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PRINCIPLES
OF
ANIMAL MECHANICS.
PRINCIPLES
OF.
ANIMAL MECHANICS.
BY THE
REV. SAMUEL HAUGHTON, F. R. S.,
FELLOW OF TRINITY COLLEGE, DUBLIN;
M. D. DUBI.., 1). C. L. Oxon.;
FELLOW OF THE KING AND QUEEN'S COLLEGE OF PHYSICIANS IN IRELAND ;
HONORARY FELLOW OF THE ROYAL COLLEGE OF SURGEONS IN IRELANI),
AND OF THE SWEDISH SOCIETY OF PHYSICIANS.
ידעתי כי כל־אשר יעשה האלהים הוא יהיה לעולם
עליו אין להוסיף וממנו אין לגרוע
"Εοικε ταύτα σοφού τινός δημιουργού και φιλοζώου τεχνήματι.
Nihil est enim, quod ratione et numero moveri possit sine consilio ; in quo
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SECOND EDITION.
LONDON :
LONGM ANS, GREEN, AND CO.
1873.
QP
303
,H37
1873
Prof. Alex. Zivet
1923
1923
DUBLIN :
PRINTED AT TÄE UNIVERSITY PRESS,
BY M. H. GILL.
Ꮲ Ꭱ E F A C Ꭼ .
MHE observations and calculations contained in this
T Book have occupied my leisure hours during
the past ten years; and they are now offered to the
public, with the view of showing the mutual advan-
tages obtainable by Anatomists and Geometers from
a combination of the Sciences which they cultivate.
Anatomists will gain by the increased precision which
numerical statements must give to their observations,
and Geometers will find in Anatomy a new field of
problems opened out to their investigation. I may
be allowed to call attention, from this point of view,
to the problem of the equilibrium of an elliptical
muscular dome, and to the use which I have made of
the hyperboloid of one sheet, of Ptolemy's Theorem,
and of some curves of the third order.
In the course of my investigations, I have met
with numerous instances, in the muscular mechanismi
of the vertebrate animals, of the application of the
principle of least action in Nature; by which I mean
that the work to be done is effected by means of the
existing arrangement of the muscles, bones, and joints,
TY
LI
415010
PREFACE,
with a less expenditure of force than would be possible
under any other arrangement; so that any alteration
would be a positive disadvantage to the animal. If, as
I consider probable, this fact should prove to be or
much wider occurrence in Nature than these instances
show, it may serve to give us some slight glimpse of
the mechanism by which the conservation of species
in Nature is secured. In Astronomy, the conserva-
tion of the Solar System depends upon certain well-
known conditions regulating the motions of the
several bodies of which that system consists ; and it
is a matter of indifference whether these conditions
were directly imposed by the Will of the Divine Con-
triver, or were the indirect result of some former con.
dition of the System. In either case, these conditions
are equally the foreseen result of the Contrivance.
If the present state of the Solar System be the result,
according to fixed laws, of some pre-existing state o.
that System, it may be said, in the language of Na.
turalists, to have been evolved out of its former state,
but in such an Evolution there was nothing left to
Chance ; it was all foreseen, and the Evolution itsel.
presided over by the Divine Mind that planned the
whole. I cannot see why there may not be in Or-
ganic Life a similar process of Evolution of higher
from lower forms of existence; but it is a Teleologi-
cal Evolution, in which every step and every result
was foreseen and planned beforehand. The Laws of
such an Evolution appear to me, in the present state
of our knowledge, to be entirely unknown.
..
PREFACE!
L
vii
,
I have explained the similarity of muscles and
bones in the vertebrate animals on simple Teleo-
logical principles, without making use of the un-
proved hypothesis of their descent from a supposed
common ancestor.
In the discussion of the Theory of Muscular
Types, and of the Laws of Fatigue and Refreshment,
I have succeeded in obtaining many new and interest-
ing applications of exact science to the problems of
Animal Mechanics.
this opportunity of returning my thanks to
Dr. Alexander Macalister and to the Rev. Richard
Townsend, who have assisted me materially in the
Anatomical and Geometrical branches of my subject;
and also, to the Provost and Senior Fellows of Trinity
College, Dublin, for the assistance they have given
me, by defraying a portion of the expenses of pub-
lication.
SAMUEL HAUGHTON.
.
11
TRINITY COLLEGE, DUBLIN,
1st January, 1873.
TABLE OF CONTENTS.
Page,
I, THE NATURE OF MUSCULAR FIBRES,
2. THE NATURE OF MUSCULAR CONTRACTION, . . 5 5
d. The muscular susurrus, .
o
b. Determination of its note,
o. o
19
f. Dr. Collongues on the note of the susurrus, . . 22
· 3. STATICAL WORK DONE BY MUSCLES, , o
24
a. Shoulder muscles (unloaded),
. . . o
h. Shoulder muscles (loaded),
fr Law of fatigue, . o fi o. o i . 29
d. Flexors of forearm, . . . . . . . 35
6. Attempt to calculate statical work, by considering the arm
as a pendulum, alternately depressed and raised,
f. Equation of fifth degree rejected,... . . . . 42
the Statical work represented by a cubical hyperbola and cus-
pidal cubic, . . . . . . .
4, DYNAMICAL WORK DONE BY MUSCLES,
. . . 44
a. Work done in rowing, . . . . .
47
b. Work done in climbing . . . . . . 57
Fr Work done in walking, i
. . . . 57
d. Work done by labourers, .
, 58-62
5. THE ABSOLUTE FORCE OF MUSCLES,
pi, 63
al, Coefficient of force of the muscles of the arm,
b. Coefficient of force of the muscles of the leg, i f i 67
6. Borelli's observations on the force of muscles, . 74
6. COMPARATIVE ANATOMY OF THE TENDONS OF THE HAND AND
Foot, AND THEIR MECHANICAL USES, e
o
A. Comparative Anatomy of the tendons of the foot,. .
t. Man, ...
. . . . . . .
b. Chimpanzee, .
. . . 92
6 Monkeys of the Old Continent, , . .
d. Monkeys of the New Continent, . . . . . 94
TABLE OF CONTENTS
TE
Page.
.
95
·
103
.
·
107
.
·
III
113
117
I 20
121
123
126
131
132
133
137
138
14!
143
144
e Carnivores, . . .
. . .
f. Ungulates, . . . . . . . .
8. Insectivores,
h. Edentates, . . . . . . . .
i. Rodents,
k. Marsupials, . . . . .
1. Birds, . . . . . . . . .
m. Reptiles, . . . . . . . . o
B. Comparative Anatomy of the flexor tendons of the hand,
a. Man, . . . . . . . . . .
. Quadrumans, . . . . . . . .
C. Carnivores, . . . . . . . . .
d. Ungulates, . . . . . .
e. Rodents, o
. .
f. Marsupials, . .
7. THE MECHANICAL WORK DONE BY THE HUMAN HEART,
a. Rev. Dr. Hales' experiments, . . . o
6. Velocity of blood spouting from a large artery, . .
C. Coefficient of capillary resistance, . . .
d. Hæmastatic pressure inside the left ventricle,
e. Work done by the heart, . . . .
Estimates of Helmholtz and Donders, . . . .
f. Experiments of Poiseuille and Spengler, . . . .
g. Time occupied in the complete circulation of the blood,
8. THE MUSCULAR FORCES EMPLOYED IN PARTURITION, . .
a. Lagrange's Theorem, . . . . . .
b. Experiments of Duncan and Tait, . . . .
6. Calculation of hydrostatical pressure caused by contraction
of the uterus, . . . . . . . .
d. Abdominal muscles, . . . . . . . .
e. Calculation of pressure caused by the abdominal muscles, .
Curvature of abdomen in young men, . . .
9. CLASSIFICATION OF MUSCLES, AND THEIR MODE OF ACTION,
a. Direct prismatic muscles, . . . .
b. Rhomboidal muscles, . . .
.
6. Penniform muscles, . . . . . . . .
Biceps humeri in nylghau and ass, i
Biceps humeri in Sambur deer, . . . . .
Mylohyoid muscle, . . .
i Triangular muscles, . . . .
Biceps femoris in lion, . . . . . . .
Pectoral muscle in angel shark, . . . . .
145
147
149
150
151
152
155
160
162
164
165
IDA
167
.........
173
179
180
TABLE OF CONTENTSRSS
Xi
.'
194
199
204
209
220
Page.
Loss of work in triangular muscle, . . . . . 183
Law of arrangement of fibres in pectoral muscle, in birds
and man, . . . . . . . . .
186
Temporal muscle and latissimus dorsi in man, . . 191
€. Deltoidal muscles, . . . . . . . 192
Trapezius muscle in man,
o o
o
193
Pectoral muscle in squirrel, o
f. Quadrilateral muscles, . . . . . . 194
g. Sphincter muscles, . .
196
Relation between tangential and radial strains, o
Iris in man, o . . . . . . .
200
Orbicularis palpebrarum in man, o o
o o 201
Discussion of the muscles of the mouth in man; .
h. Muscular fibres forming curved surfaces, . . . .
1. Ellipsoidal muscles, . . . . . . . .
Tension of fibres in any azimuth determined by the
indicatrix, . . . .
. . o
215
-Lagrange's theorem, . . . . . . . 218
Obliquus externus in man, . . . . . . 219
Obliquus internus in man, . . . . . .
Vertical and transverse tension due to abdominal muscles, 223-4
Mean curvature of abdomen in parturient women,, 221
Curvature of diaphragm, . . . . . 231
k. Muscular skew surfaces, .
.
. 232
bo Teleological postulate, . .. .
. 238
Illustrations of postulate :mmm
1. Latissimus dorsi in man, . . . . . 245
2. Latissimus dorsi in lion and tiger, ..
249
3. Latissimus dorsi in albatross, . . .
4. Hamstring muscles in lion and tiger, . .
Axis of maximum instability
5. Wing of the albatross, . .
. . 283
6.
wood pigeon, . . , 294
7. ? heron, · · · ·
295
macaw,
297
o pheasant, .
.
299
Axis of minimum stability~
10. Wing of the albatross, . .
309
II.
wood pigeon, . . . . . 312
12.
heron, .
macaw,
314
14..
pheasant, . . . . . 315
257
.
313
138
.
TABLE OF CONTENTS.
328
19
Page.
m. Quadrilateral, and skew muscles, in which the socket of
the joint does not lie in the plane containing the origin
and insertion of the muscle, . . . . . . . 320
Illustrations of Postulate--Ptolemy's theorem,
326
. 1. Adductor longus and pectineus in lion,
2. Pectinæus in man, . . . . . 329
3. Pectinæus in llama, o o o
o o 329
4. Adductor secundus (B) in llama, . . . 334
S. 19
• Master Magrath," o 343
6.
woman,
• • 346
leopard, . . . 348
Reduction of skew muscle to a single equivalent fibre, 352
Adductor secundus (B) in llama,, .
352
“Master Magrath," .
354
woman, . . • 354
leopard, . . . . * 355
Adductor secundus (a) in leopard, . .
356
Skew muscle regarded as a “supplemental contri-
vance," . o . o . . . . . 360
Transfer of force caused by skew muscles, . . . 361
10. THEORY OF HIP AND SHOULDER JOINTS, . . . 361
Hip joint of lion:
Abductors.
1. Agitator caude, . . . . o
2. Gluteus primus, . i o o o
364
3. Tensor vaginæ femoris, .
4. Gluteus secundus (a),
5. Gluteus secundus (B), . .
6. Glutæus tertius,
7. Glutæus quartus, . . . O . . . 371
8. Glutæus quintus, . . . . . . . 373
9. Obturator internus, . o . . . . 373
Confirmation of postulate from the abduction muscles
of the lion's hip-joint, . . . . . . 374-5
Adductors.
1. Adductor primus,
2. Adductor secundus (a), . . . . . 378
3. Adductor secundus (ß),
4. Adductor tertius, . .
. . 383
5. Adductor quartus, . . . . . . 383
6. Adductor quintus, o . . . .
7. Obturator externus, o . . . 386
363
376
......
TABLE OF CONTENTS.
386
387
397
399
400
401
9
406
408
Page.
Confirmation of postulate from the adduction muscles
of the lion's hip-joint, . . . . . .
General conclusions from the comparison of the teleolo-
· gical postulate, with observation, . . o
11. MUSCULAR TYPES, . . . . . . o 390
a. Larger felidæ (hip). o.o.o. o o 391
b. Lesser 1 . o o o o o o o 393
6. Larger (shoulder) . o . . o i 394
d. Lesser
395
e. Greyhound (hip),
(shoulder), o
g. Canidæ (hip), . . o
398
„ (shoulder), . . . .
i. Man and quadrumana (hip), . "
k.,
39 (foot),
i
404
(shoulder,
ng (hand), , 0
m. Struthionidæ (hip), . . .
416
0. Birds' wings, . . . . . . . . 419, 420
P. Bats' wings, o . . o . . . . 424
g. Swimming animals (shoulder), . .
425
7. Ostrich and rhea (shoulder) . . . . . 429
5. Emu and cassowary (shoulder), . . . 429
12. PRINCIPLE OF LEAST ACTION APPLIED TO THE ARRANGE-
MENT OF THE MUSCULAR FIBRES OF THE HEART,
430
Relation of lengths of proper and common fibres required by
law of least action, o o o o
o o 439
Verification of relation,
o i .
441
13. APPLICATION OF GENERAL LAWS OF MUSCULAR ACTION, .
a. Borelli's law, ó ó ó
443
6. Law of constancy of work, . .
444
Mr. Jevons' experiments,
444
6. Law of fatigue, o .o .
450
Mr. Jevons' experiments, . .
450
Author's experiments, . . .
453
Mr. Jevons' experiments, . . .
Mr. Nipher's experiments, . .
Dr. Macalister's experiments, . .
Mr. J. Gilbert Haughton's experiments,
474
Mr. Nipher's experiments, o . . o
475
Value of w, o o o o . . . . 477
442
........
408
xiy
.
Ý
TABLE OF CONTENTSPages
d. Law of refreshments a re o o
478
Coefficient of refreshment (heart), . . . 479
.. "
(walking muscles), · 479
, (shoulder muscles) 482, 3, 4
Comparison of refreshment of heart and voluntary
muscles, o q . . . . o o
484
Conclusion, . .
.
. 485
INDEX,
o
o o
487
ANIMAL MECHANICS,
1. On the Nature of Muscular Fibres. If a portion of the
muscle of any animal be macerated in water for a short time,
and gently separated into its elements by means of dissecting
needles, it will be found to consist of elementary fibres, ad-
mitting of no further subdivision by mechanical treatment.
the muscle from its origin to its insertion, and thus vary in
length from several lines to upwards of two feet.* The thick-
ness of the elementary fibres has been measured by various
observers, among whom none is more worthy of confidence
than Mr. W. Bowman, who has given the following Table as
the result of his observations on various animals :f-
Diameter of the Elementary Fibres of Striped Muscle in Fractions of
an English Inch,
From To
1. Human, ... to · Tða · • average of males, . . sta
, females, . 34
2. Other Mammals, · Tibo · • TŠE • average ....
* In the m. sartorius of Man these fibres have been found upwards of two
feet in length, and I have myself measured them in the m. bicipiti accessorius of the
Lion exactly two feet long.
t “Cyclopædia of Anat, and Phys.," vol. iii., p. 507.
ANIMAL MECHANICS.
From TO
3. Birds, i... Tood . . o .. average, ... · GÖT
4. Reptiles, . ... Tobo · · tdo .
5. Fishes, · · · · 75 ·. .. »
... ZI
6. Insects, .... 555 .. do ...
... 41
Professor Donders, of Utrecht, and Dr. Buys Ballot,* esti-
mate the cross section of the fibres of the m. biceps humeri
and m. brachialis anticus, in Man, as zdoth of a square milli-
metre. This estimate, converted into English measures, gives
for the diameter
d =4_ x 39.37 = 0.001986 = = in.
.
In this formula, 7 denotes the ratio of the circumference to
the diameter of a circle.
The same authors found for the cross sections of these
muscles the following measurements :-
m. biceps (long head) ... 530mm. square.
(short head) ... 452
m. brachialis anticus ... 614
.
Total
..
1596mm.
From these results it follows that the total number of ele--
mentary fibres in these muscles amounts to 798,000. These
important muscles are the only ones concerned in holding the
forearm flexed at right angles to the arm, and we shall have
occasion hereafter to make use of the results just given on the
authority of Donders and Ballot.
The elementary fibres are not circular in their cross sec-
tion, but polygonal, in consequence of their mutual pressure ;
andmicroscopicalexamination shows that the blood vessels which
supply the fibres are interposed at the angles of the polygons :
from this it follows that the muscles which have the smallest
fibres are best provided with blood, and are therefore ca-
* “Over de Elasticiteit der Spiren," p. 47, 8vo., Utrecht, 1863.
ANIMAL MECHANICS.
3
-
.
pable of the longest endurance of work. This conclusion
agrees with the Table of Mr. Bowman, from which it appears
that of all animals the muscles of Birds possess the smallest
fibres, and that the muscular fibres of women are smaller than
those of men; for it is well known that no animals are more
capable of long-continued muscular exertion and fatigue than
Birds; and I have myself found, by direct experiment, that the
muscles of women are capable of longer continued work than
those of men, although inferior to them in force exerted for a
short time.*
The elementary fibre is encased in a sarcolemma, or sheath,
very tough and elastic, but not endowed with the vital pro-
perties of the muscular fibre itself; and, again, many such ele-
mentary fibres, each invested by its proper sheath, are united
into greater or lesser bundles, separated from each other by
areolar tissue, sometimes containing fat, which acts as a lubri-
cating medium, to prevent friction between the muscular bun-
dles during their rapid and varying action.
Before speaking of the varieties of origin, insertion, and
action of the muscular bundles, it may be useful to say a few
words on the distinction between striped and unstriped mus-
cles, respecting which there seems to be much confusion of
ideas. On examining most muscles with the microscope, they
are found to be striped crossways with very delicate and close
parallel lines, while other muscles are perfectly free from such
an appearance. It is said by some that the distinction of
muscles into voluntary and involuntary corresponds with the
microscopical distinction into striped and unstriped muscles ;
and, if it were not for the all-important exception of the heart,
this assertion might be maintained with much appearance of
* If any man wishes for a simple proof of the inferiority of the endurance of his
muscles as compared with those of a woman, let him carry a child on his arm for the
same time that his wife or nurse can do with ease, and he will find himself much
fatigued.
B 2
ANIMAL MECHANICS.
1
UIT
truth; but since the heart is, of all the muscles of the body,
the most essentially involuntary, and yet possesses striped
muscles, we must seek for some other explanation of this re-
markable difference between muscles ; and indeed it must be
granted that, even if the voluntary and involuntary muscles
were invariably striped and unstriped, it would still remain
as difficult of explanation as before why a voluntary muscle
should differ from an involuntary muscle in so singular a
manner.
On a careful examination of the stripes in muscular fibres;
it is found that they arise from a tendency of the fibre to split
crossways into thin disks, which give an appearance like that
of cleavage in rocks to the whole of the fibres. Judging from
the analogy of cleavage, I have come to the conclusion that
the striped structure in muscles, or tendency to cleave into
disks, is due to their repeated contraction between two fixed
or nearly fixed points of origin and insertion. Mr. Sorby has
shown, by examination of the microscopical structure of cleaved
rocks, and I have shown by calculations founded on the dis-
tortion of their fossils, that the planes of cleavage are at right
angles to the lines of maximum compression; Professor Tyn-
dall also has proved, by direct experiments on compressed
wax, that the formation of cleavage planes perpendicular to
lines of pressure is an universal law of nature. Whenever,
therefore, we find in muscular fibres a distinct origin and in-
sertion, their contraction between these points will produce
the pressure necessary for the developement of cleavage at right
angles to the length of the fibres; and, in like manner, when
a muscular fibre returns upon itself, as in the sphincter ani and
sphincter vagince, its contraction produces the striped structure.
In the case of the heart, as is now well established, the mus-
cular fibres form closed circuits, coiling round each ventricle
and round the whole heart in such a manner, that their con-
traction, like that of a twisted India rubber ring, developes
ANIMAL MECHANICS.
the conditions of pressure necessary for the production of a
cleaved or striped structure. In the unstriped muscles, as
those of the bladder and uterus, of the intestinal canal and
scrotum, the contraction of the muscle is vermicular, and not
a constant strain from end to end; hence the conditions of
pressure requisite to produce cleavage are wanting, and the
muscle and its fibres remain without this distinguishing mark
of uniform and continued contraction from end to end.
It has been found, on careful microscopical examination
of the elementary fibres, that they possess not only a trans-
verse cleavage, dividing them into disks, but also a longitudinal
cleavage or jointage, separating the fibres into fibrille ; so that
the ultimate subdivisions of muscles may be regarded as ele-
mentary molecules of muscular fibre, which are to be con-
sidered as units, or ultimate atoms of muscular tissue; and
every muscle may be regarded as composed of a certain num-
ber of such atoms, variously distributed.
2. On the Nature of Muscular Contraction. — The muscular
fibres already described are capable of contracting in a longi-
tudinal direction, and in so contracting exert mechanical force
on the points of origin and insertion. It was supposed by
the older writers (as Borelli and others), that there was no
change of volume during the contraction of muscles; and this
conjecture has been confirmed in a satisfactory manner by
recent experiments; and, as this law of constancy of volume
during change of form holds true of bodies under the influence
of magnetical and electrical forces, some writers have supposed
that it furnishes a key to the explanation of muscular vital phe-
nomena.
It is certain that muscular contraction is accompanied by
developement of sensible heat (or loss of latent heat); by loss
of electrical statical tension; and by the chemical transforma-
tion of fibrine, albumen, and the higher products of animal
liſe, into urea, carbonic acid, and water, which are the main
excretions of all animal tissues: and, as it is well known that
(
ANIMAL MECHANICS.
loss of latent heat, loss of statical electricity, and change of
chemical products from higher to lower forms, are all repre-
