HENRY MOSELET, I. A. F.R.S.
CHAPLAIN IN ORDINARY TO THE QUEEN, CANON OF BRISTOL, VICAR OF OLVESTON ;
CORRESPONDING MEMBER OF TUG INSTITUTE OF FRANCE, AND FORMERLY PROFESSOB
OF NATURAL PHILOSOPHY AND ASTRONOMY IN KING'S COLLEGE, LONDON.
Second American from Second London Edition
WITH ADDITIONS BY
D. H. M A H A N , LL.D.
U. S. MILITARY ACADEMY.
V.MTIJ ri, LUSTRATIONS ON WOOD.
NEW YORK :
JOHN WILEY & SON, 535 BROADWAY
ENTERFO according *o Act of Congress, <jt\ the year 1856, Ij 1
WILEY & HA-LSTRD,
fc the Clerk's Office of the District Court of the United States, for the Southu a District
of New York.
THE high place that Professor Moseley occupies in the
scientific world, as an original investigator, and the clear-
ness and elegance of the methods he has employed in this
work have made it a standard text book on the subjects it
treats of. In undertaking its revision for the press, at the
request of the publishers of this edition, it has been deemed
advisable, in view of the class of students into whose hands
it may fall, to make some slight addition to the original.
This has been done in the way of Notes thrown into an
Appendix, the matter of which has been gathered from
various authorities ; but chiefly from notes taken by the
editor, whilst a pupil at the French military school at Metz,
of lectures delivered by General Poncelet, at that time, 1829,
professor in that school. It is a source of great pleasure to
the editor to have this opportunity of publicly acknowledg-
ing his obligations to the teachings of this eminent savan,
who is distinguished not more for his high scientific attain-
ment, and the advancement he has given to mechanical
science, than for having brought these to minister to the
wants of the industrial classes, the intelligent success of
whose operations depends so much upon mechanical science,
by presenting it in a form to render it attainable by the most
iv EDITOR'S PREFACE.
The editor would remark that lie has carefully refrained
from making any alterations in the text revised, except cor-
rections of typographical errors, and in one instance where,
from a repetition of apparently one of these, he apprehended
some difficulty might be offered to the student if allowed
to remain exactly as printed in the original.
UNITED STATES MILITARY ACADEMY,
Went Point March 8, 1866.
PKEFACE TO THE SECOND EDITION.
I HAVE added in this Edition articles : first, " On the
Dynamical Stability of Floating Bodies ;" secondly, " On
the Kolling of a Cylinder ;" thirdly, " On the descent of a
body upon an inclined plane, when subjected to variations of
temperature, which would otherwise rest upon it ;" fourthly,
u On the state bordering upon motion of a body moveable
about a cylindrical axis of finite dimensions, when acted
upon by any number of pressures."
The conditions of the dynamical stability of floating
bodies include those of the rolling and pitching motion of
ships. The discussion of the rolling motion of a cylinder
includes that of the rocking motion to which a locomotive
engine is subject, when its driving wheels are falsely
balanced, and that of the slip of the wheel due to the same
cause. The descent of a body upon an inclined plane
when subjected to variations in temperature, which other-
wise would rest upon it, appears to explain satisfactorily the
descent of glaciers.
The numerous corrections made in the text, I owe chiefly
to my old pupils at King's College, to whom the lectures
of which it contains the substance, were addressed. For
VI PREFACE TO THE SECOND EDITION.
several important ones I am, however, indebted to Mr
Eobinson, Master of the School for Shipwrights' Apprentices,
in Her Majesty's Dockyard, Portsea ; to whom I have also to
express my warm acknowledgments for the care with
which he has corrected the proof sheets whilst going through
IN the following work, I have proposed to myself to apply:
the principles of mechanics to the discussion of the most
important and obvious of those questions which present
themselves in the practice of the engineer and the architect ;
and I have sought to include in that discussion all the
circumstances on which the practical solution of such ques-
tions may be assumed to depend. It includes the substance
of a course of lectures delivered, to > the students of King's
College in the department of engineering and architecture,
during the years 1840, 1841, 1842.*
In the first part I have treated of those portions of the
science of STATICS, which have their application in the theory
of machines and the theory of construction.
