.

Iron 135

Good rope 22

Oak 14

Rr 9

** The rank which the different woods hold in this list
of Emerson's is very different from what we find in Mus-
chenbroek's. But precise measures must not be expected
in this matter. It is wonderful that in a matter of such
unqufistionable importance the public has not enabled some
persons of judgment to make proper trials."

S86. Mr, Banks (Powers of Machines, &c., p. 94)
iakeB iron at an average to be four times as strong as oak,
and 5| tunes as strong as deal or fir,

s

354 ON THE SHAFTS OF MILLS. [aPPSMD.

287* ^* According to the experiments of various authors,
the cohesive strength of a square inch of razor steel is
about 150 thousand pounds, of soft steel 120, of wrouglit
iron 80, of cast iron 50, of good rope SO, of oak, beedi,
and willow wood, in the direction of their fibres 12, of fir
8, and of lead about three thousand pounds ; the cohesiTe
strength of a square inch of brick 300, and of freestone
200 ; teak wood, the tectona grandis, is said to be still
stronger than oak.

*^ The strength of different materials in resisting com-
pression, is liable to great variation. In steel and in willow
wood, the cohesive and repulsive strength appear to be
nearly equal. Oak will suspend much more than fir, but fir
will support twice as much as oak, probably on account of
the curvature of the fibres of oak. Freestone has been
foimd to support about 2000 pounds for each square inch;
oak, in some practical cases, more than 4000.

*< The strongest wood of each tree is neither at the cen-
tre nor at the circumference, but in the middle betwe^
both ; and in Europe it is generally thicker and firmer on
the south-east side of the tree. Although iron is much
stronger than wood, yet it is more liable to accidental im-
perfections ; and when it fails, it gives no warning of its
approaching fracture. The equable equality of steel may
be ascertained by corrosion in an acid, but there is no easy
mode of detecting internal flaws in a bar of iron, and we
can only rely on the honesty of the workmen for its sound-
ness. Wood, when it is crippled, complains, or emits a
sound, and after this, although it is much weakened, it
may still retain strength enough to be. of service.** ♦

288. The cohesive force of metals has been examined
by several experimental inquirers, besides those noticed in
the extracts made by our author, and our knowledge of

* Young's Nat. Phil, toI. i. p. 151.

APPBND.] ON THE SHAFTS OF MILLS.

255

this subject has been recently extended very considerably
by the experiments of Telford, Brown, and others. Tred-
gold collected all the most important experiments on the
cohesive force of metals, (PhiL Mag. voL 1. p. 421,) and
omitting those which are given in the preceding articles,
the table is here re-arranged by him.

TABLE OF EXPERIMENTS ON THE DIRECT COHESION OF

METALS.

Discriprtoo of metal.

Force (in
lba.)that
would tear
asunder a
bar of one
inch square.

Experimentalist

Quoted from

I. 8TEBL.

Oast steel, pre-
Tionslj tilted.
Cast steel

134,256

63,065
133,152

32,973
127,632

113,077
93,964

85,797
93,069

88,972

85,900

82,839

81,901
80,833

Rennie.

Brown.
Rennie.

Brown.
Rennie.

Siokingen.

Telford.

Buffon.

MuBchenbrogk.

Idem.

Idem.

Idem.

Idem.
Soufflot

Phil. Mag. vol. liii. p. 167.

Barlow's Essay, p. 234.
Phil. Mag. 7ol. liii. p. 167.

Barlow's Essay, p. 234.
Phil. Mag. vol. liiL p. 167.

Ann. de Chimie, xxv. 9.
Barlow's Essay, p. 222.
CEuvres de Gauuiey, ii. 153,
Intro, ad Phil. Nat i. 426.

Rondelet's L'Art de Bitir, iv.
500.

Blister steel, re-
duced bj the
hammer.

Blister steel

Shear steel, re-
duced by the
hammer.

n. IfALLBABLB
IBON.

Iron wire

X von ^VITO ••••»«•••

Iron wire ..••

German bar, mark
BR, highest re-
sult

Swed]shbar,high-
eatresolt

German bar,mark
L,hiriiest result

U^ bar, highest
remit

gwiiahbar

r

s2

256

ON THE SHAFTS OF MILLS.

[appekd.

TABLB OONTINUID.

UMcnplioa of mettl.

Oosement bar,
highest result

Swedish bar, re-
duced by the
hammer.

Common round
iron.

German bar, mark
L.

Common Stafford-
shire bar.

Conmion German
bar.

Swedish bar •••.<

Oosement bar, the
same.

Welsh bar

Bar of the best
quality.

A bar of Welsh,
one of Swedish*,
and one faggoted
scrap Iron, each
gave a result of

Liege bar

Staffordshire bar .

German bar, mark
BR.

Bar (mean of 33
experiments).

Russian old sable,
mark CON.

English bar, re-
duced by the
hammer.

Welsh bar (3
experiments).

Bar of good qua-
lity.

Swedish bar, (3
experiments).

Force (in
lbf.)that
would tear
asunder a
bar of one
inch square.

76,697
72,064

71,300
69,538
69,440
69,133
68,728

66,752
66,000

64,960

Experimentalift

62,369
61,600
61,361

61,041

59,472

55,872

55,776
55,000
53,244

Muschenbroik.
Rennie.

Telford.
Moschenbro^k.

Telford.
Muschenbro^k.

Idem.

Telford.
Rumford.

Telford.

Quoted fron

Muschenbroek.

Telford.
Muschenbroek.

Perronet

Brown.

Rennie.

Brown.

Rumford.

Brown.

Intro, ad FbiL Nat L 426.
PhiL Mag. Tol. liii. p. 167.

Barlow's Essay, p. 230.
Intro, ad Phil. Nat i. 426.
Barlow's Essay, p. 230.
Intro, ad Phil. Nat L 426.

Barlow's Essay, p. 228.
Phil. Mag. z. 51.

Barlow's Essay, p. 229.

Intro, ad Phil. Nat i. 42$.
Barlow's Essay, p. 229.
Intro, ad. Phil. Nat i. 426.

(Euvres de Gaathey, iL 154.

Barlow's Essay, p. 233.

Phil. Mag. vol. liii. p. 167.

Barlow's Essay, p. 233.
Phil. Mag. vol. x. p. 51.
Barlow's Essay, p. 232.

• Tbo Swedish bar broke at a iaw.

APFBND.J ON THE SHAFTS OF MILLS.

257

TABLB CONTINUED.

Immi ipUcMi 01 meliL

Bar rfine grain). .

— (medium
fineness.)

— (coarse grain-

m. CAST IBON.

Bar, spec. gray.
7-807.

Bar, cast verti-
cally.

Bar, cast hori-
sontallj.

Bar, Welsh pig...

nr. OOPPBB.

Wire ,

Wrought copper,

ledoced hj the

hammer.
Cast, Barhary,

spec, ffray.8* 182.
CSast, ^pan,spec.
y. 8-726.

Force (m
lb8.)that
would tear
asunder a
bar of one
inch Mjuaie.

49,982
34,081

20,460

Ezperimentaliit

Quoted ftoin

68,295
19,488
18,656
16,264

Rondelet
Idem.

Idem.

grai

CSlBt

61,228
83,792

22,570
20,272
19,072

MuschenbroSk.

Rennie.

Rennie.

Brown.

L'Art de BAtir, iy. 502.

y. PLATINUM.

Platmmn wire,
^>ecmc grayity
80-847.

Fbdnmn wire ...

Vl. 8XLVXB.

Iyer wire. •

east, spec
giay. 11-091.

56,473

52,987

38,257
40,902

Sickinffen.
Remue.

Moschenbroek.
Idem.

Rennie.

Intro, ad Phil. Nat L 417.

Phil. Hag. yoL liii. p. 167.

Idem.

Barlow's Essay, p. 235.

Ann. de Chimie, zxy. 9.
Phil. Mag. yol. liiL p. 167.

Intro, ad PhiL Nat i. 417.

dOySaa

Monreau.

Sickingen.

Sickingen.
Muschenbroek.

Biclnngen.

Phil. Mag. yol. liii. p. 167.

Ann. de Chimie, xxy. 8.

Idem, p. 9.

Ann. de Chimie, xxy. 9.
Intro, ad Phil. Nat L 417.

Ann. de

xxy. 9.

358'

OM THE SHAFTS OF MILLft.

[Al

TABLB CONTINUED.

Description of metal

Force (in
lbs.) that
would tear
asunder a
bar of one
inch square.

Experimentalist.

Quoted finom

Oold cast, spec,
gray. 19*238.

VIII. ZINC.

Zinc wire

20,450

22,551

16,600

2,689

7,129
6,650

5,322

4,736
3,679

3,211

3,328

3,146
2,581

2,547
1,824

885

3,250
3,008

1,060

Moschenbroek.

Morveau.

Tredgold.

Muschenbrogk.

Morveau.
Muschenbroek.

Idem.

Rennie.
Muschenbroek.

Idem.

Tredgold.

Muschenbroek.
Idem.

Morveau.

Reunie.

Muschenbroek.

Muschenbroek.
Idem.

Muschenbroek.

Intro, ad Phil. Nat I
Ann. de Chimin?. Ixxi*

Phil. Mag. vol. 1. p. 4
Intro, ad PhiL Nat i.

IZ. TIN.

Tin wire

Ann* de CbiTnie^ Ittti-

English block,
cast.

Intro, ad Phil. Nat i.

7-295.
Cast

Phil. Mag. vol. liii. p.
Intro, ad Phil. Nat i.

Phil. Mag. vol. 1. p. 4
Intro, ad Phil. Nat. i.

Banca tin, cast,

specific gtavity

7-2165.
Malacca tin, cast,

specific gravity

6-1256.

X. LEAD.

Milled sheet, spec,
grav. 11-407.
Wire

Wire, spec. grav.

11-282.
Wire

Ann. de Chimie. Ixxi.

Cast lead

Phil. Mag. vol. liii. p.
Intro, ad Phil. Nat i.

Intro, ad Phil. Nat i.

English, spec.

grav. 11-479.

XI. BISMUTH.

Bismuth, cast,
spec.grav.9-810.

grav. 9-926.

XII. ANTIMONY.

Antimony, cast,
8pec.grav.4*500.

Intro, ad Phil. Nat. i.

APPEND.] ON THE SHAFTS OF MILLS. S59

As this table, the most extensive of the kind, exhibits
at one view the chief results of the experiments on the
direct cohesion of the metals in English avoirdupois pounds
when the area is a superficial inch, as well as references to
the works wherein those experiments are described, it will
be useful to direct the labours of future iaquirers to such
experiments as are best adapted to increase or correct our
kiiowledge on this subject. It was collected at various
times for Buchanan's information, and will be equally use-
ful to others. When experiments are not reduced to a
common standard, they cannot be compared without much
labour : in the original descriptions of these experiments,
this has not been done ; they are described chiefly as they
were made, and for further information, to these descrip-
tions the reader must be referred. If he be interested in
these researches, the works of Muschenbroek, Rondelet,
Barlow, Tredgold, and others, will afford him much inform-
ation.

289* From the simple metals we naturally look to the
alloys, some of which are of much importance. Here the
curious but important fact that the union of two metals pro-
duces a compound of greater tenacity than either of the
metals it is formed of, will be noticed.

260

ON THE SHAFTS OF MILLS.

[appshd.

TABLE OF EXPERIMENTS ON THE DIRECT COHESION OF

ALLOYS.

AUojof

Copper

Ditto...
Ditto...
Ditto,..
Ditto...

Pwte.
.. 10

8
6
4
2

Tin

Puts.
.. 1

Gun metal, hard

Brass, fine yellow

n, English.. 10
tto

T
D
D
D
D
D

T
D
D
D
D
D

T
D
D
D
D
D

T
D
D
D
D

T
D
D
D

T
D
D

tto
tto
tto
tto

8
6
4
2
1

tto

Banca... 10

8

tto 6

tto 4

ditto
ditto
ditto
ditto

Lead
ditto,
ditto,
ditto,
ditto,
ditto.

Antimony ....

ditto

ditto

ditto

tto 2ditto

tto

1

n, Banca...

ditto

10 Bismuth

tto 4|ditto

tto 2 ditto

tto llditto

tto ijditto

tto

llditto 4

n, Banca...

tto

tto

tto

tto

n, English..

tto

tto

tto

n, English.,
tto

10
2
1
1
1

8
4
2
1

1
3

Zinc, Indian

ditto

ditto

ditto

ditto

10

Zinc, Goslar

ditto

ditto

ditto

Antimony....
ditto

tto 4 ditto

Lead, Scotch 1

Ditto 2

Ditto 10

Bismuth

ditto

ditto

Force (in Ibi.)

that wmild tetr

asunder a bar

of one inch

square.

d2,0dd

36,088

44,071

35,739

1,017

36,368

17,968

6,904
7,922
7,997
10,607
7,470
7,074

11,181
9,881
12,632
13,480
12,029
3,184

12,688
16,692
14,017
12,020
10,013
7,875

12,914
15,025
15,844
16,023
5,671

10,607

10,258

10,964

9,024

1,450

3,184

11,343

7,319
5,840
2,826

flrafity of
ttie alloy.

8-351

8-392
8-707
8-723

{

7-359
7-276
7-228
7-192
7105
7-060

7-576
7-613
8076
8-146
8-580
9-009

7-288
7-000
7-321
7-100
7130

7-000

..••.■•••

10-931
11-090
10-827

Mascheobroek,
Intro, ad FhiL NH
Idem.
Idem.
Idem.
Idem.

Renide, Phil. Tiids.

Idem.

MuschenbroeL
Idenu
Idem.
Idem.
Idem.
Idem.

Muachenbroek.
Idem.
Idem.
Idem.
Idem.
Idem.

Muschenbroek.
Idem.
Idem.
Idem.
Idem.
Idem.

Muschenbroek.
Idem.
Idem.
Idem.
Idem.

Muschenbroek.
Idem.
Idem.
Idem.

Muschenbroek.
Idem.
Idem.

Muschenbroek.
Idem.
Idem.

APFEND.3 ON THE SHAFTS OF MILLS. 26l

Brass is an alloy of copper and zinc, gun metal is an
alloy of copper and tin, sometimes in the proportion of 96
parts of copper to 11 parts of tin, but perhaps more
usually 108 parts of copper to 11 parts of tin*. It will be
seen by this table that the proportion of six of copper to one
of tin, is the most tenacious compound. A proportion very
near to this is used for bearings, bushes, and some pur-
poses in machinery ; but it is too hard and brittle for many
uses. It is worthy of remark, that copper and tin are soft
and malleable metals, but when combined, they form a
tenacious, brittle, and hard alloy. Both the hardness and
brittleness is increased by augmenting the proportion of
tin.

Tables of the cohesive force of woods of various kinds
may be seen in Tredgold's Elementary Principles of Car-
pentry, Sect. II. ; also in Muschenbroek's work above quoted,
or in Barlow's Essay on the Strength of Timber.

* These numbers give the nearest chemical proportions to those in use
among founders. For further information on this subject, see the Art
Brass, Supplement to Encj. Brit.

ESSAY III.

ON THE

CONSTRUCTION AND DURABILITY

OF THB

LONGITUDINAL CONNEXIONS OF SHAFTS, DENOMINATia)

COUPLINGS.

PREFACE.

Having treated of Wheels and Shafts, both of which may
be considered as essential parts of mill- work, the next sub-
ject in point of order is the means of connecting shafts
longitudinally J denominated couplings, which accordingly
forms the subject of the present essay.

In examining this subject, there have been collected and
described a number of different methods which have been
employed in the coupling of shafts. These methods are
arranged under two classes; practical observations are
made on each coupling. These observations are the result
of Buchanan's great experience, and that of many others
well acquainted with the subject, with whom he had taken
many opportunities of conversing ; nor will these observa-
tions be altogether without benefit, should they only lead
practical men to make others more extensive, judicious,
and useful.

A number of facts relative to the subject are also stated,
which will not be without advantage. For however useful

ON COUPLINGS.

abstract reasoning may be, yet a theory excluding some ap-
parently trivial or minute circumstances, is often rendered
altogether uncertain in its application to practice. Of this
we have remarkable instances in calculations made not
many years ago by some of the most eminent philosophers
then in Europe, relative to the motion of water in canals
and pipes. The Academy of Sciences at Paris over-
estimated the quantity of water to be delivered by an aque-
duct so far, that it was, when executed with the greatest
care, found to be deficient in the proportion of five to nine.
Desaguliers made an error of five parts out of six ; and
the celebrated M'Laurin of ten parts out of eleven, in
estimating the water to be conveyed for supplying the city
of Edinburgh*.

Smeaton, who was certainly well able to appreciate
Bcience, yet seemed to place more value on the writings of
practical men than those merely of a theoretical nature,
for he says, '* I have myself always found that exact ac-
counts of buildings [_and of course, of other such toorks'}
«luch were in any degree remarkable, and actually exc-
ited, were much more instructive to my mind than si/s-
tnalical writing." t The professional learning of engineers
is now better cultivated, and they are not required to
gatber the chief part of their instruction from the practical
experiments of predecessors. No doubt different minds
require different modes of instruction ; nevertheless, a sys-
tematic course of study seems to be vastly preferable to a
desultory course, and also to have been preferred by all

fhers either of art or science. But men like Smeaton,
Professor Robisoo mentioiiB these ctrcumst&nces in tbe Encj'. Brit. Art.
Bitw. This lubject is now better understood. See Phil. Tmns. Hy-
dnoUc InveatigaUon^ subservient to an intended Croonian Lecture on the
I, Motion of the Blood, by Dr. Young. Read before the Royal Society,
f 5, 1S08.
t See Oeecription of Bddystone Lighthoiue, p. 7.

S64 ON COUPLINGS. [PBEFACE.

advanced in years, and fiill of occupation, have seldom
either inclination or leisure to follow a systematic course.
They seek for information only when compelled by pro-
fessional difficulties. They rely upon force of genius to
supply the wants of the moment, as an Indian hunter on
his exertion in the chase ; and like him they have no idea
of laying up a stock to provide for unforeseen exigencies.
How different would be the powers of a man, of equal
genius, with the advantage of a systematic course of study ;
where reasoning was joined with experiment! Might we
not then look forward to a time when theory and the laws
of nature would be merely different terms for the same
thing ? It is not however to be expected that this perfec-
tion will ever be attained while theory is confined to matter
divested of its natural properties.

But while Buchanan spoke in favour of practical works,
he did not depreciate those of science. They may be of
great mutual benefit, and while we listen with reverence
to the voice of experience*, by sound reasoning on her
dictates, we may extend and apply them to purposes more
various and useful than those to which they originally
related.

"The man of science,'* says Dr. Robison, " who
visits our great manufactures, is delighted with the inge-
nuity which he observes in every part, the innumerable
inventions which come even from individual artisans, and
the determined purpose of improvement and refinement,
which he sees in every workshop. Every cotton-mill ap-
pears an academy of mechanical science ; and mechanical
invention is spreading from these fountains over the whole
kingdom; but the philosopher is mortified to see this
ardent spirit so cramped by ignorance of principle, and

• " Experience, slow preceptress, teaching oft
The way to gloiy by miscarriage foul."

COWPBR.

PBEFACE.] ON COUPLINGS. 265

many of these original and brilliant thoughts obscured and
clogged with needless and even hurtful additions, and a
complication of machinery which checks improvement even
1^ its appearance of ingenuity. There is nothing in which
this want of scientific education, this ignorance of principle,
is so frequently observed, as in the injudicious proportion
of the parts of machines and other mechanical structures ;
proportions, and forms of parts, in which the strength and
position are nowise regulated by the strains to which they
are exposed, and where repeated failures have been the
(mly lessons."

CHAPTER I.

ON THE LONGITUDINAL CONNEXIONS OF SHAFTS,

DENOMINATED COUPLINGS.

S90. It is well known to those who are in any degree ac-
quainted with mill-work, that it is very frequently necessary
to convey motion much farther than would be practicable
by any one shaft ; it is therefore often requisite to connect
two or more shafts together*. These connexions are de-
nominated couplings J and may be divided into two classes ;
viz. 1st. Those having two bearmgs : 2nd. Those having
one bearing. CoupUngs having two bearings are men-
tioned first, because they were long in use before those
having one bearing, and because they are, generally speak-
ing, more simple in their construction.

CLASS I.

OP COUPLINGS WITH TWO BEARINGS.

291. By the hearings of shafts are meant the parts
which support their pivots, arbours, or journals. When
a coupling has double bearings, each shaft is supported by
two bridges, as represented a, b and c, d, Plate V. Fig. 1.

* For, though it be most desirable that machinery should be concen-
trated as much as possible, long ranges of lying shafts are unavoidable m
largo mills, in cotton, linen, and woollen manufactories, breweries, &c., &c.
Hence the efficient and durable connexion of these shafts is an object of
considerable importance.

»Hap. 1.]

ON COUPLINGS.

COUPLING I^FiG. I AND 2.

op THE SQUARB COUPLING.

292. This kind of coupling is formed by making the
mda of the shafts to be coupled, square. These squares
iroject beyond the journals, and are in the spaces b c and
i) E, between the bridges. One of the squares is made as
long as what is called the coiipling-box f. The use of the
»upling-box, which is made of iron, is to receive both the
iquares, so as that when the one shaft is moved, the box
ionnects it with the other shaft in such a manner that they
lUst move together. But when occasion requires, the box
iaay be slipped back upon the longest square, and so give
Sberty to take out any one of the shafts, independently of
the rest, however great the number may be. A\Tien the
iSiafts are engaged, the box is kept in its place by the
pno.

The coupling at bc is represented as engaged, and at
IE as disengaged.
Kg. 2. represents this kind of coupling upon a larger
The same letters refer to the same parts, as in

(Sg. 1.

Sometimes, instead of the section of the couplings form-
ig a square, as Fig. 2, No. 2, it is made of an oblong form,
i represented in Fig. 2, No, 3 ; but this form is more
tffficult of execution than the square. Also, instead of
laving the coupling-box solid, it is frequently made in
two pieces. Fig. 2, No. 4, in which case it embraces the
whde length of both squares, there being no occasion then
for room to slide back the box in order to disengage the
couplings.

OBSERVATIONS.

S93- Were the axes* of these shafts truly in one straight

* ^xit. — The lioe, real or imagjjiuy, that puses through onj tlung on
rttt it may revolT*.

S68 ON COUPLINGS* [CHAP.L

line, and the squares made and fitted to the hox with per-
fect accuracy, the motion would he perfectly smooth, but
in large machinery this is almost impracticahle, and eTCD
if practicable when new, would not long continue to be
the case. The brasses wear unequally, or the firaming
sinks more in one place than another. In some part of
each revolution, therefore, one or other, or both the shafts,
will be lifted off their bearings. The two adjoining jour-
nals then come to act like one twisted piece of iron, and
must obviously occasion an unsteady motion, and much
friction. This imperfection is sometimes called a Ufl
This kind of coupling has, for these reasons, been in a
great degree abandoned for null-work. But in small nur
chinery, such as in coupling the rollers of those machines
for spinning cotton called miUes, it is still used ; because
in that kind of machinery it can be executed with a very
great degree of accuracy, and is not so liable to wear out
of truth as in larger works.

294. Of this species of couplings with double bearings
are many varieties, made according to the whim of dif-
ferent workmen ; such, for instance, as is represented in
Fig. 3, which has a small projection from each angle of
the square, differing in no other respect from the common
square coupling, and liable to nearly the same faults.

COUPLING II.— Fig. 4.

OP THE BOUND COUPLINO.

295. The round coupling has the parts between the bridges
cylindrical. The coupling-box c is made to fit those parts,
and to slip backward when occasion requires, as was de-
scribed of the square coupling. When the shafts are
engaged, two bolts, de, and fg, pass through the box at
right angles to each other, and one of them through each

OBSERVATIONS.

"f the shafts ; this being done, when the one shaft is

Imored, it win evidently carry round the other along with it.
6. The effects of the round coupling are nearly the
aa the square coupling, but as the parts may be all
turned, it is more easily mad,e true at first; and when the
bolts which prevent it from twisting, wear, they may, with
little trouble, he renewed ; but, as the whole stress comes
on a small surface at these holts, both the bolts themselves
and the holes very soon wear. For this reason, after this
kind of coupling was some time tried for the rollers in
cotton spinning, it was abandoned, and the square in which
ihe strain is diffused over a greater surface, substituted.

COUPLING in.-

H^ S97' Couplings which have no coupling-boxes, are de-
nominated clutches or glands. They may, without im-
propriety, come under this class of couplings having double

Fig. 5. represents a coupling of this kind, it consists of
'«■" crosses, a a and bb, one fixed to each shaft; bb has
ifs ends bent forward, and lays hold of a a, and thus
(urns round the other shaft.

OBSERVATIONS.

i. Glands are an excellent mode of coupling for
e bearingf!, and have the advantage of throwing the
} further from the centre of motion, than in the square

^0 ON COUPLINGS. [chap. I,

coupling as commonly executed; but few workmen aie
able to execute glands with accuracy, and if this be not
the case, they make a very disagreeable movement ; it may
therefore be not improper to describe how this may be
accomplished.

299* When the axes of two journals are put as near to
a straight line as possible, by observation fix)m the work-
man's eye, it often happens, that they may be in a bad
situation for working, either because the axes do not reaUy
correspond, or, even though they did correspond, yet the
arms of the^ glands do not take hold both together, or per-
haps from both causes. To adjust these arms, observe, in
the first place, when the glands turn round, if one of the
tails be in continual contact, and the other tail always pre-
serve an equal distance from being in contact, in that case,
the centres are perfectly opposite, and axes in one line, and
consequently right ; but if the distance of the other point
of the glands vary in its distance, or becomes so irr^ular
as to free the other point of the gland, then, in that case,
the centres are wrong, and must be so adjusted, that one
point of the gland bear equally all round, while the other
preserves an equal distance. The next object is, to adjust
the points of the glands so that they be both in contact
This may be done by chipping and filing, and in that case
they will convey themselves all around in contact with each
other.

COUPLING IV.— Fig. e.

BORING MILL CLUTCH.

First Construction.

300. Fig. 6. represents a boring mill clutch ; a b c is a
round plate of cast iron firmly fixed on the shaft m, next
the moving power j de is a lever connected with the boring

CHAP. I.] ON COUPLINGS. 271

shaft N, but which is moveable in one direction on a bolt
at F, so that it may be moved to lay hold of the projec-
ticms H,H,H,H, on the plate abc, which carries round the
lever d e along with it, and so moves the boring shaft n ;
by pulling the lever backward, the boring shaft n may be
stopped at pleasure. It is represented in Fig. 6, No. 1, as
disengaged.

OBSERVATIONS.

301. This kind of coupling is applicable to such cases
only as have the motion very slow. Pressing on one side
only of the centre, although the axes of the shafts should
not be exactly on a line, there will be no lift. For the
parts, in that case in contact, slide upon one another.

SOS. It is found, however, to have a great tendency to
force the bridges o,p, on end. In order to lessen this
tendency, the projections h, h, should be made as far from
the centre as conveniency will admit.

COUPLING v.— Fig. 7.

BOBINO MILL CLUTCH.

Second Construction.

303. This coupling, like the first construction, has a
lever d e, for disengaging and re-engaging ; but instead of
being hung immediately from the end of the shaft, turns
on a bolt at f in a large cast iron plate ikl. The other
parts having the same letters of reference as Fig. 6, re-
semble them, and are for the same use. There are three
spare sets of ears, qq, &c. (which support the lever near
the point of pressure) cast on the plate ikl, to be used in
CMe of those in action breaking.

t2

^2 ON COUPLINGS. [chap. I.

OBSERVATIONS.

304. The manner in which the lever d e is hung in the
plate iKL, at a distance from the centre n, and is supported
near the point of pressure hy the ears qq, takes the stress
entirely off the holt f, and indeed off the lever d£> except
near the ears qq.

305. This, therefore, is evidently a stronger and better
clutch, and is accordingly used in horing the larger cylin-
ders, whereas, that on the first construction is generally used
for the smaller kinds of work.

COUPLING VI.— Fio. 8.

306. This coupling is constructed by having two
round cast iron plates, ab and cd, of the same size aod
form. Each of these plates has a part f, cut out,
fghi^ No. 2, and a projection l, corresponding to the part
cut out The projection of the one plate is inserted into the
opening in the other, and thus serves to engage the shafts*.

OBSERVATIONS.

SO7. This coupling is simple and durable. It is in fact
a species of glands^ (Art. 297-) It affords an excellent
mode of adjustment The pivot and hole for fixing the
plates ought to be round, and the end of the shaft turned
along with the journal. WTien the couplings are fitted
into each other, care should be taken that those parts of
the clutch which are opposite each other will take hold
together.

• A slight Tmnmtion of this coupHug has heen found a Tery good one.
It consists in making the cin^u)a^ heads toothed, so as to fit together; the
te*^ being wedge-shaped. This mode of coupling is not affected bj any
Tmnation or settlement of the bearii^s, and it is Tety doiaUe.

ON COUPLINGS.

Shafts coupled upon tliis plan, when occasion reqiiires,
are easily moved out of their places, independently of each
other.

COUPLING vir.

308. Fig. 9. Plate VI. represents the coupling link
used by Messrs. Boulton and Watt in their portable steam-
engines. From one of the arms a, of the fly-wheel, pro-
jects a strong iron pin p. On the end of the shaft r, which
is to be coupled to that of the fly wheel a b, there is a crank
c, which has the same length of an arm that the pin is
distant from the centre of the fly wheel. The pin and the
crank are connected by a link l, so that when the fly shaft
moves, it carries round the other shaft r, along with it.

OBSERVATIONS.

309. This is a very simple and durable contrivance, and
if the two shafts be only parallel to each other, they may
work with great smoothness, although their axes should be
in different lines, for the links, in that case, move without
any twisting. But if the axes be not parallel, or if they
be not in one line, a twist vn\\ take place at the link,
which may be injurious.

The crank ought to be considerably longer than those

I in general use, to prevent, as much as may be, the con-
tj^Dual drag on one side of each of the journals.

310. Fig. 10. represents a coupling, which is sometimes
used to convey motion from the fly wheel shaft a, of a

_ jteam-engine ; and is so contrived, that in case the fly
^Hioald torn the wrong way, the milUwork remains at rest.

COUPLING VIII.

27^ O^ C0UPLIN08. [chip. I.

and so prevents accidents. This ^ect is produced by
means of a joint c, on the arm b, resembling the joint of %
table, or of a pocket foot-rule. When the fly wheel turns tin
proper way, the arm d, upon the end of the fly shaft, acts
against the face of the arm b, on the mill shaft f ; and as
the joint does not yield in that direction, the mill shaft is
carried round by the fly shaft. But, if from any acddent,
the fly turns the wrong way, the arm d strikes the back of
the arm b, the joint yields, and the mill remains at rest

OBSERVATIONS.

311. This coupling may be made sufficiently durable.
Its principles are nearly the same as the boring mill clutch,
Fig. 6. In the Repertory of Arts and Manufactures, Vol
II. p. 19) there is a description of an alteration on the cat-
tle mill, to answer the purpose of these couplings, in
making the mill go in its proper direction only. Buchanan
was led to this contrivance from the accidents to which
carding machines were liable when driven by horses, and
in the year 1790 he erected several mills on that plan*.
This has since been simplified and applied to thrashing
mills.

GENERAL OBSERVATIONS.

312. All other couplings having two bearings, that he
recollected having seen, from the reasons already given, are
attended with much friction. From that fault, they have
in a great measure been abandoned, and those with one
bearing substituted.

♦ The plate referred to in this paper will serve to give some idea of the
manner in which mill-work was constructed ahout the year 1790, in this
part of the island. Cast iron was then but little used in cattle mills. See
ItUrodwHon to Buo^ Seeandy on the Skaftg ^ Milk.

CHAP. I.] ON COUPLINGS. 275

SIS. It is, however, proper here to remark, that although
the friction, and expense of erection of couplmgs with one
bearing, are considerably less than those having two bear-
ings, where the chief object is to convey motion to a dis-
tance, or where there is no great weight or lateral stress on
the shafts, yet all circumstances should be duly considered
before adopting either plan : for if heavy drums or wheels
are to be placed near both ends of the shafts, or any other
thing that will occasion much lateral pressure, it will be
advisable to use two bearings. In the next chapter, are
examined couplings having one bearing.

CHAPTER IL

CLASS II.

OF COUPLINGS HAVING ONE BEARING.

SECTION I.

314. This class of couplings, when properly constructed,
has, to a certain degree, the property of being flexible in
all directions, like the well known contrivance invented by
Dr. Hook, (Fig. 11,) called the universal joint. This joint
is sometimes constructed by a cross, as represented in the
figure, and sometimes with its four pivots fastened at right
angles, upon the circumference of a hoop, or on the sur-
face of a solid ball. The moving parts are evidently alike
in all these cases.

315. It is sometimes applied to communicate motion,
instead of bevelled gear, when the angle does not exceed
30 or 40 degrees, and where the number of revolutions is
to be continued the same ; also, where equality of motion
is not required, for as it recedes from a right line, its mo-
tion becomes irregular. The property of the universal
joint for conveying angular motion, is of great use when it
can be attained in couplings, in order to allow for the in-
accuracy which arises from the settling of the framing or
the wearing of the brasses. It need not, however, yield
further than what is really necessary for that purpose,
which is so little, that no irregularity of motion that can
be hurtful in practice can arise.

CHAP. II. ] ON COUPLINGS. 277

316. The disadvantage of the universal joint, as repre-
sented by Fig. 10, and as commonly made, is, that it has
not sufficient strength to resist great strains. There is,
however, afterwards described a modification of it, well
adapted to bear very considerable stress. (See Fig. 18, Art.

829.)

8I7. Most part of the couplings described in Chap. I.
may, with some small addition or alteration, be converted into
couplings having one bearing. Of this description is the
sqiuire coupling.

COUPLING IX.

\

THE SQUARE COUPLING.

318. It consists of a square coupling box c. Fig. 12,
which fits a square on the end of each shaft. The square
of the shaft a is close to its joumaL The square of the
shaft B is at the end furthest from its journal.

In order to support the square of b, and keep it on a
line with a, there is a round hole d, bored out of the
centre of the square of b ; to fit this hole, there is a round
projection d, from the square of a. Instead of having the
projection d, sometimes there is a hole in the end of both
shafts, into which is fitted an iron or steel dowel or pin.

The coupling box c, covers both squares. It is kept in
its place by two iron pins or bolts, g, g, one of which passes
through each shaft. These bolts also serve to keep the
shafts from withdrawing from each other.

OBSERVATIONS.

dl9. This coupling is used with good effect in conveying
motion through a great length of shafts, where there is but

278 ON COUPLINGS. [chap.il

little lateral pressure; but where there is much ktoal
pressure, for instance in drum^htyUi which give motioD to
carding engines by belts, it has been found, that the round
projection and socket soon wear and get loose. This fknlt
perhaps arose from the square being too small to admit the
projection, and the socket to be sufficiently large. But
though this be a better mode of coupling than many m
use, yet the difficulty of having the squares accurately ad-
justed is great, as is also the fitting of the inside of the
coupling box, from the ordinary mode of casting. For
these reasons, in many cases it is very liable to lifting or
straining.

320. In England, most of the rollers of those machines,
denominated mides*, for spinning cotton, have but one
bearing ; the end of one roller being squared and inserted
into the end of that next to it, as represented in Fig. 13.
But as there is a great lateral pressure, the end of the
roller is liable to wear and become wide, and occasion a
hobbling and inaccurate motion. From the accuracy re-
quired in miUe and throstle f rollers, it is not only necessary
that the axes be exactly in one line, and accurate in their
diameters, but also that they run true without hobbling or
jolting. Buchanan had never seen any method by which
this was attainable in one bearing ; even though the utmost
attention of the most accurate workmen had been bestowed

* The mule was invented about the year 1777, by Samuel Crompton,
fonnerly of Hall-in-the-Wood, near Bolton, in Lancashire, a person of yeiy
great ingenuity, and to whom the country is indebted for many other useful
improvements ; this machine probably received its name for having rollers
like Arkwright's machine, at the same time that it retained the carriage and
spindles of hand-spinning machines called common jennies.

In 1769, Arkwright obtained his patent for spinning, and in 1775, for
preparing cotton by machinery.

t ThrostUy a machine for spinning cotton, compounded from the inven-
tions of Arkwright and Crompton.

BAP. II.]

ON COUPLINGS.

®79

i accomplish this end. In Scotland, two bearings are
nerally used in the coupling of all rollers employed in
ning cotton.

COUPLING X.

OP THE BOCND COUFLlNa.

3€1. This coupling having only one bearing, is repre-
gented by Fig. 13, and has similar additions to those of the
last described coupling, Fig. 12, and which will appear
sufficiently from the figures, without further description.

COUPLING XI.

^^ OBSERVATIONS.

322. The observations which were already made on the
round coupling with two bearings, (Art. 296, Fig. 4,) are
in a great measure applicable here, as are those relative to
lateral stress, in the observations (Art. 319,) on Coupling
IX.

^( 0X3. Fig. 14 represents a coupling used in several of
■ tile mills at Manchester. It consists of a scarfed joint,
(like that used in carpentry,) and when the shafts are cn-
gaged, it is firmly bolted, as shewn in the figure.

B 324. It was probably owing to the defect relative to lateral
stress mentioned in observations, (Art. 319,) Coupling IX.,
that this contrivance was adopted. But it seems to me to
have all the defects attending the solid shaft witli more

■Mhan tiro bearings, namely, that it could not for any length

OBSERVATIONS.

S8Q ON COUPLINGS* [CHAP.U.

of time move properly on all its bearings, without a am-
tinual bending of the solid metal, which would occasicm a
waste of power, besides a great risk of breaking the shaft.

COUPLING XIL

325. The coupling represented by Fig. 15, Plate VIL,
is only a variety of Coupling XI., and, therefore, the same
observations are applicable to this also.

COUPLING XIII.

326. That shewn in Fig. 16 may also be considered as
only another modification of Coupling XI., the one shaft
being firmly fixed to the other by flanches and bolts. The
same observations are also applicable here.

COUPLING XIV.— Pio. 17.

327- Has the bearing and the joint of the coupling at
the same parts of the shaft, so that the journal is not solid,
but composed of the ends of each shaft, which are there
formed into quadrants. Each shaft having two projections,
corresponding to two recesses in the other, one of the pro-
jections is represented at a, No. 1, as engaged ; No. 2 re-
presents the coupling as disengaged.

OBSERVATIONS.

328. This is obviously so bad a contrivance, that little
need be said with regard to it. The bad effects of it
were lately brought under Buchanan's observation at an
extensive set of calico printing works. It was there ap-
plied to drive dash- wheels for washing calicoes, and had
been attended with great trouble and expense; for the

ClUr. II.] ON COUPLINGS, 281

shafts ven frequently broke at the couplings, and were
thns rendered useless. Besides, while they did last, the
couplings thus formed in the journals, acted as a kind of
cutters, producing great friction, and tearing down the brass.

COUPLING XV— Fig. 18.

329. This consists of three distinct parts, a a, BB,andcc;
A A and B B are, each of them, firmly fixed to its respective
shaft, and are so formed, that they join partly into one
another, in some degree, like the parts of a common papier
mnchi snuff-box,

c c consists of a solid ring of cast iron, into which is
screwed four strong stfcl or wrought iron pins d, d, d, d.
[lese pins serve to act as the four pivots of the universal
it already described in Fig, 11,

I A and BB are so contrived, that each of them lays hold
two opposite iron pine, the whole combined thus, form-
an universal joint. There is a space of about a quar-
of an inch left between a a and b b, in order to allow
lie joints to play; and a a and B B are commonly made
;ii)OUt It' inches diameter outside.

There is a small mortise in each pin, to receive a cot-
teril, to prevent the pin from coming out in the course of
working.

In the plate, so many different views of this coupling
given, that it ia hoped no further description will be
]uired,

OBSERVATIONS.

■ 830. Tliis coupling is somewhat expensive in its first
■cliuo ; but it being evidently, as far as is necessary in
.eh a coupling, a complete universal joint, from its dura-
JbtT and saving of power which its pliancy must occasion.

J

OH COUFLIN68. [CHAP. IL

it seems to be perbaps the best thing ci the kind that bas
yet come under Buchanan's observatioiu

COUPLING XVI.— Pio. 19, Plates VIL and VIII.

SSI. This is a kind of coupling which was in Buchan-
an's time executed at Manchester. The coupling box c is
made very long, and is square in the inside, excepting at
D£, where there is a kind of partition, with a large round
hole truly bored in it. Into this hole, each of the shafts
A and B, are accurately fitted. The round part of the
shaft B, however, is made so long, as to allow the liberty of
slipping back the coupling box, in order to disengage the
shafts. When engaged, the box is kept in its place bj the
pin B. H represents the joumaL

OBSERVATIONS.

332. The principles of this coupling are nearly the
same as those of the common square coupling, (X. Fig. l,)
(Art. 292,) but the greater length of the box, as well as
the greater strength of the round parts intended to keep
the axis true, give this last coupling very material advan-
tage.

333. This kiud of coupling has another advantage which
the greatest part of single bearing couplings have not,
viz. : when the coupling-box is shifted off, any shaft may
be taken out, without affecting those adjoining.