sentatives of force expended, or work done, it has been pro-
posed by various authors to select one or other of these ex-
ponents of work, as the means of measuring its amount under
various conditions. It seems to me that chemical transfor-
mations present the most certain and ready of all known
methods of determining the work done by the various tissues
of the body in the discharge of the several functions assigned
to them in the economy of nature.
All these chemical, electrical, and calorifical changes are
intimately connected with the supply of fresh arterial blood to
the muscles ; and any theory of muscular action that does
not explain the necessity of such a fresh supply of blood must
be erroneous.
On the electrical theory of muscular action, the contraction
of the muscle is accompanied by a loss of electrical tension,
caused by the partial discharge of the opposite electricities of the
outer and inner portions of the muscular bundles, and a supply
of fresh blood is admittedly necessary for the restoration of that
tension; the contraction being supposed due to the molecular
attraction of the muscular particles when freed from the con-
trol of their induced electricity. This theory of muscular
action certainly commends itself by the explanation which it
affords of rigor mortis on principles analogous to those of or-
dinary muscular contraction; but, inasmuch as no method has
been as yet devised for measuring the total loss of electrical
tension caused by a given amount of muscular action, it seems
to me that the electrical theory of such action must be re-
garded, in the present state of science, as a theory incapable of
yielding numerical results, which are the only tests to which
true science can submit its decisions.
Of the other two exponents of work done, namely, the ca-
loric developed, or the urea, carbonic acid, and water produced,
ANIMAL MECHANICS.
by a given amount of muscular action, I cannot treat further
at present, as I propose to confine the scope of this work al-
together to the mechanical effects of muscles. Of all the
methods of estimation hitherto proposed, the chemical (as
I have already stated) seems to me to be the most natural, and,
in the existing state of science, the most certain.
That a fresh supply of arterial blood is necessary to enable
the muscle to relax its contraction, is proved by many obser-
vations and experiments. Among these may be mentioned
puerperal convulsions consequent on uterine hæmorrhage, the
anæmic convulsions witnessed before death in horses and
other animals destroyed by venesection, the phenomena of
rigor mortis, which disappear when arterial blood is artificially
pumped through the dying arteries, and the convulsions caused.
by simple hanging
The curious reader will perhaps excuse the following di-
gression on the art of hanging.*
* I am indebted to my friend Dr. J. K. Ingram, Fellow of Trinity College,
Dublin, for the following interesting sketch of the History of the Art of Hanging in
England :-
" Hanging was a mode of execution in use among the Anglo-Saxons. Indeed,
in 'Beowulf'-which its able Editor, Kemble, believed to be a modernized form of a
poem which the invaders of Britain had brought with them from their Continental
homes-the gallows (galga) figures as an old-established institution of the Teutonic
races of Northern Europe. But it is very difficult to get any definite information as
to the history of hanging in England. It seems, however, quite certain that the idea
of immediately extinguishing the life of the culprit by a sufficient fall never presented
itself to our ancestors; their only notion was that of suspending him by the neck for
what might seem an adequate time to insure (?) strangulation. It is noticed by
Blackstone as a somewhat singular fact, that the only warrant the sheriff has for a
capital execution is the signature of the judge to the calendar, or list of all the prison-
ers' names, with their respective judgments in the margin; as, for a capital felony
it is written opposite to the prisoner's name, “hanged by the neck ;" formerly, in the
days of Latin and abbreviation, "sus. per coll.” for "suspendatur per collum.""
Originally, however, he states there was a formal precept to the sheriff under the
hand of the judge; but in none of the law books have I found a copy or exact descrip-
ANIMAL MECHANICS.
Digression on the Art of Hanging.—Death is caused by
hanging in one or other of the three following methods. :- .
1. By apoplexy, caused by pressure on the jugular veins;
2. By asphyxia, caused by stoppage of the windpipe;
3. By shock of the medulla oblongata, caused by fracture
of the vertebral column.
In the first two cases, death is preceded by convulsions,
lasting from five to forty-five minutes, which are caused by
the cessation of the supply of arterial blood to the muscles. In
the third case, death is instantaneous and painless, and is un-
accompanied by any convulsive movement whatever.
According to the original form of death punishment for
treason in England, the hanging was used as an anæsthetic,
preparatory to the disembowelling (or drawing) that always
preceded the quartering of the criminal; and the present slow
process of hanging, practised by Calcraft and others in Eng-
land and Scotland, which consists in dropping the patient
through two or three feet, and allowing him to hang until
dead, is the faithful representation of the original process
of hanging, which was intended to fulfil a purpose quite distinct
from that of the speedy execution of the criminal.
It seems to me unworthy of the present state of science
to continue a mode of execution which as at present used is
1
tion of this precept. I do not even find how long the sentence, to be ' hanged by the
neck till you are dead,' has been in use.
"" It is clear,' says Blackstone, 'that, if upon judgment to be hanged by the
neck till he is dead, the criminal be not thoroughly killed, but revives, the
sheriff must hang him again, for the former hanging was no execution of the sen-
tence.' But, strangely enough, we find in the Vision of Piers Plowman' a passage
which seems to show that the opposite of this either was, or was believed to be, the
established rule in his time :-
“ It is noght used on earthe
To hangen a felon
Ofter than ones,
Though he were a tretour.'”
ANIMAL MECHANICS.
extremely clumsy, and also painful to the criminal. Instead
of the short drop" generally used, we ought to employ the
“ long drop,” which causes instantaneous death. It has been
proved by me that the shock of a ton dropped through one foot
is just sufficient to fracture the anterior articulating surfaces
of the second vertebra at their contact with the atlas, and that
this fracture allows the shock to fall upon the medulla oblongata,
80 as to produce instantaneous death. As the result of some
consideration bestowed upon the subject, I would recommend
the adoption of the following rule:-
RULE I. “Divide the weight of the patient in pounds into
2240, and the quotient will give the length of the long drop in
feet."
For example, a criminal weighing 160 lbs. should be al-
lowed 14 feet drop. If local circumstances will not allow of
the long drop being employed, the requisite shock should be
produced by strapping a shot to the feet, so as to secure the
shock of 2240 ft. lbs. to the medulla.
Efforts have been made in the United States to give to
hanging all the rapidity of death by the guillotine, without the
painful spectacle of bloodshed. This method, which is bor-
rowed from the mode of execution practised on board ship,
consists in suddenly lifting the criminal into the air by means of
a great weight attached to the other end of the rope fastened
round his neck; the rope passes over two pulleys, one of which
is placed vertically over the patient, and at a given signal the
weight falls through a regulated height, lifting him suddenly
into the air. Sufficient attention, however, has not been paid,
even in that enlightened country, to the conditions necessary
to be fulfilled in this mode of suspension; for in many of their
executions the only care that seems to have been taken was to
make the falling weight heavier than the criminal, so as to in-
sure his permanent suspension by the neck until death termi-
nated his sufferings.
10
ANIMAL MECHANICS.
The American method of hanging, if properly applied,
seems to me to be capable of producing death by shock with
even less suffering than the " long drop;" for, although by the
latter method death is instantaneous when the shock actually
occurs, yet the mental sufferings of the criminal during the
second occupied by his fall may be very considerable. This
painful interval is altogether avoided in the American method,
provided the initial shock be sufficient to destroy the medulla
oblongata. This important condition may be effected by the
following calculations, which lead to an easy rule:-
The falling weight, acting through the intervention of the
rope, produces its effect in a manner similar to that of the
shock or collision of imperfectly elastic bodies.
Let m and m' denote the masses of the two bodies, and let
v and í denote their velocities previous to collision or shock,
while e denotes the coefficient of elasticity of the rope.
Let u, u, denote the velocities of the masses m, m', after
the shock; then it is well known* that
U =
mv + m'u – em' (v - v').
m + m
_ m + m 2 – em ( - 0)
m + m'
The vis viva lost during the shock is expended upon the
neck of the criminal, and is represented by
mv2 + m'v'? – mu? – m'u?.
After some reductions this is found to be
mm
Vis viva lost = (1 - ?) (v – v)?. (2)
This result may be applied practically to the solution of the
American problem of hanging, so as to cause instantaneous
death, in the following manner :-
* Vide - Manual of Mechanics," p. 156.
ANIMAL MECHANICS.
11
Let P denote the weight employed, and Q the weight of
the criminal ; let e denote the coefficient of elasticity of the
rope used, and V the velocity acquired by the weight Q in
falling through the height h.
If we consider the problem of the weight Q moving with
the velocity V, and causing the weight P to move through
the intervention of the rope, whose elasticity is e; the shock
produced on Pat the moment when the “ chuck” takes place
is similar to that which occurs in the collision of bodies striking
each other, and is measured by the vis viva lost during their col-
lision. The vis viva lost during the shock is, by equation (2),
I PQ
ST
S
5 P+ 8(1–e?) Va;
SP
5 (1 e) h.
(3)
but, since the work lost is half the vis viva lost, if we substitute
for V2 its value, 2gh, we find
PQ
Work lost=.
VP+Q!
For the ropes usually employed, e may be considered as a
very small fraction, and e? may be totally disregarded. The
work lost is expended in causing shock to the neck, and should,
as I have already stated, be equal to at least 2240 foot lbs.
In a case of hanging that came under my own observa-
tion, the criminal weighed 160 lbs., and was allowed to fall
through 14 ft. 6 in., which, allowing for some elasticity in the
rope, would correspond with 2240 ft. Ibs. of shock; in this
case, the anterior articulating surfaces of the second vertebra
were fractured near their posterior border (the fracture of
the bone extending to the foramina for the vertebral ar-
teries), but the odontoid process of the axis was not broken,
nor the transverse ligament of the atlas torn across. Death
in this case was as instantaneous as it would have been had
the transverse ligament given way, instead of the bone; for
the shock reached the medulla, and its consequence was
12
ANIMAL MECHANICS.
immediate and painless death. In hanging, the rope supports
the atlas, and presses it against the occipital articulations, while
the second vertebra tends to fall with the body ; and it follows
from this that either the odontoid process must give way, or
the second vertebra be broken across at its superior articu-
lating surfaces; in either case death will be immediate. The
height of the criminal just mentioned was 5 ft. 9 in. a few
days before execution, and after death he was found to mea-
sure 5 ft. Ir in., having been elongated an inch and a half by
the “long drop" of 141 ft.
From the foregoing, it therefore appears that a shock to
the neck of 2240 ft. lbs. is just sufficient to cause immediate
death. Substituting, therefore, 2240 ft. lbs. for the work lost
in the preceding equation, we find
PQh
2240 = PEQ'
or, solving for P,
P_2240 Q
Qh – 2240
(4)
From this equation it follows that, unless Qh be greater than
2240, the value of P (the weight required) will be negative;
but Qh denotes the work produced by the criminal Q falling
through the height h.
Let h, therefore, denote the “ long drop" found by Rule I.,
and the following consequences may be inferred from (4)
1. In the American mode of hanging, if the weight be let
fall through the height h, sufficient to cause death instanta-
neously by the “long drop," it would require an infinite weight
to cause immediate death ; for in this case
Q6 – 2240 =o,
and therefore Pis infinite.
2. In the American mode of hanging, if the weight be
let fall through twice the height of the “long drop," a weight
2.
t
ANIMAL MECHANICS.
13
LU
equal to that of the criminal will be sufficient to cause im-
mediate death; for in this case,
| Q – 2240 = 224;
and therefore, by equation (4)
P-Q.
For all heights intermediate between h and 2h, the weight
P must be found from equation (4), and it will always lie be-
tween Q and infinity.
In practice, twice the height of the “ long drop" would al-
ways be found convenient, and therefore the following Rule
for producing instantaneous death by the American method is
confidently recommended :-
RULE II.-"Having found from Rule I. the height of the
English long drop, use twice this height and a weight equal to
that of the criminal in the American method."*
* I have searched in vain for well-authenticated instances of fracture of the cer-
vical vertebræ produced by the usual method of hanging. Among the longest drops
that I can find recorded by competent witnesses are two observed by Dr. Charles
Croker King, when Professor of Anatomy in the Queen's College, Galway.
CASE I. (a)-A young man named Hurley was executed in Galway at 6.25, p.m.,
on the 27th August, 1853, for the murder of a young woman in Dunsandle Wood.
The rope used was ten lines in diameter; the knot was large, formed of three turns
of the rope ; and, on the noose being tightened by the executioner, corresponded to
the occipital protuberance. His weight was 101 stone, and he was allowed to drop
71 feet. These data give us as follows :-
Work done = 147 x "Ş = 1102 ft. Ibs.
In this case, as Dr. King remarks, " there was no dislocation or fracture of the
vertebral column, or injury of the ligaments or of the spinal cord.”
CASE II. (6)-On the 11th May, 1858, Patrick Lydon was hanged in Galway for
the murder of his wife. Lydon was a small man, only 5 ft. 5 inches in height; the
diameter of the rope was 10 lines; his weight was oj stone, and the drop 11 ft.
Hence we find
Work done = 133 X 11 = 1463 ft. lbs.
In this case, " that portion of the anterior common ligament of the spine which
(a) " Dublin Quarterly Journal of Medical Science," vol. xviii., 1854, p. 86, et seq.
(6) Ibid., August, 1863.
14
ANIMAL MECHANICS.
On the Nature of Muscular Contraction.- Whatever ob-
scurity there may be as to the ultimate cause of muscular con-
traction, there need be none whatever as to the modus operandi.
When a stimulus is applied to a muscle, by a nerve, by a
mechanical injury, or by electricity, there follows immediate
contraction of the muscular fibres in the neighbourhood of the
part affected, and this contraction is rapidly propagated by a
wave-like motion to the other parts of the muscular fibres-each
particle contracts for an instant of time, and then transmits the
contracted phase to the neighbouring particles, so that the en-
tire length of the muscle is never contracted together, but wave
after wave of contraction is passed through it, so long as the
action of the nerve or other stimulus is continued.
The rate at which the wave of contraction passes along
the muscles of a frog has been shown by Prof. Aebe to be
only 3 ft. per second, which is much less than the rate at which
similar waves of action pass along the sensitive and motor
nerves, which has been determined by several observers. The
following may be given as examples :-
1. Rate of action of Motor nerves of Frog (Helmholz), ... 88 ft. per sec.
2. Rate of action of Sensitive nerves of Man (Schelske), . . 97 , ,
The rate at which the wave of action runs along both sen-
sitive and motor nerves may be approximately measured by
noting what is well known to observers as the “personal
equation.”
An observer is obliged to watch with the eye, the ear, or
any other sense, a certain phenomenon, and then to record
with the hand, through some instrumentality, the instant of
passes from the body of the second to that of the third cervical vertebra was rup-
tured, so that the left halves of the bodies of the above-mentioned vertebræ were
separated from each other by an interval of onc-eighth of an inch, but there was no
displacement.”
These criminals were executed with the same rope; and death, in the second case,
was not preceded by violent muscular convulsions, as in the first case; a fact which
is readily accounted for by the excess of shock in the proportion 1463 to 1102.
ANIMAL MECHANICS.
15
the occurrence of that phenomenon. It is well known that dif-
ferent observers, equally competent, allow different but always
constant intervals to elapse between the moment of the oc-
currence of the phenomenon and the moment of recording it,
and the differences between different observers is called the
“personal equation." In other words, the time occupied by
the sensitive nerves in conveying the impression to the optic
thalamus, and by the motor nerve in reconveying the order of
the brain from the corpus striatum, is different in different per-
sons--or, the velocity of nerve action differs in different per-
sons, and a fortiori in different animals. *
Although the velocity of the wave of sensation, or of volition,
may thus vary, according to original differences in individuals,
or according to the state of health in the same individual,
there is good reason to believe that the time occupied by each
particle of the muscle or nerve fibre in going through all its.
changes of contraction and relaxation remains constant.
According to the well-known formula for wave motion-
=vt,
. (5)
where , denotes the wave length, v the velocity of the wave,
and t.the time of molecular vibration or change of state. .
From this formula it follows that, if the time of vibration be
constant, the length of the wave will vary as the wave velo-
city; so that waves of differeñt degrees of velocity may pass
along the same nerve or muscle, without any change having
taken place in the molecular conditions of contraction and re-
laxation of each portion of the muscle or nerve.
Nothing proves so conclusively as the observation of per-
sons afflicted with softening of the brain, the fact that the
* The recent experiments of Prof. Donders show that the time occupied in the
transmission of a sensation from the eye to the brain, the formation of a judgment,
and the transmission of volition from the brain to the hand, is 0.15 second. Wben
the ear is employed instead of the eye, the time is reduced to 0.09 second.
16
ANIMAL MECHANICS.
time between the observation and recording of a pheno-
menon may vary greatly in the same person according to the
state of health; but, as the principal retardation in these cases
may, and probably does, occur in the brain itself (between the
reception of the sensation and the issuing of the volition), it
would not be philosophical to adduce such cases as proofs of
a difference of velocity in the nerves either of sensation or of
motion.
Whatever be the rate of wave motion in the muscle or
nerve, the time of its molecular vibration may be determined by
the note of the susurrus caused by the muscles, and by the
note of the tinnitus produced by the nerves.
In the Croonian Lecture, read before the Royal Society, on
November 16th, 1809, Dr. Wollaston called attention to the
sound, or susurrus, produced by the muscles when in a state
of contraction. He states correctly that this sound is best
produced by inserting gently the extremity of the finger into
the ear, bringing at the same time the muscles of the hand and
forearm into strong contraction. The muscular susurrus may
also be heard easily by using a stethoscope upon a con-
tracted muscle, either of the observer's or of any other person's
body.
Having described the mode of obtaining the sound, Dr.
Wollaston correctly adds, that it "resembles most nearly that
of carriages at a great distance passing rapidly over a pave-
ment;" to which admirable description of the sound I would
add, that it bears a most striking resemblance to the deep
hum produced by the blowing fan of a large foundry.
Dr. Wollaston attempted to estimate the frequency of the
elementary muscular contraction that produces the susurrus
by allowing his ear to rest on the ball of his thumb, while his
elbow was supported by a horizontal board, in which he had
cutta number of equal notches, about one-eighth of an inch
asúnder. Against these notches he rubbed a pencil with a