In the second, of the science of DYNAMICS, and, under this
head, particularly of that union of a continued pressure with
a continued motion which has received from English writers
the various names of "dynamical effect," "efficiency," "work
done," "labouring force," "work," &c. ; and "moment
d'activite"," "quantite d' action," "puissance mecanique,"
" travail," from French writers.
Among the latter this variety of terms has at length given
place to the most intelligible and the simplest of them,,
* The first 170 pages of the work were printed for the use of my pupils in the-
year 1840. Copies of them were about the same time in the possession of
several of my friends in the Universities.
" travail." The English word " work " is the obvious trans-
lation of " travail," and the use of it appears to be recom-
mended by the same considerations. The work of overcoming
a pressure of one pound through a space of one foot has, in
this country, been taken as the unit, in terms of which any
other amount of work is estimated ; and in France, the work
of overcoming a pressure of one kilogramme through a space
of one metre. M. Dupiii has proposed the application of the
term dyname to this unit.
I have gladly sheltered myself from the charge of having
contributed to increase the vocabulary of scientific words,
by assuming the obvious term " unit of work " to represent
concisely and conveniently enough the idea which is attached
The work of any pressure operating through any space is
evidently measured in terms of such units, oy multiplying
the number of pounds in the pressure by the number of feet
in the space, if the direction of the pressure be continually
that in which the space is described. If not, it follows, by
a simple geometrical deduction, that it is measured by the
product of the number of pounds in the pressure, by the
number of feet in the projection of the space described,*
upon the direction of the pressure ; that is, by the product
of the pressure by its virtual velocity. Thus, then, we
conclude at once, by the principle of virtual velocities, that
if a machine work under a constant equilibrium of the
pressures applied to. it, or if it work uniformly, then is the
aggregate work of those pressures which tend to accelerate
its motion equal to the aggregate work of those which tend
to retard it ; and, by the principle of vis viva, that if the
machine do not work under an equilibrium of the forces
impressed upon it, then is the aggregate work of those which
tend to accelerate the motion of the machine greater or less
* If the direction of the pressure renfain always parallel to itself, the space
described may be any finite space ; if it do not, the space is understood to be
so small, that the direction of the pressure may be supposed to remain parallel
to itself whilst that space is described.
than the aggregate work of those which tend to retard its
motion by one half the aggregate of the vires vivce acquired
or lost by the moving parts of the system, whilst the work is
being done upon it. In no respect have the labours of the
illustrious president of the Academy of Sciences more con-
tributed to the development of the theory of machines than
in the application which he has so successfully made to it of
this principle of vis viva.* In the elementary discussion of
this principle, which is given by M. Poncelet, in the intro-
duction to his Mecanique Industrielle, he has revived the
term vis inertia (vis inertias, vis insita, Newton), and,
associating with it the definitive idea of a force of resistance
opposed to the acceleration or the retardation of a body's
motion, he has shown (Arts. 66. and 122.) the work expended
in overcoming this resistance through any space, to be
measured by one half the vis viva accumulated through the
space ; so that throwing into the consideration of the forces
under which a machine works, the vires inerticB of its moving
elements, and observing that one half of their aggregate vis
viva is equal to the aggregate work of their vires inertice, it
follows, by the principle of virtual velocities, that the differ-
ence between the aggregate work of those forces impressed
upon a machine, which tend to accelerate its motion, and
the aggregate work of those which tend to retard the motion,
is equal to the aggregate work of the vires inerticB of the
moving parts of the machine : under which form the prin-
ciple of vis viva resolves itself into the principle of virtual
velocities. So many difficulties, however, oppose themselves
to the introduction of the term vis inertice, associated with
the definitive idea of a force, into the discussion of questions
of mechanics, and especially of practical and elementary
mechanics, that I have thought it desirable to avoid it. It
is with this view that I have given a new interpretation to
that function of the velocity of a moving body which is
known as its vis viva. One half that function I have inter-
preted to represent the number of units of work accumulated
* See Poncelet, Mecanique Industrielle, troisieme partie.