COUPLING XVIL

334. Fig. 20, Plate VIII. represents a coupling used in
one of the cotton mills last erected, and one of the most
extensive at Glasgow. The external part, coupling box c,
is cylindrical. On the inside, the parts a, a, a, a project,

CHAP* n*] ON COUPLINGS. 28S

and are fitted into the shafts, which have similar projec-
tions, E, £, £, £ fitted into the coupling hox. Through the
centre of the coupling box there passes a bolt h h, to keep
it in its place ; and, at the further ends of the row of lying
shafts, they are kept together by working against a kind of
step formed of brass. Were this not the case, two bolts
would be necessary in each coupling, to keep the shafts
from separating.

OBSERVATIONS.

335. The advantages of this kind of coupling, seem to
be these two: Ist. The projecting parts e, e, &c., can be
accurately turned and fitted to the coupling box. 2d.
These projecting parts tending to the centre, are strong,
and little liable to wear.

SECTION 11.

OP THB COUPLINGS OP UPRIOHT 8HAPT8.

336. Hitherto we have considered couplings for lying
shafts only. But of upright shafts, little need be said;
having in general little lateral pressure, they are seldom
made with two bearings ; so that by placing any one of the
coupliiigs with one bearing, already mentioned, in a verti-
cal position, an idea will be obtained of the mode of
coupling upright shafts.

The square coupling, (IX., Fig. 12, Plate VI.,) for in-
stance, may easily be applied to an upright shaft.

COUPLING XVIII.— Pio. 21.

397* A represents the journal of the lower shaft, (which
if almost always that which has the bearing,) b is the lower
mA: of the upper shaft;, and c the coupling box.

284 ON COUPLINGS. I^CHAP.n.

COUPLING XIX.— Fig. 22.

338. A represents part of the under shaft, b the lower
end of the upper shaft, c the journal The termination of
A is made square, to correspond with which there is a
socket formed in b, which answers the purpose of a
coupling hox.

OBSERVATIONS.

339* This coupling is often used for light work, particu-
larly in flour-miUs, for connecting the feeder with the top
of the stone-spindle.

COUPLING XX.— Fig. 23.

340. A represents the lower shaft, d the upper, which is
above the joumaL Projecting and receding quadrants, the
same as in Fig. 17, serve to connect the shafts.

OBSERVATIONS.

341. This is a very good and simple mode of coupling
upright shafts. By their own weight, together with that of
wheels that may be on them, the projecting and receding
quadrants d, d, are pressed home into their holds, and are
not subject to get loose in their sockets or clutches, which
would be the case were the shafts lying horizontally. A
great many other schemes for couplings have been intro-
duced ; most of them ingenious ; but the late Mr. Tred-
gold had not examined any, which are sufficiently simple
and likely to be durable, to offer here as improvements.

CHAPTER in,

GENERAL OBSERVATIONS.

, It may be proper to observe, that the larger the parts
f the coupling can conveniently be made the better. In
ber words, the further the point of stress is from the axis,
B couplings will be the more durable. This being a
I to which too little attention in practice is paid, it
it be improper here, in a popular way, to endeavour
lain the reasons of this greater durability.
The strain on the point of stress is inversely as the
city of that point. (Essay II.) Now the revolutions
J the same in a given time, the further the point of
i is from the axis, the greater will be the velocity of
! point, and, consequently, the less the stress. Expe-
mce has taught this to those unacquainted with science ;
' every one knows that a handspike, capstan-bar, or
r similar lever, requires to be largest near the fulcrum,
i point on which it turns in raising a weight,) and may
e diminished in proportion to the distance from the centre
of motion.

344. Thus, for example, at two feet from the centre, the
Bfcresa is only one half of what it is at one foot from the
centre. This is almost evident to the feeling, from the
force the hand has to apply at those different distances.
Now, the larger the parts are, the stress must be thrown
the farther from the centre of motion, and the acting parts
will therefore be the more durable. Hence also the ad-
vantage in niachincr)' of having targe wheels and large

II

286 ON COUPLINGS. [chap. IIL

pulleys. (See Essay I., Oeneral Observations on Ae
Wheel' Work of Mills, Art. 101.)

345. It is also proper to observe, that when tbere
is a long line of shafts, the couplings, where there is only
one bearing, should, if practicable, be so arranged as that
the unsupported end of the shaft should be as far as may
be from the part subject to lateral pressure. For instance,
in Fig. 21, Plate VIII., the couplings and journals m
better as there represented, than had they been at a a, in
the middle, between the wheels.

346. The oiling of couplings is found to render them
more durable. This fact has been fiilly ascertained in one
of the most extensive cotton manufactories in Britain, in
the machinery of which the couplings were formerly very
liable to wear, but, since using oil, they have been found
sufficiently durable. The squares of couplings nearest the
journals, from accidentally getting oil, are also least worn.

347* A fly-wheel is often of use in a long line of coupled
shafts. In the vicinity of Glasgow, motion was conveyed
from a steam-engine by means of lymg shafts, to the dis-
tance of ninety-three yards. When those shafts were first
tried, from the elasticity or spring of so great* a length of
shafts, and the play of the couplings, the motion at the
further end was so very irregular, that it could not be ap-
plied to work a " calender." A fly-wheel near the " ca-
lender," connected with the lying shaft by pulleys and a
belt, was resorted to in order to cure this evil, and this
simple contrivance had the desired effect ; for the calender
ever after gave satisfaction in its work.

A table respecting the dimensions, stress, and durability
of couplings is annexed.

1

CHAP. III.] ON COUPLINGS.

487

SiS. Eacts respecttTig Couplings.

1

s

3

4

5

e

II

1

s.

1

t

■s

1

1
1
1

J

!

Is.

A, Cast iran. Square coupUog, one
bearing, (see Coupling IX. Fig
12,) coupling next the steam-en-
giiie, worn off each uigle of the
square about three-fourths of an
inch; the square was originally 5
inc&eii the Imx 10 inches long. ..
B, CmI iron. Same Une of shafts
further on ; all things but the re-
sistance the same. Some of the
omplisge not perceptibly worn
others worn off about three-eighths

IC

6
17

e

1

12
9

12

e

40

40

3S
38

50
55

50
50

7

7

5

5

5
8
4

7
7

s

8
3
2

4
31

in

10

12

6

5
8
8
6
3

■40

■20

■44

■1.'.

-02
■24
■16
-24
■12

C, Cast iron. Same kind of couplings
•e A and B, sqoaie originally 6
bchea, box 1 2 inches long, not per-

D, Cast iron. Same kmd of coupling

E, Wrou^t iron. Same kind, much

W, Cart iron. Same kind, not worn
&. Cost iron. Worn one half inch..
H, Coat iron. Not perceptibly worn
1, Cast iron. Not perceptibly worn

DXtCRIPTION OF THE TABLE.

Colonm 1 contains tlie resistance in horses' power.
Colunm S contains the revolutions per minute.
Column 3 contains the years at work.
Column^ 4 contsuos the side of the square in inches.
Column 5 contains the length of the box.

U 2

288 ON COUPLINGS. [cHAP.m.

Column 6 is found by dividing the power by the revo-
lutions per minute, which represents the ami-
parative stress, (see Essay II. Chap. IV.)

OBSERVATIONS.

I.

349* These couplings do not seem to have been durable
in proportion to their stress ; but this may be in part at
least accounted for, from difference of workmanship, and
of degrees of hardness of metal, or perhaps accidentally
getting oil.

II.

S50. Two circumstances must materially affect the da-
r ability of couplings. 1. The extent of surfaces in con-
tact at the place of pressure. S. The distance of the sur-
face of pressure from the centre of motion.

It is probable, therefore, that, all other circumstances
being the same, the durability of couplings increases in a
ratio compounded of those two circumstances, or nearly as
the squares of the sides of such couplings as have square
coupling boxes.

Thus, for example, b, in the table, is 5 inches on the
side ; admitting the above ratio to be near the truth, in
order to have the case a made in proportion to b, it should
be increased to rather more than 7 inches, for

The square of 5 is equal to 25, and as the stress on a is
double, 25 multiplied by 2, is equal to 50, the square root
of which is 7'07-

III.

351. But we may suppose it prudent to take a standard
somewhat larger than the side of b, for all the additional

CHAP. 111.3 ON COUPLINGS. 289

weight of the ports would never be felt hurtful in prac-
tiee; let us suppose 6 inches therefore a proper stand-
ud for eight horses' power, at forty revolutions per
minute.

Then the square of 6 is equal to 36, and 36 mul-
tiplied by 2 is equal to ^% the square root of which is
S*5 nearly, or the size which the coupling a ought to have
had.

IV.

352. In Observations II. and III. cases have been
Donsidered in which the revolutions per minute were
both the same ; but we conceive that velocity must mate-
rially affect durability, because the grinding or wearing,
where there is any play, must be increased by an increase
of velocity.

v.

353. With respect to durability, couplings may be con-
sidered under two distinct classes.

1. Those having boxes.

2. Those without boxes, having legs, such as glands,

&C.

The durability of the latter class wiU probably increase,
all other circumstances being equal, nearly in the ratio of
the distance of the parts of pressure from the centre
of motion. For, in glands, the pressure is commonly
Hmfined to a small space compared with that of coupling-
boxes.

SUPPLEMENTARY OBSERVATIONS.

I.

354. The durability of couplings depends upon so many
circimiBtanoes, that it is difficult to form general rules with

900 ON COUPLINGS. [CHAP.UL

regard to them. The two drcumstonces abeady men-
tioned (Art 350) are important, but others merit at least
equal consideration ; the angle which the surface makes
with the direction of the motion. Thus a square coupling
will be more durable than an octagon of the same size, be-
cause the acting surface makes a greater angle with the
tangent to the circle in which the acting part moves. A
right angle to the tangent, or the radius, wiU be the mmi-
mum.

II.

355. The durability of couplings depends greatly upon
the accuracy of the execution. For instance, a well-fitted
square coupling will have one fourth, or perhaps one third
of the surface of each side acting. But if fitted as is
common in practice, they will have little more than the
comers acting. One half of each side is the maximum.

III.

356. As square couplings of a large size are commonly
fitted, there is so little of the surface acting, that we sup-
pose their durability will be nearly as the length of the
box, multiplied by the velocity of the corners. But if
fitted as they ought to be, their durability will be as the
rectangle of the acting parts multiplied by the velocity.
Or, to simplify the case, (as the ratio will be the same,) as
the rectangle of the side, multiplied by the velocity.

These observations may perhaps suggest matter of useful
practical reflection to the considerate millwright

ESSAY IV.

ON THE METHODS OF

DISENGAGING AND KE-ENGAGING MACHINERY,

WHILE IN MOTION.

INTRODUCTION.

The subject of this Essay is so intimately connected with
that of Essay III. on Couplings, that in some cases they
are really blended ; and while a contrivance is employed
for disengaging and re-engaging machinery, it also serves
as a longitudinal connexion of shafts.

In viewing for the first time a cotton-mill, few objects
attract more attention, or excite more pleasing surprise,
than the facility with which even children stop or set
agoing particular parts of the mechanism separately from
the rest But however curious such things may be to in-
spect, it is, perhaps, no easy task, on paper, to render
them interesting to the reader. Their utility, however,
in practical mechanics, should stimulate the inquirer to
examine with attention a subject where ornament of style
is inadmissible, and where perspicuity alone should be at-
temptedy and which it is perhaps not always easy to attain.

292 OF DISENGAGING AND [eSSAY lY.

The plan followed in this Essay is similar to that in
Essay III., that is to say, in describing each method, and
making separate observations ; and the prefaitory observa-
tions to that essay are equally applicable here.

Great credit is due to Buchanan in bringing into one
point of view so many inventions, which will not fail to be
useful to the mechanic, by enabling him more easily to
compare them one with another, and more readily to se-
lect such as may be best adapted to his purpose.

357* From what has been said respecting couplings, it
may easily be understood how shafts may be disconnected
when at rest But many cases in practice require that
particular parts of a miU must be stopped, or set agomg*,
without stopping or making any sensible alteration on the
motion of the rest of the machinery. In cotton-mills, for
instance, this becomes absolutely necessary ; and there can
be no doubt, that necessity, in this case, has given rise to
many most ingenious contrivances. Previously, however,
to the invention of cotton-mills, there were contrivances for
this purpose in use ; such, for example, as the sack-tackle
in corn-mills.

358. In order to assist us in forming a judgment of the
comparative merits of such improvements, it may be pro-
per to bear in mind, a tendency attached to all matter
which is intimately connected with practical mechanics,
but on which daily experience shews too little attention is
bestowed in the construction of machinery.

The tendency here alluded to has, by philosophers,
been called inertiafj (or more frequently, though with less

* When any particular part of machinery is set agoing, it is said among
workmen to be set on^ or put in pear; when stopped, set off' or ptU out of gear.

t " A tendency to preserve in a state of rest or unifonn rectilinear mo-
tion, is a property attached to all matter, and may be considered as pro-
portional to the mass or weight of a body." (Young's Nat. Phil. Vol. I.
p. 51.)

WAY IV.]

RE-ENGAGING MACHINERY.

propriety, vis inertia,*) by which is meant, the tendency
which every piece of matter, when at rest, has to remain
i rest ; and, when in motion, to continue in motion. In
fcer words, the impossibility of instantaneoitsly producing
lotion in a body, or of tmtnntaneousiy stopping a body in
It is this tendency, therefore, that occasions
J violent shocks in attempting to set bodies suddenly
jnto motion ; and those shocks, besides tending to destroy
the machine, occasion a very great loss of power. Some
^urt of the machine must break, or at least yield to this
^■tent law.

^V359< The practical mechanic, and perhaps also the phi-
loeopher, will find it some advantage to dismiss the term
inertia from the place it occupies in science. Wlien pro-
perly understood, it simply indicates that matter never
changes its state, unless there be a change in the power or
powers acting upon it.

If you consider inertia as a power, it must be identical
with the power of gravity in one case, with the momentum
of a body in another, with the power of magnetism in a
third, with friction in a fourth, and so on throughout the
whole of the powers in nature ; but many writers write as
though it were a real power, and distinct from all these,
aad consequently lead their readers into incorrect notions

tthe subject.
It 13 easily proved, that when a body is struck by an-
ler in motion, some time is occupied in communicating
the motion from the point struck to the other parts of the
body ; and therefore, if the parts receiving the blow have
fc sufficient elasticity and cohesive power to destroy the
ble momentum of the striking body till the motion bo

' Vi» inertia is defined by Newton, (Def. 3. Book I.) lo be o power
*U)te<l in nil matter, by which it resists any change endeavoured to bs

is in its BtBto; tliut is, by which it becomes dillicnU to ultcr its stale,

nr of rett or motion.

994f OF DI8BN6AOINO AND [bSSAT IV.

transmitted to the centre of rotation, it must necesssnly
break the body struck*.

Hence any part of a machine intended to be acted upon
suddenly by a moving power should be strong but yielding,
or it may be capable of sliding with much friction on the
body to be put into motion.

360. Let us illustrate this by example. Throwing a whed
into gectTf very often occasions the breaking of the teeth,
whereas a pulley is generally put in motion by a belt so
g]^adually, that no part of the machinery can receive any
injury. In the first case, the wheel being firmly fixed on
its shaft, can yield little to inertia^ (the moving foroe.)
In the last, there is the elasticity of the belt, as well as the
liberty of slipping on the pulley to yield to the inertia^
(moving force,) till by its friction, the belt gradually
brings it into motion.

361. This subject naturally divides itself into the two
following sections :

I. Of Methods used when motion is communicated hj
means of bands, belts or chains.

II. Of Methods when motion is communicated by means
of wheel-work.

METHOD I.

THB SLIDING PULLBY. Fig. 1, Plate IX.

362. The sliding pulley is one of the oldest contrivances
for disengaging and re-engaging a machine moved by a
belt or band. The pulley p, which is driven by the mill-
work, and gives motion to the machine, is not fixed dead
on the axle a b, but has a hollow cylindrical bush made of
met^, accurately fitted to the axle, so that it may revolve
easily upon it, and slide a little backward and forward.

* Tlie rales for estimating the power of materials under these circum-
stance may bo found in Tredgold's Practical Essajs on the Strength of
Cast Iron, Sect VII.

ISAV IV.]

RE-ENCAGING MACHINERY.

295

: order to make the pulley p carry round the axle a b,
! is a cross piece or gland de firmly fixed to it. On
nc side of the pulley which is toward the cross, there is
i*ne or more teeth T ; when the puller p is moved toward
the cross de, the teeth lay hold of it, and thereby carry
round the axle. By sliding it backward, though the pul-
ley still continues in motion, the teeth t are disengaged
from the cross de, of course the axle stops, and with it
the machine to which it gives motion.

The bush c, of the pulley, projects a little upon the side

opposite the teeth, and has a groove cut in its outside.

luto this groove, a lever fg lies, but so as not to prevent

s motion of the bush. By moving the lever fo the bush

[ moved along with it, and thus serves to disengage or

age the pulley.

OBSERVATIONS.

IS63. For a long time this contrivance was applied to
I carding machines. The shock, however, proceed-
f from inertia, (Art. 35S,) occasioned by the teeth strik-
r against the fixed cross in setting the machine agoing,
was verj" great, and apt to shake the cylinders loose from
their axles. This is a very great defect, and for such pur-
poses as have been mentioned other methods more perfect
are now used.

^B 864-. The pulley or hinder a, is kept in motion by a
^Telt. It has a bush fitted to the upright axle uc, and
runs always at the same height.

On iia upper side, the binder a has t«eth t, t projecting
ward ; above is a pulley, or other piece of wood or metal

METHOD II.

THE BAYOKHT.— Fig. 2.

296 OF DISENGAGING AND [eSSAY IV.

DE, fixed to the shafit bc, through holes in which passes
the legs F, G of the hayonet*.

The bayonet is represented by Fig. 2, No. 3. It is
merely a piece of metal with two or more legs, which pro-
ject downward. The part h is made hollow to receive the
shaft, and has a groove into which ik fits, and the lifter
moves it upward or downward, serving the same purpose
as the lever fg. Fig 1.

When the bayonet is dropped downward, it lays hold of
the teeth t of the binder a, which immediately carries it
round, and this gives motion to the axle bc; all that is
required to stop it, is merely by hand to raise the lifter ik,
which has a spring or catch to keep it in that situation,
until again dropped by the hand of the attendant

OBSERVATIONS.

365. This contrivance is more perfect than the sliding
pulley, Fig. 1 ; because the pulley does not require to be
shifted in the direction of the shaft, which is often incon-
venient, and requires more space. Fig. 2. has however
the same defect as Fig. 1, of producing a shock at the in-
stant when the machine is put in gear.

This method is still much used in cotton-mills. The
figure represents one of the upright shafts of a spinning

* Instead of the legs of the bayonet passing through holes, as here de-
scribed, it is sometimes made square in the inside, so as to fit a square part
of the shaft ; which square answers the purpose of the piece of wood or
metal, viz. that of carrying round the bayonet along with the shaft. The
bayonet has sometimes one leg only ; on other occasions it is made like a
face-wheel, having a number of projecting teeth or legs. Indeed, it is almost
obvious, that all or most of the contrivances which are mentioned in this
Essay, must be variously modified, according to circumstances, or the fancy
of the artisan. I have only given the way in which they are most generally
applied. To give all the varieties would be an almost endless, and, per-
haps, useless labour.

METHOD III.

OP TSB LOCK PULLEY,— Fig. :

Bay IV.3 RE-ENGAGING MACHINEIIY. 297

frame. The same contrivance is often applied to horizon-
tal shafts.

^HiS66. This method is somewhat similar to the last de-
^^cribed, but instead of a bayonet, it has a lock at a, the
bolt of which b lays hold of the cross cc, which is fixed to
^the axle. The pulley, when disengaged, runs on a bush,
^b is the key, which is turned by means of a stop, which la
^Bade to touch a cross part on the end of the key. The
^Htrning of the key throws back the bolt, and so unlocks
^Kid disengages the pulley.

367- This contrivance is less simple than the bayonet.
Nor indeed has the lock pulley ever been generally adopted ;
a proof that it has not been found of much real use. It was
used at Manchester about the time that machines, called
mules, for spinning cotton, were begun to be changed from
being moved by hand, to receive their motion (as is now
leral) by power.

OBSERVATIONS.

TBB FAST AND LOOSE PULLBV. — Fig. 4, Plate X.

368. The pulley b is fixed on the axle a, and the pulley
c, having a bush, is loose. The belt or band which con-
veys the motion, may, at pleasure, either by hand, or by a
leTer, be shifted from the one pulley to the other. WTien

^running on the loose pulley c, the axle stands still ; when

^■h the fast pulley b, the axle moves.

S98 OF DISENGAGING AND [eSSAT IV.

No. 1, represents the pulleys as adapted to a belt.

No. 2, represents the pulleys as made for a rope. The
inner ledges are made low, in order that the rope may
slide with the greater facility from the one to the other.

OBSERVATIONS.

369* It may be proper here to mention, that in order to
make a belt run properly on a pulley, it is necessary to
have the rim of the pulley a little rounded or swelled in
the middle. The belt always inclines to that part of the
pulley which is of greatest diameter. This curious pro-
perty is found of great practical use. Until this property
was known, it was found very troublesome to get belts pre-
vented from slipping off the pulleys.

370. This contrivance of the fast and loose pulleys, i«
remarkable for its beautiful simplicity. It is attended with
no shock, and is perhaps the most perfect thing yet in-
vented for the purpose, in all cases where it can be applied

Its application in cotton-mills is now general. Never,
until it was applied, were the spinning mules found to give
satisfaction, when moved by power. They are now, how-
ever, almost exclusively wrought by power. This im-
provement has not only produced a great saving of labour,
but has been attended with another very pleasing effect
It has rendered the employment of the spinner more health-
ful than when moved by hand. The severity of the hand-
labour in a warm and confined atmosphere, had a very un-
favourable effect on the constitution.

METHOD V.-Fig. 5.

371. The undermost axle ab is kept by the mill in con-
stant motion. The uppermost axle c d has a bridge at d,

ESSAY IV.] ItE-ENGAGING MACHINERY. 399

to the form of a lever, in order to raise or lower it at
pleasure. WTien the end d of the shaft ia raised by means
of the bridge, the chain or rope e f is tightened, which
causes the axle b to move. But when d is lowered, the
chain falls slack below the pulley, on the lowermost shaft a,

1} thus disengages the shafts.
Er

OBSERVATIONS.

'2. This very simple contrivance is used iu some parts
England, in corn-mills, for the sack-tackle. It is evi-
dently applicable to many other purposes ; and it is obvious,
that E F may he either a belt, a rope, or a chain. A chain
is commonly used when applied as a sack-tackle. In that
case, the rope or chain, for taking up the sacks, is wound
id the barrel on the axle c d.

SECTION H.

I OP TBE MHTH0P8 «3ED WnEN MOTION IS COWVEYEB BV UEANS Of

73. When a mill is in motion, we may, with perfect
■ty. lift a wheel out of gear ; but in throwing wheels

V gear, when a mill is in motion, there is great risk of
ing the teeth ; it is, nevertheless, often done. Cot-
I spinners know fcojn experience, that the risk of break-
r the teeth is much lessened when the wheel to be thrown
info gear is previously set in motion by the hand. This
they do by pulling the shaft round by the belt. The reason
bvious. The inertia is lessened by this motion given
iic hand to the wheel which is to be thrown into gear.

METHOD Vr.— Fig. 6.

74. Wheels are often disengaged and re-engaged by

300 OF DISENGAGING AND [eSSAT IT.

means of one of the bridges, ab, which carries the end of
the shaft nearest the wheel, acting as a lever, haying its
jvlcTum A at one end. The other end b is moveable in
one direction, and is raised or allowed to fall, by turning
the key c d ; when out of gear the bridge rests securely on
the end of the key, as represented in No. 2, Plate XL
No. 1, represents the wheels in gear.

Sometimes the bridge is raised immediately by hand,
sometimes by a rope and pulley ; in these cases it is held
out of gear, by interposing a wedge instead of the key, or
by a catch under the end of the bridge.

OBSERVATIONS.

375. This mode of disengaging wheels is sometimes in-
convenient for want of room; it has, however, the ad-
vantage of having the wheels firmly fixed on the shafts.
In falling into gear the ends of the teeth are apt to strike
against one another, and there is often a violent shock
from the inertia.

In the figures, the wheels are represented the one above
the other ; but the same contrivance is modified, and ap-
plied, when they happen to be in any other relative
situation*.

METHOD VII.— Fig. 7, Plate XI.

376. Instead of lifting wheels out of gear, in order to
disengage any piece of mechanism, one of the wheels, in-
stead of being fast on the shaft, has a round bush like a
loose pulley, and a clutch, or bayonet, which connects it
with its shaft, in a similar manner with the bayonet and
binder. Fig. 2.

* The motion of the wheel, as it falls into gear, should be in the same
direction as it is afterwards to move in ; which will lessen the shock cod-
siderahly.

F

AY IV.3 RE-ENGAGING MACHINERY. 301

Thus the wheel a, Fig. 7, No. 1, has a busb, and works
on a round part of the shaft b. The clutch d, may slide
on a square part of the same shaft, and is disengaged or
re-engaged at ])leasure, by means of the lever ef, part of
which is represented as cut off, in order to shew the groove
man the clutch into which it acts.

^L No. 2, is an elevated section of the lever ef, shewing its
^^■Dnexion with the clutch d, by means of two iron pins
^^Bg, screwed into two slips of iron H, H, which fit the groove,
^^Bd are not affected by any oblique position the lever may
have.

OBSERVATIONS.

S77. This contrivance is sometimes made more con-
venient than is shewn in Fig. 7> by occupying less space ;
and although it is subject to a shock at setting on, yet it
is not so liable to break the teeth as in Method VI.

METHOD VIII.

I CLUTCH. — Fig. 8.

S78. This figure represents the friction clutch ; it dif.
fers from the bayonet (Method II.) in this respect, that
instead of striking on a fast cross, the bayonet or clutch
lavs hold of the ears of a screwed hoop, which embraces a
kind of drum.

AB represents part of a shaft kept in motion by the
mill; CDE a bayonet, which either slips on a square part
of the shaft ab. or passes through the arms of a cross I'q,
(as represented in the figure,) which cross is fastened to

AB.

FG is part of a shaft to be connected with the shaft ab ;
upon FG a kind of drum or pulley 11 ik is fastened; this
pulley has ledges to keep the screwed hoop j.mno steady.

I

302 OF DISEHGAOINO AND [BttAl £?.

In setting on the machine, the ho(q[) lmko is canied
round hy the hayonet or clutch ci>£» and by the fricdon of
the hoop on the drum hik, brings it into motion, in tin
some easy and gradual manner that a belt does a machiiie
driven by a pulley.

The hoop, as represented in No. 3» Plate XIL, fixim
acting more perfectly as a spring, is found to answer better
in practice than that represented in No. 1. and No. 2.

OBSERVATIONS.

379* There is a great deal of beauty in this ingenious
contrivance, it may be appUed to the largest machinery, and
variously modified according to circumstances*. It is ob-
vious that it prevents all the unpleasant and hurtful shock
so common in throwing heavy machinery into gear.

It may also be the means of saving lives, for should a
person's clothes be laid hold of by the wheel- work, in most
cases the hoops would slip on the drum, and allow that
part of the machinery to stop, without sensibly altering the
general motion of the mill. It may in this way, too, pre-
vent injury to the mill itself, which might arise from belts
getting foul, or from chips falling in among the teeth of
the wheel- work.

The friction clutch has been lately applied to frames for
spinning flax, and it seems to me, might also, with great
advantage, be applied to frames for spinning cotton water-
twist.

METHOD IX.
THE FRICTION C0NB8.— Fig. 9, Plate XIL

380. This contrivance is similar in its principles and ef-
fects to the friction clutch.

* Thus, for instance, it may be applied to the wheel and clutch. — Fig. 7*

ESSAY IV.] nE-ENGAGlNC MACHINERY. 303

On the shaft a (kept m motion by the mill) there is
fixed a hollow cone h ; on the shaft B is another cone e,
(he external part of which fits the internal part of h ; e is,
liowever, moveable like a bayonet on a square part of the
f;haft B, and may be moved outward and mward also hke a
bayonet, by a lever.

MTien e is moved forward, it rubs on the hollow part of
n, and by iriction, like the triction clutch, gradually brings
the machine connected with a into motion.

i

OBSERVATIONS.

WHBBLS ACTING BV FRICTION, — Fig. 10.

S81 . Sometimes there is added a bayonet, passing through
a hollow cone, wliich, should occasion require, gives liberty
to lock the shafts quite fast into one another.

The friction cones are sometimes applied to sack-tackles ;
one of them on this construction may be seen at Meux's
bfewery.

^^fdSQ. Sometimes wheels are made to act without teeth, as
represented in the figure ; they move one another by con-
tact, having their circumferences generally made of end
grain of wood, which may indeed be considered as forming
indefinitely small teeth.

It is evident, that wheels of this kind may work with
Uittlc noise, and be put into gear, and bear against one
Rttiother without risk of damage. They are commonly dis-
engaged and re-engaged by a bridge, acting as a lever, on
simitar principles as described in the case of Fig. 6.

» OBSERVATIONS.

383. This species of wheel-work has been used with
;. .od effect in machinery for raising coal ; it is also used,

304 OF DISENGAGING AND [£SSAT IV.

in some cases, in cotton-mills ; and has, for a nmnber of
years, been employed in a saw-mill by Taylor of South-
ampton, the principle and method of which in transmittmg
mechanic power certainly deserves attention.

METHOD XL— Pig. II.

384. The figure represents another application of fric-
tion in transmitting mechanic force, but instead of the
friction being on the hem of the wheel, as in Fig. 10, it is
here applied to the sides of the wheel.

The mode here represented, is successfully put in prac-
tice, in a tackle for raising and lowering sacks in a r^
spectable brewhouse in London, a is the axle which gira
motion to the tackle, upon which is the friction wheel b ;
upon the axis a are the friction wheels c, d, and the roller

E, round which the rope winds. The end of the axis a runs
in a socket in the end of the axis a ; the other end, in die
brass, is in the post^ In raising the sacks, the wheel c
is kept fast against the wheel b, by the lever h and catches cc.

For lowering sacks, the wheel d is kept against the wheel

F, by the lever h and weight d going over the pulley e, and
a man holding the line g in his hand, makes the friction
of D upon E more or less, as is necessary.

OBSERVATIONS.

385. This is a very ingenious and simple machine, and
as its principles might be applied in other cases, it is well
worth the attention of the millwright.

METHOD XII.

SELF-DISBNOAOINO COUPLING.— Fig. 12.

386. A represents a shaft, kept in motion by the mill ;
B c a cast iron wheel fast on the shaft a, having four pro-

IBSAY IV.] RE-ENCAGING MACHINERY.

305

f'jecting teeth, d, d, &c., of wrought iron; ef, is another
lilar wheel, with similar teeth, g, g, &c. but is loose on
} shaft H, and is made to slide on it, and to act as a kind
f bayonet. (Method II.) The teeth project obliquely, as
lay be seen in the figure.
I When the coupling is engaged, the teeth lay hold of one
lother, and the shaft h, is, by their means, carried round
nth the shaft a, but when any extraordinary stress comes
. the shaft h, the pressure on the oblique teeth forces
tck the bayonet e f, and disengages the coupling, i k l,
la bended lever, having its fulcrum at k, the bayonet is
pt forward by the weight of the part m k, of the lever,
r the ordinary stress on h.
■When the bayonet ef, is forced back, the lever is held
I by a catch, until the coupling is re-engaged by the hand
I the attendant. The coupling is represented in the figure
I disengaged'.

* In order to succeed in producing this effect, the angle bao must be

k aanewhat greater than would cause the surfaces to slide upon one another,

irheo acted upou by a pressure in the direction de, perpendicular to a c.

Then, when the machine is in motion they would actually slide apart, were

it not for the frictioD on the shaft, and the weight of the lever.

If the angle ba c be less than the angle which would cause the bodies to
■IHe, the coupling would not disengage itself by any force whatever.

Accordiiig to Coulomb's experiments, (see Brewster's Additious to Fer-
guson's Lectures, vol. ii. p, 155,) the friction of iron on iron is about one
fourth of the pressure, hence the angle bac should be greater than 15
degrees; otherwise the coupling will not disengage.

^

306 OF DISENGAGING, ETO, MACHINERY. [SSSAT IV.

OBSERVATIONS.

387. This coupling, as it prevents accidents from any
sudden stress, is found very useful where turning lathes
are driven by wheel-work.

Some good instances of self-disengaging apparatus may
be seen in looms driven by power. Respecting these ma-
chines, the reader is referred to Duncan's " Essays on
Weaving," as also to the Edinburgh Encyclopedia*.

* It will always be found, that, in engaging by wheels, the teeth will be
less liable to be stripped in small wheels than in laige ones ; becaiue in
small wheels the stroke will be made with a less d^;ree of velocity, ind
also a small wheel requires less force to put it in motion ; hence a small and
light wheel with strong teeth will seldomer fiul ihan a heavy one. Peikpi
the best arrangement will be when the wheel in constant motion is sbuS^
and that to be occasionally put in motion a laiger one, with elastic aniii
These elastic arms mig^t be made in the manner of coodi q^rings.

FtHl

ESSAY V.

MECHANISM

EQUALIZING THE MOTION OF MILLS, DENOMINATED
LIFT-TENTERS, ENGINE GOVERNORS, AND WATER-
WHEEL GOVERNORS.

INTRODUCTION.

:I3 Efisaj relates to machinery not less curious in its
construction than useful in practice ; and as some of the
apparatus is intimately connected with water-wheels, the
papers which are suhjoined, containing an account of some
experiments" and observations on their velocity, may not
be unacceptable to the reader, who will observe that when
a part of the machinery of a mill js suddenly stopped, or
euddenly set a-going, and the moving power remains the
same, an alteration in the velocity of the mill wUl take
place; it will move faster or slower. Every macliine
having a certain velocity at which it will work at greater
advantage than at any other speed, the change of velocity
arising from the above cause, is in all cases a disadvantage,
and in delicate operations exceedingly hurtful. In the
ease of a cotton-mill, for instance, which is calculated to
move the spindles at a certain rate, if from any cause the

* An account of tbeee experimenta was ori^&Uy pulilisbed in the 10th
e of the PLilosophicnl Magosine, p. ITS.

308 ON EQUALIZING THE [eSSAT V.

Telocity is much increased^ a loss of work immediately takes
place, and an increase of waste from the breaking of the
threads, &c. ; on the other hand, there must be an evident
loss from the knachinery moving too slow.

388. In steam-engines this evil is remedied by a c(m-
triyance called a governor. (Plate XIII. Kg. 1.) — "Two
baUs are fixed to the ends of rods, in continual revolution,
and as soon as the motion becomes a little too rapid, the
balls rise considerably,'' and, by the intervention of a lever,
act upon a throtUe-valoe^ ^ which diminishes the quantity
of steam admitted, and of course serves to make the motion
less rapid.

SECTION I.

THE 8TRAM-BN0INB GOVBBNOB. — ¥\g. 1.

389. IK represents a spindle kept in motion by the
enginet ; a, b the centrifugal balls ; c a and cb the rods
by which the balls are suspended. These rods cross
one another, and pass through the middle of the spindle at
ۥ There is a round pin put through the spindle and the
rods at c, whicli serves as the point of suspension for the
centrifugal balls or revolving pendulum. There is a part
of the spindle above c which is square, and nicely polished,

* A thrdttle-vdlve is fofrmed by a plate of metal, which is fixed on a
spindle passing across the middle of it. When the edge of this round metal
plate is in the direction of the current of steam, the aperture is at its great-
est extent of opening ; and as the plate becomes more oblique the opening
becomes less, until it is shut by the plate being at right angles to the
current.

The pressure on both sides the spindle Ibeing the same, this kind of vhItc
is opened or shut with more ease than amy 'other ; and is, therefore, veiy
applicable here. It is not easy to make it quite steam-tight when shut, but
its tightness is not of consequence in this case.

t This motion is sometimes produced by a rope and pulleys, but wheel-
work being more certain, is much to be preferred.

AT v.] MOTION OF MILLS. 309

t that the piece of brass m may slide easily up and down

The piece of brass m is round on the outside,

I has an external groove turned upon the upper end of

> receive the lever n o, the fulcrum of which is at p.

3 piece of brass is connected with the ball-rods by two

lort pieces and joints de, fc.

I The construction of steam-engine governors sometimes
Hers a little from that now described j but if this par-
alar construction be imdcrstood, there will be no djffi-
Bty in comprehending any other in use.

I 390. When the engine goes too fast, the balls fly oflT
I the spindle, and depress the end n of the lever, which
tly shuts the throttle-valve, and thereby diminishes the
utity of steam admitted into the cylinder; and, on the
other hand, when the engine goes too slow, the balls fall
down toward the spindle, and elevate the end n of the
lever, which partly opens the throttle-valve, and thereby
increases the quantity of steam admitted into the cylinder.
391- This apparatus being of great practical use, and as
it is applied to other purposes which I am about to de-
scribe, it may be proper here to give a rule for the number
of revolutions which the spindle i k ought to make in pro-
portion to the situation of the balls with regard to the cen-
tre of their suspension c.

In order to explain this rule, therefore, it is proper to
observe, that " there is a great analogy between the vibra-
tion of pendulums and the revolutions of balls suspended
from a fixed point. If a body suspended by a thread re-
Hmlve freely in a horizontal circle, the time of the revolution
^BU be the same whenever the height of the point of sus-
^^ksion above the plane of revolution is the same, what-

L^

310 ON EQUALIZIKO THE [ESaiTV.

Fig. % be made to revolvei they will arrange tibeoifldyitt w
as to remain very nearly in the same horicmital plsn.
^' The time of each revolution of the balls is equal to tfae
time occupied by a double vibration of s pendolmny of
which the length is equal to the height of the point of m-
pension above the plane in which they revolva'^*

392. Thus, for instance^ if the height of the point of
suspension d, Ilg. S, above the plane on which the baDs
revolve, be equal to the length of a pendnlnm whidi lu
brates seconds, the balls, in that case, should make SO le*
volutions per minute.

393. If pendulums are of the following lengths, their
oscillations in one minute of time, in Britain, are as fol-
low:

BiitiBli feet and inches. OsdHaiions.

"Feet 0 . . . Inch 1-565

0 9782

3 . . • . . 3-128

13 0-512

52 2-048

300

120

60

30

15

394. " Hence the oscillations of pendulums are in the
subduplicate ratio, or as the square roots of their lengths ;
and the lengths of pendulums are in the duplicate ratio,
or, as the squares of their oscillations. In order to find
the length of a pendulum that will oscillate a certain num-
ber of times in a minute, make this proportion : *' t as the
square of the given number of oscillations is to the square
of 60, or the number of seconds in a minute, so let the
standard pendulum, or 39'128 inches, be to the pendulum
sought.

Example. — Required the length of a pendulum which
will vibrate 20 times per minute.

* See Young's Natural Philosophy, vol. i. p. 47.
t Anderson's Institutes of Physics, vol. i. p. 250.

AT T.3 MOTION OP MILLS.

31

Feet. Inches. Feet
0x"o:'2S*"^3liOO::3..S-l.8:«.
lengtli required.

Inehe.

.-i-ia

SECTION I.

L 395, In a windmill, when the velocity is increased by
! irregular action of the wind, the com is sometimes

arced rapidly through the mill without being sufficiently
ground. There is an elegant contrivance for preventing
this, (similar to the governor of a steam-engine,) but which
I believe was much earlier xn use, called in some parts of
England a Lift- Tenter. " By means of the centrifugal
force of one or more balls, which 6y out as soon as the ve-
locity is augmented, and as the rise in the arc of a circle,
allow the end of a lever to rise with them, while the oppo-
site end descends witli the upper millstone, and brings it a
little nearer to the lower one." *

This machine is curious, and might perhaps in other

ises be usefully applied. We shall, therefore, describe
3 constructions, but both on the same principles.

L1FT-TENTBB3 FOR WINIIHILLS.

First Conttruetim,

196. This machine and part of the stone-spindle and

ning with which it is connected, are represented in Fig.

. Plate XIII.

To the stone-spindle there are fixed four arms a, a, a, a,

there are four similar arms b, b, b, b firmly attached to the

hollow cylinder c, which is loose on the spindle fg.

The pendulums d, d, d, d are hung above to the arms

* Young's Natumi PliiJi)Bo|ib_v. vol. i. p. 233.

SIS ON EQUALIZING THE [eSSATT.

A, A, A, Ay and through holes toward their lower extremi-
ties pass the arms of the loose cylinder.

When the mill is at rest, the pendulums hang vertically ;
hut, hy their centrifugal force, when the miU is in motion
they hang ohliquely ; and that ohliquity is increased in
proportion to the velocity, and proportionately raises the
loose cylinder c.

This cylinder c acts on the one end of the lever e, which
has a connexion with the clove upon which the bridge of
the stone-spindle rests, and accordingly raises or depresses
the upper millstone in proportion as the wind is weak or
strong.

Second CatuintetiaH,

397. Another modification of the same principle, (ap-
plied above the millstones ,) but having one pendulum
only, is represented by Fig. 4, and will be easily under-
stood from what has been said respecting the First Con-
struction.

These lift-tenters are drawn from sketches taken in the
neighbourhood of Liverpool in the year 1790.

SECTION III.

398. Governors are sometimes applied to water-wheels,
and made on various constructions. Smiths* bellows have
been applied to that use, the upper board rising or falling
in proportion to the velocity of the lower board, which re-
ceived its motion from the mill. But those we are about
to describe, appear to me on better principles ; and as they
have been found of very material use, we shall proceed to
describe a construction which has for several years been at
work in Cartside cotton-mill, which was erected under the
direction of the late Robert Bums, Esq., (of whom Pro-

ESSAT v.]