ANIMAL MECHANICS.
S
02
regular motion, until he made the sound so produced to coin-
cide roughly with that of the muscular contraction; and he at-
tempted to estimate the number of notches passed over in a
second. His conclusion from these comparative experiments
is the following:-
- The greatest frequency that I think I have observed was
about 35 or 36 in a second, and the least was as low as 14 or
15 ...; they appeared to be in general between 20 and 30
in a second; but it is possible that the method I employed
may be found defective, and it is to be hoped that my estimate
may be corrected by some means better adapted to the deter-
mination of intervals that cannot actually be measured."
An accidental observation made upon myself in the early
part of 1862, enabled me to fix with the precision desired by
Wollaston the rate of the muscular contraction that causes the
susurrus. In December, 1859, on recovering from a fever,
which I acquired from an English patient under the care of
Dr. Jonathan Osborne, in Sir Patrick Dun's Hospital, I found
that it had left the following trace of its visit in my system :-
Occasionally since that period, when overworked by mental
exertion, I have been subject to a singing in both ears, which
is relieved by a drink of warm milk, and by sleep; sometimes,
however, I have found the tinnitus aurium so great as to pre-
vent sleep.
In June, 1862, while kept awake by this disagreeable
noise, I amused myself with producing, by the contraction of
the masseter muscles, their well-known susurrus. To my great
surprise and pleasure, I observed that the tinnitus and susurſus
were in unison, differing from each other by several octaves.
I followed up the clue thus found, and consulted my mu-
sical friends, whom I instructed in the mode of making Wol-
laston's experiment, without, however, informing them of the
note that I myself believed to be the true sound of the susur-
rus and tinnitus. My friends have arranged themselves into
ICC
18
ANIMAL MECHANICS.
two groups, which have fixed upon CCC and DDD, respec-
tively, as the note of their susurrus--that is to say, two octaves
below bass C and D.
These notes are found on the new pianos only; and several of
my friends informed me that the C or D, the lowest on their
piano, was an octave above the susurrus ; but, on inquiry,
there was no difficulty in ascertaining what note was really
indicated. The tuning of the pianos was carefully tested with
a standard CC diapason, of 64 double vibrations per second,
used for acoustical experiments, and corresponding with 426}
vibrations in the second for the note, treble A.
Four of the observers, of whom two were ladies, found CCC,
and five, of whom three were ladies, found DDD, which notes
correspond, respectively, to thirty-two and thirty-six vibra-
tions in the second
To my own ear, the susurrus has constantly the sound
of CCC; and the tinnitus had, on the morning of 30th June,
1862, the sound of the octave above treble C, that is to
say, it was
five octaves above the susurrus,
and therefore corresponding to a rate of vibration thirty-two
times faster than that of the muscle, or 1024 times in the
second.
* This is the note given out by a well-formed hand bell, of four inches di-
ameter.
* Although the tinnitus aurium is caused by the molecular vibrations of the
nerve particles, and takes its note from them, yet I believe it to be influenced by
ANIMAL MECHANICS.
19
S
The tinnitus is altogether independent of muscular or vo-
luntary action of any kind, and is, I believe, a sign of the rate
at which nervous action takes place in the brain. .
The sound of the muscular susurrus, when fully heard,
is ridiculously like the sound of the cab wheels of London
heard in the silence of the night, when the absence of tho-
roughfare in the streets enables the cabmen to drive fats. I
have measured the intervals of the Guernsey granite pave- *
ment, and found them to be about four inches, making there-
fore three impulses in a foot traversed by the cab wheels. If
the cabs be supposed to drive eight miles an hour, the number
of impulses per second will be
5280 x 3 x 8.
3600
=35.2.
WS
Considering that the standard DDD of the susurrus is thirty-
six vibrations per second, its resemblance to the sound of cab
wheels ceases to be a matter of surprise.
In order to determine the sound of the susurrus precisely,
I procured a long wooden tube, 3 inches by 21 inches inside,
like an organ pipe, fitted with a moveable piston, so that on
applying it to an acoustical bellows, on moving* the piston
in or out, I could produce any note I pleased in the neigh-
bourhood of CCC. This pipe is tuned, by means of the piston,
to the susurrus of any observer, and then played in conjunc-
UM
the resonance of the skull which contains the brain. The following observa-
tion throws light on this subject. On the 26th August, 1863, I received a vi-
olent blow from a racket ball in the back of the head, which partially stunned
me; and I was conscious at the instant of receiving the blow of a loud singing
sound in both ears, which I attributed to the vibration of the brain case, trans-
mitted to the cochleæ aurium. This ringing sound seemed to me to have the
same pitch as the singing that occasionally occurs in my ears.
* The piston is moved in the acoustical tube by means of an endless chain
movement, made for me by Mr. Spencer, of Aungier-street.
C 2
20
· ANIMAL MECHANICS.
II
tion with the standard CCC, and the beats counted if it be not
in unison.
Let m denote the number of vibrations per second in the
standard pipe, n the corresponding number in the tuned su-
and let k be the number of beats per second counted on play-
ing the two pipes together; then it is easily shown that :
k=m-n.
(6)
As examples of this principle I take the following observa-
tions :-
Dr. William Stokes, Regius Professor of Physic in the
University of Dublin, kindly aided me in my investigations
upon one occasion, by tuning the wooden pipe to the susurrus
observed in himself, leaning both his elbows on a table, with
the muscles of the fore-arm contracted, and making the sound
of the pipe coincide with the susurrus when heard by both
ears together.
: On comparing the note he fixed on, with the standard CCC,
it was found to be higher than it to such an extent as to pro.
duce 50 beats in 15 seconds.
· Hence,
10
pe=m-32; or
3}=m-32;
and, finally,
m=35} vibrations per second.
This is DDD natural, a little flat, for DDD corresponds to
thirty-six vibrations in the second.
In July, 1863, Professor Czermak, of Prague, assisted me
at an acoustical experiment, and found his susurrus to corre-
spond with a position of the moveable piston below CCC, such
as to give 40 beats in 20 seconds.
ANIMAL MECHANICS.
Hence,
32-40 = k= 30;
showing that the pitch of his muscular susurius conresponded.
to 30 vibrations per second.
The preceding investigation of the precise sound of the
muscular susurrus was published by me in 1863, having formed.
part of a thesis for a medical degree in the University of
Dublin in 1862; and shortly after its publication, Dr. Col-
longues, of Paris, forwarded to me a copy of a work published
by him in 1862 (" Traité de Dynamoscopie," 8vo, Paris, 1862),
in which he had solved the same problem, and found an iden-
tical result from his acoustical experiments. Dr. Collongues
attempted to determine the pitch of the susurrus, with com-
plete success, by means of a tuning fork, with sliding weights,
which was put into unison with the susurrus, and then com-
pared carefully with a standard tuning fork, whose time of
vibration was determined graphically by the aid of a black-
ened revolving cylinder, which was marked with a zigzag
line by a pin attached to one arm of the vibrating fork. The
note fixed on by Dr. Collongues was Re-1, or 72 vibrations
per second, reckoned in the French fashion by half vibrations,
and corresponding to 36 vibrations per second of English
acousticians. I readily accord to Dr. Collongues the pri-
ority of publication of the exact determination of the pitch of
Wollaston's susurrus, and believe it to be a matter of some
interest to science that two independent observers, by different
methods, one in Paris, and the other in Dublin, succeeded in
arriving at a precise and almost identical value for this im-
portant physiological constant.
Dr. Collongues believes that the susurrus is caused by the
nerves, and not by the muscles, and the greater part of his
valuable book is taken up with an account of the variations of
the susurrus in health and in disease. With this part of his inves-
a
S
S
22
ANIMAL MECHANICS.
tigations we have here no concern, as we are only interested,
at present, in the rate of muscular action indicated by the
susurrus. Dr. Collongues' tuning fork, which is made by
M. Koenig, of Paris, under the name of the Biometer, is an in-
strument more portable than my organ pipes, and in other re-
spects also better suited for the investigation of variations in
the susurrus; for this reason I have copied in the note* his
description of the instrument.
Le Biomètre est un instrument d'acoustique appliqué à l'homme bien por-
tant ou malade, destiné à évaluer et à reproduire par l'unisson et à volonté les
sons perçus ordinairement par l'auscultation des doigts. '
A cet effet, le Biomètre fait entendre des sons de différente hauteur, les
traduit en notes et en nombre de vibrations, ce qui permet la comparaison du
rapport des intervalles qui existent des deux côtés du corps, soit dans l'état
normal, soit dans l'état anormal, et d'exprimer en chiffres les différents degrés de.
santé et de maladie.
L'instrument se compose d'un manche, d'un diapason avec curseur et d'un
dynamoscope.
Le manche est en caoutchouc : il comprend une poignée, une overture et une
extrémité digitale. La poignée isole le bruit de la main qui supporte l'instru-
ment, l'ouverture permet l'introduction de la tige du diapason , et l'extrémité di-
gitale reçoit le dynamoscope.
Le diapason est formé de deux branches, longues de 30 centimètres ; sur
chacune de ces branches glisse un curseur qui peut adhérer sur tous les points
de sa course, à l'aide d'une vis de pression. Chaque branche comprend huit
divisions annotées de cette façon.
Notes et gamme du Biomètre, Re-2, Mi-2, Fa-2, Sol-2, La-2, Si-2, D0-1, R.-1.
Nombre absolu de vibrations, 36, 40, 42, 48, 54, 60, 64, 72.
Intervalles de Re-2 à Re-1, 1, § $ .
2.
Intervalles de Re-i à Re-2, 2,
a
1.
Intervalle des notes,
Ire 2€ 30 40 50 60 70 8e
L'acoustique a recherché quelle relation existe entre le nombre de vibrations
de deux notes lorsqu'elles produisent tel ou tel accord.
L'accord entre deux notes peut être exprimé par le rapport de deux nombres
de vibrations, et, si le rapport est égal à l'unité, les deux notes sont à l'unisson.
Accord parfait de la gamme du Biomètre ; Re-2, Fa-2, La-2, Re-1, intervalle des
vibrations en montant i , 2.
ANIMAL MECHANICS. ·
23
The muscular susurrus varies in different persons from
30 to 36 vibrations per second, but in partially paralytic
affections of the limbs it is always reduced in pitch. The fol-
lowing remarkable cases illustrative of this fact have come
under my observation :
On the 22nd May, 1863, I saw, in conjunction with Dr.
Stokes, a gentleman engaged in the tobacco manufacture, in
good health, but affected with paralysis agitans of both arms,
probably produced by tobacco, but attributed by him to a
slight injury of the left hand, by which he lost the nail of the
thumb about three years previously, at which time, according
to his account, the trembling motion of the hands and arms
first appeared. We examined the muscles of the forearm and
arm of both sides with a stethoscope, and found a "paddle
wheel” susurrus in all the muscles, especially those of the
forearm. Both Dr. Stokes and myself compared the sound of
the susurrus estimated by the stethoscope with that of a
Re-1, La-2, Fa-2, Re-2.
Intervalle des vibrations en descendant:
Limite des sons perçus par l'auscultation des doigts:
Limite des sons graves : La-g, 27 vibrations. Limite des sons aigus: Re-1,
144 vibrations.
Manuel opératoire. On fait entrer la tige du Biomètre dans l'ouverture pra-
tiquée dans le manche. Le dynamoscope est appliqué à son extrémité digitale,
des deux branches du diapason sont tournées vers la terre. On les met en vibra-
tion en les rapprochant brusquement. Après avoir saisi la poignée du manche,
on détruit les sons harmoniques en touchant légèrement les deux branches du
diapason à leur extrémité fermée. On applique le dynamoscope dans l'oreille,
on compare le son du Biomètre à celui produit par l'un des indicateurs, et si on
ne les trouve pas semblables, on abaisse ou on élève les deux curseurs jusqu'à ce
que les deux sons produisent l'unisson. Alors, grâce à cette loi de physique
que deux sons de même hauteur ont toujours le même nombre de vibrations, on
traduit en nombre le bourdonnement vital. On renouvelle l'expérience pour
l'indicateur de l'autre main, et, après avoir obtenu sa valeur numérique, on com-
pare les deux chiffres ou les deux notes, et l'on constate l'égalité ou la dif-
férence.
24
ANIMAL MECHANICS.
tuning fork properly weighted, and considered the number of
vibrations to amount to only 6 per second, which is lower
than two octaves below the normal CCC.
A friend of mine, aged forty-five years, who was a seven
months' child, prematurely born while his mother was dan-
gerously ill of typhus fever, has had slight congenital paralysis
agitans of the right arm, forearm, and hand, all through his
life; this gentleman was carefully examined by me in August,
1863, with the result that his muscular susurrus in the arm
affected had only the pitch of 4 beats per second, although
the susurrus of the other arm is of the usual note.
I have also ascertained, by over-fatiguing the forearm by
racket playing, that the pitch of the muscular susurrus is
lowered, and that a sensible paralysis agitans of a temporary
character is set up by the over-exertion.
I have not been able to verify the continuance of the
susurrus after death, except in cases of tetanus, nor have I
succeeded in observing it in amputated limbs ; but I have no
doubt, from the published results of Dr. Collongues' careful
observations, that my failure is simply the result of a deficiency
in my own powers of observation, and that the susurrus may
continue audible to more delicately organized ears than my
own, long after death or amputation.
3. On the Statical Work done by Muscles in continued
Contraction.- Muscles may be employed either in continuous
statical work resisting pressure, or in dynamical work, lifting
weights with intervals of repose. The work done by muscles
in the latter case is easily measured by measuring the ex-
ternal resistances overcome ; while in the estimation of stati-
cal work, there is much difficulty in obtaining numerical
estimates of the work actually done : at the same time there
is no doubt as to the reality of the work, for the muscles
become rapidly fatigued when employed in doing statical
work.
ANIMAL MECHANICS.
25
If both arms be extended horizontally, with the palm
upward, either unloaded or loaded with equal weights in
both hands, after a short time, much fatigue is experienced
and the experimenter finds himself compelled to terminate
the experiment-as the feeling of fatigue passes rapidly into
pain, chiefly felt in the supraspinatus and central deltoid mus-
cles. It is necessary to keep the arms quite horizontal, for
if they be held either above or below the horizontal line
the distressed muscles are aided by the action of other
muscles.
The total work done in this case may be estimated as
follows; let
w = weight held in hand;
a = weight of arm;
a = length of arm, measured from centre of glenoid ca-
vity to centre of weight;
t = total time in seconds, until fatigue stops the experia
ment.
The muscles exert a force capable of sustaining the weight
of the loaded arm at its centre of gravity, and if we suppose,
this weight to be moved (with an unknown angular velocity)
uniformly, the total work done is measured by the weight of
the loaded arm multiplied into the arc through which the arm
would move in the time during which the experiment lasts.
Let
a = distance of the centre of gravity from the centre of
the glenoid cavity;
e = the angle through which the arm would move in the
time t;
w = the unknown angular velocity.
The total work done is represented by
(w + a) x0 = Total work;
26
ANIMAL MECHANICS.
but
(w + a) x = a (w +)
+
0 = wt.
Hence we find
Total work = wa (w + 9).
wa ' W +
(7)
2 /
The following experiments were made upon the unloaded
arms:
No. 1.-Myself.
a = 22 inches.
W = 0.
a* = 7.5 lbs.
t = 420 secs.
Total work in ft. lbs. = 2887-5 W.
No. 2.--Dr. A. W. Foot.
a = 24 inches. w = 0.
a = 7 lbs.
t = 480 secs.
Total work in ft. lbs. = 3360 W.
No. 3.-Captain John Haughton, R. A.
a = 22 inches.
20 = 0.
a = 5 lbs. 14 oz. t = 720 secs.
Total work in ft. lbs. = 3877.5 W.
No. 4.- Mr. M. H. Ormsby.
a = 24 inches. w = 0.
a = 6.5 lbs.
t = 390 secs.
Total work in ft. lbs. = 2535 w.
* The approximate weight of the arm may be found by weighing it in a
halance, whose scale is on a lovel with the centre of the glenoid cavity, and
treating it as a lever, so that the weight of the arm shall be to the weight that
supports any point of the arm inversely as the distances of the centre of gravity
and of that point from the centre of the glenoid cavity.
ANIMAL MECHANICS.
2
No. 5.- Mr. Samuel Haughton.
a = 22 inches.
w = 0.
a = 8.5 lbs.
t = 525 secs.
Total work in ft. lbs. = 4090.6 w.
If we collect together into one Table the preceding re-
sults, we obtain as follows:
TABLE I.– Work done by the Shoulder Muscles employed in
holding the unloaded Arms in the horizontal Position, with
Palms upward :-