in the body so long as its motion is continued. This number
of units of work it is capable of reproducing upon any resist-
ance opposed to its motion. A very simple investigation
(Art. 66.) establishes the truth of this interpretation, and
gives to the principle of vis viva the following more simple
enunciation : " The difference between the aggregate work
done upon the machine, during any time, by those forces
which tend to accelerate the motion, and the aggregate
work, during the same time, of those which tend to retard
the motion, is equal to the aggregate number of units of
work accumulated in the moving parts of the machine
during that time if the former aggregate exceed the latter,
and lost from them during that time if the former aggregate
fall short of the latter." Tims, then, if the aggregate work
of the forces which tend to accelerate the motion of a
machine exceeds that of the forces which tend to retard it,
then is the surplus work (that done upon the driving points,
above that expended upon the prejudicial resistances and
upon the working points) continually accumulated in the
moving elements of the machine, and their motion is thereby
continually accelerated. And if the former aggregate be
less than the latter, then is the deficiency supplied from the
work already accumulated in the moving elements, so that
their motion is in this case continually retarded.
The moving power divides itself whilst it operates in a
machine, first, into that which overcomes the prejudicial
resistances of the machine, or those which are opposed by
friction and other causes, uselessly absorbing the work in its
transmission. Secondly, into that which accelerates the
motion of the various moving parts of the machine, and which
accumulates in them so long as the work done by the moving
power upon it exceeds that expended upon the various
resistances opposed to the motion of the machine. Thirdly,
into that which overcomes the useful resistances, or those
which are opposed to the motion of the machine at the
working point, or points, by the useful work which is done
Between these three elements there obtains in every
machine a mathematical relation, which I have called its
MODULUS. The general form of this modulus I have discussed
in a memoir on the " Theory of Machines " published in the
Philosophical Transactions for the year 1841. The deter-
mination of the particular moduli of those elements of
machinery which are most commonly in use, is the subject
of the third part of the following work. From a combination
of the moduli of any such elements there results at once the
modulus of the machine compounded of them."
"When a machine has acquired a state of uniform motion,
work ceases to accumulate in its moving elements, and its
modulus assumes the form of a direct relation between the
work done by the motive power upon its driving point and
that yielded at its working points. I have determined by a
general method' 35 ' the modulus in this case, from that statical
relation between the driving and working pressures upon
the machine which obtains in the sfate bordering upon its
motion, and which may be deduced from the known condi-
tions of equilibrium and the established laws of friction. In
making this deduction I have, in every case, availed myself
of the following principle, first published in my paper on the
theory of the arch, read before the Cambridge Philosophical
Society in Dec. 1833, and printed in their Transactions of
the following year: "In the state bordering upon motion
of one body upon the surface of another, the resultant
pressure upon their common surface of contact is inclined
to the normal, at an angle whose tangent is equal to the
coefficient of friction."
This angle I have called the limiting angle of resistance.
Its values calculated, in respect to a great variety of surfaces
of contact, are given in a table at the conclusion of the
second part, from the admirable experiments of M. Morin,f
into the mechanical details of which precautions have been
introduced hitherto unknown to experiments of this class,
* Art. 152. See Phil. Trans., 1841, p. 290.
f Nouvelles Experiences sur le Frottement, Paris, 1833.
and which have given to our knowledge of the laws of
friction a precision and a certainty hitherto unhoped for.