MOTION OP MILLS.

313

WATER-WHEEL GOVBBNOB.

First Comtruetion

fessor Robinson makes respectful mention in the Encyclo-
psedia Britannica, Art. Water-works.) and which has there
A«en gireat satisfaction * ; we shall afterward describe some
H^r similar machines for the same use.

^899. The principles of this kind of water-wheel governor
are nearly the same as those of the governor of a steam
engine. It has a revolving pendulum which receives its
motion from the mill, and in proportion as the machinery
moves faster or slower, the centrifugal force acts upon the
governor, and raises or depresses an iron cross, which,

Eg on a lever, reverses the motion by the wheel-work,
h operates upon a sluice so as to enlarge or lessen the
ige of the water to the water-wheel ; this sluice is
made on the principles of the ihrotUe-valve already de-
scribed, Art. 388, in order that it may be moved by a
Htall power. So long as the machinery is moving at a
Beper velocity, this wheel-work of the sluice apparatus
remains at rest.

Fig. 5 represents different views of this machine, and
some of its parts detached. The same letter in all the
figures refers to the same part.

The revolving pendulum efgh receives its motion from
the mill-work by means of a rope giving motion to a pulley
I. The upright shaft mn is kept in constant motion by

n a letter which Buchanan hod from Mr. Bums, dated February 1B08,
TJtes to tlie following effect ;

rhe goTemor is the most useful thing For a wftter-whecl that can pos-

V be thought of, and I wish you would adopt it in your practice wherever

{doMrs to make your employer prosper. 1 am sure it ia worth a lai^

< us at CarlAidc mill, from its keeping up the speed of the machinery,

it de'viUJng the yeitr round."

314 ON EQUALIZING THE [E881T Y.

the wheel work oprb. The wheel n acts constantly into
the two hevelled wheels t and u, and makes them move in
contrary directions. They are loose on the shaft when the
miU is going at its proper speed.

But if the mill moves either too £Btst or too slow, the one
of these wheels, hy means of a clutch q, in a way to be
described, is connected with and carries round the lying
shaft D c, and, by a pair of bevelled wheels, communicates
motion to the oblique shaft b w, which again, by a screw x,
and quadrant wheel t, moves the sluice z, and by making
it stand more or less oblique, alters the area of the passage
for the water.

From inspecting Fig. 5, No. 1, it will be evident that
the box a will be raised or depressed in proportion as the
baUs E and f of the revolving pendulum efgh are further
or nearer to the centre of motion ; when the velocity is
greatest, the balls e and f by their centriftigal force will
extend themselves farthest from the centre of motion, and
raise the box a. See also Iig. 5, No. S, No. 3, and
No. 4.

To the box a is fixed a cross be. There is a forked
lever dqe^ the fulcrum of which is at^ and which turns
horizontally. This forked lever has four prongs, 1, 2,
3, 4.

^V^lcn the mill is at its proper speed, the cross works within
the prongs 1 and 2 ; in this situation of the forked lever
the clutch Q is disengaged from both the wheels t and u,
and they move on their bushes without carrying round the
lying shaft. The clutch is made to slide on a part of the
shaft which is square.

When the mill goes too quick the cross gland is raised,
and in turning round strikes the prong 3, which immedi-
atelv causes the lever to throw the clutch into the arms of
the wheel u, which then carries the clutch and shaft round
with it, and by the means already described acts on the

E88AT y.3 MOTION OF MILLS. did

shiice, and by lessening the quantity of water falling on
the wheels diminishes its speed.

On the other hand, when the mill goes too slow, the
cross is depressed, and striking the prong 4, reverses the
motion of the shaft, and so produces a contrary e£Fect on
the sluice.

400. It may be proper to remark, that the train of wheel-
work is so calculated as very much to reduce the motion at
the sluice, and it is found firom experience that this is ne-
oetrary. Were the area of the aperture ttio suddenly
changed, the effect on the water*wheel would be too vio-
lent. Every time the mill is stopped, it is proper to lift
the wheel r out of gear. The centre on which the sluice
turns should be one third of its height firom the bottom, in
order that the pressure of the water above the centre may
balance that below.

At m there is an upright shaft, which is worked by hand
when required.

WATBR-WHBBL OOVBBNOB.

Second Conttmctum.

401. Fig. 6 represents a sluice regulator as executed in
some parts of England. It differs little fi:'om that already
described, only that the lying shaft a b receives its motion
immediately firom the mill, instead of firom the axle of the
revolving pendulum, as in the first construction. (Art. 399*)
From having so minutely described that construction, it is
hoped that the attentive reader will find no difficulty in
ccmiprehending Fig. 6, firom inspecting the plate.

WATBB-WHSBL GOVBRNOB.

Third Construction.

ifiSL Fig. 7» Plate XV. represents a water-wheel go-
vernor of a very simple. cpziBtractiaQ» differing £rom the

316 ON EQUALIZING THE [e88AT V.

foregoing in this respect, that it communicates most part
of its motion by bands and pulleys instead of wheel-work.
The motion is reversed by the simple means of having one
of the pulleys a with an open band, and the other b with
a cross band.

OBSERVATION.

It is proper to observe here, however, as was already
done with regard to the governor of the steam engine, that
wheel- work is much to be preferred in point of certainty,
to bands and pulleys.

WATBB-¥rHBSL OOVBBNOIU

Fourth Constmetion,

403, This construction is represented in Fig. 8. The
revolving pendulum aklm is kept in constant motion by
the water wheel.

a, a, two wheels fixed on round sockets upon the go-
vernor spindle.

B, a clutch upon a square part of the spindle, (or what
might be better, a round with a feather upon one side ;) c, a
gland to connect the clutch with the sliding part k of the
governor, which has a groove to receive it like that of a bay*
onet. (See Essay IV. Art. 282.) d, a piece of iron which
prevents the gland from turning round, and for keeping it
from flying off^; e, a wheel working into the wheels a, a ;

F, an endless screw upon the same axle with the wheel e ;

G, a wheel upon the same axle with another screw h, which
acts into the quadrant i, upon the sluice.

The operation of this ingenious apparatus, from what
has been said of the other constructions, will, it is hoped,
be sufficiently clear. This governor was designed by Jam^
Carmichael, millwright, of Dundee.

ESSAY V.3 MOTION OF MILLS. 317

WATBB-WHEBL OOVBRNOB.

Fifth Camtructian,

404. From the inspection of Fig. 9^ and what has been
ahready said respecting water-wheel governors, this par-
ticular construction will doubtless be easily comprehended.
We need only mention that the wiper a, by means of the
forked lever bdc, acts on the clutch e. The rest of the
movements resemble those of the first construction. (Art.

3990

This apparatus, remarkable for its neatness and simpli-
city, was constructed by Hewes of Manchester*.

* A simple and not a very expensive apparatus for equalizing the exer-
tion of horses in thrashing-machines, is descrihed in the Art. Agriculture,
Supplement to Ency. Brit p. 200 ; and in Brewster's edition of Feiguson's
Lectoies, p. 201, vol. ii.

1. <i

APPENDIX.

ON THE

VELOCITY OF WATER-WHEELS.

No. I.

On the Velocity of Water- Wheels^ by Mr. Robertson
Buchanan^ Engineer : communicated in a Letter ta the
Editor of the Philosophical Magazine.

The accompanying paper was read in May 1799> ^ *
Philosophical Society at Edinhurgh, and was afterwards
published in the Philosophical Magazine.

405. There are many cases in which it is of importance
to know the proportion of power necessary to give different
degrees of velocity to a mill*. But as the construction of
mills, and the purposes they serve are various, it is perhaps
impossible to find any law of universal application. Mr.
Banks, in his Treatise on Mills i, has drawn a conclusion
which he appears to consider as invariable, namely, that
" when a wheel acts by gravity, its velocity will be as the
cube root of the quantity of water it receives."

But if we suppose a wheel raising water by means of
cranks and pumps, on Mr. Bank^s's principle, Buchanan

* It was a scarcity of water for the Rothesay mills which directed my
attention particularly to this suhject

t See Banks on Mills, pp. 17, 18, 144, 145, 146.

APPEND.] ON THE VELOCITY OF WATER-WHEELS. 319

thought it might easily be demonstrated, that by reducing
the velocity of the wheel to a certain degree, the wheel
would raise more water than would be necessary to move it
at that velocity ; a thing evidently impossible.

In this view it would seem there is no actual case in
which Mr. Banks's conclusions will hold true. But, how-
ever they may apply to other mills, the experiments of Bu-
chanan seem to prove at least that they do not apply to
cotton-mills. On the ground of these experiments, made
at different times, and with all the attention in our author's
power, (and not from any abstract consideration,) did he
presume to call in question an authority for which we en-
tertained the highest respect.

406. In January 1796 he measured the quantity of
water the Rothesay old cotton-mill required : 1st. When
going at its common velocity ; and 2dly, when going at
half that velocity. The result was, that the last required
just half the quantity of water which the first did. It is
to be observed, that in these experiments the quantities of
water were calcidated from the heads of water and aper-
tures of the sluices.

feFrom these experiments he inferred, " That the quantity
water necessary to be employed in giving different de-
Bes of velocity to a cotton-mill, must be nearly as that
velocity."

He was satisfied with this experiment, and the inference
drawn from it, till some gentlemen well acquainted with the
llieorj- and practice of mechanics expressed their doubts on
the subject. He had then recourse to another experiment,
which he considered as less liable to error than the
former.

■toy. The water which drives the old cotton-mill falls, a
little below it, into a perpendicular-sided pond, which
serves as a dam for a corn-mill at some distance below it.
To ascertain, therefore, the projiortional quantities of

y2

320

ON THE VELOCITY

[essay Y.

water used by the old mill, notmiig more was necessary
than to measure the time the water took to rise to a eertahi
height in that pond ; and accordingly, on the first of May
1798, he made the experiments noted in the following
table :

Number of experiments.

•

1

2

3

4

Revolutions of one of the upright
shafts per minute.

46

46

24

23

Rise of water in the pond in inches.

5

5

5

5

Time in minutes and seconds.

6-58

6-57

14-45

150

The first and second experiments were made with the
mill at its common velocity ; the third and fourth at nearly
half that velocity.

The time which the mill required to use the same quan-
tity of water in these experiments may be taken in round
numbers ; the proper velocity at 7 minutes, and half that
velocity at 15 minutes.

The result of these experiments approaches very nearly
to that of 1796. The difference may be accounted for by
the small degree of leakage which must have taken place
at the sluices on the lower end of the pond ; and the time
being greater in the third and fourth experiments, the
leakage would of course be greater.

408. Smeaton * and others have proved, in a very satis-
factory manner, that " the mechanic power, that must of
necessity be employed in giving different degrees of velo-
city to the same body, must be as the square of that velo-
city." But it appeared to Buchanan, that the result of
the above experiments may be easily reconciled to this pro-
position, by considering what Smeaton says immediately

* Sec Smeaton on Mills, p. 18. See his Miscellaneous Papers, p. 92.

^PEND.]

OF WATER-WHEELS.

321

terwards: — "If the converse of this proposition (says

) did not hold true, viz., that if a body in motion, in
eing stopped, would not produce a mechanical effect equal
or proportional to the square of its velocity, or to the me-
chanical power employed in producing it, the effect would
not correspond with its producing cause."" Now it is to
be observed, that Smeaton's experiments were made on the
velocihi of heavy bodies Jree from frictioyi and other causes
nf resistance. ; but in mills there is not only friction, but
obstacles to ho removed : and experiments made on friction
have proved that the frictions of many kinds of bodies in-
crease in" direct proportion to their velocity. But the velo-
city of a cotton-mill at work may be considered as a me-
chanical effect; and, if so, must correspond with its pro-
ducing cause.

409. The preceding esperiments on the Rothesay mill,
are undoubtedly correct and consistent with the principles
of motion and power, and also with the experiments of
Smeaton on Mills and Mechanic Power.

It is shewn in the additions to this essay that the me-
chanical power is as the quantity of water on the wheel,
multiplied into its velocity when the wheel, fall, and other
circumstances remain the same, and since the mechanical
effect is measured by the resistance multiphed into the
velocity of the working point when the friction is con-
stant ; if the quantity of water be diminished by its half,
either half the resistance, or half the velocity with which

is overcome, must be taken away, otherwise there

I not be an equilibrium between the power and effect.

at the same time it is to be observed, that an

•eased velocity lessens the friction of the intermediate

ichinerj', and consequently a greater effect would be pro-
Ked by the greater velocity, as appears to be the case by
1 Mechanie Powers applied to Bodies at rest. Miscellaneous

3^ ON TH£ VELOCITY [eSSAT V.

the experiments* There is not, however, in the detail of
these experiments, sufficient data to enahle ub to arrire at
any useful conclusions.

410. Roherton, an engineer of some eminence, made
ohservations on the foregoing paper, alleging that the
conclusions of Banks give most satisfactory evidence
that particular care and judgment are necessary in "*ffVrf
experiments.

It appeared to Roherton that the wrong oonchuoanB
which have heen drawn hy this and other writers on ihiB
suhject have wholly arisen from misapprehending some of
Sir Isaac Newton's fundamental principles of mechanici^
and from a love of establishing theoretical expressiape
rather than strict observations of the invariable laws of na-
ture ; expressions such as these : viz., Quantity qfUdotkn^
Instantaneous Impulse.

Taking a constant portion of time (viz., a second) to be
the measure of the velocity of a body, and an instant to be
the measure of the effect it produces ; or by taking time
as the measure of the cause, and space as the measure of
the effect. As to an instantaneou.s effect, Roherton ar-
gues that it is an absurdity in itself as well as in mechanics.
We can form no idea of a body put into motion, without
the acting power or body act upon the body put into motion
for some timey and also over some space ; and to suppose
otherwise leads us entirely out of the sound principles <tf
mechanics.

In mechanics every effect is equal to its producing cause.
In the case of a power acting on a body producing motian»
and also this body acting against another power which
tards its motion : if the causes of action and resistance
each measured by the tbne the motions are produced and
retarded, the result will be equal.

Or if they be measured by the space over which tbey
act, the results will be equal ; and this is an uid¥B(aa]l

I APPEND.] OF WATER-WHEELS. SS3

principle, whether applied to accelerating power and mo-
tion, as gravity, &c., or to machines which act constantly
and uniformly. Yet in the case of uniform motion, space
or time may be used at pleasure ; as from the uniformity
of space and time they become a common measure-
To illustrate this, suppose the body a acted upon by the
power of gravity through the space ab, in a portion of
time which we will call one, M'hen it arrives at b, ^ q
it meets with another medium of resistance, which
is ten times greater than the former ; the body a
will be resisted in proportion to the cause of action
and resistance, that is to say, if the time of action
were one second, the tijne of resistance will be one
tenth of a second, and the distance ab will be to
the distance bc as ten to one; so that whether
space or time be taken as the measure of action,
the same must be taken for the measure of the
eflfect, to have the results proportionate and equal.
But if the cause be measured by time, and the effect
by space, the results will be as the squares of the times, or,
which is the same thing, as the squares of the velocity.
^^ Thus, suppose a body in motion, with a velocity of one,
^Hm a power to penetrate into a hank of earth one foot.
^Ki the same body, with a velocity of two, strike the bank,
It will penetrate to the depth of four feet: for the velocity
is double, and the time of action is double, and therefore
the results will be compounded of both, that is as the
square of the velocity.

From the above it may be inferred, that if equal bodies
be acted upon by unequal powers, the times requisite to
produce an equal motion will be reciprocally proportionate
to the powers ; that is to say, if a power of ten act upon a
body for one second of time, and the power of one act upon
an equal for ten seconds, thev will produce equal velocities.

1

324 ON THE VELOCITT [M84T.T.

very unequal, being as ten to one : and if ihe squan nMb
of the powers producing the effects be taken, that will give
the times they take in carrying the body acted upon through
equal spaces.

But it is obvious this doctrine has no more to do wifli
the operation of machines, than simply their first startnig
from rest to the motion necessary for working. When
this is acquired, the power applied and the power of resiit-
ance balance each other, and whatever be the motioii the
machine moves at, the same power will carry it on, (if it he
upheld,) provided the machine act in such a manner as not
to accumulate resistance by the accumulation of motijon,
which is the case in forcing fluids through pipes, &c. Li
cases of this kind, the nature of the machine must be par-
ticularly kept in view, and not to adopt any law to explain
the resistance the acting body meets with, but what is am*
ply deduced from the very machine we have under cod*
sideration. But in most cases, any machine may be cod*
sidered as acting purely on a statical principle. The rais-
ing of weights, or overcoming friction, Roberton considerB
purely as acting on that principle ; and when the power of
action is equal to the resisting power, the machine is indif-
ferent to motion or rest. If the machine be at rest, the
power will not move it, being a balance to the resistance.
If the machine be set in motion, the power will keep it in
the same motion, (provided the power be upheld,) the
same as equal weights himg over a pulley, or in the oppo-
site scales of a beam. If they be at rest, they will remain
so ; and if they be put in motion, they will endeavour to
persevere in the same.

The above doctrine of a statical principle is proved in
the most satisfactory manner by the experiments made at
the old mill of Rothesay, the motion of the water-wheel
being exactly proportioned to the quantity of water ez»
pended, and therefore an exact and equal load upoii • the

APPEND.] OP WATER-WHEELS. 325

wheel ; that is to say, the buckets icere equally full when
the mill moved at its ordinary motion^ or at half that mo-
tion.

The effect, therefore, of letting more water on a wheel,
is not to lodge a greater quantity in the buckets, but to
supply the same quantity when the wheel is in a greater
motion.

Banks, however, made his experiments agree with his
theory, yet Roberton took no trouble in enquiring into
them^ alleging it would be to little purpose to have done

»r

Bus

Suffice it to say," he adds, " that the very small quan-
bties of water which Banks made use of, and the slowness
of the motion of his wheel in his experiments, gives no
ground for placing the smallest dependence on them, and
when compared with the more judicious and accurate ex-
periments of Smeaton, they dwindle into contempt."

411. Roberton further says that "Smeaton, in running
wheel at nearly three feet in the second, brought it
nearly to a maximum, and lost but about one fourth or
one fifth of the original effect (alludiug to his overshot
wheels). Banks, at his highest motion, run his wheel
about one foot in the second, and reducing it to one
half of that motion, the same quantity of water then ex-
pended, was capable of performing four times the wurk ;
and by deduction from thence, it appears plain that his
wheel, (from his own theorj',) would perform about twenty
times the quantity of work which iSmeaton's could per-
form with the same quantity of water, and about 16
more than nature ; so that the observation, (alluding
the theory of Banks,) is very just in sajing that, by re-
\g the motion of the wheel, it is demonstrable it would
more water than supply itself."

326

ON THE VELOCITY

[essay V.

ON 0VBB8H0T WATBA-WHBKL8.

412. The best water-wheel is that which is calculated to
produce the greatest effect when it is supplied bj a stream,
furnishing a given quantity of water, with a given &IL

The mechanical effect depends on the proportion of the
wheel's diameter to the height of the fall ; and on the ve-
locity of the circumference of the wheeL These are the
two principal parts to be considered in the theory of wheels ;
but there are also some other points which ought to be at-
tended to, because the effect is much decreased when they
are neglected.

Of the proportion of the radius of the wheel to the
height of the fall. — ^Let acbd be the wheel, and ea the

depth of the buckets ; then, according to experiments on
water-wheels, it appears that the rotary force of the water
in the buckets is nothing at c and d, and that it increases
nearly, if not accurately, in the direct ratio of the distance

OF WATER-WHEELS.

from c or rf, and is greatest at a. That is, the force at '
any poiot n in a direction ea, or ' perpendicular to the
radius, is as ac.

A slight consideration of the figure is sufficient to in-
form us, that the wheel will not produce the greatest effect
when it receives the water at the upper pomt c, and that
there must be considerable advantage in making the wheel
of a greater diameter, so that it may receive the water at
some point between a and c. The point which will ensure
the greatest efiect we are now to calculate.

Put c = that portion of the circumference which is to he

loaded with water ; and .r = the arc comprehended between

the point where the water flows upon the wheel and the

horizontal line ea ; also make b = the area of the stream

supplying the buckets. Then the solid which represents

<? — '^

the effective force, will be 4 5 x I ^ ; which is to be

'^ c—x '

c* — 2j*
the greatest possible ; or — ^^— = a maximum. By the

principles of maxima and minima, this takes place when
j-=c(l— ■j\)OT X = ''29'29 c. Accordingly the arc c — x
must be the quadrant dg or 90", and the arc x = 37'27°.

Hence we have this important practical maxim, A
water-wheel will produce the greatest effect when the dia-
meter of tlie wheel is proportioned to the height of the fall,
so that the water flows upon the wheel at a point about
52j degrees distant from the summit of the wheeL

If r be the radius of the wheel to the extreme part of
tfie bucket, and h the effective height of the fall, then k = r
w(l+sin. 87^,) or A = 1-605 r; for the sin. 37i = -605.
E|&lso 'GSS h~T. Therefore when the effective height of
Kdie fall is determined, the radius of the wheel is easily
^lnJculated. When the effective fall is § of the whole fall,
^ if we make h the whole fall, r = ■554'/i, or I'lOS A = the
diameter of the wheel.

328 ON THE VELOCITY [eSSAT V.

The effective height of the fall is less than the true
height, hy as much as is necessary for giving the water the
same velocity as the wheel hefore it flows upon it.

In low falls a wheel would work with advantage in a
considerable depth of tail water, provided the buckets were
of a suitable form for moving through the water, and the
effective fall made through a very accurate sweep, so that
the sweep and not the form of the bucket should confine
the water upon the wheel.

413. Of the velocity of the circumference of the whed
to produce a fnasimum effect. — It is necessary to premise,
that the velocity with which the water flows upon the float-
boards or buckets, is considered to be equal to the velocity
of the wheel, and to strike against the floats as nearly as
possible in the direction of the motion of the wheeL

Let X be that part of the fall which gives the necessary
velocity v to the water, when the effect is a maximum ;

V will then be the velocity of the circumference of the
wheel. Also, make a = that part of the fall which would
correspond to the velocity of the circumference of the
wheel when the power would be equal to the friction of
the loaded machine only ; or when the useful effect would
be nothing. Now if A be the whole fall, the effective force
of the water on the wheel will always be proportional to
A — jT, when the effect is a maximum ; and to A — a, when
the useful effect, or work done, is nothing.

Hence, v (^h ^ x — h — a) must be a maximum ; or,

V (a — ^) = a max., but v = x^, therefore jr* (a ^ jr) = a
max., which according to the rules of maxima and tninima^
takes place when a = 3x.

It is evident that the value of a must entirely depend on
the nature of the machine, for if there be many moving
parts between the power and the resistance, the firiction
will be greater, and consequently a will be less. The m^
chine must be very simple indeed, if the friction be less

JPEND.] OP WATER-WHEELS. 329

I one half the moving power, and it will often amount
I thirds of it. If we suppose it to be two thirds,

len a = o, and consequently x = t., and u = V — j^ — =

^ h.

Hence, when the friction amounts to two thirds of the
moving power, the velocity of the circumference of an over-
shot wheel in feet per second, should be 2*67 times the
square root of the whole height of the fall in feet.

Again, that part of the fall is to be determined, which

will give the water the same velocity as the wheel, and

h h

= 5, and 3jr = a, we have ^ = fi- Hence, when the

^Bnce

^picti'

iction is two thirds of the power, that part of the fall
which will give the water the proper velocity, is one ninth
of the whole height,

These results may now be usefully compared with the
experiments of Smeaton ; at the same time it is obvious
that his experiments were not adapted for arriving at
general conclusions, because the water was always delivered

I upon the same wheel ; for it is clear, from the preceding

■bivestigation, that every particidar wheel must have its

H^articular maximum.

"^ In Smeafon's experimenta on overshot wheels, the wheel
was 2 feet in diameter, therefore the height of the fall
should be 2^ feet. Now the square root of 2^ is 1*5;
and 1 -5 X 2-67 = -t-005, that is, the velocity of the wheel
should be 4, feet per second ; or it should make 38 turns
per minute. Smeaton infers that " the best velocitj' for
practice" will be when a wheel of 2 feet tliameter makes
30 revolutions per minute. (Miscellaneous Papers, p. 51.)
But his model had much more friction in proportion to the

I effective force of water on the wheel than two thirds, that
fe have here c-alculated upon. When the calculation is

!

390 ON rHE VELOCITT [eIIATT.

made aooordiiig to the fricti<m of Smeaton's model, vm

9r4i s/ h ; and the velodly of the model wheel would oome
out 3*6 feet per second, or 34 turns per minute. Thk
yelocity will perhaps apply correctly enough to overshot
wheels, where the water flows on at the summit, and to
rough made machinery ; but the former calculation is that
which I consider most correct, for the improyed kind of
wheels here pointed out It is to be understood, that the
friction allowed for, includes all the kinds of resistance and
loss of force which lessen the useful effect, as well as the
resistance of the rubbing sur&ces, properly called fiictka.
Many readers will think that two thirds of the effective
force, IB greatly too much to be lost \ it will be well if it
draw their attention to lessening the stress on every part
of machinery, and to the importance of having few rubbing
surfeces, and other causes of resistance.

414. On computing the power of overshot water-whedi.
— ^In determining the proportion of the radius of the wheel
to the height of the fall, an equation is given for the eflec-

tive force. Resuming that equation, we have i b ( -)

= the effective force of the water, and i J^; ( —\ =

its mechanical power. But the quantity of water expend-
ed in maintaining this power, will he bv. Hence, the
quantity of water expended, is, to its mechanical power, as

1 : i (1=^).

When the wheel is supplied at the summit, j: = ^ c ;
and therefore, the quantity of water expended, is to its
mechanical power, as 1 : ^ c. Or the power is equal to
half the weight of water supplied to the wheel.

The same relation takes place when jr = o ; that is,
when the wheel is supplied at the height of the axis. Hence
when the radius of a breast wheel is equal to the effective

APPEND.] OF WATER-WHEELS. SSt\

height of the Fait, its power will be the same as that of am,
overshot wheel supplied at. the summit.

Whcu the wheel is supplied at the point which pro-
dooes the greatest effect, x => -HQ-Jit c ; and consequentlv
the quantity of water expended is to its mechanical power
as 1 : 0*5857 c. This effect is greater than when
wheel is supplied at the sxunmit in the ratio of 1*1714

These comparisons will convey some useful infonuatioD
to many readers ; and they may sometimes suggest to scien-
tific wtiters the advantage of studying the actual nature
of machines i for relations so extremely obvious and simple
could never have been overlooked by any one ivho might
have condescended to examine the subject.

The power of a water-wheel may be considered under
two points of view j each of which has its peculiar use.
If we wish to compare it mth any other first mover, then we
shall have to calculate its mechanic^ power. But when
it is desirable to compute the resistance it will overcome at
the working point, the effective force should bo calculated.

415. "When the water flows upon the wheel, either at or

above the axis, the mechanical power is ^ be — - cubic

wer {

ion ^\

4

bs. \Vhen bv a the
quantity of water expended in a second, in cubic fcpt
the part of the circumference between the lowest
of the wheel, and the place where the water flnwe
it in feet, and j- the part o[ the eireiMfetw^,.. i^-_
the point which is level with the axic, and that »K-,„ i^
water flows upon the wheel in feet.

Throughout these Esvvs, the M«»-lnMf^ ■«»»• «/
horse is estimated at 300 tU. HKnia^ wiA a JtUtlu <rf sA
feet per second. Then a water-rind wiQ fce^^j
31-33 ip(c'-gjO 00*86^ i-^.^^^"^

332 ON THE VELOCITY [ESSAY V.

When the water flows on either at the summit or at the
level of the axis, the mechanical power is 31*25 bvclbR. or
it is = 0*00426 bvc horses.

When the water flows on at 52f degrees distant from
the summit, the mechanical power is 37*192 bvclhs. or =
•005 bvc horses. Since in this case, c = 127i degrees of
the circumference, we have c = 127^ x '0174533 r ; and

as r = *554 h ; and v = 2*67 >/ A ; hy substituting these
quantities, we have 122*176 bhi lbs. = the mechanical
power ; or *0l64 bhi = the number of horses, where h =
the whole height of the fall in feet, and b the area of the
apertiure through which the water flows upon the wheel in
feet.

«

416. The effective force is 31*25 be Iha. when the water
flows on either at the summit, or at the level of the axis.

When the water flows on at 52f degrees distant from
the summit of the wheel, the effective force is 37*192 iclbs.
or 45*746 6 A lbs.

OF THE POWER OF BREAST WHEELS.

417. When the water flows on below the level of the
axis of the wheel, it may be termed a breast wheel.

Let 1/ be the distance below the axis measured on the

circumference, then 5-7 — 7 — r equal the mechanical power

,. /. /. 31-25 c^fti;,, „^

m cubic leet 01 water, or lbs. When y = c the

c -^-y ^

power will be reduced one half, and when y = 2 c it will be
reduced two thirds, and so on.

If we assume that the mechanical power of an imdershot
wheel is half that of an overshot one " under the same cir-
cumstances of quantity and fall ;" * then it will be an ad-

* Smeaton's Experiments, Miscellaneous Papers, p. 49.

APPEND.^ OF WATER WHEELS. 333

yantage to employ an undershot wheel whenever the fall is
less than three tenths of the radius of the wheel. But
since the radius of the wheel may in many cases he dimi-
nishedf it does not appear to he desirahle to employ an
undershot wheel in any case, except where the quantity of
water is great and the M inconsideraiae.

ESSAY VI.

ON

CHANGING THE VELOCITY OF MACHINERY

WHILE IN MOTION.

INTRODUCTION.

The machinery employed in manufactures may be diyided
into two classes : 1st. Millwork. 2nd. Smaller Machinery.
The mechanism described in this Essay belongs to the
latter class, and has hitherto been chiefly used in cotton-
mills ; but useful hints may perhaps be taken for its apph-
cation to other valuable purposes.

It would be satisfactory to be able to record the names
of the inventors of many of the ingenious contrivances
which are described in these Essays. But the secrecy
which interest prompts in the machinery used in manufac-
tures— the same difficulties giving rise in different minds,
without any communication of ideas, to the same means of
overcoming them, and the very gradual steps by which im-
provements are usually made, render it, in most cases, im-
practicable to trace the inventions to their true sources.

It may be taken for granted that the reader is acquainted
with the common modes of altering the velocity of any par-

[8SAY VI. j CHANGING THE VELOCITY Or MACHINERY. 335

liar part of machinery, by changing the wheels or pul-
This change, however, requires that the macliincry
B stopped for some time until the alteration he made.

t many cases occur in which it is desirable to change
velocity without such loss of time. Some of those
s shall now he considered, beginning with one of the
; simple, — that of changing the speed of a turning-
he, according as the nature of the substance to be turned,
s diameter may require.

LATHS MOTIONS.

4.19. A series of pidleys gradually increasing in size
«n an axle, is moved by the mill, and on the spindle of
the lathe is a similar series, but in an opposite order, so
that the same length of belt will work on all the opposite

tlleys, according to the speed required.
These series resemble two tnmcated cones, having the
aller diameter of the one opposite the greater diameter
of the other, so that the same belt is equally tight on what-
ever pair of pulleys it may work.

This contrivance is represented by Fig. 1, Plate XVI.
applied to the spindle a b of a turning lathe, c d is part
of a shaft driven by the mill at a certain regular velocity.
When a slow motion is required, the belt works at ef;
when a greater velocity is required, the belt is shifted by
issing it to one side, to another pair of opposite puUeys.
I is hoped that the figure will make this so clear, that all
her explanation will be unnecessary.

OBSERVATION.
(420. This contrivance, very simple in its construction,
I found of important practical use in the turning of va^
i substances.

336 ON CHAN6IK6 THE [E88AT VI.

II.— ALTEBNATB 0ONB8.

421. There is another contrivance on similar principles
to that above described, which has b^n found very useful
where a motion constantly varying is required. Instead
of the opposite series of pulleys, there are two opposite
cones.

The one of these cones gives motion to the other by a
belt which by the machinery is gradually moved firom one
end toward the other of the cones.

This piece of machinery is represented by Fig. 2. ab
is the belt; c is the guide, which, receiving its motion
from the machinery, traverses the belt at pleasure, mih
any velocity which the case may require.

Thus the one cone moving at a uniform motion, com-
municates a varying velocity to the other.

OBSERVATION.

422. This piece of mechanism, remarkable for its sim-
plicity, I have had occasion to put extensively in practice,
and have found it give great satisfaction.

III.<^-ALTERATION OP VELOCITY BY WHEELS MOVING ONE ANOTHER

BY FRICTION.

423. The same eflTect as the alternate cones is some-
times produced by the rim of one wheel moving on the face
of another, by means of the roughness of their surfaces,
the inequalities of which may be considered as indefinitely
small teeth.

Thus A B, Fig. 3, is a face- wheel, moving at a uniform
rate.

CD is another wheel, which, from the face of ab, re-

lAY VI.] VELOCITY OF MACHINERV. 337

ceives a vertical motion. Accordingly as it is required to
move CD slower or faster. It is by proper contrivances
ide to act nearer or further from the centre of a b.

f

W4A

OBSERVATIONS.

42'i. As there must a twisting motion, similar to that
of edge-stones for bruising various substances, take place
here, and as it is only properly applicable to cases in
which the strain is exceedingly small, 1 apprehend that
the alternate cones is a much more perfect manner of pro-
ducing a change of velocity. These wheels, however, work
verj' well for regulating the taking up motions of the bob-
bins in machines for ro\'ing cotton by spindles. In this
;, the force required is verj' small.

Certain eases in practice require an instantaneous

of velocity ; as, for instance, when those ma-

ines for spinning cotton, called Alules, are moved by

Irer.

jThc mule is a machine different in its construction

\ that brought to a high state of perfection by Sir R.

kwright.

The mule is better adapted than the water-ficist frame
(Arkwright's Machine) for spinning all kinds of weft, and
produces finer yam than can be spun by any other machine.
For the invention of the mvk we are indebted to James
Crompton, formerly of Hall-in-the-Wood, near 13oltun-Ie-

IDors, Lancashire, This machine was, for many years,
rked by hand only, the variety of its movements render-
f it difficult to accomplish the moving of it by power of
ter or of steam sufficiently simple to be of common use.

338 ON CHANGING THE [[eSS^T VL

William Kelly at Lanark, early obtained a patent for a
mode of working this machine by power, but it was not
until a considerable time afterward that power was gene-
rally adopted. The plans which were tried were very
various, and the improvement was progressive. One
happy consequence of this improvement has been expe-
rienced ; the spinners are now found to enjoy better health
than they did when they had to labour hard^ while they
breathed in warm and confined apartments.

In order to save time after the carriage of the mule is
brought to its furthest extent, it is necessary to increase
the velocity of the spindles. This increase of velocity is
called the double speed. Various contrivances have been
adopted for this purpose, but three only shall be described;
one being performed by r(^eSf another by belts^ the third
employing the aid of wheels. This last, indeed, firom its
greater certainty, is jvhat is most generally adopted. These
contrivances will, however, serve to shew the progress of
improvement in this species of machinery.

DOUBLB SPBBD.

First Canstniction.

426. The axle ab, Fig. 4, Plate XVII. is suspended by
a cast iron frame from the ceiling of the room. This axle
is kept in motion by means of the fixed pulley c, which is
moved by a belt from the mill- work. On the same axle are
two loose pulleys d and e. (Essay IV. Art. 280.) Ropes
from these pulleys communicate with the fast pulleys f and
G, on the axle x y of the fly wheel of the mule.

The loose pulleys have catches on their sides; while
these are disengaged the mule is at rest. In order to put
the mule in motion, the smaller pulley e, by means of the
sliding guide ikl, is slipped to one side, so as to lay hold

I

;AV VI.] VELOCITY OF MACHINERY. 339

of the glfuid H, which is fixed on the axle, and carries the
pulley round along with it, and thus moves the mule at its
slower motion.

When the fly wheel w has made a suflScient number of
revolutions at this rate, the slider is moved by peans of
wheel-work and a wiper toward the fast pulley a, which
motion disengages the small pulley from the gland, and
engages the larger pulley with c, which produces a quick
motion in the fly wheel.

OBSERVATIONS.

4^. This apparatus was in use in Manchester in the
jear 1797> but the shocks produced by the catches (Essay
Art. 281.) and other imperfections, soon occasioned
disuse. But there is often much to be learnt from the
examination of machines which have been abandoned. It
is but by comparison of things of the same species that we
are able to appreciate their true value.

b

DODBLB BPEBD.

Second ConOruction.

428. This apparatus differs from the First Construction,
principally in having belts instead of ropes for communi-
cating motion.

On the axle a b, there are five pulleys, e, d, c, g, h, all
of them loose but c, which is fast. When the belt from
the mill-work is on c, the mule is at rest, because the
axle revolves without carrying round any of the loose

In order to put the mule in motion, the belt is, by means
of a sliding guide, shifted on to the pulley d, which carries

840 ON CHANGING THE [^£88AT VI.

round the pulley x along with it ; and by another belt»
moves the pulley f on the fly wheel axle xy, and thus
moves the mule at its slower motion ; afterward (as was
described of the First Construction) the sliding guide
shifts the belt from d to o, which, by carrying round h in
a similar manner, produces a quick motion in the fly
wheel w.

OBSERVATION.

429* This construction was in use in Manchester m the
year 1799 ; and as the shocks complained of in the First
Construction did not occur in this, it was found a material
step in the improvement of working mules by power.

DOUBLE 8PBBD.

Third CanstructUm.

430. This construction differs from the second^ in having
the whole apparatus attached to the framing of the mule,
and in having the aid of toothed wheels for producing the
change of velocity.

On the axle a b, Fig. 6, are three pulleys, c, d, e. The
pulley c is fast on the axle, d and £ are loose, but on the
side of E is fixed the small spur-wheel f. The larger spur-
wheel G is fast on the axle ab.

On the axle x y are fixed other two spur-wheels, h and
I, of the same size as those on the axle a b, but placed so
as that the larger wheel on the one axle shall be constantly
in gear with the smaller on the other.

When the belt (put in motion by the mill-work) is on d,
the mule is at rest ; when shifted on to e it carries the
smaller wheel f round with it, which being in gear with
the larger wheel h, moves the fly wheel axle x t at its
slower motion.

S88AT VI.3 VELOCITY OF MACHINERY. 341

On the other hand, when the belt is shifted to the pul-
ley Cy which is &st on the axle, it carries round the larger
wheel G, which is also fast, o being in gear with the
smaller wheel i, moves the fly wheel at the greater velocity,
or, as it is termed, at double speed.

OBSERVATIONS.

431. This piece of mechanism was first adopted in Man-
chester about the year 1800, and although sometimes its
parts may be somewhat differently arranged, it continues,
I believe, still in general use.

While it is firee from the shocks produced by catches, it
is also (owing to having the change of velocity produced
by wheels) free from the uncertainties of motion arising
from any change in the tightness of the belts as employed
in the Second Construction.

ESSAY VII.

ON THE

FRAMING OF MILL-WORK.

PREFACE.

The preceding Essays relate principally to the moving
parts of macliinery, but as it seemed essential to a system
of mill- work, to say somewhat on the subject of the Jram-
ing which supports the moving parts, Buchanan was in-
duced to commit to paper the following ideas on that head.

SECTION I.

432. The general principles of carpentry must obviously
be applicable to the framing of mill-work. These prin-
ciples I shall not here repeat, but beg leave to refer to
what Professor Robison has written on this subject, in the
Encyclopaedia Britannica, and to Mr. Peter Nicholson's
various writings on the subject; I shall here consider only
the peculiarities of the framing of mill- work*.

433. Mill-work, from its motion, occasions a tremor on

* See also Art Carpentry, New Supplement to the Encyclopsedia Bri-
tannica. Tredgold's Elementary Principles of Carpentry, 4to. 1820; and
Practical Essay on Cast Iron, Svo. 1822.

lESSAY VII.3 ON THE FRAMING OP MILL-WORK. 343

ail the parts of its framing, which subjects it to much more
speedy decay than the mere pressure upon carpentry.

Besides this general tremor, it is often subjected to vio-
lent sudden thrusts, from the bad action of the wheels, or
from reciprocating motions.

It ought, therefore, not only to be sufficiently strong and
stiff, but sufficiently heavy, to give solidity and steadiness.

Where the framing of machinery is not firm and well
bound, a ^ibratorj' motion in its parts, of course, takes
place ; which vibratory motion expends a considerable por-
tion of the power applied. This loss of power ia very diffi-
cult of investigation. It is certain, however, that whatever
motion of a vibratory nature is communicated to the fram-
ing and objects in contact with it, (absfracting from the
elasticity of the parts,) must be lost to the effect the ma-
chine would produce, were the parts sufficiently strong and
well bound together ; and it is to be observed, that firm
and well-bound framing is much preferable to heavy fram-
ing not so well connected in its parts. It is as certain,
that though the framing in either case may be constructed
so as to be equally strong ; yet the heavy framing, from its
vibration, will expend more of the original power than that
which is less heavy but firmly connected.

434-. Besides strength, siiffnessy and solidity, the framing
of mill-work requires to be constructed so as to be e«.«y of
repair ; and so contrived, that any particular part may be
reftaired or renewed with the least possible derangement to
the other parts of the framing.

435. There is another circumstance in this species of

ing which demands great attention. The shajis often

tquire to be restored to Ihetr true situations, from which

* ihev may have deviated by the wearing of the parts. Now

the framing ought to be so constructed as easily to admit

of this restoration of' (he shttjis, as also of any other shift-

Liiig of them which may in practice become ncccseary.