Age.
Weight.
Height.
Work done.
Time of Ex-
haustion.
No. I.
No. 2.
126 lbs.
67 inches.
420 secs.
2887.5 W.
3360.0,
3877.5”,
No. 3.
651
No. 4.
No. 5.
2535.0 ,
4090.6 ,,
480 »
720 ,
390 1
12
225
Means,.
3350·12 W. | 507 secs.
The following experiments were then made on the same
persons, holding the arms horizontally, as before, with two
pounds weight in each hand, with the following results. The
times of exhaustion were found to be as follows:
1. Myself, . . . . . . 200 secs.
2. Dr. Foot, . . . . . 210 ,-
3. Captain Haughton, · · 300 ,
4. Mr. Ormsby, . . . . 180
5. Mr. Samuel Haughton, 240 „
Using these numbers in equation (7), and substituting
2 lbs. for w, and half the weight of the arm for a, we obtain--
28
ANIMAL MECHANICS.
TABLE II.-- Work done by the Shoulder Muscles employed in
holding the loaded Arms in the horizontal Position, with the
Palms upwards.

Time of Ex-
haustion,
Work done.
No, 1.
No. 2.
200 secs.
210
No. 3.
2108.3 w.
2715.65
1890.0,
2750.0,
No.4.
180
12
No. 5.
240
»
Means,
226 secs.
2354.78 W.
I was led by several considerations to the conclusion, that
it was probable that the total work done in exhausting
muscles varies inversely as the rate of doing the work; from
which it would follow that the square of the total work varies
as the time of doing the work; this would give us the
relation
(Total work)
- = Constant.
Time
n
UL
Calculating this constant from Tables I. and II., I found
the following results :-
Product of Total Work and Rate of Work.

First set Second set
of Ex- of Ex-
periments. periments.
Ratio.
I. 12
ZZZ
niin
19852
23520
1.08
20882
16478
22224
25410
24581
19845
31510
No.4.
No.5.
1.17
I 20
0.99
31873
Mean,
111
ANIMAL MECHANICS.
29
From these results I obtained my first conception of the
Law of Fatigue, which I shall fully develope further on, with
the proofs of it, derived from more extensive and varied ex-
periments ; and I shall also prove that the coefficient w is
equal to unity.
The Law of Fatigue may be thus stated :-
When the same muscle (or group of muscles) is kept in con-
stant action until fatigue sets in, the total work done, multiplied
by the rate of work, is constant.
Thus, if W denote the total work done, and T denote the
time of doing it, the rate of work will be expressed by W
divided by T, and
1
Total Work Rate of Work = W x*
If this product be constant, we see that the work ana
rate of work may be represented by the coordinates of an
hyperbola referred to its asymptotes as axes of coordinates;
while the relation of the total work to the time of doing it,
may be represented, as in equation (8), by the coordinates of
a parabola referred to a tangent and diameter as axes of
coordinates.
As an illustration, let us suppose that a man walking at
the usual rate does not become tired until he has walked
30 miles. On another occasion, let him walk twice as fast,
then (neglecting the difference of action necessitated by the
different rates of walking) we should expect, by the Law of
Fatigue, that the man will be completely fatigued at the end
of 15 miles, having done only half the total work, in a quarter
of the original time. If the man walk three times as fast, he
will become tired out at the end of 10 miles, in one-ninth part
of the original time; and so on, the total work done always
· varying as the square root of the time necessary to produce.
fatigue.
30
ANIMAL MECHANICS.
In order to find the quantity of muscle that gives out the
preceding quantity of work, it would be necessary to know
the weights of the living muscles in each of the persons ob-
served; but, as this, except in rare cases, is impossible, we
must endeavour to obtain the result, indirectly, by observa-
tion of the muscles in the dead subject. With this object in
view, I carefully dissected the muscles in four dead bodies,
selecting those only that were well developed, and had not
experienced much wasting of muscular tissue during their
last illness. From these dissections, I obtained the following
results :-
TABLE III.- Weights of Muscles.

Supraspinatus.
Portion of Deltoid
employed.
Deltoid.
A. Female,
B. Male,
C. Male,
D. Male,
1.01 oz. av.
1.76 19
2.18 19
1.75 »
1.82 oz. av.
2.98 19
3.99
2.53 1
7.10 oz. av.
10.16
12.97
10.25 1
Means, 1'675 oz.
2.830 oz.
10-12 oz.
The only difficulty in these observations consists in estimat-
ing the portion of the, deltoid actually employed in holding
the arm horizontal; this was done by holding the arm in that
position while dissecting the deltoid, and estimating as well
as I could the fibres actually in action. It will be seen that
my estimate varies somewhat in the different cases, as fol-
lows:-
In A, the portion of Deltoid was both of the whole.
34 th
3.25th
4.05th
Mean,
Bath of the whole.
ANIMAL MECHANICS.
The portion of the deltoid employed, as found from the
means, is 3.5th of the whole; so that, combining all the
foregoing results together, it is very probable that the weight
of deltoid used in holding the arm horizontal is &ths of the
whole weight of the muscle.
If the muscle be divided into a clavicular, an acromial,
and a scapular portion, the portion used in holding the arm
horizontal is very nearly aths of that taking origin from the
whole tip and line of the acromion; but for the purpose of
future experiments it may be found easier to weigh the whole
muscle, and take oths of that weight, than to separate it into
its parts by direct dissection. The mean weight of the por-
tion of the deltoid engaged in holding the arm in the hori-
zontal position has been found to be 2.83 oz. av.; and to
this must be added the mean weight of the supraspinatus,
which may be found as follows:
TABLE IV.- Weight of Supraspinatus Muscle.

Female,
Male,
Male,
Male,
Male,
P. K. Male,
1.01 oz. av.
1.76
2.18
1.75
0.93
2.40
Mean, . . 1:671 oz. av.
This result is almost identical with that found in Table III.,
from four well-developed dead subjects. This result of Table
IV. is interesting from the fact that it contains the actual
weight of the supraspinatus, observed 40 minutes after death,
in a criminal (P. K.) executed, by hanging, for murder. As I
shall have occasion to refer to this case for other purposes, I
here give the data, as observed by myself:-
32
.
NY
ANIMAL MECHANICS· P. K. was executed on the 20th of July, 1865, for
murder :-
1. Weight in clothes, 160 lbs.
. 2. Length of drop, 14 feet 6 inches.
3. Height during life, 69] inches.
4. Height after death, 71 inches.
5. From tip of acromion to metacarpo-phalangeal articu-
lation, 28 inches.
6. From olecranon to metacarpo-phalangeal articulation,
15 inches.
7. Weight of m. biceps humeri, 5.9 oz. av.
1 m. supraspinatus, 2.4 "
9. 1 m. rectus femoris, 9.2 ,
. 10.
brain, 53.3 oz. av.*
Combining together the results of Tables III. and IV., we
have :-
1. Average weight of supraspinatus, 1.67 oz. av.
2. Average weight of portion of del-
toid employed in holding the
arm in the horizontal position, 2.83 ,
Total Weight,
4.50 oz. av.
Hence, assuming the coefficient w, to be equal to unity,
we find, from Tables I. and II.:-
Work done by Shoulder Mus)
cles employed in holding L 3350.12 = 1.468 ft. lbs.
the unloaded Arms horizon-
tally, per ounce, per second, J
-
weighing the brain of a distinguished mathematician, and found it to be 47 oz.
I leave the deduction of suitable inferences to phrenologists.
ANIMAL NIECHANICS.
33
Work done by Shoulder Mus->
cles, holding the loaded
Arms horizontally, per í
ounce, per second.
2354.8
= 2.315 ft. Ibs.
226 x 4.5
t
Having thus found the work done by the muscles of the
shoulder, I proceeded to check the result by observations made
on the muscles of other parts of the body; and after trying
many experiments on other muscles, such as the pronators of
the forearm, and the openers of the jaw, &c., I finally se-
lected as the most reliable the following experiments made
on the flexors that hold the forearm in the horizontal position
If the forearm be held, with the palm of the hand up-
wards, in the horizontal position, it is allowed by all authors
on Animal Mechanics, that it is kept in that position by the ac-
tion of the biceps humeri, and of the brachicus or brachialis anti-
cus; to these muscles must be added the supinator radii longus,
which in the horizontal supine position of the forearm acts as
a flexor cubiti, and has a resultant that passes above the axis
of the elbow joint.
On holding a weight suspended from the wrist, with
the forearm in the position described, it is found that the
feeling of fatigue commences on the outer side of the lower
half of the arm, with the supinator, then extends to the biceps
humeri, and finally reaches the brachicus, when the experi-
menter is forced to yield to the insupportable feeling of fa-
tigue, and to confess that he can hold out no longer.
In the following experiments, I have made use of the same
persons who so kindly aided me in the experiments on the
muscles of the shoulder. *
The total work done is expressed, as before, by the
equation
Total work = w (w + a) xt,
()
S
34
ANIMAL MECHANICS.
where
w = coefficient of angular rotation.
w = weight.
a = weight of forearm.
x = distance of centre of gravity of loaded forearm from
centre of elbow joint.
t = time in seconds, until fatigue sets in.
No. 1.*— Myself.
x = 8.2 in.
w = 8 lbs.
a = 3.75 lbs.
t = 480 secs.
Total work in ft. lbs. = 3854 w.
No. 2.—Dr. Foot.
x = 8.2 in.
w = 8 lbs.
a = 3:5 lbs. . t = 360 secs.
Total work in ft. lbs. = 2829 w.
=
No. 3:- Captain Haughton.
x = 8.38 in.
w = 9 lbs.
a = 2.94 lbs.
t = 720 secs.
Total work in ft. lbs. = 6003.3 w.
No. 4.--Mr. M. H. Ormsby.
x = 8.27 in.
w = 9 lbs.
a = 3.25 lbs.
t = 555 secs.
Total work in ft. lbs. = 4685.4 w.
* In all these experiments, the weight was suspended at a distance of 9 inches
from the centre of the elbow joint.
ANIMAL MECHANICS.
No. 5.-Mr. Sumuel Haughton.
x = 8.2 in.
w = 9 lbs.
a = 4.25 lbs.
t = 780 secs.
Total work in ft. lbs. = 7062. I w.
Combining together the previous results into one Table,
· we obtain-
TABLE V.- Work done by the Biceps humeri, Brachiæus
. (Brachialis anticus), and Supinator radii longus, employed
in holding the loaded Forearm horizontal, with the Palm
upward :-

Time of Ex-
haustion.
Total work
done,
480 secs.
360
No. 1.
No. 2.
No. 3.
No. 4.
No.5.
720
19
3854.0 w.
282.90 ,
6003-3 ,
4685.41
7062.1 ,
Means,
579 secs.
4886.76 w.
D
These experiments differ so much from those made on
the shoulder, both as to rate of work and amount of muscles
employed, that they are not directly comparable with them;
still it may be useful to deduce from them a coefficient of
work per ounce per second.
This coefficient, as well as those already given for the
shoulder muscles, is not constant, but varies with the rate of
work.
The weight of the muscles employed in giving out this
quantity of work, previous to their exhaustion by fatigue, is
estimated, as before, by observations on the dead subject, and
D 2
36
ANIMAL MECHANICS.
in making this estimate, we have no difficulty, as before,
in finding the portions of muscles employed, for they are
perfectly well defined; at the same time, it may be considered
that the muscles now under investigation are not in them-
selves so well adapted as the muscles of the shoulder, to give
an average result, as they differ more widely in one man as
compared with another, than the shoulder muscles are found
to do. This difference is caused by the more varied uses of
the forearm, and of the muscles actuating it, than of the
muscles of the shoulder that hold the entire arm horizontal.
The results obtained by me from well-selected subjects are
contained in the following Table :-
TABLE VI.- Weights of Muscles employed in holding the Fore
arm horizontal, with the Palm upwards :-

Biceps.
Brachiæus.
Supinator radii
longus.
Female,
Male,
2.74 02. av.
4.01 )
4.30
2.30 oz. av.
3.9
3.2
4.0
1.9
5.90
0.75 oz. av.
1.40
2.30 1
C. Male,
D. Male,
E.
Male,
P. K. Male,
1.66
»
Means,
3.54% oz. av.
3.17 02. av.
1.48} oz. av.
(D
Hence we find that the average total weight of the muscles
employed in holding the forearm horizontal is 8.2075 oz. av.
From the preceding result as to the average weight of the
muscles acting on the forearm, combined with Table V., and
assuming w to be equal to unity, we find
Work done by muscles
holding the loaded Fore 4886.76
- 1.028 ft. lbs.
arm horizontally, per 579 x 8.2075
ounce, per second, j
=
ANIMAL MECHANICS.
37
My first attempt to calculate a formula to represent the
Law of Fatigue, in the case of the arms extended horizontally,
was based upon the periodicity of muscular action, as proved
by the muscular susurrus, explained in the preceding section,
and may be thus stated :-
Each portion of the muscle contracts, relaxes, and goes
through all its changes thirty-two times in each second, and
does so in succession, so as to keep the ends of the fingers
steady ; but the amount of work given out by the whole
muscle is evidently the same as if all the particles went through
their changes simultaneously, and not in succession.
If the whole muscle acted simultaneously, the arm would
fall like a compound pendulum during the end part of a
second, and then be raised to its original level by the renewed
contraction of the muscle. It may be shown, as follows, that
in this case the arc through which the arm fälls is that which
would correspond to the centre of oscillation falling freely
for zind of a second.
For, if W denote the weight of the arm, x the distance of
its centre of gravity from the centre of the glenoid cavity, w
the angular velocity acquired after falling for any time, and I
the moment of inertia of the whole arm; we have, by D'Alem-
bert's principle, the following equation :-
Idw = Wxdt.
(10)
.
If 0 denote the angle made with the horizontal line at any
moment by the arm, and k denote its radius of gyration, and
l the length of the equivalent simple pendulum ; we have at
once from equation (10)
(11)
multiplying both sides by 2, and integrating, we obtain
: 38
ANIMAL MECHANICS.
( * - - 200 + Const. i
and as w and 0 vanish together, we have
(wul)2 = 29 (07),
v2 = 295,
(12)
(
where v denotes the velocity acquired by the centre of oscil-
lation in traversing a small arc, such as would be described
in zand part of a second, and s denotes the space passed
through by the same point.
The equation (12) is the same as that of falling bodies,
and therefore for small arcs and times, the centre of oscilla-
tion of the arm falls through the same space as it would if
the arm were released from its socket, and fell freely through
space.
Let us now use the following notation :
· 1 = radius of oscillation;
& = distance of centre of gravity from centre of glenoid
cavity ;
as = small space through which the centre of oscillation
falls in shnd part of one second ;
n = number of such falls during
t = whole time required to fatigue the muscles ;
a = weight of arm.
The total work done by the muscles, on the foregoing
supposition, in the time t, is evidently
(w + a)žnos;
ANIMAL WA
MECHANICS.
but nos varies as t; therefore
C
Total Work varies as (w + a)
and the rate of work varies as
(w + a)
Now, by the Law of Fatigue, the total work varies in-
versely as the rate of work, and hence we have
(v + a)
t = Constant ;
.
but, supposing the arm to be a simple uniform cylinder, we
have
X
i
3 (2W + a)2
4 (W + a) (3w.+ a)
Squaring, and substituting in the preceding equation, we
obtain
(2W + a)" t
(3w + aja = A,
or,
,
t,
A (3W + a)?
(2W + a)
(120)
In order to compare this formula with observation, I
made use of Mr. Stanley Jevons' experiments published in
Nature (30th June, 1870), using the following constants,
which I found to answer best :-
A = 22050,
a = 7.4 lbs.
• The comparison of observation and theory is shown in the
following Table :--
40
ANIMAL MECHANICS.
Mr. Jevons' experiments (holding weights with arm
horizontal).

t (observed).
t (calculated).
Difference.
1
34. 2 secs.
5*.7
+
5.0
19
32.5 secs.
60.3
87.4 12
147.91
218.9
321.2
84.5
,
147.
6 11
234 4 1
305.5 »
+ 2.6
+ 0.3
- 15.5
+15.7
»
1
The agreement here shown between observation and
theory is quite satisfactory, and I remained quite contented
with it, until I came to plot the curve (12a), and compare it
with the observations, as shown in the accompanying dia-
gram (page 41).
The relation between t and w represents a curve of the
fifth degree, one branch of which is shown in the figure ; the
other branch being hyperbolic in form, corresponding to
positive values of t, and negative values of w greater than
Mr. Jevons observations are shown by the small circles cor-
responding to the weights used by him, and their agreement
with the curve is evident.
Differentiating equation (120) we find
(6W + a) (3w + a)
1
(2W + a)"
(126)
The preceding equations prove that
1°. The values of t are positive, whether a weight be
held in the hand, or whether the hand be supported by a
force acting upwards.
2°. The axis of w is asymptotic to the curve at positive
and negative infinity.
GW-40
.
''
----|2wta=
:
-----3w+0=0
.
.
T
-500
-
400.
.!
!
.
-
Į
--
..
.
DIAGRAM
TUTA