Of the various elements of machinery those which rotate
about cylindrical axes are of the most frequent occurrence
and the most useful application; I have, therefore, in the
first place sought to establish the general relation of the
state bordering upon motion between the driving and the
working pressures upon such a machine, reference being
had to the weight of the machine.* This relation points out
the existence 'of a particular direction in which the driving
pressure should be applied to any such machine, that the
amount of work expended upon the friction of the axis may
be the least possible. This direction of the driving pressure
always presents itself on the same side of the axis with that
of the working pressure, and when the latter is vertical it
becomes parallel to it ; a principle of the economy of power
in machinery which has received its application in the
parallel motion of the marine engines known as the Gorgon
I have devoted a considerable space in this portion of my
work to the determination of the modulus of a system of
toothed wheels ; this determination I have, moreover,
extended to bevil wheels, and have included in it, with the
influence of the friction of the teeth of the wheels, that of
their axes and their weights. An approximate form of this
modulus applies to any shape of the teeth under which they
may be made to work correctly ; and when in this approxi-
mate form of the modulus the terms which represent the
influence of the friction of the axis and the weight of the
wheel are neglected, it resolves itself into a well known
theorem of M. Poncelet, reproduced by M. ISTavier and the
Rev. Dr. Whewell.f In respect to wheels having epicy-
* In my memoir on the " Theory of Machines " (Phil. Trans. 1841), I have
extended this relation to the case in which the number of the pressures and
their directions are any whatever. The theorem which expresses it is given in
the Appendix of this work.
f In the discussion of the friction of the teeth of wheels, the direction of the
mutual pressures of the teeth is determined by a method first applied by me to
cloidal and involute teeth, the modulus assumes a character
of mathematical exactitude and precision, and at once
establishes the conclusion (so often disputed) that the loss of
power is greater before the teeth pass the line of centres
than at corresponding points afterwards ; that the contact
should, nevertheless, in all cases take place partly before
and partly after the line of centres has been passed. In the
case of involute teeth, the proportion in which the arc of
contact should thus be divided by the line of centres is
determined by a simple formula ; as also are the best
dimensions of the base of the involute, with a view to the
most perfect economy of power in the working of the
The greater portion of the discussions in the third part of
my work I believe to be new to science. In the fourth part
I have treated of " the theory of the stability of structures,"
referring its conditions, so far as they are dependent upon
the rotation of the parts of a structure upon one another, to
the properties of a certain line which may be conceived to
traverse every structure, passing through those points in it
where its surfaces of contact are intersected by the resultant
pressures upon them. To this line, whose properties I first
discussed in a memoir upon " the Stability of a System of
Bodies in Contact," printed in the sixth volume of the Carrib.
Phil. Trans., I have given the name of the line of resist-
ance ; it differs essentially in its properties from a line
referred to by preceding writers under the name of the
curve of equilibrium or the line of pressure.
The distance of the line of resistance from the extrados of
a structure, at the point where it most nearly approaches it,
I have taken as a measure of the stability of a structure,* and
that purpose in a popular treatise, entitled Mechanics applied to the Arts,
published in 1834.
* This idea was suggested to me by a rule for the stability of revetement
walls attributed to Vauban, to the effect, that the resultant pressure should
intersect the base of such a wall at a point whose distance from its extrados is
iths the distance between the extrados at the base and the vertical through
the centre of gravity.
have called it the modulus of stability; conceiving thia
measure of the stability to be of more obvious and easier
application than the coefficient of stability used by the
That structure in respect to every independent element
of which the modulus of stability is the same, is evidently
the structure of the greatest stability having a given quantity
of material employed in its construction ; or of the greatest
economy of material having a given stability.
The application of these principles of construction to the
theory of piers, walls supported by counterforts and shores,
buttresses, walls supporting the thrust of roofs, and the
weights of the floors of dwellings, and Gothic structures,
has suggested to me a class of problems never, I believe,
before treated mathematically.
I have applied the well known principle of Coulomb to
the determination of the pressure of earth upon revetement
walls, and a modification of that principle, suggested by M.
Poncelet, to the determination of the resistance opposed to
the overthrow of a wall backed by earth. This determina-
tion has an obvious application to the theory of foundations.