344 OK THE FRAMING OF MILL-WORK. [E88AT VH.

436. But thougli the framing which supports the parts
of mills and machines should be firm, it is an advantage
that the part on which any axis rests should have a small
degree of elastic tremor when the machine is in motion.
Such tremor has considerable power in diminiRhing the
friction. It may further be observed, that framing to sup-
port machinery should be as independent of the building
as possible, because the tremor it always communicates is
exceedingly injurious.

Before proceeding further into the subject, it may be
proper to consider the bearings of shafts.

SECTION 11.

OP THB BEARINGS OF SHAFTS.

437- The bearings on which gudgeons and journals rest
and revolve, are sometimes termed Pillows^ and fi^uently
BrasseSf from being often made of that substance.

The bearings for pivots, at the lower extremity of up-
right shafts, are denominated Steps ; the parts where the
journals of vertical shafts or spindles turn and bear against
are called Bushes; and for small spindles, such as those
used in the manufactures of flax and cotton, Breasts.

It has become general to fix pillows in blocks of cast
iron. Hence the term Pillow Blacky and sometimes, cor-
ruptly. Plumber Block. In Manchester they are called
Pedest€ds.

438. The substances used for Pillows^ &c., are various,
but brass is the most common*. Other substances, how-
ever, which are cheaper, have in many instances been found
equal, at least, in durability.

* The metal our author terms brass, is usually the composition of copper
and tin, called gun metal. Gun metal is much harder than common brass,
and much more durable. Common brass is a compound of copper and
sine, and is now rarely used for bearings.

iSAy VU.] ON THE FRAMING OF MILL-WORK. 345

At the cotton works of Deanston, near Down, a water
wheel has nm nearly 30 years on pillows of cast iron, with
little sensible wear on the gudgeons, nor were they ever
found liable to heat".

The outer skin of cast iron, particularly when caat in
metallic moulds, is remarkably hard, and it is reasonable
to suppose that it would make a durable pillow, as we have
|seen is the case in the above instance.

Mr. Murray of Leeds was enabled to bore the hardest
1st iron, some cylinders of which, from the whiteness of
he grain at the places broken off by a chisel, denoted its
fuperior quality. Such iron is equally hard throughout.

A patent was granted long ago for wheel bushes of me-

, of a peculiar hardness, which proved to be nothing
' more than very hard cast iron, but the patentee had dis-
covered a mode of boring it, which Murray imitated.

Stone has often been used with good effect for pillows for
gudgeons and journals. The great objection to stones for
this purpose, is the difficulty, arising from their hardness,
of forming them into proper shapes.

At Sheffield, where the joumejTnen grinders are obliged
to keep this part of their machinery in oil, and in repair,
they liave found from long experience, that a piece of green
(unseasoned) thorn tree is exceedingly durable f . But in
general they prefer using brown paper, adding always one
ply some time after another, so as to form a kind of paste-
board ; this substance they find less liable to heat, and
much more durable than brass t.

• Heating geiieraUy lakes place from the surfaces of the journal and
pillow being too small, and sometinieB from the journal having worn too
deep into the pillow, in which last ease, in jiarticular, a great friction takes
jtloce.

t Qreen oak soaked in boiling oil, is said to be need with advantage,
r's Additions to Ferguson, vol. ii. p. 179.

} If the bearings for gudgeons were made hy screwing maay thickneases
of posteboftril together, in the same manner as the rollers of calendars are
iDule, they would be extremely durable, and have very little fricdon.

346 ON THE FRAMING OF MILL-WORK. [^£S8AT VII.

Wooden pillows ore often used. Box wood and lignum
yitsB were long in use. The latter ha^ been found an im-
proper substance for the purpose. Beech is preferable to
either, and has been used with great success for steps.
Holly has also been found to answer well for the same
purpose*.

•

FOBMS OP 8TBP8 AND OP PIVOTS OP UPBIOHT 8HAPT8.

4<S9« The most usual form is to terminate vertical shafts
somewhat like an egg^ leaving the pivot rather smaller up-
wards, to prevent it from binding in the step. (Plate
XVIII. Fig. 1.)

440. Another mode, (see Fig. 2,) is to terminate the
shaft flat at the lower end, and make a small oblong recess
in it. To this end is fitted a round piece of steel a, flat
below, excepting a groove a across it, something like that
in the head of a screw naiL The use of the groove is re-
gularly to feed in the oil. The step into which the pivot
works is made square or octagon, so as to supply oil from
the angles. Nos- 2, 3, and 4, Fig. 2, are diflFerent views of
the piece of steel a.

441. A pivot and step on this construction was deposited
with the Society for the Improvement of Arts, and was
said to have been 20 years in use in a horse-mill, without
being perceptibly worn. Buchanan applied this contriv-
ance in a case of heavy upright shafts, where other means

* Frictioii rollers are sometiiiies employed to diminish the quandtj of
friction, bat not with much adyantage in heavy machinery, becaase they are
liable to get out of order, and require very accurate workmanship. Their
effect depends on converting a sliding into a rolling motion, with only the
slide of the small axis which retains the roller in its place. Friction wheels
arc also used for the same purpose ; they differ from rollers in this — the
stress on the axis is borne by the axes of the friction wheels, but the stress
is on the surface of the rollers. The advantage of finctioii wheels is very
trifling.

AY VII.J ON THE FRAMING OF MILL-WORK.

347

3 found inadequate to prevent the action of the pivot,
t it ever after gave satisfaction in its use. It is to be
wn'ed, however, that it does not answer well in cases
I much lateral pressure, and a better mode (see Fig.
J is now in use in several mills. The steel pivot is made
Cylindrical, and fixed into the foot of the upright shaft.
The bottom of the step is convex, and there is interposed
one or more pieces of steel formed Uke a double convex
lens. These pieces of steel having a little motion, make
the relative motion of the pivot less, and consequently
ien the friction.

442. The egg-formed pivot is sometimes made separate,
wrought iron and steel, and inserted into the lower end
the cast iron shaft*.

443. The late Mr. Bramah, in the specification of his
tent machinery for surface planing, includes a mode of
Oning pivots entirely on a fluid, and raising or depressing

□ at pleasure, by means of a small forcing-pump and
> cock. See Gregory's Mechanics, vol. Ji. p. 418, 2d
Ltion.

444. The journals of upright shafts are supported some-
les by breasts, (Fig. 4,) and sometimes by bushes, in
ising through a floor.

445. The spindles of millstones usually run in wooden
hes. A block of cross elm, abc, Fig. 5, about 9 inches
meter, and 3 inches thick, forms the principal part of it,

d is lodged in the eye of the millstone. In order that
} spindle may at all times run steadily, there are three
Kes of hard wood, d, d, d, lot into grooves in the block,

' The rubting Burfaces of caat iron pWota ahouJd not have a greater
Bore upon them than one ton QpoQ a square inch, or they vill he very
ject to heat, and the friction and weai will he increased. Large vertical
^ may often be made to revolve on conical rollers, on the same prin-
e aa fnctiDn rollers; when they nro well made, the motion is very
and the friction much reduced.

346 om TBE PBAJOVG or xux^-wobk. [bmat vil

•o that tbeir three ends embraee die ^indies. TheeepieoeB
are of equal breadth throaghont, to that thej may easify
be wedged fiirwafd when dinr wear. This flimple and in-
genioiis cootriraiioe has been vor long in use. Some use
apiece ci cast iron in {ireAsrence to die Uock ci efan, to
answer the same pivpose^ and some a greased ntpe to nm
the spindle in, instead <rf die dnree pieces ci wood.

SECTION m.

09 WOODSH FSAMniO.

446. The framing used for suppcnrting die gudgeons of
a water wheel is denominated the headstock firaming.

Af Fig. 6, represents the headstock, which omtains the
pillow-block, BODE is the frame.

The frame is supported by the top of the building of the
arch or wheel-pit

The headstock rests on the frame, and is moveable hori-
zontally on all sides ; when in its proper situation, it is
kept there by a key or wedge g at each end of the head-
stock, which is supported by the dovetail form of the lower
])art of the headstock.

FRAMING FOR LYING SHAFTS.

447. Lying shafts are usually supported by bridges be-
tweon posts, a b. Fig. 7, represents part of a lying shaft,
c 1) the bridge which carries the shaft. It is raised or
lowenni by means of a key below, and another above it, in
the mortice of the posts eg and hi, through which it
passes.

The bridge is also moveable on end, and is kept in its
pn>iH>r situation by means of small keys k, k, k, k.

lAT VII.]

ON THE FHAMING OF MILL-WORK.

3-1.9

448. The posU, instead of being; each made of one solid
piece of timber, are sometimes framed of separate pieces,
as represented in Fig. 8.

LjTUg shafts, instead of being supported by posts, are
sometimes suspended from a ceiling, as shewn in Fig, 9-
The bridge is tempered by keys, &c., as when posts are
used.

J FOB UPtllOHI SHAFTS.

449- Upright shafts are generally supported by bridges
adjusted endwise, and upward and downward, like those of
the lying shaft ; hut in order that they may be moved hori-
zontally in every direction, the pedestal is contrived to
deceive keys at its ends, similar to a hcadstock. (See

Wt '"'^

^Bocrews arc frequently used instead of wedges for adjust-
ing the step.

450. Fig. 1 1 represents the framing of an upright and
^UDg shaft connected by bevelled wheels.
^■^1. Sometimes a bridge is not immediately supported
^Pposts, but by intermediate pieces, which are called
cJoves. This construction is common in single corn-mills.
Thus, AB, Fig. 12, is a bridge; cd and ef arc cloves.

452. Respecting the decay of timber, and the means of
preventing it, Buchanan refers to Dr. Parry's paper, in
the Transactions of the Bath Agricultural Society, and re-
printed in Nicholson's Journal, Vol. XX. Nos. 85, 86, 87»
;uid Repertory, No. LXIII. That paper appears well
worthy the attention of those who have occasion to con-
struct works of timber*. Kyan's mode of preserving
nber, till a better shall be proposed, now supersedes all
VschemoB heretofore promulgated.

" Blementdty Principles of Carpeutiy," Sect. X. Art. 327—

J

350 ON THE FRAMING OF MILL-WORK. [eMAT YIL

SECTION IV.

OF CAST IBON FBAMINO.

453. In a preyious part of this work, mention lias been
made of the great increase of late years of the use d
cast iron in mill-work.

Cast iron possesses great superiority over timber, for
constructing the framing of mill-work. It is not only
much more durable, but &t)m the uniformity of its texture,
may be converted into any shape, so as to give it great adU
vantage in arranging the materials with respect to strength,
and proportioning it to the stress it has to sustain. Tim-
ber, on the other hand, being stronger in some directions
than others, and of very limited breadth, is confined in its
arrangement, and requires, in certain cases, much work-
manship ; whereas, after the patterns for cast iron are once
made, any number of castings may be formed from them
with very little labour or trouble.

Those who have scientifically considered the strength
and stress of materials, know that when timber is broken
by any lateral pressure*, it is owing in a considerable de-
gree to the compression of the beam on the hollow side,
which puts the fulcrum of the ideal lever much nearer the
point of resistance than it would be in a substance less
liable to compression. Cast iron is much less liable to
compression than timber, which gives it an advantage in
withstanding lateral pressure, greater than might be ex-

* See Emerson's Mechanics, Sect. VIII. p. 93, and Gregory's Mecha-
nics, Vol. I. Book I. Chap. V.

These references must have heen made without consulting the works
quoted, as the investigation of the strength of heams is conducted by both
these writers on the supposition that the virtual fulcrum is an incompres-
sible arris at one of the surfaces of a beam ; and the one conadere the
materials to be extensible, the other inextensible.

lAT VII.] ON THE FRAMING OF MILL-WORK. 351

■jed from the mere comparative absolute cohesion of the
wtances'.

54. " Iron is generally much more uniform in its

. than wood; yet experiments shew that there is

i difference occasioned by different kinds of ore : the

ference is not only found in iron from different furnaces,

t from the same furnace and the same melting ; this may

! in a great measure from the different degrees of heat

,ch it has when it is poured into the inould."t

Banks concludes that a bar of the weakest cast
, 1 inch square, and 1 foot long, will break with about
., and that cast iron is at an average 4 times as
strong as oak, and 5^ times as strong as deal; the weight
being in all cases applied in the middle ; the beam lying
horizontally, and supported by props J,

■ls5G. The strength of any beam, to withstand any weight,
being as the breadth and the square of the depth, (see
Essay II. on the Shafts of Mills, &c.. Chap. IV. Emerson,
p. 93.) it is evident that a bar of cast iron of the same
length, must be much stronger when its tranverse section
is like Fig. 14, than when like Fig. 13. The form repre-
sented in Fig. 14 has a further advantage, that of greater
stiffness. The distinction between strength and stiffness
is not in practice generally understood or attended to. This
distinction is most easily comprehended by considering their
limits. The limit of strength, is well known to be frac-
ture, or breaking. The limit of stiflfhess, is Jiexure, or
bending. Now stifihess increases in a much higher ratio

• The idea that the yirtmil fulcrum is nearer the compressed side in cast
iron ihaa in wood, when the pieces are similarly strained, is at best an na-
aenion without a proof, either from theory or experience. And at the time
oar author wrote, the resistance of cast iron to compression was greatly
uvemtad. See Esaay on Cast Iron, Art, 63.

^^ t On^ry's Mechanics, Vol. I. Art. A. 190.

^^L% Banks on Power of Machines, &c., p. 94.

d5S ON THE FRAMING OF MILL-WORK. [eSSAT TU.

ratio than strength, viz,, as the cube of the depth •• For
example, if we double the depth of a beam, we increase its
stress only 4 times, whereas we increase its stiffness 8
times. (See Essay II, on the Shafts of Mills.)

457- For these reasons, the advantage is evident of
making cast iron framing in thin broad plates, at right an-
gles to one another, instead of imitating the solid forms of
wooden framing. This practice is called by millwrights
feathering. The plate is sometimes on one side, as repre
sented in Fig. 15, and sometimes its section is like the
letter t, see Fig. 14, the whole being one solid mass.

The common practice in making cast iron framing now,
is to imitate wooden framing, which has been found from
experience sufficiently strong in giving the same breadth
and depth of the several pieces at their point of greatest
stress. Thus, suppose Fig. 16 to be the section of the
timber at the place of greatest stress, the section of the
cast iron is made like Fig. 17» or like Fig. 18 ; advantage
is also taken of the nature of the material, to give it a
breadth varying in proportion to the stress. This variation
in shape is not always in practice judiciously done ; by at-
tending to what is said in Essay II., Chap. IV., the mill-
wright, it is hoped, will be better enabled to proportion the
parts to the stress they have to sustain t. In addition to
what is said respecting the making of patterns, in Essay
I. on the Teeth of Wheels, Buchanan says it is a good
practice to give the patterns a thin coat of oil paint ; as,
while it preserves the pattern, the paint makes it rise
more freely out of the sand.

* Young's Nat. Phil. Vol. ii. Art. 333.

t The most advantageous forms for different purposes have heen con-
sidered in the Practical Essay on Cast Iron^ Sect. III. and IV., where ex-
tensive tables of the strength and stiffness of cast iron will be found, which
may frequently save the millwright much trouble in calculation ; for he
cannot always have examples of the same construction to refer to, either
executed in wood or iron.

ESSAY Vll.] ON THE FRAMING OF MILL-WORK. 353

458. To give an instance of this variation of form in the
framing of mill-work, bridges of wood for sustaining the
shafts are usually made as represented in Fig. 19, those of
cast iron as shewn in Fig. 20.

4s59. In cast iron framing, advantage is also taken of
the properties of the hollow cylinder, of the economical
ilication of which form in nature we have so many beau-
tiful examples. (See Essay II.)

460. A headstock of cast iron for a water-wheel is re-
presented in Fig. 21, Plate XIX.

461. Various modes are used of suspentling shafts from a
ceiling. Fig. 22 represents a construction in very general
practice.

462. Fig. 23 represents the cast-iron framing of a flour
mill having three pair of mUl-stones, and to Fig. 24', a ma-
chine used in bleaching, called Squeezers.

After the process of washing by the dash-whoel, the
iter is compressed from the cloth by means of this ma-

Kfiluni
■ So

Squeezers consist of a pair of wooden rollers, which, in
mo\Tng, draw the cloth through between them. The lower
roller receives its motion from a mill, and the uppermost
is pressed down upon it by means of levers. Till of late
these rollers were fixed in strong wooden frames ; hut the
framing is now generally made of cast iron, which makes
a neater and more durable piece of work.

A represents the lower roller, b the upper roller, c D a
lever which presses upon the brass of the upper roller, e f
another lever to increase the power connected with cd.
The extremity of f is kept down by a pin j in some cases a
ht is used in place of the pin.

ESSAY VIII.

GEOMETRICAL AND PBACTICAL METHODS

lOK rataaic thb

CENTRES OF GRAVITY OF MILL WHEELS;

ILLU8TRATXD BT BZAMPLBS,

IN WHICH TWQ, THREE, AND FOUR WHEELS COMPRISE THE
SYSTEM UPON ONE AND THE SAME SHAFT.

463. There is one branch of mechanical science which
belongs essentiallj to mill-work that must be here added
to these Essays of Buchanan. We allude to Methods of
finding the Centre of Gravity of two or more bodies con-
nected together by straight inflexible rods passing through
their respective centres-
Suppose A and B to be two bodies connected together by

Fig. 1.

Q o

the straight inflexible bar a b passing through their centres,
and it were required to find the centre of gravity of those
bodies.

At the points a and b, Fig. 2, we should erect the per-

FiG. 2.

ESSAY VIII.] CENTRES OF GRAVITY OF MILL WHEELS. 355

pendiculars a c and b d of any convenient length, and through
c draw CD parallel to ab ; then we should produce ac to f,
Fig. 3, and make c e to ef as the body b is to the body a. Then
joining fd, and through e drawing eh parallel to fd, and

Fig. 3.

firom H dropping the perpendicular h g, the point g would
indicate the centre of gravity of the two bodies a and b ;

for A : B :: bg : ag
hence equating the products of the extreme and mean
terms

A. AG 3sB.bg

From which we infer, that when two bodies connected to-
gether by a straight inflexible bar, are in equilibrio, the
products of their masses multiplied by their respective dis-
tanceSy are equal

Let a = the mass of a
b ss the mass of b
d ss the distance ag
S = the distance bg
and D = the distance ab
Then according to the foregoing proportion,

ad ^ bSi but S = i} ^ d
consequently, ad^hi} -^ hd\ and

d = J , j> : also 8 = — —r-

356 ON THE CENTRES OF GRAVITY [[sSftAT Vm.

Consequently, the places of the centre of gravity is known
in terms of the masses, and the distance between their re-
spective centres. Hence the following practical role :

464. Multiph/ either body hy the whole distance be-
tween their centres ; divide the prodtLct hy the sum of
the bodies ; the quotient will be the distance from the
centre of gravity of that body opposite to the one by which
the whole distance is multiplied.

Example. Let the two bodies be respectively 4 and 7
cwt. ; and their distance asunder 24 feet.

Fio. 4.

B H A

G \ O

7 *

7 X 24
Then we have a = 4, J = 7> and d = 24, or -7 — — =

168

-r-r = 15^ feet, being the distance of the centre of gravity

from the body a.

4 X 24 96
Also -^ ry' = 11'^ ^^ ^*^^*' being the distance of the

centre of gravity from the body b.

465. The example supposes the connecting rod to be
void of weight ; but in mechanics this is never the case.
The same law must obtain, with respect to the portions of
the connecting rod, that we saw existing in the mass of
each body multiplied into its distance from the common
centre of gravity. The centre of gravity of an imiform
connecting bar must be at the middle of its length when
that bar is prismatic or cylindrical.

If ^ = mass or weight of one unit or length of the bar,

then is ^-5- = eflFective energy of one portion, and ^--r =

the eflFective energy of the other; and these, together with
the eflFective energies of the bodies a and b referred to op-

ESSAY VIlI.j

posite sides of the centre of gravity, must still be in equi-
librio ; hence

arf +^ = AS + ^
But 8 = D - (^ and therefore by substitution we obtain
(a + fi + p-D) rf = (6 + t2-) D,

which being reduced ^ves the following equations
rf_ (S&+;)d)d
2 (a + 6 + p rf)

2(o + i + pd)'
Hence the following practical rule :

466. To twice the weight of either body, add the
whole weight of the lever or connecting bar, and multiply
the sum by tfie central distance ; then divide the pro-
duct by twice the mass cmnpounded of the bodies and tlie
bar, and ths quotient will be the distance of the centre of
gravity from that body opposite to the one whose double
is employed in the first step of the operation.

Example 1. — The bar is 24- feet, and weighs 1 cwt.,
the bodies 4 and 7 cwt. respectively as before ;

Then a=4; b = T; d = ^4- feet, and;) s^jcwt. .■. d =
(a X 7 + Vt X g4)Q4 15 X 24 360

2 (♦ + 7 + A X 24) ~ 2 (4 + 7 + 1) ~ 24 ~ ^^ ^^
being the distance of the centre of gravity from a, and
therefore 24 — 15 = 9 feet, the distance of b from the
centre of gravity; for 9 + 15 = 24 feet.

Example 2. — Let the bar be of cast iron, 42 feet long
and 252 lbs. weight; the bodies at its extremities weighing
13440 and 17920 Iba. respectively.

It will be found by calculation that a = 13440 and b =
17920, are respectively 23-ifH feet, and ISrlir feet from
the common centre of gravity of the bodies.

467. When three bodies connected together by a straight

S5S an the csbtbbs or gkayitt [essay vui.

mfleiible bar, are in eqoililHio^ die product of one body
nmltqdied hj its distance from the ccmimon centre of gra-
vis of the STSleBy is equal to the product which arises
when the smn of the other two bodies is multiplied by the
distance hetmeai their common centre, and that to irhidi
the whole system is referred*.

If a = mass of the body a ;
b == mass of the hoAj b;
l> = mass of |> concentrated in |> ;
d = distance between a and b ;
fi = distance between a and/i,
and X = A H the distance between a and the common

c^itre H.

Fio. 5.

' 1

Then if h fiJls between a and />» pH := 8 — :r, and hb =
rf — x; but if the common centre falls between b sndp,
we have pn = or — S, and bh ^ d -- x\ and in either
case we have

(a + 6 + p)x =^ bd -h p8

(bd +jpS)

or JT =

(a + 6 + jo)

468. The practical rule is the following :

Multiply each of the bodies b and p by the respective
distances from a ; then divide the sum of the products hy
the aggregate of the three masses for the distance of the
centre of gravity from the first body a, to which the dis-
tance of the other bodies b and p are referred.

Example. — ^Let the bodies be 1 5, 20, 25 tons respeo
tively ; and their distance 12 and 16 feet from each other;
then it will be found that

ftc? = 700; pS = 240; mdbd -{-pS = 940

* Dr. Jaoiiesons Mechanics for Practical Men. Loodoiiy 1837, Svo.

BSAT viir.3

OF MILL WHEELS.

359

but (a + b + p) = GO; therefore z is distant from a by

-qq = 15f feet ; x — S = 3f = the distance of j: fromp ;

1 d — j: = 12^ feet = distance from b : that is to say ;
AH = 15§ feet, or the distance of a from h
pu = 3f feet, or the distance o( p firom h
and BH = 12J feet, or the distance of b from h.
Hence ah + hb = 15| + 12^ = 28 = l6 + 12 feet.
469- These results Dr. Jamieson verifies by the follow-
ing construction in his "Mechanics^ Practical Men."'

Fia. 6,

Draw the straight line ab, and from a scale of equal
"parts make Ap = 12 and ps = l6 feet j through the point
B draw the straight line bf in any direction with respect to
AB ; make be = 20, and ef = 25, the numbers which re-
spectively express the magnitudes of the bodies ^> and b
acting on the straight line a b, at the points p and b ; join
rp, and through the point e draw eg parallel to Fp, which
produce to c, and makcGD = 15, the number which cx-
iresses the magnitude of the body a acting at a, and make
be = 45, the number = sum of p and b acting at g : join

* Article, Centre of Gmvitj, pp. IS, 20.

360 ON THE CENTRES OF GRAVITT []eSSAT VHI.

c A, and through d draw dh parallel to c a; theA is h the
place of the centre of gravity of the three forces Oy p^ and
hy acting at the points a, p^ and b of the har ab ; and ah,
pay and b h, their respective distances, which if measured
from the scale will be found equal to 15f, 3f, and IS^
feet respectively.

Workmen may be informed, that in constructions of this
kind it is not necessary to take the numbers which express
the magnitudes of the bodies from the same scale as those
which express their relative distances ; for since they are
magnitudes dissimilar to one another, they cannot be com-
pared ; consequently the ratio or proportion of the numbers
is all that we require : all the magnitudes of the same kind
must however be taken from the same scale.

This remark is made because some of the foregoing
numbers express weight, others lineal measure ; in setting
off their relations we used the same scale for all ; but this
is not necessary.

470. The cases of utility consistent with this theorem
are only three ; viz.

1. When p is less than a + 6, but such that a -h j» is
greater than 6, and h + p greater than a j

2. When p is equal to a + 6 ;

3. When p is greater than a + J.

The equation of equilibrium is the following, which we
borrow from the " Mechanics for Practical Men."

d

The following examples are given to show persons un-
acquainted with algebra how they may apply the principle
now before them.

Example 1. — At the extremities of an iron shaft 22
feet long are fixed two wheels, a and 6, respectively 2 and
2^ cwt. ; and somewhere between these another wheel, />,
is fixed, 1^ cwt ; at what distance from each of the ex-

I ESSAY VIIJ.3

OF MILL WHEELS.

361

treme wheels must the intermediate one be fixed, so tliat
the whole weight may come upon the middle of the shaft,
I when it is supported by a transverse bearer h ?

I

Here a = Q; p = 1'5; b = 2*5, and rf = 22 teat, sup-
posed to be the distance between the centres of the extreme
wheels ; then since the shaft is supported on its gudgeons
■at the extremities, and on the journal at the transverse
bearer, we may consider it as having no effect upon the sys-
tem of wheels as regards the place of the centre of gravity ;
therefore by substituting the above numbers in the fore-
going equation, we have

^^ (2 + 1-5 ~ 2-5} = ?? X 1 = 74 feet :
X 1-5^ \ 3 '3 '

being the distance of the lighter wheel a from p ; but the
middle of the shaft is 1 1 feet from a or i ; therefore we
have 11 — 7^ = 3| for the distance of;* from the journal ;
id 11 + 3| = Hf for its distance from the greater
'heel b,
4'71. When the distance is known or limited by situa-
tion, as in practice frequently happens to be the case, the

gpB

'Here wo have given, as is plain from the terms on the right

I equation becomes
Here wo have give:

36S

ON -THE CBNTRES OF OEATITT

[»

hand ride of the equation, the magnitudee of the three
bodies a, p and h, acting in the same straight line, and the
distance between the middle body p, and the first extreme
a ; and we are required to find d, the distance between the
extreme bodies a and b, and that the common centre of
gravity, or the centre of the system shall fall at the middle
of that distance.

473. Suppose then for illostration of the case we take
the following

Example. — The shaft of a mill-wheel has to sustain
three wheels of the weights of % 11, and 3 cwts. ; what must
be the length of the shaft from centre to centre of the ex-
treme wheels, in order that a transverse girder placed at
the middle of its length shall release the gudgeons from
the pressure and sustain the system at rest, the distance
between the first extreme and the intermediate wheels
being 12 feeL

Here we have ^ven by the question, a = 2, f> = 7, i =
3, and S = 12; then writing these numbers for their con-
stituents in the equation, we have

2 X 7 X 12 168

rf = ,

= 28 feet.

1+7- 3 "
for the distance between centre and centre of the extreme

KS8AT VIII.j

OF MILL WRBBLS.

363

= 4-8 feet.

wheels a and h ; consequently the place we must assign as
the common centre of gravity is It feet from either ex-
:^me end, and 2 feet from the place of p the intermediate
|rheeL

I 473. To verify this result, we may compute the place of
> common centre of gravity of the two wheels p and h by
iie first problem, in which case we have
16x3
7 + 3 '
s the distance from p, consequently the distance between
the centre of the system and that of the two bodies p and b
is 4*8 — 2 = 2*8 feet ; then reasoning by the previous

tustration, we have
14a = 2-8 (p + b), that is
14 X 2 = 2-8 X 10 = 28, as before.
r, to numerous machinists who are masters of algebra, if
I put X = the distance of each extreme wheel from the
centre of the shaft, then 2 j: = the whole length of the
shaft, and jr — 12 = the distance of the intermediate

Iieel ; hence
2x + (_x -IQ) x7 = 3x
or 7.r - 84 = X ; 1. e. 6 .r = 84 ;
jrefore x = 14, and 14 x 2 = 28 as before.
474. When the weight of the axle of the wheels is given,
we may adopt 7c to express that element. Then, if the bar
be of uniform shape and density, the centre of gravity of
each segment made by the centre of the system, will occur
at the middle of its length, and the weight of the segment
^^riil be expressed by wx, and iv (d — x) respectively. The
^BpTective strength of the energies is then
^f 4 tci-*, and ^ w (d—x)\

consequently in the case of an equilibrium, we shall have
ax -\- p (J'-S) + ^wx^ = b (rf-.r) + ^ w (d-x)*,
+ 4 M^r" = b (d-x) + p (S-.r) + ^w (rf-.r)*;

364

ON THE CENTRES OF GRAVITY. [eSSAY IU.

but in either case, when the equations are properly re-
duced, we 6nd generally that

^_(g& + w d)d + gpg
2(a + i+j> + «!(/)'
The following practical example will bring this compli-
cated equation into a readable form, better than could be
1 by a rule.

FiQ. 9.

Example. — A cast iron shaft, 4 inches square, and 3Q feet
long between gudgeon and gudgeon, is required to sustain
three wheels, whose weights are 4, 7, and 6 cwt. respect-
ively, placed at the distance of 14 and 33 feet from each
other. At what point of the shait must an iipright be
placed to remove the pressure entirely from the gudgeon^
and balance the shaft with all its apparatus.

Here we have given the wheels a = 4, j) n 7, and ft b
6, and the shaft d = 36, also S = 14. Writii^ then
numbers for their correspondent symbols in the finregonig
equation, we shall have

(g X 6 + 36 «?) 36 + g X 7 X 14
^' 2 (4 + 6+7+36 w) '

Now since the material of which the shaft it nuidB 3|
cast iron, the weight of 1 foot in length, or the value of w
is 4 X 4 X 3'2 = dl*3lb6.*; conseqaently, by substi-

* THe wd^t of a bar of cast iron one inidi aqona and 1 2 inches loug,
ia 3-S Iba., the mnldpUer nnd m the qoeitton.

ESSAY VIll.] OF MILL WHEELS.

toting 51-2 instead of w in the foregoing value of x, we
shall obtain

.(^

36

-51-2) X 36 + g X 7 .

2(4 + 6 + 7+ 36 X 51-«)
18 feet very nearly.

Therefore, the place of the support is at 18 feet from

each of the extreme wheels, and 4 feet from the inter-

I'lnediate one ; but if the weight of the shaft had not been

ken into the estimate, we should have had x = 18 -j^-

hence the effect which this element produces is, to

ice the support ^ of a foot, or very nearly 6 inches more

B way than the other, a quantity which in large construc-

ttiB may be disregarded. But it was necessary to shew

gtiiat we ahould not consider the axle void of weight in our

* calculations, especially where their accuracy may be tested

by other persons who would not allow this element to be

thrown out of the equation of equilibrium.

H Of the centre of gravity of Jour or more bodies situated
H in the same right line.

^^ 475. This is but an extension of the previous case : in-
^^eed the law of continuation is so obvious, that we shall
make one example suffice for its illustration ; but to make
the way smooth, let a, p, n, b, represent the four bodies
I taken in order, from a the first, to b the last.

¥

i-+-i F

Let i denote the distance from « to 7^, and S^the dis-
tance from « to n, and d the distance from a to i. Also
X denote the distance from a to the place of the com.

centre of the whole mass.
If then the bodies a and p are on one side of the com-
mon centre, while the other two bodies » and b are situate
the other side of that centre, we shall have x ; (.r — t) ;

tanci
^niion

_4ai the other side 01 t

366 ON THE CENTRES OF GRAVITY |^£88AT TID.

(fi' — or), and (^d — x) for the respectiye distances of Ad
bodies from the centre of gravity ; consequently, by the
principle ah*eady indicated, we have

a ^ + p (^ — S) = n (8^ — or) -f 6 (rf — x) ;
which by transposition and division gives

^a + p + n+6''
And we may write this equation thus, for the benefit of
such readers as may require its meaning in words at
length.

476. Rule. — Multiply the magnitude or density of each
body by its respective distance from the beginning of the
system^ and divide the sum of the products by the sum of
the bodies for the distance of the centre of gravity sought.

Example. — Four bodies connected by a straight inflex-
ible bar, have their weights respectively, 18, 26, 12, and
30 cwt. ; and their distances from each other are.

From a to |7, I7 feet,
a to 71, 23 ditto,
a to i, 40 ditto.

At what point in the length of the bar shall the common
centre of gravity be marked ?
Here we have given

a = 18; 71 = 12; S = 17; , ^

^ = 26; 6 = 30; S' = 23 ; ^^^ ^ = ^'

Let these values of the elements of the system be sub-
stituted in lieu of the symbols in the foregoing equation,
and it reads

r = ^^ X 17 + 12 X 23 -h 30 X 40 _ ^^ ^

18+26 + 12+3 ^"^

feet from a\ 5*3 feet from p ; iV from n ; and I77
from b.

477. We shall now exhibit the principle of continuation

BSSAT Vni.3

OF MILL WHEELS.

S67

by a geometrical construction, in which the reader may
trace with great facility the combinations involved in the
equation we have just worked out for him.

Let A B be a straight line passing through the centres of
the four bodies, a, p, n, b.

Fig. 10.

Make a b = 40 feet, taken from a scale of equal parts.
On the straight line a b set off Ap and a n equal respect-
ively to 17 and 23 feet, taken from the same scale as ab.
Then are the points a, p, n, and b, the positions of the
four bodies, the weights of which constitute the elements
of the question, and whose common centre of gravity we
shall now trace by completing the construction of the dia-
gram.

Through the point b draw the straight line b f in any
direction at pleasure ; make b e proportional to the weight
of the body n, and ef to that of the body b ; join fti ; and
through the point e draw eg parallel to rn ; then is the
point G the common centre of gravity of the bodies b
and 71.

Next produce e g to c, making g d proportional to the
weight of the body p, and do = bf, or proportional to
the sum of the bodies 6 and n i join cpf and through the

bb2

368 ON THE CENTBES OF GRAVITY [bSSAT TQI.

pdnt D, draw oh parallel to c^. Then is the prant a the
common centre of gravity of the three hodies, p, n, b.

Finally, produce dh to the point e, making hi pn^-
tional to the weight of the body a, and ik equal to go, cr
proportional to the sum of the bodies p^ n, and b ; ysa
KA, and through the point i, draw ix parallel to ka \ thai
shaU the point x on the line ab be the common centre
of gravity sought'.

For we have bf = n + 6-; an = d — fij ep = &}

, i (rf - 8) + B (y - 8). ^,

n + 6 *

_ hd+pt + ng.
n + ;» + 6 '

Ki =p + n + 6; and hx = \ t-E — '*" "1 an eqiia-

( o +;» + n + b i

tion which is identical with that from which we demosD-
strated the example, and which we shall now turn to ac-
count in the solution of another bearing immediately on
the subject of this essay, and with which also it is our in-
tentioD to bring it to a close.

p + M + 6 ; g;> =

+ P + n + 6;

as n + i: HK

Example. — Four cast iron wheels, the weights of which
are respectively -t, 5, 6, and 7 cwts., are fixed upon a shaft
at the scraral distances of 8, 10, and 12 feet apart, taken

* In Dr. Jainiesoii's " Mecluuiin for Practic*] Men,' there ia ta elc«u>t
dratoiutntian of thii coattnictiou.

ESSAY VIII.] OF MILL WHEELS. 869

in order ; then, if the shaft have no influence upon its po-
sition, what is the distance of each wheel from the common
centre of gravity of the system ?

There are here supplied by the question, seven terms of
its elements, and the eighth is to be found thus, agreeably
to the foregoing equation ; viz.,

a = 4;jE> = 5;n = 6;6 = 7;S = 8;8' = 18;
and d ^ SO; hence, if we substitute these numerical values
for their corresponding symbols, we shall have

^ 5x8+6x18+7x80 ,^,

X = = lO Ti

4 + 5+7
feet from the body a; 8i^ fi^m p^ li\- from n; and
18A fi^ni 6.

The numerical operation deducible from the geometrical
construction, furnishes three elegant proportions. For by
gunilar triangles, beg and BPn, we have

BF : Bn : : EF : no ; that is
(n + b) : (d - O : : 6 : no = *IfLl^; which

arithmetically becomes

6 + 7 : 80 - 18 : : 7 : MG = ?| = 6-|.

Again, in the similar triangles hdg and pcG, we have
GC : Gj) : : DC : pH ; that is

(p + n + b) : — i^ ^ r^^ -^ : : » + ft : oh

^^ ^ n + ft ^

p -h n -{• b
arithmetically is written

5 + 6 + 7 : ^ + (18 - 8) : : 6 + 7 :i>H =^ = ll|

Finally, in the similar triangles hio; and hka, we have
the following proportion,

370 ON THE CENTRES OF ORAVITT f ESSAY Vm.

HK : HA : : Ki : Ax; that is,

(a + p + n + 6 : fifL±-£l±^l : : (« + n + 6) : a j;

I /> + » + 6 J

which by equating the product of the extremes and means
gives the elegant equation that preceded the example, and
which arithmetically is written out thus :

(4 + 5 + 6 + 7) : (^ + 8) : : (5 + 6 + 7) : ax = ^;

consequently by division the fraction becomes ax = 16^
feet, the same as before.

478. Thus we have traced fit)m principles of the greatest
sunplicity the theory of the common centre of gravity for
so much of null- work as has reference to systems of wheels
arming the same axle. We shall close this article by re-
marking in reference to bevel gear, that

1. The centre of gravity of the surface of a cone is the
same as the centre of gravity of its triangular section.
And the centre of gravity of a right cone is situated at
f ths of the axis from the vertex : or ^th from the base of
the cone. The same holds good of any pyramid whose
base is a polygon.

2. The centre of gravity of the surface of a conic frus-
tum is the same as the centre of gravity of the trapezoid
formed by a plane passing along the axis. And the centre
of gravity of the conic frustum, when its height and the
diameters of the two ends are given, is detemuned by the
following equation,

Where h is the height of the conic frustum ; r is the
radius of the less end ; r the radius of the greater ; and S
represents the distance of the centre from the less end.
The practical rule is this :

I ESSAY Vltl.]

OF MILL WHEELS.

371

Tq the gum of the sfjxiares of the rndii of the two ends
add their product, then multiply the sum by 4, aiid reserve
the result for a divisor.

To three times the square of the radius of the greater
, add the square of the radius of the less end, toge-
• with twice tfie product of the radii, and multiply
%^ sum hy the height of the frustum for a diimlend.

Tlien, divide the dividend by the reserved divisor, and
the quotient will express the distance between the centre of
magnitude of the less end, nnd tlie centre of gravity qftlie
istwn.

S. The centre of granty of the surface of a cylinder is
the same as the centre of gravity of the parallelograni
made hy the plane passing through the axis.
_ 4. The distance of the centre of gravity of a circular
^birc from the centre of the circle is a fourth proportional to
^nhe length of the are, the radius of the circle, and the
Htdiord of the arc.

^1 5. The ordinate of a common parabola is a mean pro-
portional between the abscissa and the parameter of the
axis ; and the position of the centre of gravity is in the
axis of the figure, and at the distance of three fifths of the
jscisaa from the vertex.

y 6. The centre of gravity of any semiparabola occurs in
ke ordinate of the axis, passing through the centre of gra-
Hty of the whole parabola,

7. The distance between the vertex and the centre of
gravity of a parabolic conoid, is equal to two thirds of the
axis.
^B There are several other figures that occur in mill-work,
|Bint the discussion we have entered into has spun out much
^twyond the limit we had assigned to it, and we are there-
fore compelled to refer the reader to other treatises, which
Ipter into the composition, revolution, and properties of

372 CENTRES OF ORAVITY OF MILL WHEELS. [»S8AT YUI.

bodies in motioiiy for all such matter as should be known to
complete the education of a sound millwright.

479- The tables of squares and cubes which are annexed
will be acceptable, as also the square roots and cube roots
of all numbers from 1 to 1000, which have been taken
from Hutton's ** Course of Mathematics,'' and will be found
very useful on many occasions.

■

■

373

Squ»e.

Cube.

SquueRoot.