100%
1
.
.
"
.
i
0
1
2
4
5
::.?...:::
10
. 14:
42
ANIMAL MECHANICS. ·
3º. The line 2w + a = 0, is also an asymptote to the
curve.
4º. The time t has a maximum corresponding to the value
of w given by the equation 6w + a = 0.
5º. The value of t becomes equal to zero4 and the curve
is a tangent to the axis of w, when 3w + a = 0.
The first three of these conditions are fulflled by observa-
tion, for-
1°-2°. The time of holding up the arm must be always
positive, and must become zero, when the weight is infinite.
3°. When 2w + a = 0, the arm is supported by a force
equal to half its weight applied at the hand, and being there.
fore in a condition of statical equilibrium, can be held up for
an infinite time. .
t
The last two conditions, however, are inconsistent with
observation, for
4º. The value of t, for all positive values of w, must
increase continuously as w diminishes to zero; and for negative
values of w can have no maximum but infinity, corresponding
to the condition for statical equilibrium, 2u + a = 0.
5°. The value of t can never be zero, except when w is
infinitely great.
From the preceding investigation, I became satisfied that
my first idea, viz., that the arm dropped and was raised again
during repeated short intervals, was erroneous, and that a
simpler form of curve would represent better the relation
between t and w. This curve l'afterwards found to be a
cubical hyperbola, which represents the observations equally
well, and is free from all the preceding objections.
So far as the observations, themselves are concerned, they
can be represented by any curve of an hyperbolic form, and
ANIMAL MECHANICS.
our difficulty is to find what kind of curve is best. Having
found that a curve of the fifth degree will not answer, I shall
now show that the relation between w and t may be repre-
sented by a cubical hyperbola.
By equation (7),
Total work = wa (w to me;
wa
and
.
Rate of work = wa w +
2/
But, assuming the truth of the Law of Fatigue, the product of
these two expressions will be constant. Hence
21 wf-
t - cons
or,
w*a! (w + q* t = constant,
(0-4):- 4.
w
+
(120)
This equation represents a cubical hyperbola, whose
asyınptotes are
t = 0,
20 + - = o.
31
This curve is free from the objections made to the curve
of the fifth degree, for t becomes zero when w is infinite, and
increases continuously, as w diminishes, and becomes infinite
when w+ * = 0, which corresponds to the case of statical
equilibrium ; and t has no intermediate maximum between
zero and positive infinity. In fact, if we differentiate the
equation (120), we find
dt
dw
4A
(2W + a)3
12d)
ANIMAL MECHANICS.
This equation shows that the tangent to the curve does
not become horizontal for any value of w, except infinity.
An interesting consequence may be drawn from equation
(120) by seeking the relation between the useful work done
and the weight w. The useful work inay be represented by the
product of w and t; hence if u denote the useful work, we find
U = wt,
bat
** Twith you
- 2)
Hence
IL
-
(120)
W
+
+
From this equation, which denotes a cuspidal cubic, I shall
hereafter show that the useful work is a maximum when the
weight held in the hand is half the weight of the arm.
4. On the Dynamical Work done by Muscles alternately
contracted and relaxed. In the preceding section, I have dis-
cussed the work done by muscles kept in a state of continued
contraction until tired out; and I have called this work
“Statical,” because the muscles appear to be at rest, and
really do undergo no movement of translation in space, al-
though a very rapid and active molecular movement takes
place within their substance, to which the work done is al-
together due.
In the present section, I shall consider the more usual
kind of work done by muscles, to which the name “Dyna-
mical” is given, the amount of which is measured by external
movements of various parts of the body, and by the removal of
external resistances through spaces easily measured.
In most kinds of labour which are recognised as effective,
and which are usually employed, almost all the muscles of the
upper and lower extremity are employed, together with the
ANIMAL MECHANICS.
45
muscles of the lumbar and dorsal vertebræ; and previous to
instituting a comparison of the work done in this kind of
labour with that done by muscles in continued contraction
(which has been already discussed), it is necessary to have an
approximate estimate of the total weight of all the muscles
of the upper and lower extremity in a healthy man of ave-
rage weight.
In order to obtain this estimate, I made a series of obser-
vations on the weights of the muscles of two well-developed
male subjects, of 71 inches and 67 inches in height, re-
spectively, and who had died of acute diseases (nephritis and
pneumonia), that terminated life without much loss of sub-
stance by wasting.
The results thus obtained were the following:
Muscles of Lower Limb.
1. Hip Joint, . . 77.33 oz.
2. Knee Joint,. • 55.28 ,
3. Ankle and Foot, 32.91 ,
4. Lumbar, .. 14.81 ,
Muscles of Upper Limb.
1. Shoulder Joint, . 47.30 oz.
2. Elbow Joint, .22.94 ,
3. Wrist and Hand, 17.33 ,
4. Trunco-scapular, 19.41 ,
180.33 oz. av. |
106.98 oz. av.
Doubling these results we find
C
Muscles of Lower Limbs = 360.66 oz.
Muscles of Upper Limbs = 213.96 ,,
Total .... 574.62 oz.
If we compate the muscles of these two subjects with
those of P. K. (p. 32), dissected in the prime of life, we find
reason to believe that 50 per cent. should be added to the
muscles of even the best selected subjects, in order to bring
them up to the standard of life.
Thus,
46
ANIMAL MECHANICS.
m. rectus femoris (selected subject), · · · 6.1 oz.
m. rectus femoris (P. K.), ...... 9.2 , .
Percentage to be added,. · 50.8
2. m. biceps humeri (selected subject), .. 3.24 oz.
m. supraspinatus
. . . 2.18 ,,
5.42 02.
..... 5.9 02.
m. biceps humeri (P. K.),
m. supraspinatus
T16. suprusperccUS
"
.
.
.
.
2.4
%
8 3 02.
Percentage to be added, · 53.1
We may therefore safely assume 50 per cent. to be added
to the muscles of a carefully selected subject in order to bring
them to the standard of life.
Hence,
Total muscles of upper and lower limbs, . 574.62 oz.
Add 50 per cent. on ditto, ... 287.31
Total, ....861.93 oz. av.
The work done by the muscles of the human body, in ex-
ertion that calls into play nearly all the muscles of the body,
may be divided into excessive work and ordinary work. The
first kind, or excessive work, produces an injurious effect upon
the respiratory and circulatory systems, rather than upon the
muscles themselves; so much so, indeed, that death (which
sometimes is the consequence of excessive muscular work) is
found to be caused, in this kind of over-exertion, by pulmonary
apoplexy, or by rupture of an artery, produced by the effort
made by the lungs and heart to supply fresh blood to the over-
worked muscles.
Example No. 1.---As a first example of excessive muscular
work, I shall take the case of an eight-oared outrigger pro-
pelled at the rate of one knot in seven minutes. The mea-
ANIMAL MECHANICS.
47
surements by means of which the following calculation is
made, were kindly furnished to me by Dr. E. A. Parkes, and
by Mr. Maclaren, of Oxford.
The Resistance offered by the water to the motion of the
boat is divisible into the following parts, due respectively to,
1. The Distortion of the particles of water;
2. The production of Currents ;
3. The production of Waves ;
4. The production of Frictional Eddies.
In the case under consideration, the first three causes of
Resistance may be neglected, in consequence of the “fair”
form of the boat, and of the limited speed at which she is
driven; and the whole resistance may be regarded as due to
the production of Frictional Eddies.
In discussing the amount of Resistance due to this cause,
I shall adopt the principles laid down by Professor Rankine, *
from which it appears that
The Eddy Resistance = f w cie | gºds ;
се
where ds denotes the element of the boat's skin ;
9, the ratio which the velocity of gliding of the water
over that element bears to the speed of the boat;
C, the speed of the boat ;
g, gravity;
w, the specific gravity of the water, or weight of one
cubic foot;
f, the coefficient of friction (-0.0036).
In this equation, Sqºds is the augmented surface of the boat's
skin, and is supposed to sum up together the skin resistance
and that due to the excess of water in front, and to the defi-
ciency of water behind.
** Treatise on Shipbuilding,” Fol., Mackenzie, London, 1866, pp. 78 et seq.
48
.
ANIMAL MECHANICSY
From the value of the coefficient of friction employed
(which is deduced from Professor Weisbach's experiments on
the flow of water in iron pipes), it follows from the preceding
equation that at ten knots the Eddy Resistance is nearly equal
to one pound avoirdupois per square foot of Augmented Surface ;*
and varies, for other speeds, as the square of the speed.
: The whole difficulty of the calculation of the eddy re
sistance turns upon the calculation of the augmented surface,
which is effected by Proſessor Rankine, on the assumption
(conformable to repeated experiments), that the augmented
surface and its resistance are the same as those of a Trochoidal
ribbon, whose length is the length of the boat on the plane of
floatation, whose breadth is the mean immersed girth of the
boat, and whose coefficient of augmentation is
1+ 4 sin ?w + sin *w,
where w is the angle of the greatest obliquity to the horizon
formed by a tangent to the Trochoid.
Applying the foregoing principles to the sections of the
Oxford eight-oar when loaded with its crew, I have found the
following results :-
1. Length of plane of floatation = 52 ft.
2. Girth of central immersed section = 31.5 in.
(Mean immersed girth) = frds = 21 in. = 1.75 ft.
3. Sine of obliquity = 1.
Hence the coefficient of augmentation is
[ +4 ()+ ()* = 1.254;
and the
Augmented surface = 52 * 1.75 * 1.254 = 114.11 sq. ft.
The speed of the boat is assumed to be one knot in seven
* It is exactly maths of a pound.
+ The weight of the boat is 350 lbs., and the average weight of the crew is
11 st. 4 lbs. oach, and the coxwain is 8 stone.
ANIMAL MECHANICS,
49
minutes, or ºm knots per hour; hence, by the rule already laid
down, the resistance per square foot of augmented surface is
- 30 x
6
1
100
625 lb. av.
625 - 36 x
644 49
644
and finally, the total resistance is
36 625
114.11 X2 X =
2 = 81.36 lbs.
49 644
This resistance is overcome through the space of one knot
(1000 fathoms) in seven minutes ; and therefore
The Total Work done -81.36 x 6000 -
U = 218 ft. tons.
2240
The work done per man is 27.25 ft. tons in seven minutes,
or nearly 4 ft. tons each minute.
A good idea may be formed of the rate in which the mus-
cles give out work in a boat-race, from comparing this work
with the average daily work of a labourer. At many kinds
of labour there are 400 ft. tons of work accomplished in ten
hours. The oarsman therefore performs, in one minute, the
hundredth part of his day's labour; and, if he could continue to
work at the same rate, he would finish his day's task in 1h. 40m.,
instead of the customary 10 hours.
The work done, therefore, in rowing one knot in seven
minutes, is, while it lasts, performed at a rate equal to six
times that of a hard-worked labourer.
The rate of work in this case is four tons per minute ; and
if we assume (p. 45) 575 oz. av. as the weight of muscles em-
ployed, we find the rate per ounce per minute to be
81.36 x 6000
8 x 7 x 575
= 15.169 ft. lbs. per oz. per minute.
50
lill
ANIMAL MECHANICS.
The experiment detailed in the note, * made under the
direction of Mr. Maclaren, was communicated to me after I
had made the preceding calculation, and seems to confirm the
result arrived at as to the resistance overcome in this kind of
labour.
* Mr. Maclaren's Experiment." An eight-oared racing boat, weighted with
sandbags to represent an 11 st. 3 lb. crew (the weight for which she was built),
and steered by an 8 st. coxwain, was towed over part of the Oxford course, where
the water is straight, broad, and deep.
“The four-oared boat by which she was towed was itself towed by men on
the bank, and kept in a straight course by a coxwain. The eight-oar was kept
as nearly as possible in a line with the four-oar by the coxwain placed on board
for that purpose.
"The towline from the four to the eight was fastened to the bow-oar's thwart
in the eight, exactly on a line with the keel, and the strain measured by a dy-
namometer (a Salter's spring-balance), interposed between the end of the tow line
and the four-oar.
The distance travelled, ........ 560 yards.
The time occupied, ......... 6m. 20%.
The average strain on the dynamometer, · · 7 lbs.
“There was a light side wind, but not enough to ruffle the water, or seriously
interfere with the experiment. The strain was measured when the keels of the
two boats were as nearly as possible in the same straight line. The course was
against the stream, which is very slight.”
In this case, the velocity expressed in knots per hour is
560 x 60 x 3 = 2.65 knots per hour.
19 X 2000
In the case already considered, the velocity was assumed to be one knot in
seven minutes, or
= 8.57 knots per hour.
60
.
Hence it follows (assuming the resistance to increase as the square of the ye.
locity) that the resistance at one knot per seven minutes would be
1857 13
| 265 /^!
6 x 7 lbs. = 73.21 lbs.
Professor Rankine, in a letter addressed to me, 12th September, 1866, sug-
gests that I ought to have reduced Weisbach's coefficient from 0.0036 to 0.0030,
ANIMAL MECHANICS.
51
Example, No. 2.-The most effective mode in which the
labour of man can be employed is in lifting his own weight
through a height for a number of consecutive hours. Captain
Coignet, a French officer of engineers, invented (1835) a
hoist or box-lift, applied to the lifting of barrows of earth
from an excavation of about forty feet deep, which was in-
tended to be used in the formation of earthworks for fortifica-
tions. This box-lift consists of a strong rope passing over a
large pulley, and having suspended at each end a cage, or en-
closed platform. The barrow load of earth is brought by an
assistant to the foot of the hoist, and placed in the cage, which
is supposed to have just descended to the lower level. At
the same moment, the labourer, with an empty barrow, steps
because the skin of the boat is smoother than the skin of iron steam-boats, and he
has found that the friction in iron pipes is diminished to about five-sixths, by
lining them with smooth pitch. This would reduce the result of my calculation
from 81.36 to 67.80 lbs.
In consequence of my publication of the foregoing results, Mr. J. Scott Russell
· published in “The Practical Mechanic's Journal,” November, 1866, an addi-
tional calculation of the resistance of the Oxford eight-oar, the result of which
is 74.23 lbs.
The preceding calculations and observations give-
Haughton, ..... 81.36 lbs.
Rankine, ...... 67.80 ,
Scott Russell .... 74.23 ,
Maclaren, · · · · 73.21 ,
Mean, .i. 74.15 lbs.
From the preceding results, we may obtain the following rates per ounce
per minute of the muscles engaged :-
Rate per ounce per Minute of Muscle employed in rowing a Boat-race.
Foot pounds per ounce per minute.
15.169 ..::· Haughton.
12.640. . . . . Rankine.
13.840 ..... Scott Russell.
13.650 . . . . . Maclaren.
Mean, . . 13.825
E 2
52
ANIMAL MECHANICS,
i
into the upper cage, and descending by his weight, lifts the
full barrow through forty feet, to the higher stage, where it is
at once removed by a second assistant. The labourer then
leaves the cage in which he has descended, and returns to the
higher stage, by mounting a ladder; and again steps into the
upper cage, which is now ready to commence the descent
afresh. It has been found, by repeated trials, that a labourer,
aided by two assistants, to bring and remove the loaded and
empty barrows, is able to work for eight hours a day, during
which he lifts his own weight (70 kilos.) through a height of
13 metres, 310 times.
From these data we find
70 x 13 x 310
Work done in one day - 4
= 910 ft. tons.
311
Converting this result, as before, into work done, expressed
in ft. lbs. per ounce of muscle, per minute, we obtain
Rate of work, per ounce, per
1 910 X 2240
minute, of muscles em- y = 60 v
ployed in climbing, ...)
=7.389 it. lbs.
The example just given may be confirmed by the follow-
ing:-A friend has informed me that he performed the ascent
of Monte Rosa, from Zermatt, 9000ft., in nine hours, his weight
being 14 stone. Here we have
Rate of work, per ounce, per
1_14 14 x 90006894
minute, of muscles em- } =-
ployed in climbing, ...)
)
9 60 x 676=5.683ft.Ibs.
Example, No. 3:—It is recorded in the Memoirs of the in-
comparable Martinus Scriblerus, how, in his eager pursuit
of knowledge, he met with an extraordinary misadventure,
through the ignorance of his assistant, Crambe. Having se-
cured the body of a malefactor, he hired a room for its dissec-
S
ANIMAL MECHANICS:
53
tion near the Pest Fields of St. Giles, at a little distance from
Tyburn Road. The body was carried by night, without much
difficulty, on Crambe's back, who found it easy (being both
young and lusty) to travel along the level road, carrying on
his legs the double weight, of the malefactor, and of his proper
self. As soon, however, as he commenced to ascend the
staircase, the wonderfully increased exertion (20 to 25 times
as great as before) began to tell upon Crambe, and, as the
accurate narrator records, upon the corpse also; in conse-
quence of which Crambe dropped his burden in disgust and
fright, and allowed it to roll down the staircase, while he him-
self ascended breathless into the upper room, where Martin,
scalpel in hand, eagerly awaited the arrival of his expected
subject.*
Next to the exertion of lifting his own weight, the sim-
plest mode in which the force of man can be employed is in
transporting the weight of his body, loaded or unloaded, along
a horizontal road; and the work done in horizontal transport
of loads may be readily shown to be about a}th part of the
work done in lifting the same loads vertically through heights
equal to the horizontal distances.
In order to obtain the proportion between the work done
in the horizontal and in the vertical transport of weights, we
must examine the mechanism of simple walking.
In the act of walking, it is found that the leg which rests
upon the ground supports the centre of gravity, which is first
lifted by the thrust of the other leg against the ground, and
afterwards falls forward, while the suspended leg swings as a
pendulum, and becomes in its turn the support of the centre
* It is true that Mr. Pope gives a different account of the cause of this cele-
brated disaster, and one which acquits Crambe of the charge of having proved
unable to carry the malefactor up a flight of stairs; but I feel no hesitation in
asserting that my own explanation of the misadventure is preferable.
ANIMAL MECHANICS.
of gravity, which is made to ascend and descend, as before,
like an inverted pendulum, by the thrust and subsequent fall
of the other leg. The forward movement of the body, or its
horizontal transport, is effected by the pendulous movement
of the legs, caused by the successive raising of the centre of
gravity; and, consequently, the work done in walking one
step, is simply the work of lifting the man's weight through
the height, to which his body is lifted at the commencement
of each step.
It has been ascertained, by means of an apparatus in-
vented by Messrs. William and Edward Weber,* that in ordi-
nary walking, when the length of the step is 28.740 inches,
the mean elevation and depression of the centre of gravity of
the whole body is 1.248 inch; and it was also found that
this proportion remained the same for different lengths of
steps.
From this observation we find,
Vertical elevation 1248
Horizontal transport - 28740
I
23.03
Hence it follows that a man walking 23.03 miles along a
horizontal road has done as much work as if he had lifted his
body up a vertical ladder through a height of one mile.
Under such unfavourable circumstances, it is not to be
wondered at, that poor Crambe's courage failed him, when he
commenced the ascent of the staircase, with the corpse upon
his back; and that, finding himself unable to ascend with the
alacrity required by the occasion, he quietly let go his hold,
and rushed breathless into the presence of his master, being
completely overcome by his unwonted exertions.
The work done in the horizontal transport of burdens may
* Mechanik der menschlichen Gewerkzeuge, p. 238. Göttingen, 1836.
ANIMAL MECHANICS.
55
be calculated also from a totally different set of observations
made by the Messrs. Weber. *
When the body is at rest, in the standing posture, the
line drawn from the centre of gravity of the trunk to the
centre of the horizontal line joining the centres of the acetabula
of the pelvis must be vertical; and when, in walking or run-
ning, the body is carried horizontally with a velocity v, this
line must incline forwards, with an inclination such that it shall
be in the direction of the resultant of the momentum of the
horizontal velocity and the weight of the body.
Thus, if O B be
agiven time, and 0 A
the horizontal space
bethe vertical space,
through which the
which on the same
trunk (balanced, like
scale represents the
a jugglers'pole, on the
weight of the body;
pelvis) is carried in
if there were no re-