In the application of the principle of Coulomb I have
availed myself, with great advantage, of the properties of
the limiting angle of resistance. All my results have thus
received a new and a simplified form.
The theory of the arch I have discussed upon principles
first laid down in my memoir on " the Theory of the Stability
of a System of Bodies in Contact," before referred to, and
subsequently in a memoir printed in the "Treatise on
Bridges" by Professor Hosking and Mr. Hann.* They
differ essentially from those on which the theory of Coulomb
is founded ;f when, nevertheless, applied to the case treated
* I have made extensive use of the memoir above referred to in the following
work, by the obliging permission of the publisher, Mr. Weale.
f The theory of Coulomb was unknown to me at the time of the publication
of my memoirs printed in the Camb. Phil. Trans. For a comparison of the
two methods see Mr. Hann's treatise.
by the French mathematicians, they lead, to identical results,
I have inserted at the conclusion of my work the tables of
the thrust of circular arches, calculated by M. Garidel from
formulae founded on the theory of Coulomb.
The fifth part of the work treats of the "strength of
materials," and applies a new method to the determination
of the deflexion of a beam under given pressures.
In the case of a beam loaded uniformly over its whole
length, and supported at four different points, I have deter^
mined the several pressures upon the points of support by a
method applied by M. Navier to a similar determination in
respect to a beam loaded at given points.*
In treating of rupture by elongation I have been led to a
discussion of the theory of the suspension bridge. This
question, so complicated when reference is had to the weight
of the roadway and the weights of the suspending rods, and :
when the suspending chains are assumed to tte of uniform
thickness, becomes comparatively easy when the section of
the chain is assumed so to vary its dimensions as to be every
where of the same strength. A suspension bridge thus
constructed is obviously that which, being of a given
strength, can be constructed with the least quantity of
materials ; or, which is of the greatest strength having a
given quantity of materials used in its construction.!
The theory of rupture by transverse strain has suggested
a new class of problems, having reference to the forms of
girders having wide flanges connected by slender ribs or by
open frame work : the consideration of their strongest forms
leads to results of practical importance.
In discussing the conditions of the strength of breast-
summers, my attention has been directed to the best positions
of the columns destined to support them, and to a comparison
* As in fig. p. 487. of the following work.
f That particular case of this problem, in which the weights of the suspending
rods are neglected, has been treated by Mr. Hodgkinson in the fourth vol. of
Manchester Transactions, with his usual ability. He has not, however, suc-
ceeded in effecting its complete solution.
of the strength of a beam carrying a uniform load and sup-
ported freely at its extremities, with that of a beam similarly
loaded but having its extremities firmly imbedded in
In treating of the strength of columns I have gladly
replaced the mathematical speculations upon this subject,
which are so obviously founded upon false data, by the
invaluable experimental results of Mr. E. Hodgkinson,
detailed in his well known paper in the Philosophical
Transactions for 1840.
The sixth and last part of my work treats on " impact ;"
and the Appendix includes, together with tables of the
mechanical properties of the materials of construction, the
angles of rupture and the thrusts of arches, and complete
elliptic functions, a demonstration of the admirable theorem
of M. Poncelet for determining an approximate value of the
square root of the sum or difference of two squares.
In respect to the following articles of my work I have tc
acknowledge my obligations to the work of M. Poncelet,
entitled Mecanique Industrielle. The mode of demonstration
is in some, perhaps, so far varied as that their origin might
with difficulty be traced ; the principle, however, of each
demonstration all that constitutes its novelty or its value
belongs to that distinguished author.
30,* 38, 40, 45, 46, 47, 52, 58, 62, 75, 108,f 123, 202,
267,t 268, 269, 270, 349, 354, 365.
* The enunciation only of this theorem is given in the Mec. Ind., 2me partie,
f Some important elements of the demonstration of this theorem are taken
from the Mec. Ind., Art. 79. 2me partie. The principle of the demonstration
is not, however, the same as in that work.
\ In this and the three following articles I have developed the theory of the