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1-687401

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25

125

2-2360680

1-709976

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36

216

2 '4404897

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121

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3-4641016

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169

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13-2287666

5-593445

176

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5-604079

177

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5-614673

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6-738794

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13-7840488

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13-8202760

6-768966

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13-8664066

6-768998

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13-9283883

6-788960

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14-0366688

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17-0587221

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806404

724160702

200600481

9-647736

800

808201

720572000

200833287

9-661316

000

810000

720000000

30-0000000

9-664898

001

811iU)l

731432701

30-0100020

9-668468

002

813604

733870808

30-0333148

9-662040

003

816400

730314327

30-0400584

9-666609

004

817216

738703204

30-0005028

0-669176

906

810026

741217026

30-0832170

9-672740

006

820836

743077410

30-0008330

9-676801

007

822040

740142043

30-1104407

9-679860

008

824404

748013312

30-1330383

9-688416

000

820281

761080420

301490200

9-686970

010

828100

763571000

30-166-2063

9-690631

Oil

820021

760068031

30-1827766

9-694009

012

831744

768660628

30-1993377

9-697616

Old

833600

701048407

30-2168809

9-701168

014

836300

703661044

30-2324329

9-704608

016

837226

706000876

30-2480669

9-708286

016

830060

768676206

30-2664919

9-711773

017

840880

771006213

30-2820079

9^16806

018

842724

773620632

30-2086148

8^18886

1

m

m

^

^

1

Nmnber.

Square.

Cube.

Square Root

CuW RooL

919

844361

776151550

30-3130128

0-722383

I

920

846400

778088000

30-3315018

9-725888

■

921

848241

781220061

30-3470818

9-729410

■

922

850084

7H3777448

30-3644529

0-732930

■

923

851020

786330407

30-38091 61

9-730448

■

924

853776

78888)1024

30-3973083

9-739903

■

925

865C25

701453125

30-4138127

9-743476

w

92fl

B57476

704022776

30-4302481

9-746085

927

859320

700507083

30-4466747

0-750493

928

801184

700178752

!M)-4e30024

9-753998

928

863041

801765089

30-4795013

9-757500

930

804900

804357000

30-4050014

9-76 lOOO

931

800701

800954491

30-5122920

9-764497

932

808024

B00557568

30-5286750

9-767992

933

870489

812160237

30-5450487

0-771484

934

872350

814780504

30-5614136

9-774974

936

874225

817400375

30-5777097

9-778461

93G

876006

820025850

30-5041171

9-7«2946

r

037

877060

822656953

30-6104557

9785428

M

938

870844

825203672

30-6267857

9-788908

■

039

881721

827036010

30-0431069

9-792380

■

940

883600

830584000

30-6504194

9-795861

941

8854B1

833237621

30'6767233

9-799333

M2

887304

835896888

30-6930185

9-802803

043

880249

83850 IB07

30-7083051

9-806271

044

801136

8412323(14

30-7245830

9-809736

045

803026

8439(18625

30-7408523

9-813198

ft

840

804916

846590536

30-7571130

9-816659

■

047

806800

B49278123

30-7733651

9020117

■

»4S

898704

851071302

30-7896086

0-823572

■

04»

900({01

854070310

30-8058430

0-827026

■

950

902500

857375000

30-8220700

9-830475

■

951

904401

860085351

30-8382079

9-833923

w

9S4

900304

062801408

30-8544972

9-837360

053

008200

865523177

30-8700901

9-840812

954

010116

068250004

30-8808904

9-844253

055

012025

870083875

300030743

9847692

'

056

013030

873722810

30-0192497

9-851120

957

015849

B70467493

30-9354166

9-854501

958

017764

870217912

300515751

0-857902

959

919081

881074070

30-9077251

9-801421

900

921600

884730000

30-0838608

0-864848

981

923521

887503681

310000000

0-868272

002

925444

800277128

310101248

9-871694

903

927309

003050347

31-0322413

98751 13

964

02i»29e

80584 i:f44

31-0403404

9-878530

905

931225

808632125

31-0644491

9-881945

966

033150

901428006

31-0805405

0-886337

967

035089

004231063

31-0066-230

9-888767

908

937024

907030232

31-1I26084

9-892174

069

938961

909853209

31-1287048

9-895580

392

Number.

Square.

Cube.

Square Root.

Cube Root

970

940900

912673000

311 448230

9-898983

971

942841

915498611

31

1608729

9*902383

972

944784

918330048

31

1769145

9*905781

973

946729

921167317

31-

1929479

9-909177

974

948676

924010424

31'

2089731

9-912571

976

950625

926859375

31

2249900

9-915962

976

952576

929714176

31

2409987

9-919351

977

954529

932574833

31

-2569992

9-922738

978

956484

935441352

31

2729915

9-926122

979

958441

938313739

31

►2889757

9-929504

980

960400

941192000

31

•3049517

0-932883

981

962361

944076141

31

3209195

9-936261

982

964324

946966168

31

3368792

9*939636

983

966289

949862087

31

3528308

9-943009

984

968256

952763904

31*

3687743

9*946379

985

970225

955671625

31

3847097

9-949747

986

972196

958585256

31'

4006369

9-953113

987

974169

961504803

31

4165561

0-956477

988

976144

964430272

31'

4324673

9-959839

989

978121

967361669

31'

4483704

9-963198

990

980100

970299000

31'

4642654

0-966554

991

982081

973242271

31

4801525

9-969909

992

984064

976191488

31

'4960315

9-973262

993

986049

979146657

31

5119025

0-076612

994

988036

982107784

31-

5277655

0-070050

995

990025

985074875

31

5436206

0-083305

996

992016

988047936

31

5594677

0-986648

997

994009

991026973

31

5753068

9-989990

998

996004

994011992

31

5911380

9-993328

999

998001

997002999

31*6069613

9-996665

*^* In this edition some errors in the table of Dr. Hntton have been
corrected. The most extensive and accurate tables of the powers of num-
bers, are those published by Professor Barlow, of the Royal Military
Academy, entitled New Mathematical Tables^ which also contain tables of
prime numbers and factors, so useful in arranging the numbers of wheel
work.

APPENDIX B.

IRODUCTION OF THE SLIDE PRINCIPLE

TOOLS AND MACHINES EMPLOYED IN THE
PRODUCTION OP MACHINERY.

BY JAMES NASMYTH.

The striking and rapid progress which has within the
last thirty years taken place in the perfection of all descrip-
tions of machinery, not only as regards a more complete and
sound knowledge of the principles of mechanical or con-
structive science, as exhibited in the general arrangement
of the parts, but more especially in respect to the increased
perfection of the workmanship, which is now so generally
met with in the vast variety of machines which are yearly
sent forth, as it were to proclaim new triumphs over mat-
ter, cannot but lead us to endeavour to find a cause for
so remarkable and important a feature in the history of
mechanism.

481. In pursuing this inquiry, we shall find that the accu-
mulated experience and skill in constructive science, which
has resulted from a continually increasing demand for ma-
chinery, will only throw light on one portion of this in-
teresting subject ; inasmuch as increased experience alone
ill not sufficiently account for the almost mathematical

394 NASMTTH ON TOOLS

accuracy and precision which we find existing in, and con-
ferred on, the forms of the various details, whether of the
most delicate or ponderous machines ; to have produced
which, were it even possible by manual dexterity and
labour, would have entailed so vast an expense in construc-
tion, that neither in respect to quantity or price could we
have ever hoped to be able (even with our present
mechanical population increased tenfold) to have kept pace
with the demand which has resulted from the increased
perfection and facilities of production realized by improved
mechanism.

482. Viewing abstractedly the forms of the various details
of which every machine is composed^ we shall find that they
consist of certain combinations of six primitive or ele-
mentary geometrical figures, namely, the line, the plane,
the circle, the a/Under, the cone, and the sphere ; and that,
however complex the arrangement, and vast the number
of the parts of which a machine consists, we shall find
that all may be as it were decomposed and classed under
these six forms ; and that, in short, every machine, what-
ever be its purpose, simply consists of a combination of
these forms, more or less complex, for the attainment of
certain objects and performance of required duties. It
therefore follows, that the more near to absolute mathema-
tical truth we can have the forms of those parts, the more
perfectly will the machine perform its duties.

483. Up to within the last thirty years, nearly every part
of a machine had to be made and finished to its required
form, by mere manual labour ; that is, on the dexterity of
the hand of the workman, and the correctness of his a/e,
had we entirely to depend for accuracy and precision in
the execution of such machinery as was then required;
consequently, the enormous expense which was incurred in
such attempts, even in the production of comparatively
simple machines, in most cases proved a fonnidable barrier

AND MACHINES. 395

to the supply of such as the increasing wants of civiliz-
ation rendered desirable, and when at length the success-
ful efforts of Watt and Arkw-right produced such an
entire revolution in the steam engine and cotton manu-
facture, and so disclosed to mankind such vast mines of
wealth in the latent powers of production, and capabilities
of every country, and as the only obstacle to the attainment
of so desirable an end consisted in our almost entire de-
pendence upon manual dexterity for the formation and pro-
duction of such machines as were required, the necessity
of more trustworthy and productive agents rendered some
change in the system imperative. In short, a sudden de-
mand for machinery of unwonted accuracy arose, while the
stock of workmen then existing were neither adequate in
respect to number or ability to meet the wants of the time,
and but for the introduction of the principle which I am
about to describe, we never could hav(! attained to one-
thousandth part of the bright objects which were then dis-
closed to view, and which have since been so wonderfully
Bid amply realized.
484. The principle to which I allude consists in the sub-
' stitution of a mechanical contrivance in place of the human
hand, for holding, applying, and directing the motions of a
cutting tool to the surface of the work to be cut, by which
we are enabled to constrain the tool to move along or across
the surface of the object with such absolute precision,
that with scarce any expenditure of force, and indeed, in
most cases, none at all on the part of the workman, (as
^ahall be seen presently,) we are enabled to produce any
^kf the before-named clementar)' geometrical forms with
^K degree of accuracy, case, and rapidity, as compared with
^Hie old, imperfect, hand system, as may well be considered
Hp mightj- triumph over matter ; and the more justly so, '
^^hen we behold the vast results which improved machi-
nery is enabling us to bring about, ali of which may, in a

etosi

fri

39G

NASMYTH ON TOOLS

more or less direct manner, be traced back to the
of power which we have acquired hy means of the
introduction and appUcation of the slide rest principle.

AND MAcnmEs. 397

485. How it. has liappenetl that the inestimable merits of
lis contrivance have not been more justly appreciated, and
as it were unobserved, it is difficult to account ; it
ly be that its beautiful simplicity has been overlooked in
the glare of dazzling results which it has produced ; it is
only by considering how we could "get on" without its im-
portant help, that the real value of this admirable con-
trivance appears before us in its true light.

48(j. It is not indeed saying at all too much to state, that
its influence in improving and so extending the use of ma-
linerj-, has been as great as that produced by the im-
ivemcnt of the steam engine in respect to perfecting
lufactures and extending commerce, inasmuch as with-
it the aid of the vast accession to our power of producing
perfect mechanism, which it at once supplied, we could
never have worked out into practical and profitable forms
the conceptions of those master-minds who, during the last
centurj, have so successfully pioneered the way for
ikind ever after attaining the otherwise latent trea-
sures of the material world, even although opposed by
time, space, and the elemental 1 regret much that my
limits will not permit me to trace in detail, through all
their ramifications, the almost infinite benefits which have
been conferred on mankind by our having (through means
of this admirable slide rest principle) obtained a most
complete and signal triumph over the material world.
steam engine itself, which supplies us with such un-
ided power, owes its present perfection to this ad-
'able means of giving to metallic objects the most precise
and perfect geometrical forms. How could we, for in-
stance, have good steam engines, if we had not the means
<rf boring out a true cylinder, or turning a true piston rod, or
ing a valve face? It is this alone which hius fiiniished
with the means of carrying into ])riictice the accumulated
of scientific investigation in mechanical subjects,
u D

me (

^^pral

398 NASMYTH ON TOOLS

487. With a view to render the preceding remarks
more generally understood, I have given the annexed sketch,
in order to illustrate the advantages of the slide rest prin-
ciple, as a substitute for manual labour and dexterity, in the
case of the turning lathe ; the more so, as it was in this
form and application in which its admu*able merits became
first known to the mechanical world.

488. Fig. 1. represents the system of hand turning in
general practice previous to the introduction of the slide rest
Here it will be seen that the workman has no other means
of applying and guiding his tool to the work in the lathe,
than his mere unaided muscular strength, the expenditure
of which, in the case of turning large objects, would be so
great, that he could stand it for no length of time ; and
even if he were able, he would have to depend on his
strength and dexterity alone for producing even the hum-
blest and most plain class of work. By such means of this
nature as were generally practised before the introducticm
of the slide rest, we could only attain to any thing hke
true work, by an almost infinite expenditure of labour, for
with the utmost care on his part he could not avoid occa-
sionally cutting a little too deep, the consequence of which
would be that he would require to go all over the rest of
the surface, in order as it were to lower it to the level of
the accidentally too deep cufe just named ; in most cases, in
so doing, he would make the work too small, or have occa-
sion either to leave the mark in the bar, or else alter all
his measures to suit the bad results of depending on the
chance of his dexterity. It will be seen that the workman
in Fig. 1. rests, or obtains support for, the end of his tool,
so as to resist the force of the cut, by placing it upon " the
rest'* R.

489. Now let us just suppose that instead of holding his
tool with his hands, that he had it bolj:ed firm to this same
rest, and that while it was cutting a shaving from the bar

I

tf

AND MACHINES. 399

lathe, that lie had means of sliding the rest with its
tool along the bed of the lathe, parallel to the axis of the
work, it is evident that, in so doing, we should be able to
turn the bar quite true ; and if a screw was provided for
the purpose of giving this sliding motion, we should then
have a slide rest ; exactly in such manner was this truly
admirable tool introduced to the mechanical world. On
reference to Fig. 2, it will be at once seen that these ob-
jects are attained in a very simple manner. The tool is in
this case held fast and firm by a species of iron hand or
"rice, while it is constrained to move in a definite direction
by means of the slide s, (see Fig. 3,) the sliding motion
l)eing communicated by the hand of the workman to the
screw handle h, the required depth of cut being regulated
by the under slide k, operated upon in like manner by a
screw and handle ; so that by the separate or comhined
motion of these two slides, the point of the tool can be
made to traverse along or across the work as required, with
an expenditure of power on the part of the workman so
trifling as scarce to be appreciated ; and with such a degree
of definite and precise accuracy will the tool by these means
move, that, after setting the tool to work, he needs not to look
at it 80 long as he simply keeps turning the screw handle j
and by a very simple contrivance, which we have en-
deavoured to exhibit in Fig. 3, x, the attendance of the
workman ia entirely dispensed with by the introduction of
le self-acting principle, by which the revolution of the
irk in the lathe is made to supply the place of the hand
the workman. As may be seen at x, Fig. 3, by simply
fixing to the work in the lathe a piece of iron as at o, and
placing on the end of the screw s of the upper slide a star
wheel X, it is evident that at each revolution of the work
in the lathe the end of the iron finger x will come in con-
tact with one of the teeth of the wheel, and move it round
at each turn, bringing the next in succession into a
D n 2

400

NASMYTH ON TOOLS

situation so as in like manner, at each revoluticm of the
work, the screw wheel x is moved round, and the tool by
that simple means slid by successiye steps along the sur£ace
of the work ; here, then, by this simple adaptation, we
have not only done away with necessity for a dexterous
workman, but have entirely removed all necessity of at-
tendance whatsoever during the progress of the tool
over the slide length of the surface of the work.

'Tu--

^ -A-

This will in some degree convey an idea of the nature of
the self-acting principle, by the adoption of which we are
enabled to elevate to so high a degree the productive

I
I

■ mi

K'

AND MACHINES. 401

powers of our workmen and machinery. There are a vast
variety of modes of attaining this self-acting motion, but
the one above alluded to will be sufficient, the more so as
it is the most generally employed, and most simple.

490. It was this holding of a tool bymeans of an iron hand,
and constraining it to move along the surface of the work
in so certain a manner, and with such definite and precise
motion, which formed the great era in the history of me-
chanism, inasmuch as we thenceforward became pos-
Besscd, by its means, of the power of operating alike on
the most ponderous or delicate pieces of machincrj' with a
degree of minute precision, of which language cannot convey
an adequate idea j and in many cases we have, through
its agency, equal facility in carrying on the most perfect
workmanship iu the interior parts of certain machines,
where neither the hand nor eye can reach, and nevertheless
we can give to these parts their reqiiired form with a de-
of accuracy as if we had the power of trajistbrming
Ourselves into piginy workmen, and so apply our labour to
the innermost holes and comers of our machinery.

4>91' It would be blamable indeed (after having en-
deavoured to set forth the vast advantages which have
been conferred on the mechanical world, and therefore on
mankind generally, by the invention and introduction of the
slide rest) were I to suppress the name of that admirable in-
diridual to whom we are indebted for this powerful agent
towards the attainment of mechanical perfection. I allude to
the late Henry Maudslay, engineer, of London, whose useful
life was enthusiastically devoted to the grand object of im-
proving our means of producing perfect workmanship and
machinery ; to him we are certainly indebted for t/ie slide
fsi, and consequently, to say the least, we are indirectly
for the vast benefits which have resulted from the intro-
duction of so powerful an agent in perfecting our ma-
chinery and mechanism generally. The indefatigable care

402 NASMYTH ON TOOLS

which he took in inculcating and diffiising among
workmen, and mechanical men generally, sound ideas of
practical knowledge and refined views of construction, has
rendered and ever will continue to render his name iden-
tified with all that is nohle in the amhition of a lover of me-
chanical perfection. The vast results which have sprung firom
his admirable mind, is his best monument and eulogium.

492. The vast practical advantage which resulted from
the substitution of ** the slide rest '' in place of the hand
in the process of turning, had its natural efifect in causing
its adoption and application to other important processes
in constructive science. So striking and certain were the
effects and advantages as respects the superior quality and
cheapness of the work produced by its means, that it soon
induced a very marked change in mechanical designs, in-
asmuch as this, that many improved arrangements in me-
chanism had been kept back from the vast expense attend-
ant on the employment of certain forms in the parts, such
as perfectly true cylindrical rods or circular or flat surfaces,
which the important aid of the slide rest now renders so
c^heap, (comparatively speaking,) that every practical en-
gineer, in making out his design in detail, had only to keep
in mind the vast capabilities and powers of the slide rest,
to enable his fancy to luxuriate in the introduction of the
most perfect geometrical forms, as not only attainable in
practice, but actually the cheapest forms through whose
agency he could attain his object. I have every reason, in-
deed, to call the introduction of the slide rest a great era
in the history of mechanism, as every piece of machinery
which was produced by its agency, bore such evident marks
of superiority, as very rapidly and extensively proclaimed
to the mechanical world that a great step (leap forward, I
should rather say) had been made, and in proof of it, we
have only to look around us at this day to see what is doing
by improved machinery, to place beyond doubt what I have

AND MACHINES. 403

i as to this era in mechanism — " the introduction of
the slide rest."

493. Were I to attempt to trace in detail tho almost in-
finite application of the slide rest principle, I should re-
quire to describe almost every machine which is employed in
giving definite forms to materials ; but as such would he in-
compatible with my limita, I shall confine myself to one or
two of tho more generally used and important applications ;
and in endeavouring to do so, I shall, for the sake of clear-
ness, avoid those minute details which, although most fre-
quently combined with the slide principle, yet are so sub-
ordinate, and so frequently varied according to the taste of

tthc engineer, that it is best to strip them from the simple
ilustrations I have endeavoured to give, so as to leave, as
t were, more prominent and conspicuous iJte principle of
[he machine.

494. I cannot properly introduce to the attention of my
peaders a more worthy and truly important immediate de-

iendant of " the slide rest " than the plmii/ig macluTie,
which has done more within the last 10 or 15 years for
reducing the cost, and for extending the use of perfect ma-
chinery, than had been the case by all the improvements

1 mechanism for the last century.
¥J5. There is no form which is so frequently reijuired
and essential to any piece of mechanism as the plane sur-
fece, or rectangular prismatic forms generally,

496. The vast expense attendant on the production of
such, by the tedious and unsatisfactory process of chipping
and filing, caused every engineer to avoid by all means any
arrangements which rendered such forms necessary, how-
ever essential they might be to the perfect action of the
machine. It is quite laughable to observe, in any old piece
of mechanism, the niggardly use of those important forms

ffr*im the above obstacle. The introduction of the
machine at once altered the entire system, inas-

rea
^^chi]

V and

404 NASMYTH ON TOOLS

much as forms and arrangements became practically pos-
sible, which formerly the engineer dared not think of using.
This was simply following out in the plane surface, what
the slide rest had produced in the turning lathe as regards
cylindrical forms ; and the result was, that not only was
the machinery produced by its agency most strikingly su-
perior, by its direct influence, but also as the planing ma-
chine enabled us to produce improved tools at so very much
reduced cost, that mighty principle in all affairs, (namely,
cause and effect tearing each other alternately.) The first
planing machine enabled us to produce the second still
better ; that again produced a better still ; and now shde
rests of the most perfect kind came streaming forth from
them, and they, again, assisted in making better still ; so
that in a very short time a most important branch of en-
gineering business, namely, tool-making, arose, which had
its existence not merely owing to the demand preexisting
for such improved tools, but in fact, raised upon a de-
mand as it were of its own creating, and all this caused by
the slide rest, and its offspring, the planing machine. One
has only to go into any of those vast establishments, which
within the last 10 years have sprung up for the purpose of
supplying the demand for machinery, and we shall find
that nine-tenths of all the fine mechanism in use, and in
process of production, is through the agency, more or less
direct, of the slide rest and planing machine.

497- Figure 4 represents the general arrangement of
parts existing in most planing machines. It consists of
two principal parts, namely, the bed b on which the table
T slides by certain mechanism backward and forward, so
that any piece of work, w, being bolted to it, partakes of
the same, as if it were a part of the table t ; the table t
being constrained to move in a perfectly straight line to
and fro, by its sliding on the two angular ridges, c c.

AND MACHINES.

/%|A^

498. Over the table T is fixed " a slide rest " s, which
is held fast by being bolted to the two upright standarda
N N. This slide s has a transverse slide d, which serves to
hold the tool in such a manner that it may be lowered down
and adjusted so as to cause the tool to take a cut more or

406 NASMYTH ON TOOLS

less deep as desired, which adjustment is performed hy the
handle l, so that every time the tahle and the work fixed
to it moves to and fro, the tool in the down slide d, is by
certain apparatus moved each time a little way across the
table, so that by a repeated series of sliding backwards
and forwards of the table, the tool is made to traverse the
surface of the work, and in so doing it transfers the per-
fectly true figure of the slide s, on to that of the surface
of the work w, and so produces a perfect plane surface. I
trust an inspection of the figure will do more to render this
clear, than any further attempt at description.

499* As to the means of giving motion to the table, as
also to the screw of the slide s, it is not required here to
enter into such details, as they vary so much according to
the fancy of different makers, who have each their peculiar
fancy as to the best arrangement.

500. An inspection of the figiure will, I trust, satisfy any
one that this machine is derived from the slide rest, for
the slide s is nothing more than a slide rest, held to its
work by the two standards nn, while the work w repre-
sents a surface on the lathe, which is made to move in a
straight line, in place of a revolving motion, as it would
have done had it been a cylindrical surface being turned
in the lathe. This, indeed, is my main object in giving
this figure, as it serves to show that it is to the slide rest
system that we are indebted for the planing machine, how-
ever varied the constru^ve details of such planing ma-
chines as we mejt y^idx may be, yet we shall find that they
all embody the above principal arrangements, and are all
slide rests for turning, i. e. planing^a^ work.

501. Again, in the case of the screw-cutting machine,
we shall find (Fig. 5) that it consists simply of a slide re^t,
which receives its sliding motion from the revolution of the
spindle or work in the lathe. I have chosen the latter, as
it tends to render the arrangement more distinct.

AND MACHINES.

407

Here we have the slide rnrt s, whose tool-holder is slid along
1^ means, of .the screw b> which receives its motion from the
work in the lathe> by means of the wheels w w, by which
it is evident, that as the work x revolves in the lathe, a
revolving motion will be transferred to the screw s, and the
pmnt of the tool will» ob sliding almg, have a spiral or

<»l

408 NA8MYTH ON TOOLS

screw on the work ; and according to the respective disp
meters of the wheels w w, so shall we have a screw formed
on X, more or less fine m the pitch of the thread, accord-
ing to the proportions of the respective diameters of the
wheels ww, as in the figure w or the work, is twice the
diameter of w or the end of the slide screw. The pitch
of the thread on x will he twice as wide as on s, and as 8
and X are revolving in opposite directions, we shall have a
right hand screw on the one, and a left hand screw on the
other, or the reverse, according to the nature of the guide
screw s ; and by placing an intermediate wheel between w
and w, we shall then cause them to be either both right
hand screws, or both left, as the case may be ; the depth
of cut is given in succession, by the set or transverse ad-
justing screw N.

502. Again, in the case of the wheel-cutting machine,
we have the slide rest in full existence. See Fig. 6.

503. All wheel-cutting machines, however complex they
may be in their minor arrangements, consist of two essen^
tial parts, the slide rest s, which holds the revolving cutter
R, and the spindle t, on which the wheel w, which has to
be cut, is fixed. This spindle is made part of the dividing
wheel D, by fitting into a socket or chuck, so that when
the head d is moved roimd in successive steps, or according
to the required divisions on the face of it, which is set off
or divided and held fast by the index point or holder e, it
is evident that whatever be the di>dsions or fractions of the ,
divided circle d, we move round step by step ; the same
will be most faithfully transferred to the wheel w, which
we desire to cut or divide into teeth ; and by means of die
slide rest s, we slide the revolving cutter across the bee or
edge of the wheel w. It is likewise evident, that we must
thereby cut a tooth every time we slide the cutter across,
after each division is taken in succession by the ahifting gf
the head or dividing wheel d.

AND MACHINES.

This is a very meagre descripdon of the principle of a
moBt important machine, in which, as in innumerable other
iiHtances, the slide principle enables us to produce with such

410 NASMYTH ON TOOLS

facility, results in the form of workmanship, whose mathe-
matical accuracy throws all hand work utterly into the shades
not only as to absolute precision, but also economy of pro-
duction.

504. As before said, were I to endeavour to trace in
detail the countless applications of the slide principle firom
its first appearance before the mechanical world, as intro-
duced by the late celebrated Henry Maudslay, and follow
it down to the present time, a thousand pages would not
give space for all that might, with such truth and justice,
be said on the advantages which mankind have been and
are now deriving from the slide rest, and its lineal de-
scendants.

505. Some Observations respecting the Form of Tools
employed for Turning and Planing Irony Brass,
^c.y toget/ier with some Remarks on the Hardening
and Tempering of sicch Tools.

Hitherto, so far as I am aware, the form of tools em-
ployed in turning or planing iron, &c., has not either re-
ceived that attention which the importance of the subject
calls for, nor has any attempt been made to reduce the sub-
ject to such plain and general principles of which it is not
only capable, but when so treated, then only adapted to be
of service to those in whose hands the management of such
tools is for the most part entrusted. Indeed, so much
practical importance attaches to this subject, that the
quality as well as the quantity of work produceable firom
turning lathes and planing machines, entirely depends
upon the skill of the operator in giving to his tools the
proper form. There are many excellent workmen, wh0| by
a species of intuition, have acquired the art of giving to
the tools either the true form, or so near have they got

AND MACHINES. 411

true principlp, that by holding to and repeating
again and again that tbrm which they found the best, they
are enabled to produce the required resuh. But even
with such, when a case occurs in which they have to go a
little out of their usual routine, they are then as much " at
sea" as if they knew nothing about the matter. This
ftrises from no other cause than the want of the knowledge
of the general principle, which would guide them to the
true form, whatever be the case ; and moreover, now that
slide lathes and planing machines are becoming so very
common in the workshops of engineering establishments,
and that such machines, from their automaton power, no
longer require regularly bred mechanics to attend them, it
becomes more than ever necessary to reduce the subject to
those simple principles to which it is capable, so that
ithe subject may be brought within the range of the sup-
•^sed inferior capacity of a humbler grade of men, from
whom we want no more than careful attention to secure the
Iwst results from those surprisingly productive machines.
Wo shall now proceed to the subject of these remarks, and
with that view shall take, in the first place, the most sim-
ple case.

The chief, and indeed the only point which we require to
consider, is the direction in which we wish to cut or pene-
trate the metal. Suppose, therefore, the plane a b is the
surface of a plane of metal, from which we wish to cut off

Iavings, in the direction a b, either hy a u moving against

I
I

I

%

li

\g)

41 S NASMTTH ON TOOLS

the tool, or the reverse, namely, the tool moving against it^
for it is the same action in either case. Suppose we were to
employ such a tool as No. 1 ; in this case we should have
little or no penetrating quality in the form of the tool,
which would in consequence not cut, but rvh off the par-
ticles, or crush them off by sheer brute force. The reason
of this is, that we have given it so very blunt or obtuse an
edge at the point of cutting, that by their coming against
it at right angles to its face, the whole force which moves
the plane a b will be consumed in merely rubbing off (not
cutting) the particles of metal.

Next, in the case of No. 2, which looks more like a tool
that would cut, we shall find that there again we should
fail to produce the required result, and also encounter other
evils. In this case we still have no more penetrating pro-
perty in the direction a b, for the force of the tool is still
m the same position, with regard to the surface to be cat»
as in the instance of No. 1, that is, it is at right angles to it^
so that we have no advantage here ; and what is far worsen
we have from this tool a penetrating quality, in a direction
quite opposite to that which we desire, namely, in the di-
rection c D. In moving the surface a b against this tool.
No. % we should, on attempting to take a cut, find that
the penetrating quality in the direction c d, would imme-
diately exhibit itself in a series of saw, teeth-like marks,
more or less deep, according to the strength of the cut and
that of the tool, which indeed would, on account of its form,
not preserve its point entire for a moment, but would be
snipped off with little or no force, because the cross section
of metal at its point is scarce measurable. This is the
most usual error in the forming of tools, that is to say, be-
cause they look sharp, that is thought sufficient ; forgetting
altogether the direction in which the strain is to be ap-
plied, and in consequence not providing sufficient metal
a cross section in the direction of the strain.

.^ND SIU'HINES.

413

^

If we look to No. 3, we shall find that all these requi-
jritcs are provided. In the first place we have a high de-
gree of a^-uteness in the direction of the cut, namely ab ;
then as to strength, behind the point we have all the nietaJ
from E to F to give the point E the requisite support ; in
short, as regards strength, we have as much more strength
in the case of No. 3, over No. % as the distance e f is
greater than c. No. 2. Besides this great strength which
we have in the case of No. 3, we have also another advan-
tage of great moment, namely, the entire absence of all
tendency to chatter or produce a rippled surfece, fe acting
as a most complete stop to ajiy risk of digging into the sur-
fcce which we are planing or turning, which would in-
evitably be the case with No. 2, supposing the point to be

tpable of resisting the force, which it could not. The very
of the shaving in the case of either of these tools,
'flTOald exhibit the relative advantages of each. In the case

No. 3, they would be most complete curls, as may be

dent from the form of the tool.

In No. 1, therefore, we have strength, hut no acutcness

either direction,

In No. 2 we have acuteness, it is true, but in a direction
quite opposite to that in which we require it, and 7to
strength.

In No. 3 we have acuteness entirely in the direction
in which we require it, and the greatest degree of
strength.

Wo may therefore establish from this attempt at inves-
ligation, the following principle, namely, that in forming
and setting a tool to cut any surface, we have only to attend
4o placing it so that the end of it forms the least possible
angla with the surfiire to be cut, and wlxatever degree of
acuteness be considered proper, let the keenness be given

hollowing out the surface e c, as given here.

414

NASMYTH ON TOOLS

No. 3.

I again repeat the principle, namely, that in forming the
cutting tool, what we have to attend to is, to let the end of
the tool he as nearly parallel to the surface to he cut as
possible, and any acuteness that may be required shall be
given to the surface on which the shavings slide ; the very
same holds good in the case of turning tools, and indeed
in every tool, from a razor or carpenter's chisel up to the
most enormous and powerful tool in a lathe or planing ma-
chine. In the case of turning, we may sec the application
of the " principle '* very clearly exemplified.

No. «3 as a turning tool.

Here we see No. 3 as a turning tool, ab being a portion
of a cylindrical bar in the lathe ; £ f should be as near as
possible a tangent, that is, at right angles to the radius of
the curve.

AND MACHINES.

415

No. 2.

In the case of No. % employed as a turning tool, we
should not be able to preserve its point for an instant, as
will be evident from the small cross section at c.

When scraping is all that is necessary, which is a last
finish just before preparing the work to be polished, No. 1
may be employed with advantage, as in that case its low
penetrating quality in both directions becomes of much ser-
vice, but then it is not desired to employ it as a cutting
tool.

s^i^-^^^^;,

■K^N^;^§:^#.

In the instance of a common joiner's plane, we shall find
the same principle carried out most fully ; e f is the plane
iron, A B being as before the surface to be cut. In the case
of this tool, an artificial end is given to the cutting tool,
by means of the sole of the plane, which gives the requisite
non-penetrating quality in all directioiis, except Uiat in

416

NASMYTH ON TOOLS

which we require to remove the material, or take the cat,
namely, ab.

A

The same again is seen in the action of a chisel or hat-
chet. It will be observed that the bevelled surface of the
chisel is always placed outwards, and the flat surface placed
next to the wood which we are about to cut, so that the
angle between the face of the chisel next the wood, and
the sur^c of the wood, shall form the least possible an^
with it.

AUo, in forming drills, we shall find the very same prin-
ciple in action, as has been given in the foregoing ex-
amples. Thus, H being the end view of a drill, the e^gv
UP should be the least possible prominent, or oat of t^

AND MACHINES.

417

plane of the sutfiace of which they are the edges, o s heing
less prominent than o p, so that there may he as little pe-
netrating quality at the edge op as possible. A drill so
formed, will cut the smoothest holes without any chatter-
ing, which is so commonly the case when the edges are
bevelled very much back, as given at r. Such a drill would
very soon lose its edge, and would cut a very rough hole
besides.

In order to give great keenness to the edge of the drill,
we have only to apply the same principle as before stated,
in respect to turning tools, by hollowing out a groove at x,
on each cutting face.

Pig. 1.

Face Tool applied to the Gauge,

FlO. 2.

K

Right hand To(»l

ZX i

Face Tool, |

1 / •

Lf/t hand Tool, / \

V 1

The above is a sketch of a very convenient and simple tool
gauge, for enabling any one to ascertain whether a tool is
ground or formed to the proper angle. It consists of a planed
plate of metal, ab, on whose surface there is at one end fixed
a conical steel pin c, whose taper or angle formed by the
sides of the cone with the surface of the plate ab, is just
that which is proper for the cutting face of the tool c, be-
ing a cone given in a very simple universal gauge for every

418 NASMYTH ON TOOLS AND MACHINES.

kind of tool, such as seen in Fig. S. By using this ganger
all difficulty of forming the tools to the proper angle, is at
once removed.

And the same gauge will answer for every kind of
planing or turning tool whatsoever, and of whatever size.
A B may he ahout 15 inches long, hy 5 wide, and about |th8
of an inch thick ; these dimensions are by no means abso-
lutely requisite, but will be found generally usefiiL

The angle formed by the sides of the cone, and the sur-
face of the plate, should be about three degrees.

GENERAL

EXPLANATION OF THE PLATES

PLATE I.— ESSAY I.

On thb Teeth of Wheels.

Chart shewing the horses' power to which the teeth of wheels of certain
pitches, working under different circumstances, are equal, — is described
on the Plate, and in Chap. V. Arts. 163—179.

PLATE IL— ESSAY II.
On the Shafts of Mills and other Machines.

Fig. 1. — No. 1, No. 2, and No. 3. — Old mode of fixing gudgeons called

laid-in-pudgeons^ Art. 187, pp. 178, 179.
Pig. 2. — No. 1 and No. 2. — Cross-tailed gudgeons. Art. 188, p. 179.
Fig. 3, Fig. 4, and Fig. 5. — Parts of cross-tailed gudgeons, Art. 188,

p. 179.
Fig. 6. — A hollow cast iron shaft. No. 2. An end view, to shew the

manner of fixing the gudgeons. Art 192, p. 181.

PLATE III.— ESSAY II.

Fig. 7. — No. 1, No. 2, No. 3, and No. 4. — ^A feathered shaft, with dif-
fefooi seetjonsy Art 193, p. 181.

T.'-7

4520 EXPLANATION OF THE PLATES.

Fig. 8. — No. 1 and No. 2. — Square shafts, Art. 193, p. 182.

Fig. 9. — No. 1 and No. 2. — Stress on shafts. Art 194, pp. 182, 183«

Fig. 9. — Water wheel with teeth on the shrouding, to give motion to Ae

mill, Art. 104, p. 183.
Fig. 10. — The cliief strain on the vertical shaft is that which tends to twiit

it. Art. 194, p. 183.
Fig. 11. — Example of a compound strain on the shaft, Art. 194, p. 183.
Fig. 12.— Shaft of a water wheel. Art. 194, p. 183, and Art 195, p. 188.

PLATE IV.— ESSAY II.

Fig. 1 . — Effect of position in causing a greater or less degree of stnin on

shafts. Art. 195, p. 183.
Fig. 2. — Stress on shafts from the action of the moving power. Art 196,

p. 184.
Fig. 3, Fig. 4. No. 1, No. 2, No. 3; and Fig. 5.— Effect of position in

causing more or less stress on the sliafts. Art. 197, pp. 184, 185; and

Art. 198, pp. 18.5, 186.
Fig. 6 and Fig. 7. — To explain the effect of the place of a wheel on a

shaft. Art. 199, p. 186.

PLATE IV. A.— ESSAY II.

Fig. 1, Fi«^. 2, Fig. .'3. — Represent an improved method of fixing gndgeons
on wooden shafts Art. 207, p. 1J^3, &c.

Fig. 4. — Plan of a sliaft and wheels to show how the stress on a shaft may
bo determined, Art. 207, pp. 1 93, 1 94.

Fig. .*>, Fig. fi, and Fig. 7. — Figures to show the stress on shafts and gud-
geons in different circumstances. Art. 210, p. 19,3.

PLATE v.— ESSAY III.

On the Longitudinal Connexion of Shafts, denominated

Couplings.

Fig. 1. — Represents three shafts connected hy couplings, supported by

double bearings. Art. 291, p. 2GG.
Fig. 2. — No. 1 and No. 2. — The square coupling, witli double bearings^

Art. 202, p. 207.
No. 3. — A different modification of the square coupling, Art. 292,

p. 267.
No. 4. — A coupling box made in two pieces, Art 298, p. 29T*

EXPLANATION OF THE PLATES. 421

Fig. 3. — No. 1, No. 2, and No. 3.— A variety of the couplings with

doable bearings, Art. 294, p. 268.
Fig. 4. — No. 1, No. 2, and No. 3. — The round coupling, Art. 295,

pp. 268, 269.
Fig. 5. — No. 1 and No. 2. — Clutches or glands. Art 297, p. 269.
Fig. 6. — No. 1 and No. 2. — Boring-mill clutch, first construction. Art.

300, pp. 270, 271.
Fig. 7. — No. 1 and No. 2. — Boring-mill clutch, second construction, Art.

303, pp. 271, 272.
Fig. 8. — No. 1, No. 2, and No. 3. — Coupling having circular plates, Art

306, p. 272.

PLATE VI.— ESSAY III.

Fig. 9. — No. 1 and No. 2. — Coupling link used by Messrs. Boulton and
Watt in their portable steam engines, Art 308, p. 273.

Fig. 1 0. — No. 1 , No. 2, and No. 3. — A coupling sometimes used to con-
nect the fly-wheel shaft of a steam engine with the mill-work, and is so
contrived, that in case the fly should turn in a wrong direction, the mill-
work remains at rest, Art 310, p. 273.

Fig. 11. — The universal joint. Art. 314, p. 276.

Fig. 12. — No. 1, No. 2, and No. 3. — Square coupling, having one bearing.
Art. 318, p. 277.

Fig. 13. — No. 1 and No. 2. — Mode of coupling rollers used in machinery
for spinning cotton. Art. 320, p. 278.

Fig. 13*. — No. 1, No. 2, and No. 3. — Cylindrical coupling, having one
bearing. Art. 321, p. 279.

Fig. 14. — Bolted coupling, used in some mills in Manchester, Art. 323,
p. 279.

PLATE VII.— ESSAY IIL

Fig. 15. — No. 1, No. 2, and No. 3. — A variety of the bolted coupling,
Art. 325, p. 280.

Fig. 16.-^Another modification of the bolted coupling. Art 326, p. 280.

Fig. 17. — No. 1, ^o. 2, and No. 3. — Quadrant coupling, Art 327, p. 280.

Fig. 18. — No. 1, No. 2, No. 3, and No. 4. — Coupling forming an uni-
versal joint, Art. 329, p. 281.

Fig. 19. — No. 1. — Elevation of coupling, executed at Manchester, Art
331, p. 282. No. 2, and No. 3, are on the next Plate.

422 EXPLANATION OF THE PLATES.

PLATE VIIL— ESSAY III.

Fig. 19. — No. 2 and No. 3. — Sections of couplings czeeated at Mm-
Chester, Art. 331, p. 282. No. 1, is on tibc preceding Plate.

Fig. 20. — No. 1, No. 2, and No. 3. — Cylindrical coupling, having projec-
tions, Arts. 334, 335, pp. 282, 283.

Fig. 21. — Square coupling (having a coupling-box) for upright ahafta, Alt.
337, p. 283.

Fig. 22. — No. 1 and No. 2. — Square coupling for upright ahafta, honng a
socket instead of a box, Art. 338, p. 284.

Fig. 23. — Coupling for upright shafts, having projecting quadranta, Alt.
340, p. 284.

Fig. 21'. — Lying shafts, shewing the situation of the couplings with regpid
to the wheels. Art. 348, p. 286.

PLATE IX.— ESSAY IV.