no
mg
sistance caused by
setting the forward
foot upon the ground,
or by the air; the
body would be
obliged to lean for-
ward through an an-
gle 0 A B, such that
mo
(13)
tan i = tan OA B= =
mg 32.2
* Mechanik der menschlichen Gewerkzeuge, pp. 233, 4.
56
ANIMAL MECHANICS.
The angle i has been carefully measured for three diffe-
rent velocities by Messrs. Weber, with the following results :-
Velocity.
Inclination of Trunk.
Feet per Second. Miles per Hour.
1. 2.667 ... 1.818 ... 2° 43'
II. 6.385 ... 4.353 ... 7 40
III. 15.513 .. . 10.577 . .. 19 13
1
If there were no resistance to horizontal motion, the slope
of the body would be found by calculation from equation (13);
or, assuming the slope to be given, as measured, we find the
velocity from the equation
v = 32.2 x tan i
(14)
Solving this equation for the three inclinations of the
trunk measured by the Messrs. Weber, we find-
Velocity of Feet per Second. Inclination of Trunk.
I. 1.528 ..... 2° 43'
II. 4.334 . . . . . 7 40
III. 11.224 . . . . . 19 13
11
From the preceding results, it follows that the observed
velocity is in excess of the velocity calculated on the hypo-
thesis of no resistance for fixed inclinations of the trunk, by
the following quantities :
I. (2.667 – 1.528) = 1.139 ft. per second.
II. (6.385 - 4.334) = 2.051
III. (15.513 - 11.224) = 4.289
The proportion of resistance overcome to weight lifted, in
the three cases, is therefore,
ANIMAL MECHANICS.
57
Velocity. Proportion of Resistance to Weight.
Miles per Hour.
1.139
I. 1.818 ....
32.2 28.27
2.051
II. 4.353 . . . . .
32.2 15.70
4.289 1
III. 10.577 · · · ·
32.2 7.51
From these results it follows that a man walking at the
rate of 1.818 miles per hour should travel 28.27 miles before
he has done the work represented by lifting his body through
one mile of vertical height.
If we interpolate between the results I. and II., to find the
resistance at the pace of 3 miles per hour, we find it to be -
20.65,
a result that agrees well with that already found from direct
ineasurement of the vertical elevation of the centre of gravity,
at each step in ordinary walking.
As an example of the rate of work that may be attained
in carrying loads horizontally, I shall give the following, to
which my attention was called by Professor Rankine, and
which is founded on data obtained by his father, Lieutenant
David Rankine :-
A labourer, supposed to weigh 150 lbs., carrying a load of
126 lbs. for 30 seconds, can travel at the rate of 11.7 ft. per
second (8 miles an hour) ; what is the rate of work done, per
ounce of muscle, per minute?
At 6.385 ft. per second, the resistance is measured by a
loss of velocity of 2.051 ft. per second.
At 15-513 ft. per second, the resistance is measured by a
loss of velocity of 4.289 ft. per second.
Therefore,
15.513 - 6.385: 11.700 – 6.385 :: 4.289 – 2.051: X
9.128 : 5.315 : : 2.238 : X = 1.303
or,
ANIMAL MECHANICS.
And, finally,
1.303 + 2.051 1
1. 32.2 9.60
In other words, at the pace of 11.7 ft. per second, the re-
sistance to horizontal motion is g.both of the weight, accord-
ing to the results of the experiments of Messrs. Weber. Hence
the work done per ounce, per minute, in Lieutenant Rankine's
experiment, is-
Work – 276 x 11.7 * 60
9.6 x 575 =35.119 ft. lbs.
From the foregoing example, it appears that the rate of
work done in carrying half one's weight at 8 miles an hour is
more than double the work done in a boat race, although it
is considerably less than the work done in statical work, by
muscles sustaining weights, as already described.
The following examples, among many others, may be
given of the daily amount of work given out, in ordinary la-
bour, by the muscles of man :-
Example, No. 4.-The useful work, measured by the bricks
lifted per day, done by a Dublin hodman, amounts to 181 ft.
tons; to which must be added the fatigue work done in lifting
his own weight and that of his hod up and down the ladders.
I have observed the hodmen in Dublin to make 40 ascents per
day of ladders 40 ft. high ; if, therefore, we assume the weight
of the labourer and his hod to be 170 lbs., we have,
Work done in mounting = 170 x 40 x 40 = 272,000 ft. lbs.
; descending = one-half, . . 136,000 ,
408,000
1
Hence,
1 408000
Fatigue work = ? = 182 ft. tons.
2240
Useful work, . . 1819
Total, ... 363 ft tons.
ANIMAL MECHANICS.
Example, No.5.--The paviours of Dublin make 78 blows of
the rammer in 2m. 458., and then rest for 3m. 30$; their ram-.
mer weighs 5 st. 9 lbs., and is lifted through 16 inches each
stroke; and their labour is continued each day for 10 hours.
Let us calculate from these data the work done per day.
The time of labour is to the time of rest as 165 to 210;
hence, the time of labour is to the whole time as 165 to 375;
therefore, in the day of 10 hours the number of minutes of
labour is
10 x 60 x 165-264 minutes ;
375
but in 2 minutes 79 lbs. are lifted 78 times through 16 inches.
Hence the work done in 2 minutes is
79 x 78 x ] = 8216 lbs.;
and the work done in a whole day of 10 hours, is
8216 x 204 = 788736 ft. Ibs.;
23
which, divided by 2240, gives us
Daily work of paviour, ...
352 ft. tons.
Example, No. 6.-In the construction of the Oxford and
Birmingham railway, a “box-lift” was employed in the cut-
tings, identical in principle with that used by Captain Coignet
(Example, No. 2), and the average result of more than a year's
observation on average workmen, in all weathers, was taken,
with the following results :-
Work done, ..... 1510 ft. lbs. per minute.
Daily work (10 hours), · 404 ft. tons.
On the same railway a "swing-lift” was used, in which
part of the work of the labourers was employed, not in rais-
ing materials, but in shifting the position of the “lift.” Ob-
60
ANIMAL MECHANICS.
servations made for more than a year, in all weathers, on the
daily labour of average workmen, gave the following results :-
Work done, ... 1046 ft. lbs. per minute.
Work, ... 280 ft. tons per day (10 hours).
The mean result of an ordinary day labourer, employed
with the box or swing-lift, is, therefore,
"Box or swing-lift," .... 342 ft. tons.
Example, No.7.—A very accurate estimate may be formed
of the average day's labour, from observations formed on the
horizontal transport of weights, interpreted by the rules al-
ready laid down for the conversion of horizontal into vertical
transport.
(a.) If a pedestrian be employed in walking only, without
load, it is found that 20 miles a day, with rest on Sundays,
give full employment to his strength, and that this amount
cannot be exceeded with prudence. From this assumption,
if we suppose the man's weight, including clothes, to be
160 lbs., we find
a. Horizontal transport of man ) 160 x 20 x 5280
walking without load, D 2240
= 7543 ft. tons.
(6.) Coulomb observed the work done by porters employed
to carry goods 2000 metres, returning unloaded, and found
that each carried, in the course of the day, 348 kilos. in six
journeys, loaded with 58 kilos. at a time. If we suppose
each porter to weigh 70 kilos., we find the following result
for the horizontal transport :-
Useful work, 2000 x 348 = 696000 kil. met.
Fatigue work, 2000 x 2 x 70 x 6 = 1,680000 se
Total, . . . . . . . 2,376000 kil. met.
ANIMAL MECHANICS.
61
.
Dividing this result by 311, in order to reduce it to foot-
tons, we find
b. Work done by porters, alternately loaded
and unloaded, . . . . . . . . 7640 ft. tons.
(c.) On questioning a number of pedlars who travelled,
always loaded with their packs, Coulomb found that they were
able, with a load of 44 kilos., to travel 19000 metres per day.
Assuming their weight at 70 kilos., as before, we find-
Useful work = 19000 x 44 = 836000 kil. met.
Fatigue work = 19000 x 70 = 1330000 ,
Total, ... 2,166000 kil. met.
Converting this, as before, into foot tons, we obtain-
C. Work done by loaded pedlars = 6964 ft. tons.
Collecting together the preceding estimates, we find
a. Man walking without load, ...... 7543 ft. tons.
b. Man alternately loaded and without load, 7640 ,
c. Man loaded, . . . . . . . . . . . 6964 »
Mean, ... . 7382 ft. tons.
If the horizontal transport be effected at the usual rate of
3 miles an hour, we must divide the preceding result by 20.65
(as already explained), in order to convert it into weight lifted
through height. Hence we obtain, as the mean of the work
done in the horizontal transport of weights,
Work done in horizontal transport | 7382 -
of loads, . . . . . . . :) 20.65
8 ft tona
From Examples 4, 5, 6, 7, a very close approximation may
be made to the daily work of a good labourer.
ANIMAL MECHANICS.
Day Labourer.
Foot tons.
Kilogrammeters.
1. Hodman, · · · 363 · · · 112413
2. Paviour, · .. 352 ... 109008
3. Navvy, ... 342 ... 105910
4. Pedlar, ... 358 ... 110866
Mean, .. 353.75
109549
If the day's labour be estimated at 10 hours, and converted,
as before, into work done in ft. lbs. per ounce, per minute, we
find
= 2.297 ft. lbs. per oz. per min.
575.X 600
Collecting together into one Table the rates of Statical and
Dynamical work of muscles already described, we obtain-
Day Labourer - 353.75 X 2240
TABLE VII.—Statical Work done by Muscles.

Muscles Used.
Rate of
Quantity of Duration of Weight of Rate of Work in
Work in foot Labour in Muscles in Work in foot Kilogram-
pounds. 1 minutes. | oz. av. 1 pounds per meters por
oz. per min, Hectueram
per minute.
1206
32.3
1. Unloaded muscles ,
of shoulder, .)
2. Loaded muscles of
shoulder, . is
3. Loaded. muscles of
forearm,. . .1
8.29
4.05
15.7
22.5
842
46.2
41.5
3170
9.29
20,2
TABLE VIII.--Dynamical Work done by Muscles of the whole
Body.

Kind of Work,
Rate of
Quantity of | Duration of Weight of work in footl Kilogram-
Rate of Work in
Work in foot Labour in | Muscles in
pounds. minutes.
| pounds per meters per
oz. av.
oz. per min. Hectogram
i per minute.
7.4
15.17
7.39
480
575
1. Boat race, ....1 61,040
2. Coignet’s lift, . . . 2238,470
3. Running with load, 10,091
4. Day labour, ... 792,400
575
35.11
17.I
600
575
2.30
I. I
ANIMAL MECHANICS.
63
The last two columns of Table VIII. measure the respi-
ratory distress produced by different kinds of labour; and
show that Coignet's lift is more than thrice as severe as
ordinary labour; and that running at 8 miles an hour with a
load equal to half the weight of the body is more than twice
as severe as pulling a boat race.
5. On the Absolute Force of Muscles. We have already
considered the amount of work given out by muscles, when
employed either statically or dynamically; and it now re-
mains to investigate their absolute force when employed for
an instant, or very short space of time. ..
Example, No. 1.-Flexors of the Forearm. In order to de-
termine the force of the flexors of the forearm, I held my
arm vertically, and the forearm horizontally, with the fist shut
and in supination, and found that 39 lbs. was the limit of the
weight I could just raise, when suspended at 12 inches from
the axis of the elbow joint; this gives
The moment of Force applied = 41 12= 502);
estimating the forearm itself as equivalent to 2 lbs. suspended
at 121 inches from the joint.
I endeavoured to ascertain the moment of the muscles by
dissecting a very finely-developed subject,* who had been a
blacksmith by trade, with the following results :-
Cross Sections of Muscles.
1. Biceps humeri, .. 1.914 sq. in. = 12.346 sq. cent.
2. Brachiæus, ... 1.276 , = 8.230 »
In order to find the perpendiculars let fall upon the direc-
tions of the muscles in my own arm, I selected another sub-
* This was the same subject on which I made the measurements of the leg
muscles described subsequently.
64
ANIMAL MECHANIC
ject, in which the distance from the axis of the elbow joint to
the ungual articulation of the middle finger was the same as
in my own arm, viz. 16.55 in., and found as follows:-
1. Perpendicular on Biceps humeri, 2.06 in.
2. Perpendicular on Brachiæus, . 1.07 »
Hence, if K denote the coefficient of muscular force, per
square inch of cross section, we have
The moment of the muscles = K (1.91 x 2.06 +1.28 1.07)
= 5.304 K;
and, equating this to the moment of force applied, we obtain
502) = 5.304 x K,
and,
K = 94.7 lbs. per sq. in. = 6.658 Kilos. per sq. cent.
Considering that the arm examined after death was that
of a blacksmith, it is possible that I have somewhat over-
estimated the cross section of the muscles, as compared with
my own; but, on the other hand, it must be remembered, as
I have shown in pp. 45, 6, that a large percentage should be
added to dead muscles, in order to bring them up to the
standard of life.
Other writers, especially Donders and Mansfelt, have
given measures of the lengths and cross sections of the biceps
and brachiæus, which I have added in a note, * partly on account
One.
* Over de Elasticiteit der Spieren. Utrecht, 1863.
Av, of Eight.
1. Distance from axis of elbow to insertion of biceps, 39 m.m. 45m.m.
2. Length of fibres of long head, ........ 158 - 160
3. Length of fibres of short head, ......... 202 204
4. Length of the whole long head, ........
5. Length of the whole short head, ....... 355 366
6. Distance from axis of elbow to middle of insertion of
brachiæus, . . . . . . . . . . . . . 32 30
7. Length of fibres of brachiæus, ........ (51-143) (48-134)
8. Length of whole brachiæus, . . . . . . . .
400
237
ANIMAL MECHANICS.
65
of their intrinsic value, and partly for the purpose of compa-
rison with my own.
It will be observed that their measurements of the. cross
sections of these muscles are smaller than those given by me,
and would give, therefore, a larger value for the coefficient K.
MEASUREMENTS OF DONDERS AND MANSFELT.
Cross Sections of Biceps humeri and Brachiæus.
Millims. Sq. in.
1. Biceps humeri (long head), ... 530 0.821
(short head), ... 452 0.701
2. Brachicus, . . . . . . . . 614 0.952
1596 2.474
Example, No. 2.-Flexors of the Leg. In order to measure
the force of the muscles flexing the leg, I placed the observer
lying upon his face upon a table, with the legs extended over
its edge, and having fastened down the thighs, I observed the
maximum weights suspended from the heel that could be
conveniently lifted, and found that 34 lbs. was the limit; to
this must be added 3 lbs. for the weight of the leg, supposed
suspended at the heel, which was measured as 161 inches
from the axis of rotation of the knee-joint. The perpendicu-
lars let fall upon the directions of the several muscles flexing
the leg were then measured :-
9. Distance from origin of brachiæus to axis of elbow, . (183-31) (190-37)
10. Length of radius, · · · · · · · · · · · 235 238
11. Length of humerus, ........... 342 336
12. Section of long head (biceps), ..
... 530 sq.m.m.
13. Section of short head, ........... 452 .
14. Section of brachiæus, .... .... 614
Cross section of an elementary bundle
50 sq. m. m.
Total number of elementary bundles in the cross section of the flexors of forearm
. = 798,000.
ANIMAL MECHANICS.
un
LI
Perpendicular.
1. Biceps femoris (long head), .. 0.95 in.
(short head), · · 0.56 ,
2. Semitendinosus, . . . . . . 0.40 ,
3. Semimembranosus, ...0.65 ,
4. Gracilis, . . . . . . . . 0.25 »
5. Sartorius, ........ 0.00 ,
The cross sections of the muscles that bend the leg were
found to be in the same subject*-
1. Biceps femoris (long head), .2.59 sq. in.
1 (short head), ; . 1.14
2. Semitendinosus, . . . . . . 1.87
3. Semimembranosus, .... 2.25 m
4. Gracilis, . . . . . . . . 0.89
5. Sartorius, ..... 0.59
9.33 sq. in.
Hence we find, for the determination of K (the coefficient of
muscular contraction per square inch of cross section),
r 0.95 2.59
+ 0.56 x 1.14
+0.40 X 1.87
37 * 16]. = K x
+ 0.65 * 2.25
+0.25 x 0.89
( + 0.00 x 0.59
610.5 = K *
r 2.460
+0.638
+ 0.748
+ 1.462
+ 0.222
+ 0.000.
5.530
* This was also the subject in which I had measured the muscles of the
forearm, described in Example, No. I.
ANIMAL MECHANICSTO
67
.
and, finally,
K = 610.5 – 110.4 lbs. *
5.53
The cross sections of the muscles were found, by cutting
them across with a sharp scalpel, and marking out their sec-
tion on cardboard, and afterwards weighing the marked por-
tions, the weights of which were then compared with the
weight of a known number of square inches of the same card-
board, and so the cross sections in square inches calculated.
I give here, for the purpose of illustration, the actual sec-
tions of two of the muscles of the leg:-

Rito!
..
ONLAP
.
.
.
ir
.
C
MINOR
X
POPUOTI
#
+
NEW
I
ureth
.
.
.
.
.
ST
COREAN
LA
UP
.
2212202
It
YS
#
.
08
..
..
"
OOO.
FIG. 2.-Sernitendinosus.