On Methods of Disenoaoing and Re-enoaoino Machinsbt, while

IN Motion.
Fig. 1, and No. 2. — The sliding pulley. Art. 362, p. 294. «
Fig. 2.— No. 1, No. 2, No. 3, and No. 4.— The bayonet. Art. 894,

p. 295.
Fig. 3.— No. 1 and No. 2.--The lock pulley, Art. 3G6, p. 297.

PLATE X.— ESSAY IV.

Fig. 4. — No. 1 and No. 2. — Fast and loose pulleys, or dead and live pul-
leys, Art. 3G8, p. 297.

Fig. 5.— Sack tackle, Art. 372, p. 299.

Fig. 6. — No. 1. — Wheels, in gear^ having a moveable bridge. Art. 371, p.
298. No. 2, is on the next Plate.

PLATE XL— ESSAY IV.

Fig. 6. — No. 2. — Wheels out of gear ^ Art. 374, p. 299. No. 1, ia on the

preceding Plate.
Fig. 7. — No. 1 and No. 2. — Clutch for engaging the wheel ▲ with the abaft

B, Art. 376, p. 300.
Fig. 8.— No. 1 and No. 2.— Friction clutch, Art 378, p. 301. No. 3 ia

on the next Plate.

EXPLANATION OF THE PLATES. 423

PLATE XII.— ESSAY IV.

Fig. 8. — No. 3. — Hoops of friction clutch, Art. 378, p. 302. No. 1 and

No. 2 are on the preceding Plate.
Fig. 9. — No. 1 and No. 2. — Friction cones, Art. 380, p. 302.
Fig. 10. — No. 1 and No. 2. — Wheels moying hy contaction. Art. 382, p.

303.
Fig. 11.— Sack tackle. Art. 384, p. 304.
Fig. 12. — No. 1 and No. 2. — Self-disengaging coupling. Art. 386, pp.

304, 305.

PLATE XIII.— ESSAY V.

On Mbchanism fob Equalizing thb Motion of Mills, denominated
Lift-Tbntebs, Engine Govebnobs, and Watbb- Wheel Govebnobs.

Fig. 1. — Steam-engine governor. Art 389, p. 308.

Fig. 2. — The balls a b and c being made to revolve, are all found on the

same horizontal plane. Art 392, p. 310.
Fig. 3. — LifWtenters for wind mill, first construction. Art 396, p. 311.
Fig. 4.— -Lifl-tenter, second construction. Art. 397, p. 312.
Fig. 5. — No. 1.-— Elevation of water-wheel governor, first construction,

Art. 399, p. 313. No. 2, No. 3, and No. 4, of Fig. 5, are on the next

Plate.

PLATE XIV.— ESSAY V.

Fig. 5.— No. 2, No. 3, and No. 4. — Water-wheel governor, first construc-
tion. Art. 399, p. 314. No. 1, of Fig. 5, is on Plate XIII.

Fig. 6. — No. 1 and No. 2. — ^Water-wheel governor, second construction.
Art. 401, p. 315.

PLATE XV.-ESSAY V.

Fig. 7. — Water-wheel governor, third construction. Art. 402, p. 315.

Fig. 8.— No. 1 and No. 2. — ^Wator-wheel governor, fourth construction,
Art 403, p. 316.

Fig. 9. — No. 1, No. 2, and No. 3. — Water-wheel governor, fifUi construc-
tion. Art. 404, p. 317.

PLATE XVI.— ESSAY VL

On Changing the Velocity of Machineby while in Motion.

Fig. 1. — ^Turning lathe motion. Art 419, p. 335.
Fig, 2.-* Alternate cones, Art 421, p. 336.

4S4 EXPLANATION OF THE PLATES.

Fig. 3. — Wheels moving by contaction, Art. 423, p. 336*

Fig. 0. — No. 1 and No. 2. — Mechanism used in cotton spinning odled

double speed, third construction, Art 480, p. 340.
Fig. 4 and Fig. 5, are on Plate XVII.— Art 426, p. 338.

PLATE XVII.— ESSAY VL

Fig. 4. — Double speed, first construction. Art 426, pp. 338, 339.
Fig. 5.— Double speed, second construction. Art 426, pp. 338, 339*

PLATE XVIIL-KSSAY VII.

On tbb Fraiiino of Mill-Wokk.

Fig. 1. — Pivot of upright shafts, Art. 439, p. 346.

Fig. 2. — No. 1, No. 2, and No. 3.— Pivot of upright shaft Yamng a sted

foot. Art 440, p. 346.
Fig. 3. — Cylindrical steel pivot. Art. 441, p. 347-
Fig. 4. — No. 1, No. 2, and No. 3. — Journal of tlpright shaft supported by

a breast, Art 444, p. 347.
Fig. 5. — Millstone bush. Art. 445, p. 347.

Fig. 6. — No. 1, No. 2, and No. 3. — Headstock framing for supporUng gud-
geons of water wheel, Art. 446, p. 348.
Fig. 7. — No. 1, No. 2, No. 3, and No. 4. — ^Wooden framing for lying

shafts, Art. 447, p. 348.
Fig. 8. — Framed post. Art. 448, p. 349.
Fig. 9. — Framing for lying shafts suspended from a ceiling, Art 448, p.

349.
Fig. 10. — No. 1 and No. 2. — Bridge and pedestal of upright shaft, Art

449, p. 349.
Fig. 11. — Wooden framing of an upright and lying shaft connected by borel

wheels. Art 450, p. 349.
Fig. 12. — Framing having cloves, Art. 451, p. 349.
Fig. 13 and Fig. 14. — Transverse sections, showing forms of pieces of timber

and of iron. Art. 456, p. 351.
Fig. 15. — Feathering, Art. 457, p. 352.
Fig. 16, Fig. 17, and Fig. 18. — Represent advantageous forms of sections,

Art. 457, p. 352.
Fig. 19. — No. 1 and No. 2. — Bridge made of wood for sustaining lying

shaft. Art 458, p. 353.
Fig. 20. — No. 1 and No. 2.— Bridge made of east iron for nwtsining lying

shaft, Art 458, p. 353.

EXPLANATION OF THE PLATES. 425

PLATE XIX.-ESSAY VII.

Fig. 21 Headstock for water wheel, Art. 460» p. 353.

Fig. 22. — Mode of snspending lying shafts by cast iron, Art. 461, p. 353.
Fig. 23. — No. 1 and No. 2. — Cast iron framing for supporting 3 pair of

flour-mill stones, Art 462, p. 353.
Fig. 24. — No* 1 and TSo. 2.—- Machine palled iqu$ezer$j used in bleaching,

haying cast iron framing, Art 462, p. 353.

EXPLANATION

OF

THE ADDITIONAL PLATES

OF THIS EDITION,
NUMBERED AND DESCRIBED FROBI PLATE XIX.

PLATE XX.

Represents several diagrams, shewing the theoretical mode of describing
tlie teeth of wheels ; to accompany (the Appendix A.) Professor Willis s
Paper on that subject, pp. 149—157, and 166—172.

PLATE XXL

Bbahah's original Slide Rest.

Fig. 1. is an elevation; Fig. 2. an end view; and Fig. 3. a plan of
Bramah's slide tool, which was first used in the year 1 794, and was the
workmanship of the late Mr. Maudslay, made by himself when in the
employment of the late Mr. Bramah.

The work to be turned is placed in the usual way between two centxe
pieces, one is shewn in Figs. 1 and 3, by Cy where it is secured. The square
bar dy on one end of which is fixed the tool, is then made to advance by
means of the screw e, working through a nut, secured to the square bar d
by a small set screw ; this nut piece can be moved along the bar according
to the length required, and by turning the handle e, the bar <f, and conse^
quently the tool, is brought close to the work which it is intended to turn.
The frame for carrying this bar slides between two V's on the frame b, by
the motion given to the handle and screw ; thus it will be seen, this restbas
two motions, the one at right angles to the other.

The slide rest and frame h are moveable along the traversing bar a,
according to the length of the work, and when placed in any particular
position may be secured by the handle and screw underneath.

This rest is very different to those in use at present, as will be
hereafter.

EXPLANATION OF THE PLATES.

PLATES XXII. AND XXIII.

BuAMAU's Laths fdh tubkino Sphebes.

The Iidl or sphere _; is fixed in the usual way between the two centres
of the lathe, having projecting pieces on it for that purpose. The muidril
is driven by a leather strap working on the circumference of the pulleys or
riggers d; oa one end of this mandril is the chuck /, having a pin on its
face, called the driver ; when this begins to rerolve, it cornea into contact
with a carrier fixed on one of the projecting pieces of the sphere, which
t necessarily drag it round with It, and thus communicate the reTolving
Lnotion ; there is a ECt screw on this carrier to fix it by, as is shewn in
I Figs. 1 and 3.

The tool I, Fig. 2, is fixed to the moveable slide b, by two adjusting

ind moves on its centres k' h', describing the circle of the sphere ;

L'tiib slide works in a V on one side of the bed a, and on a Rat smooth sur-

I Ace on the other. When the sphere is turned, the back centre screw ^ is

i loosened, which at once liberatcx it. There is also an adjusting screw for

the bock centre of the mandril, which at times requires tighleniug. By

withdrawiug the slide, used in spherical turning, and fastened to the sliding

carriage by four screws, a common band rest can be put in its place, when

e lathe is required for ordinary purposes.

PLATE XXIV.

Gbbat Bokino Lathe, by Messhs. Nasmytr, Gabkbll and Co.

e two lathes already described are very much smaller tlian that shewn

y this Plate; the work they are capuble of turning is altogether of a different

md.

Figs, 1, 2, and 3, severally represent an elevation and two end views,

will he seen that this lathe is suited to various speeds, depending on

B dituneCera of the piUleya or driving riggers rf, which are put in motion by a

' stmp. The hcadstocks a, and the bed A ore well secured to the

s foundation, by means of strong wrought iron holding down holts for

X purpose ; on the face-plate c, to which is fixed the work to bo turned,

aa internal wheel worked by a small pinion hung on an intermediate

jbaft, which receives its motion from the driring shaft, by the spur wheel

Bid pinion e on its opposite end; long mortices are shewn on the face-plate

D Fig. 3, through which the work to be turned is clamped, and revolves

fith it ; / is the slide rest for carrying the tool, placed on the two sUuidards

J tfKil being made to advance, by a long screw the length of the

"■■ '111- handle seen in Fig. 1, it is made to travel idong the shding

-.r*

428 EXPLANATION OF THE PLATES.

bed. It has also a second motion at right angles to tliat already described.
The tool is fixed to the rest by four screws, Fig. 1. The headstock a it
raised on a carriage b^ to which it is bolted. This hithe is of large dimeonMUy
and is capable of boring cylinders of great diameters; it is also well adapted
for turning locomotive engine wheels.

PLATE XXV.
Facb-turnino Lathe, by Messrs. Nasmtth, Gabkbll, and Co.

This Plate differs but little from the last described. There is only on
rigger or pulley a, on the mandril or shaft, and the face-plate e has fbor
adjusting screws for securing the work to it Instead of driving this plate
from the pulley-shafti it is put in motion by a small pulley worked hy a
strap, from the main shaft of the building, in a similar way to the pulley d^
making the one quite independent of the other. The internal wheel, and
the pinion, are similar in every respect to those described by Plate XXIY.,
which is also the case with regard to the slide, the slide-rest, and tbe
headstocks a a.

The foundations to which the bed b is bolted are of stone bedded in
concrete.

PLATE XXVL

Foot-Lathe, by Mr. F. Lewis, Manchester.

The bed ^, Figs. 1 and 2, is placed on the two standards b b, forming at
the same time centres for the crauk-spindle n to work in, which receiTes
its motion from the foot-board m' by the two connecting links oa. On the
spindle n^ is a pulley and also a fly-wheel, for regulating the motion; (the
latter being used at times as a pulley for quick speeds;) these convey it
through tlie upper pulleys c, by a leather strap, to the mandril or spindle d^
on the end of which is a chuck or face-plate. The headstock ^ fonns soit-
ablc bearings for tliis spindle to revolve in, having a back centre eciew to
tighten it by. The following headstock r slides along the bed to any position,
according to the length of the work to be turned, which is placed between tbe
two centres //; tliat in the latter head:*tock is adjustable, being fixed to a
cylinder tf working in it, and moved either backward or forward by the handle
or wheel (^\ which, as it revolves, works a screw fixed to it and k^C steady
by the bolt underneath. Fig. 1 . When the work has been properly eentevad,
the cvlinder <; is made secure in the headstock by the small handle and acasw A.

The slide / for carrWug the tool is advanced along the bed hy a pimsa
working in the rack i", put in operation by a small handle placed osi As
square end of the spindle •" : a motion at right an^es to this ia oteiaad ^

I

JBC

ft!'

Lfotc

EXPLANATION OF THE PLATES. 420

die sliiley, worked in the game way by the ecrew /, and lastly, a circulof
motion may bo given to the loni by the womi and worm-wheel /. The
tool is placed in the tool-box «, and there Rsed by the two set eerews; thia
box nmy be placed in any position, and then Becored by a set-screw m.

An index y' is graduated on the beadstock g, (at the purpose of tnrning
spheres.

The rack i' is secured to three small brackets bolted to the bed of the
lathe:

This inaohino is capable of turning a bar 3 feet long, and 6 in. diameter,
and spbericsl cups, halls, &c., &c., 5 in. diameter, having an index to turn
either conioally or longitudinally.

As will be seen by the title, this lathe is put in motion by the foot of the

an working it, and unltke tlic last two, it is only adapted for turning
onall work.

PLATE XXVII.

Sbven-Feet TuKNiNo-LAinB. by Mr. F. Lewis, Manchesteh.

In the usual way, the top part of the fixed bed o. Fig. 2, is of the V
form, upon which the lieadstucks bb and the slide h are moveable ; that for
carrying the driving apparatus is secured to it by strong bolts and nuts ; the
■haft of the driving pulleys c works in the two bearings of the headstock; one
d, for the purpose of keeping it always tight and to avoid any play,
rhich would be the case were it parallel, after having worked any length of
tightened against the eone-piecc of the beadstock by nuts on the
ipodte end of the shaft. To the face-plate g, die work is fixed by four
jaws rrrry moving in Vs on ihe hack of the plate, and radiating to the
centre, to which they can bo approached or drawn from, by a small handle
the square end of the screw, by which means they are made to
, (lie V'b according to the size and nature of the work to he turned ;
motion is eommunicalod to the face-plate by the spur pinions and wheels
theintermediatcshofty, and thence to the pinion working the internal w^hecl
OB tlie lace-plute, consisting of sis segments screwed to it. By this arrange-
ment the Bjieed of t}io face-plate is very much decreased, and it slioutd he
understood that though the pUte ff is on the main sliaft,- it is not lixcd to
it by keys, hut runs loose, and revolves at a niucli decreased velocity ;
B triangular carriage is bolted to the headstock's side fur supporting the
igB of the intermediate shaft.

the opposite beadstock the oyltnder q and centre n moves, being slid
it by turning the wheel and screw /, according to any required
ition for centering Uie work, trhich, when done, is firmly secured bv the
tightening rings and nuts y".

430 EXPLANATION OF THE PLATES.

Tliis headstocky and also tlie slide, is moved along the sarface of tlie bed
a hy pinions working in a rack o, fixed to the three bmckots p^ on iti
side. Tlic pinions arc made to revolve hy a lever turning the wheels next to
them. Independent of tlie motion of the slide along the top of tho bed,
it has three others, viz. : the frame j can swivel round on the carriage t, to
any angle ; the slide X* can travel along the frame J by a small handle fixed
on either end of the screw k'y and lastly, a motion at right angles to this
is obtained by the slide on which the tool is placed, being also worked by a
screw ; the tool is fixed by the bolts l\ which hold it firmly in its plaoe«

On the end of the main driving shafl e is the centre fi, and between
this and the other, the work to be tiuncd is centered in the usual war.

The tires for locomotive wheels, and also wheels of a diameter of seven
feet, can be turned either internally or externally by tliis lathe.

PLATE XXVIII.
Lathk for turning Gun-Barrkls, by Messrs. G. and J. Rbnnib.

The object of this machine is to perform an operation entirely different
from that ])rodiiced by the several motions of the common turning lathe,
where the work is either of a cylindricid or elliptical figure. In this case,
the curve required to be turned, on the whole Icngtli of the musket barrel,
is of an irregular form. This, as will bo soeu by Figs. 1, 2, 3, and 4 of
this Plato, is clfootod hy a vory simple ooutrivuucc, a description of which
fo]low.s : —

Tho l)od // is secured hy tliroc bolts a' u! t\ to the side frames or stand-
ards a a, which hnvo also fixed to thorn a tank o for receiving the wngte
oil and tho iron turnings from the barrel hoing turned ; on the top of the bed
is ])lacod tho hoadstock fnuno /, for carrying the several jiarts of the ma-
oliinery ; and on tho mandril if arc the tight and loose puller's for giving
motion to the machine : the centre c' of this mandril is adjusted by the
hack centre c\ By the two sets of spur wheels and pinions a double
motion is produced, the forward hy / and the back>i'ard by^, which are
alternately worked by means of the small clutch ^, thus giving a revolving
motion to the square threaded screwy for advancing the tool irame / along
the barrel to he turned ; this motion is made self-acting by means of two
small bars pp fixed to the clutch lever, having projecting pieces on their
sides, with wliich the frame / comes in contact, and pressing against them,
draws them and engages the clutch in the wheels for giving the oppoaitc
motion, while the object of the handle h' is to disengage it by hand. The
spindle on which these small wheels arc fixed runs freely in brasa bearings
fitted to the heodstock f .

I

I

on

EXFLANATION OF THE PLATES. 431

On the opposite end of the machine is the hcodstock m, haTing bo ftd-
justnhle centre d, worked by the screw and handle m", and may be fixed
in orv required position by a small set screw ?n"'; this bead can be moved
kloiig die bed b, to suit the length of any barrel n, and by the cross bar and
traits m' may be secured to it. The musket barrel to be turned is slid on
the long bar or mandril n', which it fits very tightly. One end of tliis
maiHlril is then dropped in the chuck e for carrying it ronnd, while it is
centred at both ends in the centre pieces f'e'.

The most curious part of this mnclunc is the frame I for holding the
tools and likewise the barrel of the musket being turned ; which latter
operadon is rendered rather difficult from the irregular shape of tlie barrel.
Fig. 3 shews a section, looking at the fmme I, which consists of a bock
plate, on the face of which is a broAs plate with the eccentric grooves
shewn by tlie dotted lines struck from different centres ; between the
latter plate and the back plate are four dies sliding in V's, having ribs
irorking in the above grooves, which, not being concentric, press them
against the barrel, and thereby hold it firmly while the tools V are turning
it; but as the diameter of the barrel alters, it is necessary to loosen tlie
dies, in order to allow the frame /, of wliich they form part, to slide along
the bed on its V's. This is thus effected : a curve bar or template k, suit-
ftblo to the curve required by the barrel, is firmly screwed to the side of the
bed, its npper edge i, Fig. 3 and 4, being of the shape of a V, on which
the lower part of the rack f" slides, the upper part working at the same
lime in the V's screwed to the projection given to the back plate / by three
•crews. Thus the motion produced by the curve of the bar it is transmit-
led throngh tlie rack and the segment V to the grooved plate on which
the latter is fixed; the result of this is the tightening or looseoing the four
dies, which must naturally be the case, as they have ribs working in the
grooves, which they are compelled to follow. By the weight suspended
from the vertical rack, any irregular motion lliot might take place is en-
tirety obviated, it being kept firmly n'orking on the upper surface of
'Sa.ii curved bar it. while the small handle is provided for the purpose of
raising it by band when required. There are two tools l\ Fig. 3, filed to
one of the four dies already described ; tlie one farthest from the frame / is
for rough turning, while the other follows it, giving the finishing cut.

A small bearing is provided on the bed for supporting the outer end of
le screw j^ and the standards are well secured to the floor by bolts for
"QlKt purpose.

This machine is capable of turning and finishing three barrels per hoar ;
and similnr machines tire in use at tlte Hoval Armoury of Enfield, fitted up
Llovd and Co., and twelve of them, in conjunction with a com-
f mncliinery fur nmkiiig musket^ arc at work in the Imperial

432 EXPLANATION OF THE PLATES.

Armourj at Constantinople, constnicted by Messrs. Bemue, bendes ma-
chines for tlie French Goverament and for his Highness the FmIm of

PLATE XXIX.— Figs. 1 and 2.
Portable Hand Drill, by Messrs. Nasmtth, Gaskbll, and Co.

The utility of the little machine shewn by Figs. 1 and 2, is its porlir
bility ; and it is found very conycnicnt in drilling holes in sach pieces of
machinery which if required to be brought under a larger machine would
cause great delay and iuconyenience.

The frame a carries the upright drilling spindle b^ the top of which is
a screw e' for raising it by the handle wheel e by hand, while the revolTing
motion is communicated to it by the two small bevel wheels for conveying
it at right angles. When required to drill a hole in any piece of machinery,
it is first of all set in its proper place ; after tliis is done, the handle c, or
small fly wheel, is turned round for working the drill, and by a slow re-
volving motion given to the upper handle ^, the drill, while working, is
made gradually to descend. It can, if required, be secured to its woxlc.

Fig. 3.
Foot Duill, by Messrs. Nasmyth, Gaskell, and Co.

The diifcrencc between this machine and those described by Plates XXX.
and XXXI., consists in the mode by which the upright drilling spindle is
made to rise and fall. In those referred to, this is made self-acting,
while in this case it is performed by the pressure of the foot of the man
superintending the drilling of the hole.

This machine is driven by the riggers or pulleys ^, the one running loose,
while the other is fixed to the spindle for conveying the motion by means
of the upper and lower sets of pulleys f, by which the speed can with
great ease be made to ^-ary considerably ; this is done by altering the posi-
tion of the leather strap shewn by the dotted lines. The motion is then
carried at right angles, to the drilling spindle by Uie bevel wheels, for pro-
ducing the requisite revolving motion.

A moveable table ^, for carrying the work to be drilled, is fixed to the
frame a, in which it slides, and can be raised or lowered by the wheel and
screw // ; this is found of great use, as the size of work may very mudi
vary.

By means of the footboard y^ working as a lever on its fnlcmm, the
drilling spindle is made to rise and faHl ; the pressure of the foot on the

EXPLANATION OF THE PLATES. 433

boBrdycauaing therody to rise, which by the upper lever fixed to tlio frame
of the machine depresses the spindle d, while it is revolving ; as soon bs
the pressure is taken off, tlie counterbalance weight/" causes the drill to

» ascend and take its former position, where it is kept until again used.

PLATE XXX.
Wall Sidb Drilling Machine, by Mbssrh. Nasmyti

The eeverol parts of this mnchinc con»st of a frame fixed to the side of
le wall of the building, ngEin§t which arc bolted the frames ib, for carry-
iitg the upright drilling bar e, put in motion by the driving pulleys or riggers
c, according to the Hpccd required, which is regulated by the diameter of
the pulley on wbicli tho atrap works. Independent of this, it has also a
double motion, obtained by the two spur wheels pad pinions d; this second
motion is fully explained in the reference to the Plates XXXIII. and
XXXIII A.

This machine is made self-ncting by the spur wheels and pinions //
mniig the drilling bar as they revolve ; the njiper port of the screw g has a
cross guide h, slidtug up and down between the two upright parallel

Tho table k, for carrying the work, is made to slide on the bedy, similar
to Ae bed of a planing machine, one of its sides being of the V form, while
the other is a Hat smooth surface ; this table is advanced by a chain fixed
to it at one end, and works round the rollers m.

In the drawing, a small cylinder n is being bored.

This machine is well adapted for boring holes fur the pivots of engine
and oU parallel holes.

PLATE XXXI.

I DoDBLE Pillar Drill, by Messbs. Nasmyth, Gaskbll, and Co.

I The principle of this mochino is the same as that last described, witli the
f a fen- of its parts, which are of larger dimensions, and for

The fmroe-work o for supporting the different parts of the machinery,

Bts on two upright pillors ; on this fmme is a small shaft for carrying the

hring pulleys b and pinion c, conveying the motion to the upper or inlcr-

iate shaft by the spur wheel, whence it is taken at right angles to the

g bar by the bevel wheels rf; the bar e is raised and lowered by the

D the upper jiart of this bar, which screws itself up in a nut ; when

uirad to bo lowered, the handle or whcoiy is worked ronnd by hand.

434 EXPLANATION OF THE PLATES.

setting in motion the upper spur wheels, thereby commnnicatiiig it to At
upright screw and drilling bar.

The moveable bed or table hy for carrying the work, travels along the bed
g ; it has two motions, the one at right angles to the other, and by the long
screws and handles i i is brought in any convenient position.

Its principal adaptation is for boring the holes for receiving the tabes in
locomotive boiler plates, and in such coses when any number of holes are
required.

PLATE XXXII.
Radial Dhilling Machine, by Messrs. Benj. Hick and Son, Bolton.

An entirely ditfcrcut arrangement of a drilling machine from those
rally used, may be seen by this Plate ; in all ordinary cases, the drilling bar
or spindle is stationary, that is to say, it has no lateral motion, being
only able to rise and fall in its bearings ; in this case, the whole drilling
tackle is made to slide along a radial bar or carriage, whereby the drill can
be brought over the work into any required portion within the limitB pre-
scribed by the radius of the arm b.

The arrangement of this machine consists of a strong upright column a»
bolted in a most substantial way to tlie stone foundations ; a screw c worid
up and down in the internal part of this column, according to the he%fat
required for the work ; this is made secure when raised to its proper posi-
tion by a uut d^ tightened by the four pins on its circumference; the apper
part of the screw lias a collar, upon which the nidial bar rests, which at the
same time is capable of revolving on this centre ; a carriage for supporting
the small square slmft, the bevel \>'liec]s, and the fast and loose puDejB
placed horizontally, is fixed to the upper part of the radial bar by bolts
and nuts ; on the other extremity of the shaft y, is a second pair of bevel
wheels, for conveying the motion at right angles, thus causing the drilling
bary to revolve, on the lower end of which is fixed as usual the drill it.
The two upright sujiports h are bolted to the travelling frame ; these carry
the apparatus for raising and lowering the drill, which consists of two small
chains fixed to the top of the diilling bar, working round rollers, and also
two others having weights suspended from them, and running over the chain
])ulleys, that with the large weight brings tlie drill down, while it is drilling,
which, after it has performed its duty, is disengaged by a lever and rod, ao
that its weight is neutralized, when that of the smaller ball comes into
o])eration for raising the drilling bar to its former position. On the lace of
the radial bar is a rack and pinion /, which, by means of the wheel handle
l\ causes the drilling frame to slide along the surface of tlie bar 6, whiob, as
already stated, is regulated by the position required for the drill, while tbe

EXPLANATION OF THE PLATES.

435

■qoara epiudlo or sbntl g slides through the wLocl, nnd also the bearing fixed
on the bar. Th? motion Is conininnicsted by a leather stm|) on the fast
pulley e, tranBmitling it through the bpve! wheels to the drilling bar.

PLATES XXXIII. AND XXXIII. a.

Upright Dhillino and Borino Machine, bv Mh. F. Lewis,
Manchester,

In most drilling machiaeB, the motion ia conveyed at once to the npright
I bar, by fixing the driving polleya ou it, which is the case in those already de-

eribed, os will be seen by rcrerriiig to the Plates ;
[ mediate shaft or spindle, placed horizoutally, re
[■■trap on the pulleys or riggers c, and is conveyed a

r of small bevel wheels k ; a double motioi
[ tnacbine whereby the epceJ can be much altered,

n this iustaiicc, an inter-
«ives the motion by a
,t right angles bv means
1 is provideil to this
independently of the
F difierent diameters given to the driving pulleys. The mode by which tbia
alterauon of speed is obtained is thus : there ore two shafts, one having b
wheel working in a pinion, and the other a pinion working in a wheel, each
pwr being of the some diameters, Fig. 2 ; it must be understood that the pul-
bys e and the pinion shewn in Fig. 1, run loose on tho shaft on which Uio
[ ftontwhceliskeyed, andarethore st-cnrcd. Now suppoang a slow speed bo
LveqniTcd, it is only necessary to throw in gear the bnck wheel and pinion,
■ sliding the shall in a groove where it may be kept fixed by a pin </",
The result of this is, that the pulleys, running loose on the front
I, convey the motion through the jiiuion fixed to them to the back
iteel, and hence through the bock pinion to the front wheel, conveying
1 through the bevel wheels to the drilling bar e. An esamplc will
better illustrate this : — Let it be supposed tlmt tlie pulleys and pinion are
Buking twenty revolutions per minute; the wheels being IC inches in
diameter, and the pinions 6 inches in diameter ; the speed of the bock shaft
wonld be 1\ turns per minnle, and consequently that of the wheel and front
abaft would be reduced Co about 3^ turns, thus making the respective velo-
cities about 5 J to 1. When a <(uicker speed is required, the back shaft,
wheel, and pinion ore disengaged, (by the same meaiis it was engaged, and
secured by the pin cT',) the front wheel is then made fast [o tlie pul-
leys by a screw d' on its face, and both revolve at the some velocity. Tho
k wheel and pinion should be always disengaged previous to fixing the
(Ont wheel to the pulleys, otherwise great damage would be done to the
JKth of the wheels.

e self-ucting moliou in this machine is the same aa those generally
I, whidi is llirown ofTnnd on by the small clutch k and lever i, imd may

43G EXPLANATION OF THE PLATEB.

be worked by hand by the wheel /; the screwy works in tlie niit^% and

by the small spur wheclsy^ raises the drill bar.

The table m for carrying the work to be drilled or bored, is raised or
lowered according to circumstances, by the two pinions n VForking in the
racks fixed to the \v^\ plate ; this is effected by the handle o, wheel and
pinion /?, and is kept from returning by tlie paul and ratchet wheel q pro-
vided for that purpose ; the plate r is bolted to the wall through four snv^
The frames h for supporting the several parts of the machinery have snitsUe
bearings on them for carrying the shafts and spindles ; they are ivcll secured
to the wall by strong bolts and nuts.

This machine can bore from lialf an inch to 14 inches in diameter, with
a depth of two feet, and tlie table m vnUl admit of a wheel 3 feet 8 inches
in diameter being placed on it.

PLATE XXXIV.

Slotting or Key oroovino Machink, ry Messrs. Nasmyth, Oasrell,

AND Co.

This machine is of a very simple construction ; the motion being under-
ground, and conveyed to it by the small shafts to a dog or crank wheel ^
giving the slotting bar a reciprocating motion by means of the connectiiig
rod r, to which it is fixed ; the lower part of the slotting bar works in a
cross guide. To the sliding ])late ^/, the wheel e to bo grooved is fixed ; the
cut is pcrfomu'd by the descent of the slotting bar, the length or depth of
which is regulated by a moveable crank ])in on the face of the plate b» It
is necessary to have access to the underground work, and for that purpose
the stairs f are shewn. A machine u])on this principle for cutting the
teeth of wheels M'as, we believe, used by Messrs. Bolton and Watt, at Soho.

PLATE XXXV.
Slotting Machine, by Messrs. Sharp and Roberts, of Manchestbr.

The persj)ective view shewn by this plate represents a machine for slot-
ting or poring small work, such as the straps of connecting rods, or cuttine
the key grooves of wheels of small diameters.

The table / for carrying the work, is a circular plate of about two feet in
diameter. It has two horizontal motions at right angles the one to the
other, working between V's, and these are made self-acting by means of the
ratchet wheels o, o, worked by the cam d on the main driving spindle c,
which, 08 it revolves, strikes against the lever k, and thereby gives them a

EXPLANATION OF THE PLATES.

4.37

I

I

progrcBiiTe motion ; the pauls pp aJvance it in proportion to the leverage
i>r distance from tlie fulcnim given to lever i; this may be altered at pleasure
by clianging tbe position of the rod r. The Beif-acting progregs of the
table / is thus made to vary from ^\ to -^^ of an inch per revolution of the
entnk wheel g. The spiral spring ./' serves to put! up the rod after it has
been forced down by the cam rf, and bring back the paiils for a new stroke.
The use of the other pauls is to prevent the ratchet wheel from running
bock while tliis latter operation is being performed. Another ingenious con-
trivuncc in this machine consists in a third or circular motion of the plate /,
which turns on its centre, and linving its circuraferencc equally divided hy
tfie notches m, may be moved ronnd an equal angular distance at a time,
■od by the spring or catch n is retained in a Bxed position while the
tool is nt work. Tliis arrangement is found to be very advanti^oiis in
cutting the key grooves of wheels, being required to be perfectly equi-
distant.

The riggers / arc worked by a strap from the main shaft, and next to
tfaem is hung a tly wheel e, to regulate the motion ; these and the emnk
wheel g are securely keyed to the spindle c. The reciprocating motion is
then carried to the slotting bar b, hy a smal! connecting rod h; it, are
groove-pieces fixed to the frame in which this bar works ; the length of the
stroke may be altered by slidiug the cnmk pin «, in a groove provided on
ibo plate _o, but its longest stroke cannot exceed 8 inches. The different
parts of this machine are fitted to the frame or standard a, having on its two
inteniol sides V's, in which the table frame slides, which may be either raised
or lowered by the handle underneath. The screw handle q and also the
huidle on the ratchet wheel, serve to bring back the tables after they have
advanced hy the self-acting apparatus. The action of this machine is
to that of the morticing machine at Portsmouth.

PLATE XXXVI.

Machine fob cuttino Kev Ghooves in Wberls, bv Messrs,
Nasmvth, Gase^sll, and Co.

Another descri
iffering merely in detml from

iiotti

ng machine is represented hy this Plate,
last described, which is limited to the
e of the work it is capable of performing by the two sides of the
indard frame preventing the admission of large wheels. In the (iresent
ichine, a wheel of any size can be grooved, the whole of the machinery
eing uudcmeath instead of above the table.
The four small colunms a, support the table b, on which there is a bed
for ifac alido to work in ; on this, the dividing pktc m b placed, having iU

438 EXPLANATION OF THE PLATES.

ciTcainfercncc notched in a similar way to the machine deacribed in Fbte
XXXV. The spindle e, is driven by the rigger e, and baa a Hj wlied d^
to regulate its motion ; on tlie end of this same spindle is a pinion, wotkii^
in the crank spur wheel y^ connecting the motion to the slotting bar by tbe
rod k ; the position of the crank pin ^, may bo altered to any convenient
length of stroke by sliding it along the groove shewn on tbe face of tbe apor
wheel /. Thus a reciprocating motion is given to the bar t^ baring on Hi
upper end the tool for cutting the grooves, represented by the dotted Ibifli
on the large spur wheel being cut. This machine is made self-acting by
eccentric or cam on the spindle y^ which raises and lowers the small 1
by means of upright rods working the ratchet wheels jy and paola t; tbe
ratchet wheels are fixed to long square threaded screws (the length of tbe
bed), and work through a nut fixed to the underside of the sliding tabk^
which may be inclined or placed at an angle suitable to the taper reqnired
for the key. The handle / prevents the circular dividing plate us from
changing its position while the groove is being cut^ after which it is dis-
engaged from the notch by hand, till it meets a second one, eqnidistsntly
divided.

Between the columns tlie diagonal stays are placed, to give strengtb and
stability to the machine, there is also a cross plate running from one side
frame to the other, through which an aperturo is made to allow tbe slotting
bar t to work through.

The tool on the top of the slotting bar can be altered at pleasnre, by un-
screwing the small screw which secures it in its place.

It is from the advantages derived by the principle of this machine, wbicb
admits wheels of unlimited diameters, from its underneath motion, tbst
Messrs. Nasmyth, Oaskell, and Co. have founded their patent

PLATE XXXVII.

Large Slotting Machine, by Messrs. Nasmyth, Gaskbll, and Co.

Among the variety of slotting and key-grooving machines already do-
scribed, there has been none similar to this, cither in the general arrange-
ment of its parts, or its capabilities as to the magnitude of the work it is
capable of slotting ; it is altogether a much larger machine, and of simple
contrivance. A cmnk is shewn in the drawing, having its sides pored off
by the tool ; however, it is not solely confined to cranks, but may bo used
for any machinery that can be placed within the limits prescribed by tbo
two columns.

The arrangement of this machine consists of a rectangular finme a.
Figs. 1,2, and 4, upon which arc placed the plummer Uoclu for euryiiur

EXPLANATION OF THE PLATES. 139

ibo shaft bearings; tho two columns o' o' resting on tlie top of the
bed b sapport this upper eiilabloture frame a. Like most slotting ma-
chines, the plate for carrying tlic worlt lias three different motions, the two
first at right angles the one to the other, while the third consists of a circu-
lar motion, which is required when the work to be operated upon is drcu-
iar. TIic first is longitudinally along the bed b ; this is done by working
the ratchet wheel and screw e", thus drawing the slide e ; the second at
right angles by the ratchet wheel and screw d" working the slide <^ in a
transverse direction ; and lastly the third, which con^sts of a circular plate
having on its circumference the worm wheel c, made to revolve on its
centre by working the worm or endless screw and the ratchet wheel c*. The
screws ^ and if severally work in nnts fixed to the under sido of the slides,
one of which may bo seen, by the dotted lines shewn by Fig. 3 in the plan

of the bed-plate. Fig. 4 shi
tool or slolilng bar i ; the n
i from the driving Bboft, is
which is a spur wheel and pir
to the large spur wheel i, t

(pvih

plan of the driving gear fur working Uio
being conveyed to the riggers or pulleys
icatcd to another set of pulleys upon
n, conveying it through an intennediate shall
which is fixed the connecting rod_/; by this
connection tlie wheel A answers Loth this purpose and that of the crank,
lying the alternate or up and down motion to the slotting bar k.
different arrangements that may be ^ven to the wheels and driving rig-
pulleys I, independent of i!ic double motion, similar to that de-
ibcd by Plate XXXIII. and XXXIII. a., a great variety of speeds may
bo obtelned, which is of importance, some ports of machinery requiring
a much greater velocity than others, white being opemted upon.

The hollow bar k is guided at one end by passing through the upper part
of the frame o, which is all one piece ; this may be seen in Figs. I and 4 ;
and the length of the stroke may be altered to suit the work by clianging
tlie position of the connecting rod _;' on the slotting bar it, sliding it in a
■iDove provided for that purpose.

sThe three different motions already described are rendered self-acting by
k|nn on the spur wheel h striking as it revolves agmnst the lever shewn by
the dotted lines in Fig. 1, which commimicate with the levers, rods, and
small bevel wheelsy, and ultimately givenself-aclingniotion to the different
Handles arc placed on llic three ratchet wheels, for working them

;. 2 shews the tool / in the a
|r which the third motion is put <

t of paring the circulor part of a crank,

1, and the other two are consequently

e frame is strongly secured to the stone foundutioj:
Iag-<kiuii bolts.

440 EXPLANATION OF THE PLATES^

PLATE XXXVII. A.
Slotting and Paring Machine, by Mb. F. Lewis, Mani

The machine represented hy Figs. 1 and 2 of this Plate, thoo^ different
in detail from those preceding it, is the some in principle ; this neccM
rily must be the cose, where either the operation of slotting, paring, or key-
grooving is to be performed, which severally require an alternate or reci-
procating motion, whereby the introduction of the crank or dog wheel is in-
dispensable.

The dra\\ing shewn by this Plate, is a slotting machine of great power,
by which work of considerable size may be operated upon, the table for
carrying the work being about 3 feet in diameter, while the stroke, wbidi
may be varied, can be extended to 2 feet 6 inches.

In the side elevation will be seen the frame a, to which is connected the
whole of the machinery for working the different parts, according to the
nature of the work required. On the front of this frame are two fiices^
Fig. 2, to which are fixed the V pieces, two of which can bo adjusted hj
the set screws, according to the wear of the slide g. Tlie speed may he
altered to suit the proper velocity for the work, by changing the str^i on
the riggers 6, these receiving the motion from the main shaft ; the fly wheel
regulating that motion, without which it would be liable to jerks, and mnch
variation. By the wheel and pinion e/, the speed of the spindle on which
is placed the crank or dog wheel ^, is much reduced ; the short rod f con-
nects the up and down motion to the slide y, for carrying the tool f ; andhy
loosening the screws h h^ any tool can be applied. The spindles for carrying
both the wheel and pinion dd run freely in bearings fixed to the frame a.
On the top of the bedy, the sliding table k is moveable, and on its upper
surface, the circular table / slides in a contrary direction, this latter table has
also a revolving motion on its centre, by working the worm on the circum-
ference of the worm wheel, wliich is done by turning the handle shewn for
that purpose. A self acting motion is given to this machine, by a pin on
the rim of the upper spur wheel d striking as it revolves against the lever
seen in Fig. 2. by the dotted lines. On the lower end of Uus lever, is a panl
for advancing the ratchet wheel, the boss of which is a brass nut, which,
as it turns, screws up the square-threaded screw, and draws with it
the table k on the adjustable bed j. At a convenient distance on this lever
is placed a small rod, connected at tlie bottom with a series of levers, which
being put in motion, work the slide frame, and give also the circular motion
connected to it by the two small wheels and paul, thus making the three
motions connected with the sliding frames for carrying tho work entirely
self acting, which is also the case in those machines already described
by the preceding Plates.

EXPLANATION OF THE PLATES. MJ

aperture through

Tbat part shewn as circultir on the main fmmc
krhich tiie spindle e U passuil.

PLATE XXXViri.

ScBEw-CnTTisG Machine, bv Mb. Fox.