.
.
.
..
.
.
ht
eners
..
•
Fig. 3.--Short Head of Biceps.
* As it was not convenient to procure a good subject destroyed by a violent
death, I made use of a powerful man who had died of cholera, and who had been
a blacksmith by profession. Now, it is natural to suppose that the muscles of
the arm of a blacksmith are more developed than those of his leg, so that their
cross section would be relatively too great, and the coefficient derived from that
cross section, therefore, probably too small; a circumstance which may help to
account for the smaller coefficient, 94.7 lbs., found from the cross section of the
muscles flexing the forearm.
It is well known that after death by cholera, life continues in the muscles,
and manifests itself for some hours by movements, and by the existence of the
muscular susurrus. Both of these facts I have repeatedly verified. It appeared
to me, therefore, that such a subject as I selected was one well suited to the pur-
pose of my observations.
F 2
68
ANIMAL MECHANICS.
The perpendiculars let fall upon the directions of the
muscles were measured by stretching strings from the origin
to the insertion of the muscles, and measuring, by means of a
compass, the perpendiculars let fall upon these strings from
the axis of the joint.
The weights of the muscles themselves were as follows:-
0%.
1. Biceps humeri, . . . . . 4.22
2. Brachiæus, . . . . . . 5.04
3. Biceps femoris, .... 10.74
4. Semitendinosus, . . . . 5.17
5. Semimembranosus, ... 7.25
6. Gracilis, . . . . . . 2.98
7. Sartorius, . . . . . 5.66
From the preceding facts, we may infer the total force of
the flexors of the leg, by multiplying their total cross section,
9.33 sq. inches, by the coefficient of contraction, 110.4 lbs.
Total force of flexors of leg = 9.33 X 110.4 = 1030.03 lbs.
The whole investigation illustrates well the difficulties
that surround researches in Animal Mechanics. I was obliged
to make the observations and measurements on three different
subjects :-
(a). The maximum weights were observed by trials made
upon myself and a friend of a similar height and strength,
with results closely agreeing.
(6). The measurements of the arms of the levers were
made upon the skeleton of a Frenchman, of the same height
and length of bones as myself.
(c). The cross sections of the muscles of the arms and
legs were measured on a blacksmith who had died of cholera,
and who, when living, was a much more muscular man than
myself, with probably, also, a greater developement of muscles
ANIMAL MECHANICS.
69
of arm than of leg. It is very difficult to obtain exact results
from observations necessarily involving so many sources of
error; and yet I venture to think that the mean of my two
sets of observations cannot be far from the true contractile
force of the muscles :-
Coefficient of flexors of arm, . . 94.7 lbs.
Coefficient of flexors of leg, , . 110.4 ,
Mean, ... 102.55 lbs.
It has been already stated, on the authority of Donders
(p. 2), that the cross section of the flexors of the forearm con-
tains 798000 muscular fibres, having each a diameter of zdoth
of an inch; from which number, and the preceding data, we
may readily find in grains, the
Muscular force of con-
3.190 X 102.55 X 7000 = 2.87 grs.
traction of a single } - 3
muscular fibre, ?
798000
Other observers have found coefficients for he force of
human muscles, referred to the unit of cross section, which
differ somewhat from the coefficient deduced above.
Weber made some experiments on this subject, from which
he deduced the force of gastrocnemius and solæus to be
1 kilogram per i sq. centimeter.
Henke and Knorz corrected the calculation of Weber, and
found that his result should have been
4 kilograms per sq. centimeter.
Their own experiments gave for the biceps, brachialis inter-
nus, and supinator longus, of the right arm-
8.991 kilograms per sq. centimeter;
and, for the mean of the two arms,
8.187 kilograms per sq. centimeter.
ANIMAL MECHANICS
Similar experiments made on the gastrocnemius and solæus
gave them
. 5.0 kilograms per sq. centimeter.
Koster has recently published in the Nederlandsch Archief
voor Genees en Natuurkunde) an account of similar experi-
ments, from which he deduces, from the gastrocnemius and
solæus,
which number he afterwards reduces to
9.5 kilograms per centimeter.
02
The preceding observations, collected together, give us
the following results :-
1. Flexors of arm, . . 8.991 kilos. per centr. (Henke).
2. , , · · 8.187
Mean, .. 8.589
1. Extensors of foot, .
2. ?
5.90 kilos. per centr. (Henke)
11.60 , (Koster).
Mean,
.
.
8.75
My own results, converted into French measures, are,
Flexors of arm, .. 6.67 kilos. per centimeter.
Flexors of leg, ... 7.78
Mean, .. 7.225
(D
The mean of all the observations made by Henke and Kóster,
on the muscles of the arm and foot, gives -
Coefficient of -8.669 kil. per centr. = 123.04 lbs. per sq. in.
contraction = 8
ANIMAL MECHANICS.
71
I believe that the difference between this result and that
found by me, viz. 102.55 lbs. per inch, is a real difference
arising from the fact that these observers used the cross sec-
tions of muscles as they occur in ordinary dead bodies,
whereas I took care to use the cross sections of muscles in a
strong man, who had died quickly by cholera, and in whom I
supposed the muscles would be found in a state more nearly
approaching that in which they occur in the living body.
If we compare the cross sections of the biceps and bra-
chiæus found in ordinary dead bodies by Donders and Mans-
felt (p. 65) with those employed by me (p. 63), we find
Cross Sections of Muscles of Arm.
Donders and Mansfelt. Haughton.
Biceps humeri, .1.522 sq. in 1.914 sq. in.
Brachiæus, ... 0.952
1.276 ,
.
IL
2.474
3.190
If we wish to ascertain the coefficient of muscular con-
traction per square inch of cross section of living muscle, we
must alter the coefficient of force per square inch of cross
section of dead and wasted muscle in the proportion of 2474 to
3190. This will give us, as the coefficient of living muscle,
from the experiments of Henke and Koster,
Coefficient of contraction
traction
123.04 x 2474
123.04 2474 = 95.42 lbs. per
of living muscle, S 3190
sq. inch.
This result comes very near that found by me as the co-
efficient of the flexors of the forearm, viz. 94.7 lbs.
I have proved (pp. 42, 43) that the same muscles, when
moderately worked, give out equal quantities of work before
becoming fatigued; but this statement requires to be modi-
fied, when applied to the amount of work performed suddenly
72
ANIMAL MECHANICS.
by a muscle, during a single contraction. When human
muscles are moderately used, their maximum force per square
inch of cross section is found to be :
IU
102.55 av. lbs. per sq. in., or 7.225 kilos. per sq. centimeter ;
but the muscles of all animals are capable of being stimulated
for a time to perform an amount of work much greater than
that done under ordinary circumstances.
The following results, obtained by Matteucci and Helm-
holz, from experiments made upon the muscles of the calf of
the frog, exhibit this principle very clearly. It will be ob-
served that the total work done in a single contraction of the
muscle increases with the weight it is compelled to lift.
TABLE IX.- Work done in a Single Contraction by the Gastro-
chemic Muscles of the Leg of a Frog..

Weight lifted,
Height
Work done,
Observer.
1.412 mm.
Matteucci.
14.57
51.21
cr A w Nm
10.32 grms.
40.32
70.32 "
100.32 )
180.32
1.270
1.130
0.840
0.765
99
9
79.46
84.27
137.94
»
Helmholz.
The most complete series of experiments on the gastro-
cnemius of the frog are those made by Mr. Baxter, who found,
as the result of the examination of 60 frogs, of whom 26 were
females-
Total weight of muscles, . . 262 grs.
Total weight lifted, ... 159360 grs.
From these experiments we find the average force of the
159360 = 2666
= 2656 grs. = 172.II grms.
60
ANÍMAL MECHANICS.
73
Other observers (Rosenthal) have found, as the maximum
force of the gastrocnemic muscles of the frog, from 1000 to
1 200 grms., corresponding to a coefficient of contraction of
muscle, per centimeter of cross section, of from 1.8 to 3
kilograms per square centimeter.
Mr. Baxter's average result given above corresponds to a
similar coefficient per square centimeter of only 371.51 grms.
Borelli has given* many estimates of the force of the flexors
of the forearm, leg, jaw, and other muscles ; but as these esti-
mates are open to several objections, as well as his measure-
ments of the lengths of the arms of the levers engaged, I must
refer the curious reader to his original Propositions, which
may be justly regarded as a remarkable attempt, considering
the imperfections of the mechanical philosophy of his time,
to anticipate the more precise results demanded of animal
mechanics in our own day. The following remarks may be
made upon Borelli's observations :-
1. He has not been sufficiently careful to avoid the action
of muscles different from those supposed to be exclusively
employed.
2. He has not measured, with sufficient care, the arms of
his levers.
3. He has committed the common error of doubling the
* De Motu Animalium Pars prima, Props. xxii. . . . xxvii. Props. xl.
xli. liii. lxxxvii. lxxxviii.--Rome: 1680. 2 vols. 8vo.
I assume the pound used by Borelli to be 5000 grs., for the following rea-
sons :-
The Cantaro piccolo is 106 lbs. av., or 150 lbs. of 12 oz. each ; this would
· give 4957 grs. to the Neapolitan pound.
Gold and silver, in the time of Borelli, were weighed by a pound of
4950 grs.
The Roman pound was 5234 grs.
There is some difficulty in ascertaining what pound was used by Bo-
relli, but it is clear that it did not differ much from 5000 grs., and was there-
fore gths of the pound avoirdupois.
74
ANIMAL MECHANICS.
tension of the muscle, which resembles the tension of a
stretched cord.
4. He has misunderstood the mechanical conditions of
equilibrium of a many-jointed lever, or funicular polygon.
5. From not being acquainted with the composition of
forces, he has misunderstood the action of penniform and ra-
diating muscular fibres.
Errors į and 2 may be regarded as errors of observation ;
but 3, 4, and 5, are mechanical blunders;—and I have there-
fore calculated the following forces from Borelli's experiments,
correcting all his erroneous mechanical inferences. These
forces may be regarded as Borelli's results, including errors of
observation only:-
TABLE X.-Borelli's Observation of Force of Muscles.

400 lbs. av.
214 %
186
OX
678
814
1. Biceps humeri and Brachiæus (Prop. xxii.), . . .
2. Biceps humeri (Prop. xxiv.), . . . . . . .
3. Brachiæus (Prop. xxiv.), . . .
4. Biceps femoris, Gracilis, Semimembranosus, and Semi-
tendinosus (Prop. xxvii.), ..
5. Vastus externus et internus, Cruræus, (Prop. xl.)..!
Solæus (Prop. xli.), . . . .
Solaus Chorse) (Prop. xli. Scholium), .
Solæus (swan) do. do., . . . . .
7. Glutæi (Prop. liii.), . . . . . . . .
8. Solæus (Prop. liii.), :,: .
..
.
9. Flexor pollicis longus (Prop. lxxxvi.),.
.
1o. Temporales et masseteres (Prop. lxxxviii.), i.
407
2000
43
180
6. On the Comparative Anatomy of the Tendons of the Hand
and Foot, and their Mechanical Uses.--The maximum force
exerted by muscles has been shown in the preceding section
to be proportional to the cross sections of those muscles, and to
amount to upwards of 100 lbs. av. per square inch of section.
The relative values of the maximum efforts resisted by the ten-
dons of the various muscles may be more readily ascertained by
comparing together the cross sections of their several tendons
ANIMAL MECHANICS.
75
C
instead of the cross sections of the muscles themselves. As the
cross sections of both muscles and tendons must be measured
after death, and as it is not always convenient to kill subjects
suitable for observation, it is obvious that the tendons, which
are less liable to be wasted by natural disease than the muscles,
are capable of affording most valuable relative measure-
ments, if we are allowed to assume that the cross sections
of the tendons are proportional to the forces resisted by
them.
It may be readily shown, a priori, that this must be the
case, by means of the following considerations :-
The principle of economy of force, or of material, in nature,
would lead necessarily to the principle that each tendon con-
veying the effect of a force to a distant point should have the
exact strength required, and neither more nor less ; for, accord-
ing to the doctrine of Final Causes, it was originally contrived
by a perfect Architect, and according to Lamarckian views it
must have perfectly accommodated itself to the uses to which
it is applied. According, therefore, to either view, if the ten-
don be too strong, it will become atrophied down to the proper
limit; and if too weak, it must either break, or be nourished
up to the requisite degree of strength. It seemed to me de-
sirable to prove this fundamental proposition in animal mecha-
nics by direct observation; and I selected for this purpose
the tendons in the leg of several of the large running birds
(Struthionide), with the result, that the cross sections of any
two muscles tending to produce a similar effect, and subjected
to similar friction in their tendons, are directly proportional
to the cross sections of those tendons.
I shall select as an example the case of the Flexor hallucis
longus and Flexor digitorum communis perforans of the Rhea,
whose tendons unite into a common tendon half way down
the posterior side of the cannon bone of the bird.
The areas of the cross sections of these muscles were found
76
ANIMAL MECHANICS
to be as 245 to 160; or the lesser was 65 per cent. of the..
greater.
Two equal lengths of the dried tendons were then weighed,
and found to be in the proportion of 845 to 495, which was
assumed to be the proportion of their cross sections. The
lesser of these numbers is 59 per cent. of the greater; a re-
sult that seems to be as near to the former result derived
from the muscles as can be expected in this class of experi-
ments.
A. Comparative Anatomy of the Tendons of the Foot.
The flexor muscles of the toes which we have to compare
are three in number, viz. :
Flexor hallucis longus.
Flexor digitorum longus.
Flexori longo accessorius.
This latter muscle, when it exists, always acts as a guy
or force directing the line of force of the long flexor of the
toes; but in many animals it is absent, if the tendon of the
long flexor does not require to be assisted to keep in the
proper direction. The accessory muscle seems also to serve
as an addition to the force of the Flexor hallucis, whenever
the tendon of this muscle unites with that of the long flexor
of the toes ; for the fibres of the accessory are parallel to the
tendon of the Flexor hallucis, and pass into it, uniting it with
the tendon of the. Flexor longus by means of broad slender
sheets of fibres. I have found the following arrangement of
tendons in various animals :-
(a.) Man. In the human foot, the Flexor hallucis, gene-
rally, has a tendon that divides into two branches, the largest
of which forms the flexor tendon of the great toe, while the
smaller branch proceeds to form a portion of the flexor tendon
of the second toe. The long flexor of the toes has a tendon
branching into four parts, and crosses the flexor tendon of
ANIMAL MECHANICS.
77
4
.
the hallux at an angle of 24°. The four branches of the long
flexor tendon are distributed to the second, third, fourth,
and fifth toes; that which goes to the
second toe uniting with the lesser branch
of the hallux tendon to form the flexor of
the second toe.
This arrangement, which is subject to
many variations, is shown in Fig. 4, which
represents the tendons seen from below,
in the sole of the foot, but displaced from
their natural positions in order to exhibit
their mechanical arrangement.
Example, No. I.-In order to obtain the
relative strengths of the various tendons,
I weighed equal lengths of those tendons,
W, Fl. dig. Fl. hall.
with the following results:

«ILS
Fig. 4.
In the sole of foot.
Flexor hallucis longus, ...... 27
Flexor digitorum longus, · .... 26
Total, .
·
.
·
In the toes.
1. First toe, . . . . . . . . .
2. Second toe, ......
3. Third toe, ...
4. Fourth toe, ... ...
5. Fifth toe, . . . . . . . .
.
·
·
81
.
.
Total,
.. 66
It will be observed that the sum of the cross sections of
the tendons actually distributed to the toes exceeds the
78
ANIMAL MECHANICS..
sum of the cross sections of the two great flexor tendons at
the heel, the chief increase taking place in the tendons of the
flexor digitorum longus. This increase is due to and represents
the force exerted by the accessory muscle, which joins the
flexor digitorum at its point of subdivision into the tendons
of the separate toes.
The relative strengths of the tendons are, therefore,
1. Flexor hallucis longus, · · 41 per cent.
2. Flexor digitorum longus, :40
3. Accessorius, · · · · · · 19
Total,' . . 100
If the whole force employed in flexion be called 100, as
above, it will be distributed to the toes in the following pro-
portions :-
1. First toe, ... ... 41 per cent.
2. Second toe, ..... 15
3. Third toe, . . . . . . 18 »
4. Fourth toe, ...... 13 ,
5. Fifth toe, . . . . . . 13 ,
Total, . . . 100
Example, No. 2.-In this case, also, the Flexor hallucis was
found to supply the whole tendon of the first toe and part of the
tendon of the second toe; while the Flexor digitorum supplied
the remaining portion of the second toe and the whole tendons
of the third, fourth, and fifth toes.
The relative strengths of the several tendons were ascer-
tained to be
ANIMAL MECHANICS.
.
41 per cent.
18
1. First toe, . . . . . .
2. Second toe, ......
3. Third toe, . ..
4. Fourth toe, ......
5. Fifth toe, ·
.
·
·
15
9
.
·
Total, . . .
100
1. Flexor hallucis longus, s.
2. Flexor digitorum longus, · ·
3. Accessorius, . .....
Total, . . .
47 per cent.
36 ·
17 ,
100
Example, No. 3.—This case resembles the two preceding
in the distribution of tendons, the relative strengths of which
were found to be as follows,
37 per cent.
17
1. First toe, . . . . . .
2. Second toe, ....
3. Third toe, ...
4. Fourth toe, . . . . . .
5. Fifth toe, ...
.
15
15
14
Total, .
100
1. Flexor hallucis longus, .. 41 per cent.
2. Flexor digitorum communis,. 38 ,
3. Accessorius, . ..... 21
Total, . . . 100
Example, No. 4.-In this case the flexor hallucis sup-
plies the whole of the tendon of the first toe, and part of the
80
ANIMAL MECHANICS.
tendons of the second and fifth toes (Fig. 5); while the flexor
digitorum longus supplies the whole tendons of the third and
fourth toes, and the remaining parts of
the tendons of the second and fifth toes. 419
This arrangement is shown in the par- 51
tially displaced tendons of Fig. 5.
The relative strengths of the several
tendons were found to be as follows :-

1. First toe, .. 47 per cent.
2. Second toe, 14
3. Third toe, . - II
4. Fourth toe, . II ,
5. Fifth toe, · · 17 ·
Total, . . 100
Fl. dig.
1. Flexor hallucis longus, . .
2. Flexor digitorum communis,.
3. Accessorius, . .....
Total, . .
F1. hall
Fig. 5.
42 per cent.
41 »
17 .
100
Example, No. 5.- In this case the flexor hallucis supplies
the first toe, the second toe, and part of the tendons of the third
and fourth toes; while the flexor digitorum communis supplies
the remaining parts of the tendons of the third and fourth toes,
and the whole tendon of the fifth toe; sending also a small
branch, previous to its own trifurcation, to the tendon proper
of the first toe. This arrangement of tendons is shown in
Fig. 6, and is a deviation from the human type, in which the
whole tendon of the first toe is usually supplied by a single
muscle, viz., the flexor hallucis longus.
ANIMAL MECHANICS.
The relative strengths of the several tendons in* this case
were found to be as follows:-
1. First toe, ..... 39 per cent.
2. Second toe, .... 15
3. Third toe, ·
15
4. Fourth toe, ...· 192
5. Fifth toe, ... .. 12 12

I0O
1. Flexor hallucis longus, . 46 per cent.
2. Flexor digitorum longus, 47 ,
3. Accessorius, &c., ... 7 ,
IOC
If we collect together (disregarding
the differences of distribution of tendons)
Fl. dig. Fl hall.
the foregoing examples, we find-
Fig. 6.
TABLE XI.—Long Flexors in Foot of Man.*

No. 1. | No. 2. No. 3. No. 4. | No. 5. I. Mean
per cent. / per cent. per cent. per cent. per cent. | per cent.
41.0
15.8
1. First toe,
2. Second toe, .
3. Third toe,
4. Fourth toe, .
5. Fifth toe,
II
II
15.6
14.6
13.0
100
100
IОО
100
100
100
42
1. Fl. hall., . .
2. Fl. dig., : :
3. Access., &c.,
40
med os
41
43.4
40.4
16.2
Ιοο
IOO
100
100
100
100
* It is well known that the prairie hunters, on soft ground, always turn
their feet well inwards, in order to compel the smaller toes to take their due
82
N
II
ANIMAL MECHANICS.
The distribution of the flexor tendons of the foot has been
carefully studied by Mr. Turner (“Trans. R. S. Edin.” vol. xxiv.
p. 181), with the following results :-Out of fifty feet dis-
sected, the flexor hallucis gave off a slip; to the second toe
only, in eleven instances; to the second and third toes only,
in twenty instances; to the second, third, and fourth toes
only, in eighteen instances; and in one instance it gave slips
to the second, third, fourth, and fifth toes.
The flexor digitorum longus, in nine instances, gave off a
slip, previous to its subdivision, to the flexor tendon of the
hallux.
Flexor hallucis of Man (Turner).
Instances. Per cent.
1. Slip to long flexor of second toe only, . II. 22
2. Do. second and third toes only, ... 20 · 40
3. Do. second, third, and fourth toes only, · 18 · 36
4. Do, second, third, fourth, and fifth toes, I. 2
Total, ... 50
100
Prof. Schultże, of Rostock, also (Zeitschrift für wissen
schaftlichen Zoologie XVII., 1.) has published the results of
a similar examination of 100 human feet, as follows-
Flexor hallucis of Man (Schultze).
1. Slip to long flexor of second toe only, , 32 per cent.
2. Do. second and third toes only, ... 58
3. Do. second, third, and fourth toes only,. 10
4. Do. second, third, fourth, and fifth toes,
Total, . . .
100
share of the work of grasping the ground in walking; if this precaution be ne-
glected, it is found that great fatigue is caused, on a long march, in the flexor
muscle of the hallux, which, when the toes are turned outwards, is compelled
to do nearly the whole work required.
ANIMAL MECHANICSI
L
83
.
I
127
He found also, that in twenty-nine instances the flexor
digitorum longus sent a slip to the tendon of the hallux, and
that this occurred most frequently in the cases in which the
flexor hallucis sent slips to the second, third, and fourth toes,
as if to compensate for the loss of power thus occasioned in
the hallux.
If P' denote the force applied by a muscle to one extre-
mity of a tendon, and P denote the force conveyed to its
other extremity, or insertion, the forces P' and P will be
equal, if the tendon experiences no friction in passing from
the muscle to the insertion of the tendon; but P will exceed,
or fall short of P, by a force equal to the friction F, accord-
'ing as the muscle overcomes an external resistance, or external
forces overcome the resistance of the muscle. We shall,
therefore, have
P' = P + F
(15)
in the two cases supposed.
In both cases P' is proportional to the cross section of the
muscle, or inherent force of muscular contraction; and P,
which represents the external force applied, must always be
less than the force sufficient to break the tendon. Solving
the foregoing equation for P, we find
P-P' FF;
from which it follows, that the safety of the system requires the
tendon to be made always strong enough to resist the force
P + F, or the sum of the muscular force and friction combined.
Hence it follows that the cross section of the tendon will
always indicate a strength, sufficient not only to resist the
force of the muscle, but the sum of that force and the friction
experienced by the tendon.
In order to ascertain the proportion of the cross section
(or force) of a given muscle, to the cross section (or strength)
of its tendon, I obtained permission from Professors Morgan
and Bevan, of the Royal College of Surgeons in Ireland, to
G 2
84
D
ANIMAL MECHANICS.
make use of the right arm and hand of a well-developed sub-
ject (male) brought into the dissecting room of that College,
in March, 1868.
I first ascertained the specific gravities of muscles and
tendons, with the following results :-
Muscles.
Sp. Gr.
Biceps humeri, · · · · · 1.050
» » ... 1.054
Brachiæus, . . . . . . 1.053
Mean, ... 1.0523
Tendons.
Scapular tendon of biceps, . .
Radial tendon of biceps, . .
1112
1.119
Mean, ..
1.1155
From these data it was easy to determine the cross sec-
tion of either muscles or tendons, by weighing a known length
of either one or other. The following Table was thus readily
constructed :-
TABLE XII.--Cross Sections of Muscles in an Adult Human
Male Subject.