Tlie machinery required for this purpose consists of a bed a anpjiorted by

o end frames; upon these are fixod the requisite hearings or plummer

•oks, i and d, for carrying the two spindles c and e, — the latter being

driven by the riggers i, which, by the small double conical friction clutch _/,

ore mode to work or revolve iu opposite directional, thus keeping the ma-

^^ chine in constant operation. By mcaus of the spur-wheel and long pinion k,

^H|4lic motion ia conveyed to the Upper or working spindle of the chuck or

^^Bbnx ni, for holding the screw or bolt to be cut, while passing through the

^^Bfes of the outer frame, (which ore tightened by the Iiandic and screw I

^^Bressing against the two springs _;',) thus working itself through the

^^Hfiea, and drawing with it the spindle c and upper wheel h, the length

^^'reqiiired. On the bos wi, Fig. 2, is shewn a screw and handle for

the purpose of tightening or slackening the screw before and afWr

the opcradon of cutting. The machine is put in motion by turning

^»^ Btuall handle and eccentric y, ou the top and bottom of the upright

^^hods, which press tlie triction clutch against cither the one or the other

^^hf the two riggers, whicli, from running loose on the spindle e, can be made

I^^lo revolve in either direction.

When it ia required to tap nuts, they are placed in the outer head, and
Bccnrod by the wheel I, which in the operation described above contained
the dies, the tap is then placed in the bos m, and aa It revolves cuts tlie
proper threat) in the nut.

kThis machine may also be used as an horizontal drilling machine, the
<rk to be drilled being secured in the head. Fig. 3, and the drill In the box
the screw above the riggers being then tightened, presses forward the
11, which as it advances drills the required hole.

The spindle which carries the box m, might he made to slide independ-
ently of tlie wheel and pinion h h, simply by having a lon^tudinal groove
T keyway cut on its circumference.

PLATE XXXVIII. A.
I Labos Screw -Cutting Macrini!, bv Messrs. Nasuvth, Qabkbll,

AND Co.

I This machine is a contrivance for the purpose of cutting screws of large
Samctero, nnd couiusts merely of two side tVames or standards for carrying

442 EXPLANATION OF THE PLATES.

the sercnil parts of the required machinery. Unlike moat machnieB, tUa lias
neither heel nor tahle, the frames hcing connected together by the two atnmg
wrought iron holts dd^ (which also answer the purpose of guides,) and the
two stretching holts for steadying the lower port of the standarda, the upper
part of the standards aa liaving hearings in which the driving abaft woifa
hy means of the pulleys />. This machine has a double motion, ramihr to
that described in the drilling and boring machine, Plates XXXIII. and
XXXIII. A., which, as it is there shewn, is obtained by two pairs of spur
wheels and pinions c. In this case, when the quick speed is required, the knrcr
shaft, pinion and wheel is slid through its bearings in a simihir way to die
second motion of a crane or crab, a projection being ^ven to the end of die
diaft for that purpose ; and when the slow motion is used, the lower shaft
is again brought into the position shewn by Fig. 1, the wheel coming against
the collar on the upper pinion, and the collar of the lower pinion against the
upper wheel ; these collars prevent them from going any farther, while die
moveable stop, Fig. 4, keeps the lower spindle in its proper position.

The chuck f^ Figs. 1 and 2, forms part of tlie shaft for canying die
pulleys ; and in it are fixed the dies for cutting the thread on the sdew,
which may be taken out at pleasure, and others put in their place, by
loosening the set screws ^ ; a second chuck or frame. Fig. 5, (for aappoii-
ing the screw, at its head, between the dies,) slides along the guide bolto dd^
as the thread is being cut on the screw ; when this operation is fiidahed, die
round headed screw, Fig. 5, is slackened, and consequently the screw is
liberated and another put in its place.

In cases where nuts arc to be tapped it is only necessary to reverse the
operation by putting the nut in the chuck f^ and the tap in that marked e,
where they are secured as already stated in reference to the screw.

The arrangement of this machine is extremely simple in constroction,
and it is capable of cutting the threads of screws of considerable diameters.

PLATE XXXVIII. B.

BOLT-SCKEWING MACHINE, BY' MeSSKS. BeNJ. HiCK AND SON, BOLTON.

The object of this machine is the same as that described by Plate
XXXVIII. A., and its arrangement and principle are in cxcry respect si-
milar, with one exception ; instead of a double motion, as is the case
in that above referred to, this has a backi^-ard and foni*ard motion
given to the hollow shaft, which is thus effected : — there are three driving
pulleys, c^ Cy d; when the strap runs on r, the machme is put in motion bj
the spur-wheel and pinion c\ thus communicating it to the chuck /^ for
holding the steel dies or cutters ff ; as soon as the whole length of the acfew
lias been cut, the macliine is required to be reversed, which may be done

EXPLANATION OF THE PLATES.

' shnft, tliiia drawing tlic

by running llio strap from the pulley c on to that
wlieel d' is set in motion by a pinion on the
on to the lower driving spindlu
former position ; and kstly, when the machine n
on to the circmiiference of the centre pulley c,
spindle, revolves u-ithout producing any effect on
machine. The upper bolts b, for steadying the t'
for the sliding frame. These and the other parts, e

of d, whereby the internal

cylinder, revolving freely

w back t<

; rest, the strap is passed
licL, being loose on the
le working parts of the
' frnmcB, form alsw guides
I already sfdd, m

IS the machine last described, which renders all further description useless.
PLATE XXXrS.

&ELF-ACTIMQ NUT-CUTItNO MacUINE, BV MbSSOS. NASIiyiH,

Qaskell, and Co.

rtl
The machine shown by this Plate is supported on a fiarao nmilor to that
described by Plate XXXVI., where the table or beJ,^ is fixed to the four
columns, which ore much strengtliencd by the diagonal crosses,

• The spindle driven by the riggers or pulleys e from the nuun shaft runs
freely in brasses fitted in the heads dd; on one end of this spindle the
■teel uutter a is fixed ; the slide y*', on which is placed the dividing plate or
Llock, is advanced between the two V's bv a screw, having at one of its
extremities a ratchet wheel i, worked from the main spindle by means
of a small strap, the two small bevel wheels conveying the motion
at right angles, thus advancing the ratchet wheel, which, as it turns,
vorka (he screw through a nnt fixed to the under side of the slide /',
thereby making it self-acting ; _/ is a small handle for the purpose of bring-
ing bodt the slide by hand, after it has performed its work. By this ma-
chine a very great saving of time is effected in planing or siding the faces
of nuts, witli the utmost accuracy, and with an almost incredible saving of
time, for in ordinary cases the work performed by this machine was en-
tirely done by cliipping aud filing. The mode of working it is simply
by fixing the nut as it comes from the forge on a mandril, aud tho
(iktter in a hole on the block A, where it is securely fastened by a small nut
re, it is then advanced to the cutler by the slldey, where it receives a
ectly smooth face ; the block is then disengaged from the handle c in
0 notch y, and made to revolve till it comes to the following notch, where
\ u agiun secured. The block is divided into six and eight equidistant
portions suitable to either square or six-»ided nuts.
The small tank I contains water, kept constantly falling on the cutter a,
T the purpose of keeping it cool.

The spindle upon which the nnls ace to be cut may be suited to any
s isliewn by tho spindle tn.

414 EXPLANATION OF THE PLATES.

PLATE XL.

Machine for CiTTiNr. the Teeth of Wheels, bt Mb. F. Lewis,

Manchester.

Bv this machine, wheels of the following description maj be cat: lrt»
common spur- wheels ; 2n(Uy, conical or bevel wheels ; and 3dly, worm-
wheels.

a a represent the tr^'o side frnmes, the one of a V shape for the
slide ^ to work on, while the other is a flat smooth snr&ce. The mafthmeiy
supported by these frames may be divided under two different heads, tie. :
I St, that required for giving the revolving motion to the wheel to be col;
and 2ndlv, that which is requisite for working the cutter in the Tarioaa
positions it assumes.

1st. Machinery for turning the wheel. The spindle b has on one ex-
tremity the handle b\ which works round the circumference or rim of a
plain wheel, shewn bv the dotted lines ; this wheel or rim has on its cir-
cumference two notches^ equal to the \i-idth of the handle b\ in one of wUdi
it falls, and is there kept fixed during the operation of catting one tooA.
On the opposite end of this spindle is a small spur change- wheel e, working
through a second or intermediate one, that fixed to the H'onn spindle^
the worm d then conveys the motion through the worm-wheel e to the
upright spindle /i on the upper end of which is placed the wheel to be cat
This spindle revolves in brass bearings fitted to the end frame.

The change- wheels and pinions ccc may be altered to regulate the speed
and consequently number of tcech and pitch given to any required wheel,
this, as will be seen, is very easily done by unscrewing the screws which
connect the bearing pieces of the two pinions and intermediate wheel to the
side frame, when wheels of ditferent diameters can be put in their places ac-
cording to the required motion. This at once regulates the distances of the
bearing pieces, after which they must be well secured to the frame by again
tightening the nuts.

2dly. Machinery for working the cutter ;/i, consists of a cross slide ^ tra-
velling on the top of the frames a, a^ already described ; this is moved back-
wards and forwards by a screw working in a nut, and a small handle, whidi
is not >een in the drawing ; on this slide the frame with the two headstocks
/<f h is securely screwed by a strong bolt and nut ; I'T are the centres by which
the frame i i is fixed to that Ia.st described, having also two heads Bunilar to
those marked /i h ; on the back of this frame 1 1, is a bolt and nat wUdi
becomes a centre for the sliding frame it, and by which it is fixed to it. The
following motions are the result of these different combinations: IsL Tlw
frame ^ is advanced by the screw. 2dly. The two heads fonnii^tfaeheMt-

Iiangf

EXPLANATION OF THE I'l.ATES. 'i-iS

stock Frame h can be moved horizontally on the frame ff either to the right

or left on its centre, 3illy. In the sanae way the frame ti can he mode to

Iiang forward or fall bock on its two centres i'i'. And+thly. That markedy

■s a position either te the right or left on its centre, while the alide i,

in fact carries the cutter and its necessary connections, is made to rise

d fall by a small pinion working in the rack ^ and receiving iis motion

by the handle k", there ore two V pieces ou the fntaiBj, in whicU the slide

works up and down ; these are filed by four screws. The pulley / receives

^^t8 motion from the main shaft by a hand, and conveys it to the cutter by two

^■■^ of wheels and pinions, by which the speed of the cutter m is regulated ;

^^b the slide it is filed the bearing pieces for the three spindles ; o and p re-

^^besent an adjusting screw and index, for regulating to very great accuracy

^Hh> position of the cutter m, which may eoaily be taken out and replaced by

^Hniother by unscrewing the small screw n.

I The frame o a is firmly stayed hy bolts and nuts shewn on its side*.

It must not be forgotten that when the headatock frames A and J have
assumed their new positions dcscrihed by the second and fourth motions, there
is fixed on both the sides of the screws or holts upon which they turn, two
smaller holts working in slols or grooves, by which they are firmly bolted,
' otherwise the frames might slip, and this would cause great damage to the
Drk being operated upon.
1 , Any one of the different positions given to these frames may be uhtained

lependently of the others, or they might he need all together.
L The following description will shew the mode by which this machine is
cn-lced for the different kinds of wheels : —

s present position it is regulated for cutting the teeth of a common

r wheel, which is securely fixed to the top of the upright spindle /"ond

e to revolve as already described hy tiie handle b', which works from

Utcb to notch for every tooth, the change-wheels ccc being regulated to

the retjuired number ; hy this operation of the handle, the distance moved bv

the wheel to be init is always the same, thereby ensuring the utmost accu-

^^^y, Tbe whole of the machinery resting on the frame g is advanced till

^Hb cutter comes close to the wheel ; the shding fnune it, to which is con-

^Hfpcted the cutter, (which is now put id motion by the pulley and small spar

^^rhtsels,) is lowered by the handle working the rack and pinion k' till tt

comes in contact with the tooth to be cut ; after the porfonuancc of thin

operation the frame p is drawn back by a screw and handle already de-

libcd, when the nlieel is again made to revolve the distance of one tootli,

i the operation of the cutter repeated.

In the case of a bevel-wheel, the working of the machine is similar, the
Y difference being the position assumed hy the cutter-frame, whi(^ is
B to incline forward at any angle suited to tlio bevel by llic third motion

446 EXPLANATION OF THE PLATES.

described; this is again altered when it is required to cut the teeth
of a worm-wheel, simikr to that shewn in this Plate by e, where the teeth are
seen at the angle of the worm ; for this purpose the machine assomes the
position described bj the fourth motion ; and lastly, i^hen used to cut the
teeth of skew berel- wheels by the second and third motions combined.

It is unnecessary to describe the utility of this machine , or its acca-
racy, when small wheels with an extremely fine pitch are required. One
has now been in use for some length of time at the Bank of England, where
it is found to be very useful in cutting the teeth of the small wheels re-
quired in that complicated and ingenious machine for marking the numbers
on the bank notes, the numbers changing as fast as a man can feed the
machine.

The following Plate shews a larger machine of the same description, which
may be better understood, and has the same letters of reference.

PLATE XL. A.

Machine for cutting the Teeth of large Metal Wheels,

BY Mr. F. Lewis, Manchester.

This Plate represents two geometrical views of the last described ma-
chine, by which the different motions vrill be understood with greater
facility. It is in every respect similar as to its working parts, with an
additional self-acting motion for working the cutter, which operation in the
last case was performed by hand. It is also adapted for cutting the teeth
of much larger wheels from the length of the bed or frame o, upon which
the cutter frames slide. The letters of reference and description are the
same as those described in Plate XL. ; consequently a repetition ^ill
be useless, it will only be necessary to describe its additions, consisting of the
self-acting motion obtained by the bevel wheels s communicating the motion
of the driving pulley / to the upright spindle f, on the top of which is the
spur wheel and pinion working the worm w, and consequently the worm
wheel w' fixed on the same spindle as the handle k^' and pinion k\ thus
raising the cutter frame ^, on the back of which is fixed the rack. The
worm wheel ?/ is seen in Fig. 2 by a dotted line behind the spindle f. The
handle X:" serves to raise the frame k by hand after the tooth is cut by the
self-acting motion, which is found to be a great improvement in the capa-
bilities of this machine. The two side frames a a are connected together
bv the stretching bars v provided for that purpose.

From the great diameters of the wheels this machine is capable of carrying,
which are fixed to the face plate o by the bolts and nuts o'o\ it is evident
that were it not for the adjusting screw or stay /), a considerable d^ree of

EXPLANATION OF THE PLATES. 447

motion would be felt at that part of the circumference where the tool is
operating, and great inaccuracy would be the result ; by the stay /?, this evil
effect is entirely obviated, as it may be adjusted to any required height.

Wheels of the follo^ving description and sizes may be considered within
the limits of this machine : spur, bevel, worm, and skew bevel wheels, five
feet diameter in iron, and 10 feet in wood, having a breadth of 14 inches
with any pitch or number of teeth.

PLATE XLL

Machine for cutting the Teeth op Wooden Wheels Models or
Patterns, ani^also those op Iron, by Messrs. Nasmyth, Gaskell,
AND Co.

This machine is beautifully drawn in perspective with the usual felicity
of Mr. Nasmyth, and intended for the same purpose as those described
by Plates XL. and XL. a. Instead of the machinery being fixed to a table,
it is here made to slide on the bed 6, supported at both ends by the stand-
ards aa.

It is driven by the leather strap c on the rigger, which receives its mo-
tion from the main shaft. The bevel mitre wheels d then convey it at right
angles to the large band pulley^ whence it is carried to the small pul-
ley fixed on the cutter spindle^, adjustable by the set screws/ both at top
and bottom. The spindle e, and also the driving spindle upon which the
rigger is fixed, are fitted to carriages bolted to the bed b. The small pul-
ley / is for the purpose of keeping the band / tight, as the position of the
cutter frame alters ; it consists of a weight hung over a tightening pulley,
the weight / falling as the bandy* slackens; the small column for carrying
the pulley being fixed on the slide ^, wliich can be moved the whole length
of the bed b ; on this slide is shewn a circular plate, having a centre on
which the upright frame h is made to turn. By means of the wheel
and handle ib, the cutter frame t can be raised or lowered in the frame last
described, by loosening the two nuts shewn on the back. The strength of
the frame h is much increased by the rib shewn in the drawing.

It is obvious from the foregoing description that by this arrangement the
following motions are obtained : 1st, a longitudinal motion along the bed ;
2dly, a circular motion ; 3dly, a transverse motion, which is required for
cutting the whole width of the tooth, the frame being worked by the screw A'
and handle hf^ ; and lastly, the motion necessary to adjust the position of
the cutter to the centre of the wheels whose diameters vary ; this is per-
fonned by the handle k.

The dotted line shewn in the drawing represents a very large spur wheel

o o 2

448 EXPLANATION OF THE PLATES.

pattern of mBlio<rany being cut ; it is securely fixed to the diiidc plate
on tlio sliaft Oy which is made to revolve in the two heads hj ■
wheel on the same sliaft, worked by the worm n and handle for that pnrpowL

The mode of working thi.s machine is almost similar to those hwt de-
Rcrihed ; the wheel being accurately clucked, the cutter fraine is adyaneed
along the bed till it approaches the wheel, when the handle and screw If
being turned, will bring the cutter to the surface of the teeth. This
done, the handle and screw //' are moved in a contrary direction, till the
wheel, being cut, revolves the distance of one tooth, by the smaU handle
and worm w.

The frame p in the adjoining figure may be put in the place of that
shewn by /, when the machine is required to cut the teeth of iron wheels,
in which case a quicker velocity is required, as will be seen by the band
wheel and also the spur wheels and pinions; the spindle for connecting
these are all adjusted by small set screws.

PLATE XLLa.
Machine for cutting the Teeth of Wheels, by M. M. Olavst.

In the year 1 8»39 a patent was taken out by the inventors of this
cliiiie, for the pur])ose of cutting the teeth of iron and wooden wheelsi
f'itlior spur or hovel, by mechanical nicnns. It is, as will be seen by the
Platr, alto;rt*tlior on a ditforfiit principle from those described, which are
certainly wry superior in capabilities.

In the nrraiigenient of the ])lan and section. Figs. 1 and 2, the cast iron
table a is supposed to he su])ported on suitable standards for that purpose.
On this table or bed are fixed two ]ilat«'s lib^ with bearings for the spindles
or cylindrical bars cr ; a cap is fitted to these bearings and secured to each
of tlieni by two screws. The two cylindrical bars are united together at
their extremities by the two connecting links dtl; these bars are made
square in the middle part, and the two V pieces it are fixed to this square
])art of the bars by screws, in which the rest // for carrying the tool n is
made to slide backwards and forwards; it is adjustable by screws. On the
s(piare ])art of the smaller spindle is fitted a frame in which a carve k is
dropi)e(l, while the steel sj>ring / presses against a pin, and consequently
the curve X* against a screw y, directing the tool to tlie required shape of the
tooth. This i*< <lone by the bar ^, Fig. 2, on one extremity of which the
tool is fixed, and the screw J at the other. By a contrivance in this ma-
chine, the advance of the tool is thus made self-acting; two Btofm pp are ad*
instable in the mortices Figs. ], 2, and 3, and are regulated according to the
j<Jvfnec n>qnired, while the long square-threaded screw m works in a hosi on

iron

EXPLANATION OF THE I'LATES. 449

the fnune in which the curve k is situate, its other extremity being tixed to
the frame o' for carrying the ratchet wheel o, prevented from turning cither
one w-ajf or lie other by the springs or paul« jy.

Tliis machine is thus put in motion ; [he wheel to be cut being 6xed on
its projier centre, which in tliis case differs from those already described,
^m its hciiig secured to a part of the machine itself. An olleniate rao-
is conveyed either by a crank or other suitable means to the rodey,
conse<inently the two cylindrical hars slide in their bearings bbbb,
iwing along with them the cutter on the surface of the tooth, the form of
which is regulated, as before mentioned, by the curve k; with these parts
tlic screw m to which it is attached, and also the ratchet frame and ratchet
wheel are moved ; till the lower part of thermmeo'. Fig. 3, comes in contact
with the stop /J, against which it strikes, thereby throwing it from its verti-
cal position to that of an angle, carrying with it the ratchet wheel to which
it is 6xed, thus working the snuure- threaded screw and ultimately ad-
vancing tlie tool n ; the opposite stop /* is so regulated as to bring it to its
original vertical position when in its turn it strikes against it. By the
handle m' the tool is brought biitik by hand.

Figs. 5 and 6 shew the adaptation of this machine for the imrpose of
cutting the. teeth of conical or bevel wheels; consisting of an adjustable or
jding centre pin in a long mortice fixed to iIik bod.
From the above description, it la doubtful whether this mnehine has ever
practice or not, owing to its evident imperfection in producing
ly thing like accuracy of workmanship.

PLATE XLII.

iBTicAL BoR/No Machine, by Mbssb3. Nasuvth, Gabkell, and Co.

\ The many advantages derived by this arrangement of a vertical boring
machine over those where the work is placed in an horizontal position, may
perhaps be unknown to persons unocqnainted with the general character of
machinery; it may not be unadvisablc to point out a few of the principal
features shewing the suiioriority of this machine.

lu the lirst place, tlie arrangements of its ports; the manner in which
the cylinder is placed, namely, its vertical poaitiun, thereby doing airay en-
■ely with all the injurious effects produced by the weight of the body
I planed or bored; thus obviating all tendency to distort Its figure,
bch is the case whcrr the operation is performed by the horisoutal sys-
I, and where the sides are bulged out from the weight of the upper part;
h VMy he better understood bv forming a cylinder of tliin paper, which
Q he found to uiden in the middle and assume un oval form from its own

p«isli..

150 EXPLANATION OF THE PLATES.

Tliis nltcration of fonn is foutid to be quite sensible wben the cylinden
urc of large diameters. Another great ad\*antage of this system of verticd
boring, is avoiding all risk of flexure in the boriug bar, upon which the
cntter wheel or head is fixed for carr}'ing the boring tools; this bar his «
tendency to bend down in the centre to a curve, instead of keeping a per-
fectly straiglit line, transferring the figure assumed by the bar to the sorfaee
of the cylinder ; but this will much depend both on its length and diameter.

Another advantage of this machine is, that the cutters are kept dear of
the borings, which fall to the bottom of the cylinder as fast as they are cnt.
By tliis superior arrangement all these objections are entirely remoTed,
thus avoiding all the tendency gravity has in altering the tmeneas of the
cylinder or the bar ; added to these, the power requisite to bore the cylinder
is found to be much less than in those placed horizontally, a very desirable
object in a large establishment.

A short description of its several parts will enable the reader more foUj
to understand the ad^intages already alluded to.

Fig. 1 represents a cross section of this machine, and Fig. 4 a plan
shewing its position in a comer of the building where it is placed. In
these two ^-iews it will be seen that the driving part of the machinery is
situated below the ground line on suitable strong foundations, in which it is
inclosed. These parts arc rendered accessible by the steps /, which are
found to be neccssar}' in cases where the machinery is likely to get ont of
cirdcr, a j)rociiution never to be neglected.

The two ri^ircrs XX" receive their motion from the main shaft by means
of a leather stnip, one of these runs loose on the shaft, and the strap is
iliroun on it when the niaeliinc is not at work ; this is done at pleasure
with the greatest possible facility ; by a bevel wheel and pinion^ it is then
conveyed through the shaft / to the endless worm ?/, working in a large
worm wheel o, which is fixed on the great vertical boring bar a, whereby
a verv easy motion is obtained, and all jerks avoided. It will be seen by
the series of wheels in Fig. 4. how much the speed of the boring bar is
reduced. The shaft / is placed at an angle, and works in a bearing or
]>hnnnier Mock and a step //, both of these being made of brass.

The vertical bar is made in two parts a and c, the upper one a for cany-
in^' the cutter head or borin<r wheel r, while to the lower one is connected
the driving apparatus ; they are coupled together by the upper one resting,
as is shewn in Fig. 3, in a socket on the top of the lower one ; a steel key
/is then driven in, which entirely prevents it from turning; the toe of the
l)ar r rests in a step or socket shewn by Fig. 5 ; the entire weight of this
bar and its appendages is thrown on the hardened cast steel disks «, which
are constantly ke])t supplied with oil. Botli extremities of the bar e are
rendered adjustable to the greatest possible accuracy by means. of the anall

EXPLANATION OF THE PLATES.

451

let BcrewB yy. Figs. 3 and 5, which, by heing tightened, press against the
micttl brass Begtnents, the upper one forming part of the great base or
»ot plate i, which is niHterially strengthened by mx strong rihs on its
nnJer wde, Tlie cross beam _? is well fitted to the sockeU /, built into
e wall of the building, where they are boiled by strong bolts. Figs. 1, 2,
d 4: It has an addidonal stay in the bolt n.

There are four standards or supports, dd. Fig. 1, for carrying the cylin-
itr to be bored, which can be altered to any convenient positioii by un-
screwing the bolts which fix them to the base plate. After the cylinder
been properly placed in its right position, it is fixed to these supports
by clamps e and bolts ; and thns rendered quite immoveable.

In the boring bar a is a deep socket m. Fig. 1, which alloy's the bar to
■lide ap and down by means of the screw p and the nut /; upon the
lower aide of this socket is a flange m, upon which the cutter head or
wheel r rests, receiving its motion from the bar by means of a nut,
■nawering both the pnrpose of nut and key. By the dilTerent arrange-
ments of the sun and planet motion of the wheels on the upper part of
ibe bar, any degree of motion can he given to the screw for the descent of
the cutter wheel. After the cylinder has been once bored through, the
«utter wheel is raised by means of a small crane, and the chains, Fig. 2,
ud by the peculiar arrangement of the nut / in the socket vi, the cutter
wheel can he drawn up the cylinder without turning the screw p, as it
leaves the nut behind, which is afterwards
Weight to raise but that of the nut. The <
aew or finishing cut, after which the cylindi
'tme. The position occupied by the c
and the bar lifted in and out with

wed up, there being no other
rs are tht-n set afresh to the
may be connidered perfectly
enables the cylinder to be placed,
perfect case ; while the space

occupied by this machine is very small compared with those where the
work is performed horizontally; it is, however, important that the base
plate fi should be well secured to the foundations by strong bolls.

The speed of this machine may very easily be varied, by having different
sized riggers or pulleys on the driving shaft which conveys the motion to the
riggers kk. Figs. 1 and 4.

One of these machines may be seen in the erecting shops at her Majesty's
Dock Yard, Woolwich.

PLATE XLIII.
Gbeat Boblno Machinb, Bif Messhs. Nasmvth, Gaskei.l, anu Co,

The mafliine ropreaenled by this Plate is, with few oxcopti<)ns, the naiua
I ibiU last described, where the cylinder to be bored is placed in a varti-

452 EXPLANATION OF THE PLATES.

cal position, wbereby numerous advantages are derived, as Already ex-
plained.

The motion is communicated by tbc driving pulley c to a bevel innioa
working the bevel wheel d; the shall on which this wheel is fixed, has on
its opposite end a worm for communicating the motion through the worm
wheel to the upright shaft f and boring bar a, having on its circmnferenoe
the grooves a' in which the cutter head is moveable, sliding up and down ac-
cording to the progress of the work ; X- is a tool carrier fixed to the cutter head.
The foundation plate h forms a bearing for the upright shaft, the lower end
of which rests in the step g^ while the cylinder / is secured by the damps
jj to the supports i i fixed to the foundation plate. These parts are in
every respect similar to the boring machine shewn by Plate XLII., hy
which they are more fully described.

Two strong piers of masonry m' support the entablature m, (for carrying
the self-acting apparatus for raising and lowering the cutter head A,) to
which it is bolted by strong holding-down bolts mf\ This apparatus consists
of a rack n worked by a pinion, the motion being transmitted from a tml-
lion wheel through two s])ur wheels and pinions o ; the whole of this upper
machinery revolves ^ith the boring bar, with the exception of tho intenial
wheel or screwed hoop /?, the consequence of which is the small trullion
wheel is made to turn on its axis by the tread of the wheel p in which it
works, and thereby ultimately raises the cutter head 6, the two side
slings connecting it to the upper frame /, to which is fixed the rack n.

Tins machine is of the largest dimensions, and was made for the purpose
of boring tho large cylinders, 10 feet in diameter, for the Great Western
Steam Navigation Company's vessel the Mammoth, now in progress at
their works at Bristol.

PLATE XLIV.

Vertical Boring Machine, by Messrs. Benjamin Hick and Son,

Bolton.

By this combination of three distinct machines, the following different
operations may be performed, viz. : boring, drilling, and iace-grinding ; it
is so contrived that the entablature h^ supported by the four columns aaao,
carries the u])per parts of the three different machines, consisting of the
requisite driving machinery for communicating to them their respectiTe
motions.

That in the centre. A, is a vertical boring machine for boring cylin-
ders of large diameters, which are fixed in the usual way on the six
moveable supports h by the clamps t; in addition to which it is rendered

EXPLANATION OF THE PLATES. 453

perfectly ste&dy by the circular frame or ring /, sliding up and down in
grooves on the back of the two middle columns, the adjusting screws y
being tightened when the cylinder is properly placed under the centre of
the boring bar Cy which receives its motion from the leather strap and pul-
leys e/, whence it is conveyed through the bevel pinion and wheel e on the
upright sh&hfy upon which is also keyed the spur pinion for driving the
wheel ff fixed to the lower part of the boring bar c, and working in the
step m ; the rack and wheel k gives the cutter head the requisite feed
while boring out the cylinder. The six supports h are made to slide in
grooves on the foundation plate n, according to the different diameters of
the cylinders being operated upon ; these, when properly placed, are bolted
to the plate n.

The second machine, B, is a vertical drilling and boring machine for work
of smaller dimensions than the machine A. It is shewn in the drawing
boring out the centre of a crank x, fixed to the travelling table /», and
slides on V's on the frame t/y which has also a motion at right angles, on
the bed o fixed to the foundation plate. The drilling bar 1/ is lowered by
the screw u^^ according to the feed, its motion being conveyed to it from
the pulley q and strap / to a spur wheel and pinion not shewn in the
drawing, the pinion is on the same spindle as the pulley, and the wheel on
that of the reversed cone tt, by which it is carried to the square-threaded
screw u'' by two pairs of small bevel wheels », fixed on an horizontal
spindle and working in suitable bearings on the carriages u'^' v^'\ The
apparatus for raising the drilling bar is the same as that described by Plate
XXXII., which consists simply of two small chains fixed to the bar t/
and working round the pulleys, its opposite end being attached to a
weight t?.

The third machine, C, is for the purpose of grinding up the fiEu:es of rings
for metallic pistons, conical valves, &c. ; the travelling table p of this ma-
chine is in every respect similar to that of B, and upon it is placed the
piston ^ to be ground ; the upright rod, receiving its motion from the pulley
and strap /, is kept in a vertical position by the cross frame «, while the
shaft r and grinding plate are connected to the lower end of the rod,
and are occasionally raised for examining the surface being ground.

The different motions given to these machines are quite independent the
one of the other, by which means any one of them can be worked sepa-
rately. The whole is. placed on a suitable strong foundation of stone.
After the large cylinder is bored, it is raised from its position by a crane
placed on the floor above.

4f54f EXPLANATION OF THE PLATES.

PLATE XLV.
Machine for planing Iron, by Nicholas Forq, Clockmakxb.

This Plate represents an elevation of a planing machine, which was in-
vented as early as the year 1751, for the purpose of planing the pump
harrels used at the Marley Water Works near Paris. These pumps con-
sisted of i;«Tought iron segments bound together by strong hoops or strspi;
for tliis purpose M. Forq erected this machine near Mauberge in Fiance.

It consisted of a small spur wheel e placed between two trundle wheeb
A and t in the same vertical plane. The lower trundle t was put in motion bj
manual labour being applied to the ninch handle p ; on the same spindle
was fixed a fly wheel c to regulate the motion, while the upper tnmdle
wheel h communicated it to a laige spur wheel b above, one half of its dr-
cumference only being funiished with teetli. On the axis of this wheel h
a wooden pulley a of a very large diameter wbb fixed, over its periphery
an endless bond or cord/" was made to pass, and also over the im»t1|gr po].
leys klmn op^ to which was fixed the cutter or tool fromc. This labot or
cutter was then drawn backwards and forwards, between two laige panllel
bars of square iron placed horizontally, for giving tlie alternate motion to
the tool, which formed a kind of cross, the cutting surface being suited to
the circular figure of the interior of the barrel, and bored or planed it
regularly.

The bearings of the «j)in(llcs were fixed on cross timbers, in a similar
wav to the fnimc for currvinir tlie work boin^r planed.

The segments forming this barrel hud been previously planed separately,
by a similar process to that already described ; its parts having been well
forged to the required sha])e j)revious to their admittance in the planing
machine, when all the joints were made perfect.

By this machine M. Forcj seems to have prepared all the barrels he re-
quired, as he says, with the greatest accuracy, which varied in their
diameters from 10 inches to 4 feet, and from 7 feet to 10 feet long.
The inventor asserts that he had seen nine barrels 7 feet long, of which
eight had a diameter of 10 inches, and the remaining one 15 inches, filled
with water for three months, and were perfectly water-tight.

A pump barrel, 10 inches diameter and 7 feet long, consisted of nine
staves or segments, held together by 1 2 uTought iron hoops three inches
wide, the extremities terminating with collars two inches broad. The
hoops were half an inch thickness of metal.

This apparatus is evidently incomplete for intuit of the attachment of the
planing tool, and the reciprocatory motion, which is not shewn in the original
drawing.

EXPLAKATION OF THE PLATES.

PLATE XL VI.— Flo. 1.

Mlllwhioiit's Planino Machtkb, dy Messrs. Nasmvi

AND Co.

Tliis Pittlc reprpscnta two iliffcrent modes of planing. Pig, 1 shews the
tool moveable; Fig. 2, the tool fixeil. By the first arrangement many od-
vantagcs arc derived, such as planing the parts of very large and heavy ob-
jects, whose great weight and size would otherwise render them quite
inadmiseible id the machines of ordinary construction, where the work has
to be fixed on a sliding bed and moved along the machine with it. It is
evident by this contrivance tliat a very great saving of power is effected,
since ihe only moving part of this machine consists of a small traversing
fnune for carrying the slide and tool.

By referring to Fig. I, it nill be seen that tlijs machine rests in a slip or
pit mode for the purpose, with steps to descend into it. The two side
f^Bmes a are bolted to the ground, and stretching bolts connect them above.
There are also two cross carriages for carrying the shaft b, and a bed plate
hftring a series of apertures through which any piece of work can be
wcureO, they can be raised or lowered to any height by four side screws.
The ftnme e for carrying the slide ivorks liockwards and forwards along the
edges of tlie bed a, and by nicaus of a screw and handle rf, its position can
easily be altered.

It is evident from the above description, that in eases, such as planing the
on large heavy shufbs, lliis machine is found to be very useful
ml economical, for the surface to be planed is but very small com-
d with the lougth and bulk of the work which otherwise would be
veiling along tlie bed, occa-siuning both waetu of time tmd power. The
ingth given to the cut can be altered at pleasure.

PLATE XLVI.— Fio. 2.

Plahino Machine, by Messrs. Nasuvth, Gaskell, i

> Co.

Tlie |>erspective drawing shewn by Fig. 2, represents a planing machine
iF ordinary constniction, ivherc the tool is fixed and the work moveable.
It consists simply of a bed a uhoiit nine feet three inches long, upon which
e iniyelliiig Inbh' b works backwards and forwards. The wheels for work-
ing this table arc so arranged ae to bring it bock after the work ha« been
It a niueh greater speed, thereby saving time to a very great extont.
e two side frames ee have on their faces the upright slots or groovds ifif,
D wliicli the cross frame </ is raised and lowered at pleasures to reguUto ita

456 EXPLANATION OF THE PLATES^

height according to the magnitude of the work being planed. TUa is
easily performed by the handle wheel and large screw k woildng in a
nut on its back. The sliding frame e for carrying the tool ^ is then
brought to its proper situation by means of the screw and handle/^ after
which the machine is set to work by running the leather strap from a looae
pulley or rigger to one fixed on the shafl.

The table b has the usual mortices, by wliich the work is clamped to it
by strong bolts.

PLATE XLVn.
Planing Machine, by Messrs. Nasmyth, Gaskell, and Co.

The only difference between the principle upon which this machine wmksy
and that last described by Fig. 2, Plate XL VI., is in the motion by which the
sliding table is brought back after it has travelled the length of ihe bed.
From its great similarity, it is termed the mangle motion, and by the great
spcc<l obtained, is found very useful.

Tlie bed a rests on six columns, connected together by diagonal inimet
to increase its stability ; and the sliding tabic b running back\i'ards and fbr-
^urds on this bed, has the usual mortices in it. The upright frames H
are fixed to the bed by bolts ; in the two grooves of this frame, the cross
frame y is made to slide up and down to suit the work to be planed, by means
of the long screw and handle /, after which it is securely fixed to them.
The sliding frame for carrying the tool can be placed in any situation along
the frame y, by turning the handle and screw //, by which it may be brought
in close contact with the work ; it can also be placed at an angle by loosen-
ing the screws in the short mortices shewn on its face. Tlie tool can be
taken out and replaced by another, by two small adjusting screws, by
which it is fixed.

The motion is communicated to the machine by a leather strap working
on the riggers c. The spindle for carrying them being supported on the top
of a small column by its side ; on the opposite end of the spindle is the
small pinion working alternately inside and outside of the tnillions on the
large wheel </, the distance travelled by the table being regulated by a
stop on its periphery ; this is effected by the spindle sliding backwards and
forwards in the groove ^, situated on the side of the machine. On the
shaft of the trundle wheel, is a large wheel, round which an endless chain
is placed, ])assing over two small pulleys y^, which pulleys arc fixed to the
frame by a screw ; one of these only can be seen, the other being on the
opposite end of die machine ; the chain is fixed to both ends of the bed,
and communicates the alternate motion to the sliding table ; dose to the

EXPLANATION OF THE PLATES. 457

gioove bearing e, (which forms part of the diagonal frame,) is a Bmall rod ff
connected to a spindle by a series of levers and bell crunks, by which the
Ujinght Toii IB made to rise and fall nllernatcly, n'orkiiig the ratchet wheel
fixed to the screw A, therehv giving a self-acting motion to the cutter aJong
the frame _/. This niHchino is capable of planing work about 8 feet long,

I by 3 feet 6 inches wide. The length being regulated by the distanco
Invclled by the sliding tabic.
The frame for carrying all the machinery is well bolted to tie floor.

PLATE XLVIl. A.
Planing MAcamB, by Mbssrs. Nasmyth, Gaskbll, and Co.

^mie
^lobe

This description of planing machine is the some as that represented by
Kg. 2, Plate XLVI., where the work to he planed is filed to n traTelling
table d, moveable along lie surface of a bed a, and made to travol both
backwards and forwards by a self-acting apparatus, which is also connected
with tlie tool o, the motion being in the first instance communicated to
Mther one or the other of the two pulleys e or/; that marked e is fixed on
a shaft y*, and conveys the motion through the wheels and pinions c" to a
n working the rack h, screwed to the under side of the table d. This

riBTelling table has a number of mortices, by means of which the machinery
to be operated upon is clamped, while the bed a on which it slides has one
a V form, and the other a flat smooth surface, Fig. 2. The
two upright side-frames i/b are Bccnrcd to the bed n, and are connected to-
gether at the top by a cross frame c ; these two frames have long vertical
mortices in which the sliJingcrosaframe/ismovenhleaccording to the dimen-
sions of the work placed on the table, and can be raised or lowered by the
wheel nnd square- threaded screw /' ; a second frame m is also moveable along
the cross frame / by the ratchet wheel, square- threoded screw, and wheel m',
suitable to any required position, while a third motion is obtained by the slide
1^^ n (for carrjing the tool «) being moveable on the frame m by the wheel and
^^Mcrew «'. By the following arrangement the reversing motion of the table
^^knd that of the frame m and tool are made self-acting. On one side of the tra-
I^Kvelliug table are two moveable studs _/_;, (regulated according to the distance
ihe table is intended to travel,) which coming in contact with the lever i"
strike against the upper part /, and communicate the motion on one side

Erod and lever k, and also to the upright rod i, to which is fixed the
ible stud and pauI i^ for working the ratchet wheel m' and screw, oa
y described, the counterhaionue weight k causing tlie paul to rise for
'wed stroke. Two guide pieces ore screwed to the frame i for the
it rod t to work in. On the opposite side of the lever i" is n short

458 EXPLANATION OF THE PLATES.

connecting rod (for reversing tLe motion of the travelling table) fised to
the bell crank working the rods shewn by the dotted lino in Fig. S under
the table, which ultimately transmit the motion to the disengaging ^ipa-
ratus, consisting of an upright lever, counterbalance weight, and cross bar J',
with two short projecting bars for the leather strap to work between, and
by which it is passed from the pulley e to that of/^ whereby the motion if
immediately reversed, the spur wheel and pinion f causing the pinion to
work the rack ^ in a contrary direction, there being three wheels in one
instance and two in the other. Both the pulley f and pinion f are fixed
to a cylinder working freely on the spindle /.

The travelling table is generally made to return at a much greater velo-
city than that required while the tool is acting, by which means consider-
able time is economized.

When the machine is at rest, the leather strap runs on the loose pnlley g
without producing any effect on the working parts of the machine, in which
case the disengaging bar i^ is situate in the position shewn by the pkn
Fig. 3.

A frame p and standard q for suppordng the shafts / and the seTenl
parts of tlie machine is fixed to the side of the bed a, and the tool o can
be changed at pleasure by unscrcT^ing the screws which hold it in its place.

PLATES XLVII. B. AND XL VII. c.

Planing Machine, driven by Steel Belts, by Benjamin Hick

AND Son, Bolton.

These two Plates represent a planing machine of a different description
from those generally used, and of unusual size, and o^\ing to the peculiar
motion given to the tool, it cuts both ways, on Whitworth's patent plan. It
is shcu-n in Figs. 1 and 2, planing the faces of the ports of a cylinder for
a steam engine.

The arrangements of its parts consists of two upright side framea a a,
fixed below the surface of the ground, on which are placed the cross and
longitudinal bearers b /;, for carrying the work to be planed. On the top of
the frames a a are bolted two long side frames dd supported also by the
three standards eee; the sliape of these two frames dd^Bs will be seen by
Fig. 2, is of the V form, and upon them the cross table c for carrying the
tooiy works ; there are two bearings on this table for supporting the hoUow
cylindrical bur, upon wliich the frame i' is moveable, (by meana of a
passing through the hollow bar,) according to the position of the too] j
the face of the work being planed ; a long key is provided on the

EXPtANATlON OF THE PLATES. 459

Aneai bar, to prcTent tlie carriage i" from altering its pontion by tnming
on it.

On the front port of the frame i" is a circular bos, which moves at each
end of tlie cut ; in it the small cyliniler for carrying the tool revolves ; tliis
front part of the frame i' can Lc placed at any angle, by means of grooves
on its facen, two small bolts and iiiita on either side fixing it, when pro-
perly adjusted, while the tool is kept in contact with the fiu» of the work
by the handle and screw above it.

The working of this machine is as follows : the motion being commnni-
cated to the pnllevB by the leather straps //, the one for working the ma-
chine forward, and the other for working it in a contrary direction, convoy
it through the two spur wheels and pinions ffff, to the puUeye fi, fixed on
two strong shatls extending across the extreme ends of the machine under
the floor, one of these pulleys being situate at each comer of tLe machine,
uid round them the steel belts work, the ends of which are attached to the
projecting levers m on the sides of the long cross bar i, and secured by
tightening screws; the operation of reversing con be effected at any point
by levers and a rod, extending the whole length of the bed, and Iios a
moveable stop for that purpose.

The friction of tlieso belts upon the pulleys, drives the machine, and is
snfficient for the heaviest cats, its only moveable parts being the carriage c
and its appendages, while the object under operation of planing is a fixture,
by which means the same amount of power is necessarily required for eitlier
a heovy or a light casting ; the surface it is capable of planing being 30 feet
long, and 9 feet ft inches wide ; when large surfaces such us lathe beds are
to be planed, the plolform b is raised to a level with the floor.

^ole. — A similar tool was also contrived, some years ago, by the editor of
thU work.

^^ The pocnlinrity of this machine is its adaptation to two different pnr-
poBes, so closoly connected to one another, tliat in the construction of boilers
of any description, or in fact to any work, where wrought iron plates are
used, either to be cut or punched, it is found very useful ; by the arrange-
ment of the parts of this machine, tlils is effected at the same time and by
the same motion.

tThe large cast iron frame or standard d, carries the shaft r, on one end
wbioh is fixed an eccentric, Pig. I , for nuong and lon-ering, or giring tho

PLATE XLVIIl.

PUKCBINO AND PLAIE-CUTTINO MaCRINB, BY MBaSKS. NASUVTH,

Gaskbll, and Co.

460 EXPLANATION OF THE PLATES.

alternate motion to the slide /*; on the other extremity of this aame ihaft,
is the large spur wheel c, put in motion bj the pinion on the driving shaft
on which is hung the fly wheel by for steadying the motion^ and also the
two pulleys or riggers aa^ the one keyed to it, while the other is aDowed to
run loose, when the machine is not working. The outer carriage jp supports
the hearing of this latter shaft.

On the upper part of the frame d is the steel cntter ^, secured to it ; a
similar one is placed in the reverse position on the top of the sliding frsme
fy die lower part having fixed to it the punches h for punching the plateSi
which work freely in the dies i screwed to the frame d; there is a stop J
on the under side of the jaws of the frame, for preventing the plate from
rising, as it has a tendency to rcmun fixed to the punches, which it is ohiiged
to quit, when it comes in contact with the stop.

From the above description, the mode of operation will be easily under-
stood. The motion being given to the fixed pulley from the driving shaft of
the building, is then communicated to die slide by the pinion and wheel,
causing the eccentric shaffc e to revolve, thereby giving the up and down
motion to the punches, and also the cutters ffff alternately, which in the side
elevation. Fig. 2, is shei^-n cutting a plate. The slide y works in a V slide
fixed by six screws, die steel cutter being also fixed by three of a similar siae.

The punching operation performed by this machine is made self-actiiig^
by the following arrangement. The plate to bo punched is secured in the
usual way by clamps to the travelling tabic ky its four wheels running on a
railway of triangular bars, on the bed plate ; two carriages o are fixed to the
under side of the travelling table, for supporting the notched bar n bolted to
them.

As the large spur wheel c revolves, the pin 7w comes in contact with the
lever connected to the rod /, by which tlie bar n^ and consequently the
table, is advanced. This done, the table is again drawn back to its former
position, by the chain shewn in Fig. 1 , when the operation is repeated.

Messrs. Nasmyth, Gaskell, and Co. have made subsequent improvements
in this machine, by altering the position of the apparatus for cutting the
plates of boilers, to the opposite side of the punching apparatus, while the
frame work is so adapted as to carry the driving parts in the centre, the
same shaft working both machines.

PLATE XLIX.— Fios. 1 and 2.

Plate-bending Machine, by Messrs. Fairbairn and Co.

The variety of forms given to different boilers, according to the natUK
of the steam for wluch they are intended, renders the machine shewn by

EXPLANATION or THE I'l.ATF.S. i-G]

I and 2 of this Plate quito indispensable to b boiler manufaotarer- It
will easily be «een by ekantining these drawings, buiv a t^at wrought iron
pUlu may l>e brought to any required curve, by passing it between the
rollers A 6, which are regulated by the large adjusting screws ^y.

The driving and loose riggers a a have the motion comiuunicated to them
"hy a leatlier strap, from a shaft worked by the engine connected to the esta-
blishment, the spur wheel e on an intenaediate shaft, then conveys it by a
n to the large wheel, on which is fixed one of the rollers b, the other
being worked by the two pinions on the opposite side of the machine. The
fiy wlioel (1 is placed on the driving or rigger shaft, and by its great weight
^vee a. regular steady motion to the different working [>arts of the machine.
The roller bearings aje adjusted by the large square-tlireaded screws working
in the two side frames e for supporting them. Much additional strengtli is
^ven to this machine, by the stretching bolts//, which bind it together.

By placing a handle on one of the arms of the fly wheeltl, this machine
inight be put in motion by manual power.

PLATE XLIX.— Pigs. 3 and 4.

Vice fob ccttino Boilkr or othbb WBonoHi Ibon Plates, by Messrs.
Nasuyth, Gaskrll, and Co., Manchester.

Figa. 3 and 4 aliew a simple contrivance, by which n-rought iron plate
■n be held secure in a frame, while the edges are cut with a chisel. The
Trnme conaals of the upper and lower parts of a vice a and c, the fates of
which are hardened steel, iwtween these a wrought iron plate d is placed,
where it is securely clamped by tightening the two nuts on the large squarc-
btreaded screws ft, keyed through the frame, which also connect the ma-
diine to the foundation of the building.

The chisel t for cutting this plate is shewn by Fig. 4, by which the work
IB performed in a very perfect way.

The lower vice frame « is much strengthened by tlie ribs shewn by Fig. 4,
which give it a Krm bearing on the ground.

PLATE L.— Fios. I and 2.

PuNCHiNd Machine, by Mhssrs. Kinmond, HuTTrtN, and Stbrl,

UUNDRE.

These two figures represent an elevation and an end view of a machine
T punching holes in wrought iron plates for boilers and other purposes.
The Rtrong frame J, which carries the whole nf the machiner)-, is maile

462 EXPLANATION OF THE PLATES.

of cast iron, and is Very firmly fixed to the floor of the bnildiiig. The pat
leys a receive the motion from a line of shafts ; one of these runs looM^
and tlic 8tra]) is thrown on it when the machine is at rest* The qpeed
given to tlie pimching spindle £/ is reduced hy two pairs of spur vdiedb wdA
pinions c c ; upon the end of the spindle^ is fixed the eccentric or Gamy ivindi,
as it revolves, raises and lowers the slide h for carrying the punch e, 'wUdi
works in a die y^ large enough in diameter to fit it, and through thia die
passes the circle of the plate cut out hy the punch.

A very uniform motion is obtained by phicing the heavy fly wheel i oe
the main driving shaft.

The different ports of this machine can, without difficulty, be taken eat
and replaced by otlicr suitable to larger or smaller work.

PLATE L.— Fios. 3 and 4.
Messrs. B. Hick and Son's Mandbil for rxpandino Kxnqs.

The mandril in a lathe, is that part upon which tlic work to be turned k
placed or fixed, consequently different mandrils arc required to suit the va-
rious kind or forms of macliincry to be turned ; by tliis contrivance of Mr.
HickK, rings of very different diameters may be turned ; a few words will
give an idea of the mode in which this is performed. The spindle or man-
dril a is of a cylindrical form, having a square-threaded screw at one end ;
on tlie larger or middle part of this mandril arc four grooves, in which the
conical pieces care made to slide, and have on their circumference the ring d
to be tunied ; a tightening cone ^ of a cylindrical shape is then placed on
the screw, and by screwing the nut, presses it against Uie four pieces c, thos
expanding them till the ring becomes quite securely fixed on their circum-
ference.

By this means, rings of various diameters may be turned, within the limits
intended by the mandril.

PLATES LI. and LIL

Machinr for Punching Boiler Plates, by Messrs. Maudslat, Sons,

AND Field.

Fig. 1 is an elevation, Fig. 2 a plan, and Fig. 3 a side view of a punch-
ing machine on a very improved principle, whereby the plates required for
boilers and other purposes may be punched T^ith the greatest possible accu-
racy, insuring at the some time very superior workmanship and gml dis-
])atch. It is usual in all ordinary punching machines, fiist of all to nnk
out the rivet holes in the plate, by a template, with white pamt^ end dwn

EXPLANATION Of THK I'LATES.

i63

l

to place it as nenr as the eye will pennit under the punch ; by the coDtriv-
ttace of this machine, this operation is entirely dispensed with, it being only
Seccseary to fix the plate to a travelling table, and then to adjust the various
Jiarts of the machine to the proper distance reciuired between the rivets.

The large cast iron frame /I carries the several parts of the machinery,
and also the two plummer blocks for supporting the hearings of the lying
Aaft a running the whole length of the building, for the purpose of work-
ing other machines ; this frame is securely bolted to the wall, which, added
to its own weight, gives it great stability.

On the shafts a, connected together by the two coupling boxes, is placed
the crank b, the motion being communicated through the crank pin to a
connecdng rod c, to which is attached the upper lever d, being always at
work, while the shaft a is revolving ; the fulcrum of this lever is on the
(tKtae p. A lower lever e for raising and depressing the punching frame /
has also its fulcrum on the same frame ; this last lever working only when
the punching operation is being performed. On the top of tlie frame p is
A lever and long rods e' for engaging and disengaging the machinery. The
mde view represented by Fig. 3, shews the machino at work, and by draw-
ing down the lever e", connected by the rods to the eounterbolance weight,
which will allow it to remain steady in any position, they are disengaged
^ra the pin on the lever </, which at once cesses to communicate the mo-
tion to the lower or punching lever e ; thus it will cosily be understood that
parts of the machine constantly at work, are the lying shaft a, the
crank b, the connecting rod c, and the upper lever d.

The alidcyfor carrying the punclies, works between V's, Fig. 2; one is
fixed to the frame yi by adjusting screws, both on its face and sides, and by
the curious shape given to the end of the lever e, tliia slide is made to rise
and fall, the counterbcdance weight and lever p being connected by two
short links ; on the bottom of the slide is screwed a small frame for carry-
ing the punches, wliich may be taken out and replaced by others. The dies i
in which the punches work are placed in a frame bolted to the frame p, and
by unscrewing the small adjusting screws, these niny be taken out and re-
placed by others suitable to the different sized punches that may be required.
On tlie under side of the frame p is screwed a small stop, by which the
circle punched from the plate ia forced out os the slide rises.

in the front of the machine is placed a long table, supported by the car-
I, consisting of two columns and diagonal frame, having bolted to
^em two long bars, upon which the moveable table is made to slide
tilis table, as nill be seen bv Fig. 2, him a number of holes by which thi
jlaxe to ho punched is secured by clamps ; tl is odviuiced by the rope or chain

passing under the pulley. Figs. 1 and 3, and over a tiecund one hung from
iling of the building, to which is ctiuuccted ti weight sullicicntly hi

holes by which the ^

by the rope or chain I

icund one hung from ^fl

;ht sullicicntlv heavv ^^^M

4G4 EXPLANATION OF THE PLATES.

to draw fon^'ard the table and plate fixed to it ; afler traTeUing the kagdi
of the table o, it may be brought back by taming the windi handle wai
spindle /, on which is a pinion, for working the rack, fixed to the under ade
of Uie moveable table m.

By a very ingenious contrivance, forming part of this machine^ the met
holes of boilers may be pimched to very different pitches, a thing Teiy mtA
wanted in such cases as the repairing of old boilers, or replacing an oU
plate by a new one, where it is of the utmost importance to have As
rivet holes coinciding with the greatest accuracy, which may he hetts
understood by supposing tliat in the length of a plate, one, two, or dnw
additional rivet holes may be required, which distance would have to hd
equally divided in the whole length of the plate. A description of this pvt
of the machine will fully shew how this operation is performed. On die
end of the punching lever e is screwed a small plate Hith a pin/^, adjostahle
by a screw working in the short mortice. Fig. 3 ; on the frame p is holled a
fulcrum piece for the lever and counterbalance weight y; to this lever a long
rod is connected at its lower end, passing under the table m, and the lever
r is fixed to it, as will be seen by the dotted lines in Fig. 2 ; it has also two
stops or projecting pieces / / on its surface. A second lever « has its fnl-
crum fixed to the table o, one end of tliis lever being connected to the lever
r, while the opposite end has on its surface two small pins s' / ; the fnlcram
«''of this lever is adjustable by the two small screws shenn in the plan,F%i
2. A long notched bar k is fixed to the under side of the table m by
screws, and the bar t has its fulcnim f also fixed to it; this latter
bar t is moveable on its centre, and may be placed at any angle by an sd-
jnsting screw working in a mortice t'\ where it maybe fixed in any required
position, as will be seen ; it is by the angle given to this bar, that the re-
gular incrcatie of distance is obtained between the rivets. This is effected in
the following manner : as the lever e works on its centre, the pin / to
which it is fixed, strikes as it descends on the end of the lever/; its motion
is then communicated to the long rod, passing under the table, for working
the lever r backwards and forwards, it being fixed to the rod ; as the latter
lever moves, it alternately engages and disengages the stop or projecting
pieces r / from the notched bar, allowing it at the same time to slide for-
ward the distance between the rivets ; while the bar t placed at the required
angle, and working between the pins / s' on the lever «, immediately affects
the disuuice tnivelled by the lever r, which it shortens ; by this arrangement
the machine is made quite self-acting, but it may also be worked by the
handle on the end of the rod J.

When it is required to change the punches h for different siied holeSi it is
necessary, when replaced, to adjust them to the greatest possible aeemacj in
the dies i ; this could not be done without stopping the whole line of diaft

EXPLANATION OP THE PLATES.

4S5

t, were it not for a provision tiiaile for tLat purpose ; in bucIi a caec, the

■in the upper part of theeonoectitig rod c is withdrawn, a block of wood is

D placed between the ^me and the lever d, whose fulcrum is on the

e of the frame p, upon which it rests ; it is of course understood that

e punching lever d ia at rest, being disengaged from the pin e', the result of

is the working of the long mortice of the connecting rod up and doHii

Llie pin, without communicating any motion whatever ; and by the lover

rorked gently by hand, and shewn dotted in Figs. 2 and 3, the puucliing

rr e can be miscd and lowered, communicating its alternate motion to

slide fmme, by which means the punches ore with great facility ad-

Stcd in the dies f.

3 jaws in the frame /> arc much strengthened by placing a square
;ht iron bar between them, in places provided for the jiurpose.
im llie above description, it uill be seen, tlint the advantages pos-
i«l by this machine, more than compensate for the increased number
3 parts, which are but few when compared with its superiority over
e of ordinary construction ; there are several at work in the boiler shop
r Majesty's Dock Yard, Woolwich,

PLATE LII. A.

Steam Punching Maciiinb, by M. Cave, Paris.

The mode of applying the motive power to this machine is altogether on
a different principle from the others contwned in this work, to which it is
either conveyed through wheels, pulleys, straps, or bands, driven from the
shafting running through the building. In this case it is worked by a small
steam-engine, connected to and forming part of the machine. In the steam
^linder a is a solid piston and piston rod b, accurately fitted, the cylinder
Lbeing bored out in the usual way ; (^ is a shde valve, worked bv the rod d on
e faces of the ports c' and c", the former for tie admission of steam to the
f ojrlinder, and the latter for the exhaustion, whence it is carried through a
pipe to any convenient outlet ; the rod d works steam tight throui;h the
etuffing box of the slide case, which contains the steam brought by the
a pipe e from the boiler. This latter pipe is connected to the valve
iwng, by flanches bolted together.
[ On the lop of the piston rod A is a cross head 6', on either side of which
e the links yy connected to the punching lever y"; two small rollers &' b'.
Fig. 3, are placed on the outer ends of the cross head which slide up and
down in the guides ,/""/"", whereby the parallel motion of the piston is kept

I perfectly true. On one end of the punching lever _/ is the connecting rod
t, for conveying the altoriiatc or reciprocating motion of the piston tltrough

I

^Lbeing
»liefi
^Fojrlint

4>G6 EXPLANATION OF THE PLATES.

the crank / to the fly wheels ira, by which it is r^ahted. Oil die
end of the lever f (whose fulcrum is on the frame A) is the
cylinder for holding the punch, both of which have projeetiiig pins ob
for connecting them together by the wrought iron links ^ ^^ for
the parallelism of the punch, these links being adjostable by the
above. The punch n is connected to the lower end of the cjlinder $\^%
key, as shewn by Fig. 8, while the die o in which it works can be ahcnii
according to the size of the punch used, by the two small adjoBtiiig tovn
shei^-n in Fig. 1 . A stop p prevents the plate from rising after it has bees
punched, the circular pieces punched out falling through the apertme « pn^
vided for that purpose.

The strong frame h for canying the several parts of the machine, is »-
curcly fixed to the stone foundation ; on the front part of this finame is s
cap (fixed by six bolts and nuts, Fig. 4, representing a sectional plan of tlw
frame) which is tightened according to the wear of the cylinder g^ ^riuch
becomes considerable after having been at work any length of time ; by tUi
means, any irregularity in the motion is entirely obviated.

The long lever^ is fixed to the rod d for alternately opening and ■l»w**iiig
the ports (f and c'^ by the slide valve, which operation is performed by Ae
lever y^ as it rises and falls, striking against the pins cTaT, the counteibafaneB
equalizing the weight of the opposite side of the lever ; whose falerom is
screwed to the upright guide frames f'\ and projects consideiably ow
them, Fig. 1 . The handle /' is connected to the lever y, for starting or
stopping tlie engine by hand ; and by means of the pinsy' on the two up-
rights fixed to the upper side of the frame h^ the lever j is secured when
the machine is at rest.

In its present position the working of this machine is as follows : steam
being admitted by the pipe e to the slide casing, passes through the stesm port
</ to the under side of the piston, which it presses up, causing the levery* to
rise between the guides /*" /*", and assume the position represented by the
dotted lines, the connecting rod, crank, and consequently fly wheel following
their respective motions. In this new position, the levery moving on its ful-
crum, must necessarily depress the punching cylinder g and punch w, thereby
communicating to it the requisite alternate motion. In its ascent, the lever
y* strikes against the upper pin d\ and raises the valve over the steam port
r, which is no sooner done tlian the steam that has performed its duty
rushes through the exhaust port (the communication being made between
the two) and through it makes its esca})e, when the piston, rod, lever, &e^
fall by their own gravity to tlieir original position, (having merely to over-
come the pressure of the atmosphere,) till the lever on its descent again
strikes on the lower pin, which inunediately opens the steam port ^ to die

EXPLANATION OF THE PLATES. -tdj

posidon shewn by Pig. i, which has already been described, — when tho
operBtion is agaia repented.

This machiDC may be used for catting the edges of plates, byadoptiug the
cutter ahewu by Figs. 5, G, and 7-

AU the parts of an liigh pressure steam-engine ore necessary, as already
described, to work this description of punching machine, oud a higher ve-

Ihwdty to the punch must necessarily be given than by the ordinary metiiod.
I Tt

BivBTiNo A

PLATE LIII.

INK, BV Messrs. Paihbacsns and Co.

The drawings shewn by Figs. 1 and 2 very much reeemhle the com-
in punching machine, the motion being in every respect similar. The
work performed by this machine is in all ordinary cases done by ma-
nual labour, which operation necessarily requires the services of three men,
the bolder on, to hold his hammer or tool inside the boiler against thu
of the rivet, while tlie other two beat out the iron to the conical form
to its opposite end; this is attended with a very disagreeable noiac,
[nite unoToidable, but which is entirely done away with by the nse of lliia
machine, which performs with almost instantaneous pressure, what is done in
all ordinary cases by a long series of impacts. Fig. 1 is an elevation, and
3 a plan, where it will be seen that the motion is commuuicated to the
[ht and loose riggers a a by a leather strap ; the speed at which these re-
is much too great for the purpose intended by tlus machine, consc-
.tly a pinion and large spnr wheel b is required, by which it is reduced as
istol, thatis while the pinion shaft is making six revolutions, that on which
luge wheel is hung, and also the cam, is revolving at only one. On the
pinion and rigger shaft is placed the fly wheel c, for giving a unifunn motion
to the working parts of the machine. The riveting lever e is then mode
to rise and fat! hy the action of the cum d, the face of which is steeled, wliilo
on the end of the lever e is the steel roller /, revolving as the cam works
gainst it; hy this contrivance the friction, which would bo considerable, is
materially obviated ; the riveting lever works easily on its fulcrum A, the
two short links connecting it to the riveting too! j, sliding backwards and
forwards in the socket bearing, by which it is kept working in a perfectly
troe and horizontal direction.

The various parts of this machine are connected to the side frame •;,
made of cast iron, and on it are placed the plmmncr blocks for carrying iJie

I On the sole plate is fixed the riveting block i, agmnst which Uial pait

4G8 EXPLANATION OF THE PLATES.

the boiler, chimney, or other work k being rivets is placed ; it is Im^
by the block and chain shewn in Fig. 1 .

The operation is thus performed : the rivet being put through the holeslif
the attendant workman, previously punched by the punching tw^Ki^ft^ it
brought round so that its head will fall into the recess on the projecting
part of the riveting block ; the machine is then put in motion hy rhenging
the position of the strap from the loose to the fixed pulley, which bsii^
communicated to the riveting tooiy, gives the required shape to the

The whole is placed on a suitable foundation / for giving solidity and
bility to the machine, and a pit is made to allow the wheels to wotIl in.

It is stated by Messrs. Fairbaim, that with the attendance of two
and two boys to the plates and rivets, this machine can ^tl in the finnest
manner eight rivets of three quarters of an inch diameter in a minute^
whereas by the common process, three men and a boy can only rivet op 40
per hour, the qiumtity then done in the two cases being in the piopoitian of
480 to 40, or as 12 is to 1, exclusive of the saving of one man's lahonr.

PLATE LIV.
Double Gkindino Machine, by Messrs. Nasmyth, Gaskell, and Oo.

The object of the machine represented by this Plate, is to grind up the
f\u;e8 of the different parts of machinery, when a great surface is reqnired
to be made ]>erfoctly smooth ; to accomplish which, this contrivance has
been used.

Figs. I, 2, and 3, severally shew a side elevation, an end elevation, and
a plan of a double face grinding machine, the one side being a repetition of
the other. To the two cast iron cross frames a a, are bolted two large
plunimer blocks for carrying the main shaft, having at each extremity the
circular frames divided into twelve compartments, in which are placed the
grinding stones f^ each being adjustable by the small set screws m round its
circumference. On the top of the cross frames a, are placed two longitn-
dinal frames b h^ made also of cast iron, for supporting the long bed frames c c,
and also the self-acting apparatus furnished to this machine. Two motions,
the one at right angles to the other, are given by the slides d working along
the beds r, and also the face plates e for carrxnng the work, by which it is
brought into contact ^ith the grinding stones. Pits are made to allow the
wheels, ior carrying the stones, to work in.

Tlie self-acting motion given to the work being faced, by means of which
it slides along the bed c while the grinding stones are revolving on their
axes i-^ thus obtained : on the main shail next to the driving riggers or pul-
leys ^, is a worm /, which, as it revolves, works a worm wheel rejnresented

EXPLANATION OF THE PLATES. 469

by the dotted lines in Fig. 2, tbos commimicatmg ihe motion to the upright
spindle ; from this it is carried by the bevel wheels to the spindle running
horizontally the whole length of the machine, having at each extremity three
small bevel wheels. The action of this apparatus is thus, supposing the slide
to be travelling in the direction towards the small bevel wheels, two of
which are required for the purpose, while the third or outer one runs freely
on the spindle, without producing any effect, the small clutch being dis-
engaged from it ; on the travelling slide d is fixed a stud or pin A' ; a long
rod h of the same length as the bed e, is moveable in two stud bearings
fixed to it As the slide d travels, the pin A' comes into' contact with
a second stud or pin adjusted to any position on the rod h^ according to
the length of the motion required, which must naturally press it forward,
and thereby throw out the clutch on the end of the spindle, which being
shifted from one bevel wheel to the other, disengages that which had been
at work before, while it engages the outer one, that had been running
loosely on the spindle ; by this curious contrivance, the screw for working
the slide revolves in a contrary direction, and instead of drawing the slide
d towards it, sends it back. A counterbalance weight n is connected to
the extremity of the rod h^ for keeping it in a steady position while this
operation is being performed.

The tappet wheel k fixed on the end of the screw for advancing the
other slide e, is also worked by a pin on the same rod A, whereby the
work is advanced to the face of the grinding stone ; it is on the upper part
of the slide e that the work is fixed. A substantial foundation, consisting
of stone work, is prepared for receiving the two frames a a, and to which
they are firmly bolted down by strong holding bolts.

Another mode of performing this same operation might be adopted, by
fastening a whole grindstone into the chucks, and passing a bolt through two
surface plates of two feet diameter each, one on the middle part of each
face of the grindstone, by which means they would be more effectually se-
cured in their places.

f
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ii

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; ■

INDEX.

Action, manner of . .192

Alder, strength of . . 250, 251
Alloys, strength of . • 260

Alteration of velocity by friction 836
Alternate cones . . . 531
Anderson .... 485
Angle defined ... 71
Arc of a circle defined . .128
Arkwright, invention of cotton

machinery .
Ash, strength of
Asp, strength of
Axis, defined
Axles, hollow
■ solid .

252. 394. 428

. 251. 253

. 253

. 410

. 231. 236

. 353. 369

Banks, John .

Barlow, Peter, experiments

Bayonet

Beams . . • .

Bearings of shafts .

Beech, used for patterns .

— — — pillows .

strength of .

Bending
Bevel gear
— wheels .
Birch, strength of .
Bismuth, strength of
Blocks .
Bodies of shafts
Bone, strength of .

. 351

by 259

. 295
338. 352

. 456

. 128

. 346

. 251

. 222

. 51
51.62

. 253

. 260

. 344

. 221

. 252

Page
. 270

. 273
. 253
. 346
. 347
253. 261
258. 260
. 344

Boring mill clutch .
Boulton and WaU .
Box-wood, strength of .

used for pillows

Bramah, Joseph
Brass, composition of
—strength of .
Brasses ....
Breadth of teeth of wheels

95. 97. 200

Breasts 347

Breast wheels, power of . . 332
Brewster, Dr. ... 64

Brick, strength of . . • 254
Britain, manufactures established in

172. 177
Brown, B., experiments by 255, 256 '
Bucket water-wheels, weight of 204
Buffon, experiments by . • 255
Bums, Robert . • • 236
Bushes 347

Camus, on the teeth of wheels

6. 10. 80. 65

Capstan bar .
Carmichael, James •
Cast iron, introduction of

bearings .

■ gudgeons

framing .

■ pinions .

• 285

. 107

. 274

. 344

198. 205

350.854

50.56

shafts 287.240.850.858

472

INDEX.

CSast iron staves

■ strength of
— ^-^— trundles .

Page
23. 113.204

. 251

. 35

274. 422

. 251

2

Cattle mills .

Cedar, strength of .

Centres, line of

Centre of gravity of shafts loaded

with 2, 3, or 4 wheels 354. 372
Charts, construction and use of

129. 131
Cherry tree, strength of . . 253
Chord of an arc defined . . 73
Circle defined ... 72

Circular arcs for describing teeth

of wheels .... 23

, Willis's, for the same

148. 151

Cloves 349

Clutches and glands . . 269

Clutch, boring mill . . . 270

second construction . 271

Cogs, term explained . . 1
Cohesive strength of bodies 250. 262
Conductor .... 37
Cones, alternate . . . 336

defined ... 75

friction . . . 302

proportional . . 53

Connection of shafts . .267
Copper, strength of . .250

alloys of . . . 260

Com mills, numbers for . . 215

tackle for . . 299

Corollary defined ... 73
Cotton mills . . . .264
Coulomb's experiments on friction

Coupling box .

link

Couplings, with one bearing
— ^ with two bearings

306
267
273
276
267

Couplings for aprigbt d

with gknds

— — — durability of
— — — • round

■ self-disengagiiig

square . . 267.

Crab tree, strength of
Crompton, Samuel .
Cross-tailed gudgeons
Cubical parabola
Curves, epicycloidal
Cycloid, form of teeth
Cylinder cutters of a defined
diameter . . . 75.
Cylinder, solid, strength of

hollow, strength of 229.

Cylindrical shafts .

27S

a04
277

278
17i

54

155
197
241

iron .
Cypress, strength of

- hollow, of

240
251

Dash wheels .... 280
Deal, strength of . . , 253
Decay of timber . . . 349
Desaguliers, Dr. ... 88
Diameter of pitch line • • 55
Disengaging machinery . .291
Donkin's table of radii of wheels 114
Double speed, method of obtain-
ing 426

. 278
. 97
. 306
. 290
79.83

Drum shafts .
Du'Buat
Duncan, John
Durability of couplings
■ wheels

Eclectic Review, cxtiuct from

173, 174
Edge stones . • • . 337
Egg-formed pivots . • . 847
Elder, strength of . • . 258

INDEX.

473

Page

Elm, streDgth of . . 253. 257
Emerson, W. . . 253. 351

Epicycloids, properties of • ?• 17

mode of describing 8. 16

. 8.13

. 12

54. 57. 59

. 11

16.17

16

17

175

176

exterior

interior

spherical

Exterior epicycloid •
Epicycloidal curves

lengths of

■ areas of

Essay on the shafts of mills
■ how treated

Face wheel . . . .336
Fast and loose pulley . . 297
Feathered shafts . • .181
Feathering .... 352
FeeUng, a term . . . 222
Fen ton, Murray . • .10
Fen wick, Thomas . . .191
Figure, best, for teeth of wheels 14
Figures iUustrating the slide rest

398. 405

Fir, strength of

shafts

Flanks of teeth
Flexure
Flour mills
Fly-wheels

shafts

Force defined

immediate,

and horses .
Fracture
Framing of mill-work

' for lying shafts

— — upright shafts

for flour mills

cast iron •

wooden •

251. 253
. 243
. 37
. 221

284. 353
. 212
. 216

. .77
of men

. 89
336.351
342. 344
. 348
. 349
. 353
. 350
. 348

Free-stone, strength of .
Friction of teeth of wheels

cones .
clutches

loss of power by

rollers

wheels acting by

. 254

29.67

. 302

. 301

. 329

. 346

. 303

Galileo 228

Gear 51

spur . . • .18

bevel . . . .51

Generating circle • . . 7. 50

Glands 269

Gold, strength of . • 257, 258
Governors, principles of • . 312
steam engine . . 308

water-wheel . .313

first construction . 313

second construction 315

third construction . 315

fourth construction . 316

fifUi construction . 317

windmill . . 311

Gravity, centre of, rules for find-
ing that of shafts loaded with

2, 3, or 4 wheels . 3.54. 372
Gregory, Dr. Olintfaus . . 232
Grindstones, spindles of . . 210
Gudgeons, mode of fixing . 200

diameters of . . 197

cast iron . . 205

malleable iron . 207

strength of . 198

viratei^wheel . 199, 200

stress and friction of 184

pressure, greatest, of 194

table of . . • 202

of cast and

wrought iron • . . 210
Gun metal .... 860

474

INDEX.

Halley, Dr., on qiicjcloids
Hatton on clock work
Haxel, strength of .
Headstock framing
Heating, to prevent
Hempen rope, strength of
Hewes, of Manchester
Hollow axes .

cylinders .

shafts
Holly, strength of .
— used for pillows
Hook, Dr.
Horn, strength of .
Horse mill, or oxen

power .
^— — ^— performance

men and horses •
Hatton, Dr. Charles

Imison .

Inequalities of teeth
Inertia .

Interior epicycloid .
Intermediate spindles
Internal pinion
Involute teeth

properties of

length of .

Page

17
125
253
848, 849
345
253
317
231
241
181.240
253
846
277
252
190
88
of, hy

90
372

27
28
292
8
206
47
44. 64
63
44

Iron bar, strength of 251. 255, 256
cast, strength of .257

■ for strength of mate-
rials ... . 83. 88

■ demonstration of power

of 254

sliafts — see Shafts.

Ivory, strength of . . .252

Jacks, form of teeth for . . 48
Jamiesou's, Dr., Mechanics for
Practical Men . . 77.359

Jennies, common
Johnson, Dr. •
Joints, aniversal
Journals

— table of

cast iron

— ^ of shafts,

Joumejrs or Joomals
Jujeb, strength of •

Kelly, William

Kyan's preserving timber

La Hire
Lantern .
Lateral stifihess

strength

stress
Lead, strength of .

allo3r8 of

Leaves, defined
Lemon, strength of
Lift

tenters .

— — — for windmills

Lignum vitffi .

Line of centres

Lock pulley .

Locust tree, strength of

L}ing shafts .

278
48

m

SIS

su
sia

SIS

S7

857

. MS

. 71
1

cast iron
malleable iron

251. S5S

• S0O

. l.«
. S51

. 868

. 805

. 311

• diff

8

. 897

. 851

849.348

. 839

'. 841

Machines, power of, men and

horses • • • • 90

Machinery, remarks on . .178

changing velocity of 334

Malleable iron shafts . . 849

strength of . 850

Materials for patterns . .188

Maximum effect 90» 91

^5

pechanics for Practical Men
Bchaniciil power explttined

78

- substitute for the

79.83
311.347

S42. 344

humELn hand
Men, strength of
Metals, strength of .
Meux and Co.
Mill-wheel work, shafts
■ •-*— stones, on 6xing
^mf^ — Telocity of
^^^•^ work, on fruntog
^fkotnentuin defined
^Hlorvcau, Gayton . 257, 258

^piotioD, of the methods nsed,
when motion is conveyed to
wheel-work . . . 299

mechanism for eqiiaJ-

iztng the motion of mills ■ 305

— uniform . . .323

Holherry tree, strength of . 251

chan^g velocity of
ly, rule for teeth

. 278
. 337

ihro^k's experiments

myth's remarks on the intro-
' dnction of the sUding principle
in tools and machines employed
in the production of machi-
nery 393

Newton, Sir Isaac ... 76
Nicholson, Peter ... 332
Nicholson's journal . . 349

Numbers, for arranging, for wheel
work 115.117

- for horse engines

Oak shafts . .841

Odontograph, Willis's . . 168
Oil, effects on couplings . . 286
Overshot wheels, experiments on 232

, theory of . 326

^~^—^— , power of . 330

■f velocity of
-, weight of

328

Parabola
Parallel lines .

motion

Parent, experiments by

Parry, Dr.

Pasteboard used for

Pattern teeth .

Patterns, on making

Pedestals

Peel, Williams, and

Pendulums

Perpendicular

Pillow block .

Pillows .

beatings

friction of .

internal

to find the fignre

wheel, &C.

Pitch .
table of

Pitch pine, strength of
Pi tot, experiments by
Pivots, fonns of

pressure on .

Phine tree, strength of
Planing machine
Platinum, strength of
Plumber block
Plum tree, strength of

27
. 134, 125

46,47
of the

21,25

257
347

-176

Polygon defined . . .

73

Round conpling

M.n

Pomegtnnatc tree, etrengtb of .

251

Romford. Const .

. w

Poplar, strength of .

251

RDpp,ofH»cfaMter

. M

Power, nature of . . 78, 77

. mechanical .

78

Sack tackle .

. M

hortea'

88

ScrewB ....

. M

of water wheel .

78

SemicaUd puobok .

. m

330

Shafts ....

. m

PrimitiTe ndii

3

geometrical figurea ,

39*

how made till of Ikte

. m

Principal diameter .

55

framing for npri^t

. SM

Proportions of melalg

261

on the bodies of

. HI

3

53
S8

lateial atifibess and html

n^ . . .

2

table of.

. 241

75

proportion of .

. 211

Pump mochinery, numhcre of .

123

wrought iron, to ttn

Mb-

Pnlley, fast and loose

297

teral streas of .

. Ml

lock ....

297

cast iron

181. ssr

eliding . . .

294

cyUndrical 17».

S88.I44

hoUow .

. MO

Quince tree, strength of.

251

square . 223, 224, 2X5

stress upon .

. 183

Back and pinion, tcetli for

48

to rcwst torsion

232.235

Radius, to find tlic, of pinion .

50

ivooden . 179,

180. 242

Radii of wheels, table of . 114

,115

Sickengcn

. 257

real . . . 3. 24.. 40

-Sliver, strength of .

251. 257

Red fir, strength of.

253

Slide-rest principlo ,

. 898

Re-eiignging machinery .

300

Sliding pulley

. 294

Ronnie, George 256

,257

. 192

RciKJrtory, No. 73 .

349

Smcaton, John

. 283

Revolving pendulum

309

Solid axles .

353. 369

Robcrton, John, on teeth of wheels

Soufflet ....

. 255

103

Southern

. 210

no rlinft''

183

Spindles

Spmdle grindstone .

i,,r... .r„.u.,~u

. 847
. «0

Robison, Professor . 64. 233

269

Rondelct, experiments by

857

Spurgear . . .

18,18

Rothsay cotton mill, experiments

wheeh, . . .

. 62

on

318

- 877

INDEX.

477

Square shafts.
Squeezers
Staves .

cast iron

strength of .

indefinitely small

when to be used

Staye-formed teeth .
Steam engines

governors

Steel pivots .
strength of

Page

267

353

J. 18

37

113

19. 21

37

19

176

308

. 347

, 308

. 346

. 226

Steps, form of

Stiffness

Strain, measured by horses' power 87

Strength . . . .226

of gudgeons . .198
of horses . . .113

■ of iron . . . 250

■ of men . . 89. 94
of shafts . .182
of staves . . .113

' of teeth ... 53

■ of timber . . . 253

Stress, lateral .... 228

on shafts and gudgeons

179. 197. 212
on teeth of wheels 86. 108

Tables of gudgeons

202. 205, 206. 210

of journals . . . 216

of pitches of wheels . 114

— ^— of shafts of cast iron 240, 241

solid . 240, 241

hollow 240, 241

— of squares, cubes, &c. 242

of wheels ... 95

how to be con-
sidered, &c. . . .175
Tamarind tree, strength of 257

Tangent of a circle
Teak wood, strength of .
Teeth, principles of
method of forming

Page

73
254

4
5

described by circular arcs 23
of bevelled wheels 55. 62

95.97

18. 22. 25

5. 8. 16

66.70

. 28

breadth of .

epicycloidal . *

form of

friction of .

inequalities of

involute

to describe the teeth of a

wheel for a trundle, by means

of a circular arc .

indefinitely small, as of

50

23

one cylinder rolling on an-
other ....
— to determine the breadth

of

of wooden . . .

figure of, when the staves

of the trundle are cylinders of

a finite diameter .

— ^- length of

strength of, in

111
112

power

thickness of

wear of

wooden

21

44

horses'

83. 106

109,110

. 29

29.112

. 255

. 278

. 308

. 251

. 260

. 260

Telford, Thomas .
Throstle
Throttle valve
Timber, strength of
Tin, strength of

alloys of .

Tools, the making of and tem-
pering .... 410
Torsion, on shafts subject to

231. 243
Tremor, effect of . . 343,344

1 1

I

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t

9

I.

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I ,

I'

i

■• ■!■■:■

r

INDEX.

*79

Wooden abafts
— — - teeth, breadth of
' thickness of

wheels, weight of

Wrought ironr—see Iron,

Yew, strength of .

242
112
110

204

258

Young, Dr. T. • 5. 65

•— letter from, to Bucfa&mui

66.71

Zinc, strength of .
— Indian .
Goslar •

. 260
. 260
. 260

THB BND.

O. WoodftU SDd Son* Prtattm^ Aflgil Ooort, SklHMT Sliittt

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