Muscle.
Weight of
Muscle.
Length of Weight of
Muscle used. Muscle used.
Cross Section.
9
1. Biceps humeri, . . 3.48 oz. av. I 3 inches. | 713.8 grs.
2. Palmaris longus, . .10.22
39.4 19
3. Ext. carp. rad. longr., 1.26 ,
310.6 ,
4. Ext. carp. rad. brevr., 0.81
2 » 215.3 »
5. Biceps humeri (long li
head), · · · ·
0.895 sq. in.
0.148
0.584
OV
0.405
503.92
0.379
.
0.228
0.234
.
7. Fl. carp. rad., . . .1*0.53
8. Ext. carp. uln., ..
9. Fl. digit. subl., , . 2.21
10. Fl. dig. prof., · · 2.74
11. Ext. oss. met. poll., 0.52
12. F. carp. uln., ·
13. Brachialis anticus,
00 on nio om
.
0.212
0,618
0.768
0.223
0.182
0.923
97.6
736.0 ,
ANIMAL .
TIR
III
MECHANICS.
85
TABLE XIII.--Cross Sections of Tendons in an Adult Human
Male Suliject.

Tendon.
Length used.
Weight. .
Cross Section.
2 inches.
,
17.87 grs.
6.3
31.4
18.6
0.0317 sq. in.
0.0055
0.0223 22
0.0220
17.90
0.0212
1. Biceps humeri (radial
tendon), ..
2. Palmaris longus, ...
3. Ext. carp. rad. longr., ..
4 Ext. carp. rad. brevre, .
5. Biceps humeri (scapular
tendon), ..
6. Fl. pollicis long., ...
7. Fl. carp. rad., .
8. Ext. carp. uln.,
9. Fl. digit. subl.,
1o. Fl. digit. prof.,
ii. Ext. oss. met. poll., . .
12. Fl. carp. uln., . . . .
13. Fl. dig. long ped., ..
14. Fl. hallucis long., . . .
:: NNNN - W N w Wara
8.2
13.1
5.6
0.0145
0.0155 »
0.01991
0.0665 1
0.0928
0.0289
37.5
52.3
16.3
14.3
0.0254 »
0.02409
0.0260
• The cross sections of the tendons of the foot, Nos. 13,
14, are estimated from the dried tendons of the subject
No. 1, Table XI. (figured in page 77), by a process some-
times usefully employed in these investigations. The tendon
is dried at 212° F. and weighed, and from that weight the
original cross section, when quite fresh, estimated as follows:
A portion of the tendon of the gastrocnemius muscle of a
large Pyrenean mastiff weighed, when quite fresh, 21.25 grs.,
and when dried, at 212° F. weighed 9.1 grs. From this ex-
periment we may infer the quantity of solid matter and water
in fresh tendon-
Solid matter, . . . . . 42.8 per cent.
Water, . . . . . . 57.2
100.0
I found also, by careful measurement of the cross sections
of perfectly fresh tendons, that a tendon originally one inch in
86
ANIMAL MECHANICS.
length, and weighing, when dried at 212° F., five grains, had
an original cross section of 0.0401 sq. inches.
If we unite Tables XII. and XIII. into one Table,show.
ing the proportion between the cross sections of the muscles
and their tendons, we obtain the following results :-
TABLE XIV.- Ratio of Cross Section of Muscle to that of
Tendon in Man.

Muscle.
Ratio of Cross
Section of Muscle Force of Muscle.
to that of Tendon.
28.2
26.4
1. Biceps humeri, ..
2. Palmaris longus, ...
3. Ext. carp. rad. long., ..
4. Ext. carp. rad. brev.,
5. Biceps humeri (long head),
6. Fl. poll. long., ' • • •
7. Fl. carp. rad., .. .
8. Ext. carp. uln., ...
9. Fl. dig. subl., . ....
10. Fl. dig. prof., ..
11. Ext. oss. met. poll., . .
12. Fl. carp. uln., ..
13. Brachiæus, . . .
26.2
18.4
18.0
15.7
15.1
10.7
91.78 lbs. av.
15.18
59.89
41.53
38.86
23.38
24.00
21.74
78.76
22.87
18.66
94.65
63.38
com :
From an examination of the foregoing Table, it is evident
that the muscles which experience least resistance have the
largest coefficient of cross section as compared with their respec-
tive tendons. Thus the biceps humeri has a coefficient of 28.2 as
compared with its radial tendon, and a coefficient of 18.0 only,
as compared with its scapular tendon, which is exposed to the
friction of the head of the humerus. Again, the ext. oss. met.
poll., whose tendon winds round the radius, and has the duty
imposed on it of binding down the tendons of the radial ex-
tensors of the wrist, has the coefficient of 7:7, as compared
with 26.2 and 18.4, the coefficients of the comparatively free
tendons of those extensors.
ANIMAL MECHANICS.
87
If we solve equation (15) for F, the friction experienced,
we shall have
F= P - P,
where P denotes the strength of the tendon necessary for
safety, and P' denotes the actual force of the muscle.
Assuming, for the present, that the cross section of the
free muscle is to the cross section of the free tendon in the
proportion of 28.2 : 1, as in the case of the biceps humeri and
its radial tendon, we find the following frictions for the several
muscles :-
TABLE XV.–Comparison of the Strength of Tendon, the
Force of Muscle, and the Force consumed in Friction, in
several Muscles of Man.

P
Muscle.
Friction,
Strength of
Tendon.
Force of
Muscle.
Friction, &c.,
per cent.
&c.
-
-
--
91.78 lbs.
91.78 lbs. 0.00 lbs.
11.011
0.0 percent.
6.2
41.53 »
7.1
34.7
36.6
1. Biceps, . . . .
2. Palmaris long., ,
3. Ext. carp. rad. long.,
4. Ext. carp. rad. brev., 63.62 »
5. Biceps humeri (long?
head), · · · ·
6. Fl. poll. long., . .
7. Fl. carp. rad., . .
8. Ext. carp. uln., . .
9. Fl. dig. subl., .
16. Fl. dig. prof., .
II. Ext. oss. met. poll., | 83.56
12. Fl. carp. uln., . . | 73.45 ,
13. Fl. dig. long. ped., . 1 69.40 ,
14. Fl. hall. long, · · | 75.19 »
23.38 ,
24.00 ,
isan oo mtu mco i co
22.45 »
| 18.55
| 20.82
| 35.81
128.93
74.5
»
In order to compare the relative cross sections of the mus-
cles and tendons of other animals with those of man, given in
Table XIV., I destroyed, on the 21st March, 1868, a large
Pyrenean mastiff bitch, by strychnia, and proceeded to inves-
88
ANIMAL MECHANICS.
tigate the relation between the cross sections of her muscles
and their respective tendons, with the following results :-
A portion of the gastrocnemius muscle, two inches in
length, weighed in air, 1382 grs., and in water, 53.7 grs.,
giving a specific gravity of 1.040.
A portion of the tendon of the same muscle was found to
weigh in air, 21.25 grs., and in water, 2.80 grs, giving a
specific gravity of 1.152.
From these data, the following cross sections of muscles
and tendons were readily calculated.
S
TABLE XVI.-Cross Sections of Muscles in Pyrenean Mastiff.

Muscle.
Weight of
Muscle.
Length of Weight of
Muscle used. Muscle used.
Cross Sections.
2 inches.
6.22 oz. av.
0.35
0.28
.
1. Gastrocnemius, ..
2. Fl. carp. rad., ; . .
3. Fl. dig. long. (pedis),
4. Ext. carp. rad., . .
5. Fl. carp. uln.,*
6. Fl. hall. long., ..
7. Biceps humeri, ..
8 Fl. dig. subl., : . .
9. Fl. dig. prof., · · ·
10. Ext. carp. uln., ..
1382.0 grs. 2.631 sq. in.
74.3 »
51.31
332.0 ,
92.5 ,
357.5 »
477.3. -
167.6 ,
355-3 ,
95.3 ,
.
ooooooooo
.
2.11
1.06
2.75
0.47
.
.
The cross sections of the corresponding tendons were found
to be
* The flexor carpi ulnaris consists of two muscles, of which one takes its
origin from the front of the inner condyle of the humerus, and the other from
the olecranon of the ulna.
ANIMAL MECHANICS..
89
TABLE XVII.-- Cross Sections of Tendons in Pyrenean Mastiff.

Length used. Weight of do.
Cross Section
2 inches.
30.27 grs.
5
29
5
1
8.6 32
6.5 99
14.
09
4.9
12
1. Gastrocnemius, ...
2. Fl. carp. rad., .
3. Fl. dig. long. (pedis), .
4. Ext. carp. rad., ...
5. Fl. carp. uln., ....
6. Fl. hall. long., . . .
7. Biceps humeri (scapular
tendon), · · ·
8. Fl. dig. subl., . . . .
9. Fl. dig. prof., ....
10. Ext. carp. uln., ...
13.3
بتا یا کہ یہ " ماته
0.0520 sq. in.
0.0059 19
0.0045
0.0160
0.0056
0.0228
0.0449
0.0251
0.0830
0.0197 »
19.6
14.6
36.2
,
»
1
11.5
29
From the preceding Tables we may easily construct the fol-
lowing :-
TABLE XVIII.— Ratio of Cross Section of Muscle to that of Ten-
don in Mastiff.

Muscle.
Ratio of Cross Sections.
50.6
.
48.0
43.3
39.5
31.4
.
1. Gastrocnemius, . . . . . . . . .
2. Flexor carpi radialis, . . . . . . . . .
3. Flexor digitorum longus (pedis), . . . . .
4. Extensor carpi radialis (long. et brev.),
5. Flexor carpi ulnaris (olecranalis), i
6. Flexor hallucis longus, . ..
7. Biceps humeri (scapular tendon), . .
8. Flexor digitorum sublimis, . ...
9. Flexor digitorum profundus, ....
16. Extensor carpi ulnaris, .;..
.
29.8
.
.
20.2
12.7
.
109
.
9.2
If we suppose, as we have already done in the case of
human muscles, that the strength of the tendon is intended
to be divided between the pull exerted upon it by the muscle
and the strain caused by external forces; and also suppose
that 50.6* represents in the mastiff the proportion of cross
O
* This coefficient iş taken from the gastrocnemius muscle, and exceeds that
taken from the radial tendon of the biceps humeri in man; this difference is, no
90.
ANIMAL MECHANICS.
sections of free muscle and free tendon, we may readily obtain
the following Table :-
TABLE XIX.- Comparison of the Strength of Tendon and Force
of Muscle in the Pyrenean Mastif".

Dſuscle.
Strength of
Tendon.
Force of Muscle.
Friction, &c. &c.
1. Gastrocnemius, ...
2. Fl. carp. rad., . . .
3. Fl. dig. longus, ..
4. Ext. carp. rad., ..
5. Fl. carp. uln., . . .
6. Fl. halī. long., ...
7. Biceps humeri (scapular
tendon), ....
8. Fl. dig. subl., .
9. Fl. dig. prof., · ·
10. Ext. carp. uln., . .
100
100
100
100
100
100
37.9
41 I
IDO
60. 1
100
100
39.9
25.1
21.2
18,2
1
100
81 8
1
The results exhibited by the preceding Table are similar
to those shown in Table XV., for human muscles. In both
cases, the muscles whose tendons are exposed to the greatest
amount of friction, are those which transmit the least force to
their points of application. Thus, in the mastiff, the flexor
sublimis and flexor profundus lose, by friction, a much greater
portion of their. original force than the comparatively free
tendons of the radial and ulnar flexors of the wrist; and more
than the radial extensor of the wrist, while the ulnar extensor
of the same joint resembles the flexors of the fingers in its
proportionately large loss of force transmitted.
The following observations were made on the muscles and
tendons of a Wallaby Kangaroo :-
The gastrocnemius muscle weighed 2.93 oz. av., and one
inch of its central portion weighed in air, 356.9 grs. ; in water,
S
doubt, real, for the dog's tendon is stronger than that of man; at the same time
it is worth while to notice the similarity of the coefficients of the scapular head
of the biceps.
ANIMAL MECHANICS.
91
25.3 grs. Three inches of its tendon weighed in air, 31.4 grs.,
in water, 4.4 grs.
The flexor digitorum longus (pedis) weighed 0.83 oz. av.;
one inch of its central portion weighed 96.3 grs., and three
inches of its tendon weighed 21.7 grs.
From these data we obtain-
Specific gravity of kangaroo muscle, .. 1.076
Specific gravity of kangaroo tendon, . 1.163
- From the preceding data we obtain

Kangaroo Muscles.
Cross Section of
Muscle.
Cross Section of
Tendon.
Ratio.
1. Gastrocnemius, •'
2. Fl. dig. longus, . .
1.313 sq. in.
I
0.0356 sq. in.
0.0246 -
36.9
14.4
0.354
Assuming, as in the case of Man and the Mastiff, that the
friction is zero in the muscle that has the highest coefficient
of cross section as compared with its tendon; we find in the
kangaroo-
Fl. dig. long.–Strength of tendon, . . 100
Force of Muscle, ... 39
Friction, &c. . . . . 61
It appears from the whole of the preceding investigation
that the cross section of a muscle does not bear a constant
ratio to the cross section of its tendon, unless the friction
exerted upon the muscle and tendon be also constant; and it
is also possible that the strength of the tendon may be some-
what greater than what would be absolutely necessary to
counterbalance the force of the muscle and friction; this ad-
ditional strength forming, in fact, a surplusage difficult to
calculate, but one which cannot be supposed very large, if
the principle of economy of material in nature be admitted.
92
ANIMAL MECHANICS.
On examining Tables XV. and XIX., and the result just
stated for the long flexor of the foot of the kangaroo, it appears
that the friction of the tendons and muscles, and the surplusage
of strength provided against any sudden accident, may con-
sume as much as 75 or 80 per cent. of the strength of the ten-
don, leaving only one-quarter or one-fifth of that tendon to be
strained by the contraction of the muscular fibres. This, how-
ever, can only occur in very extreme cases, and the force ex-
pended in such cases reappears as heat, and so assists the
general purposes of the body, which requires its temperature to
be maintained at 100° F. in order to perform its functions. In
the great majority of cases the muscles, in performing their
usual duties, act in such a position of the tendons as to cause
the minimum amount of friction, and to produce the maximum
of external effect; the surplus strength of the tendons being
provided as a safeguard against strains that rarely occur, and
then only as accidents.
(6). Chimpanzee. In this animal the flexor hallucis sup-
plies the whole tendons of the first and fourth toes, and part
of the tendons of the second and third toes; while the flexor
digitorum supplies the remaining parts of the tendons of the
second and third toes, and the whole tendon of the fifth toe.
This arrangement would seem to indicate a concurrence in
action between the first and fourth toes, which must corre-
spond to some peculiarity in the animal's mode of grasping
objects; while the fifth or outer toe retains its indepen-
dence.
(c.) Monkeys of the Old Continent. - The Macaques of Asia,
and the long-tailed Monkeys of Africa, such as the Mangabeys,
are characterized by an arrangement of flexor tendons of the
foot, which is shown in Fig. 7, representing the flexor tendons
of the foot of the Bruh, or Pig-tailed Macaque (Macacus Ne-
mestrinus), seen from below, and drawn aside from their natural
positions, so as to exhibit their mechanical arrangement.
ANIMAL MECHANICS.
93
3
Flexor Tendons of the Foot in Asiatic and African Monkeys.
In these monkeys, the flexor
hallucis supplies part of the tendon
of the first toe, and the whole ten-
dons of the third and fourth toes;
while the flexor digitorum longus
supplies the remainder of the ten-
don of the first toe, and the whole
tendons of the second and fifth
toes. This arrangement produces
a remarkable concurrence of ac-
tion in the following combinations
of toes :-
a. First - Third Fourth.
b. First . Second . Fifth.

-
Fl. hall.
Fl. dig.
I have often seen the Rhesus
Macaque avail himself of the first
of these combinations to catch a
flea, with a dexterity that showed
Fig. 7.
how superior his foot is, considered as a prehensile organ,
when compared with that of Man.
The relative strengths of the several tendons in the foot
of the Macacus nemestrinus, measured by the weights of equal
lengths, were found to be as follows :-
Macacus Nemestrinus.
1. Flexor hallucis longus, . 54 per cent
? Flexor digitorum longus, · · 44 ·
3. Accessorius, 8c., .. . . 2
Total, .... 100
»
ANIMAL MECHANICS.
'1. First toe, . . . . 25 per cent.
2. Second toe, . ...
3. Third toe, . ..
4. Fourth toe, ... 25
5. Fifth toe, . . .
Total,
..
. 100
The weights of the several muscles in the Macacus Nemes-
trinus were found to be
1. Flexor hallucis longus, ... 0.66 oz. av.
2. Flexor digitorum longus, . : 0.36 ,
3. Accessorius, · · · · · · 0.05 ,
SORRIS
11
MINNASTO
WAR
SAASAASA
(d.) Monkeys of the New Continent.— The Capuchin and
Negro monkeys of South America
exhibit, in the arrangement of the
flexor tendons of the foot, a type
quite different from that of the
monkeys of the Old World. This
arrangement is shown in Fig. 8,
which represents the tendons of
the foot of Lagothrix Humboldtii,
or Negro monkey of South Ame-
rica.
In the monkeys of the New
World, the flexor hallucis longus
supplies part of the first toe, and:
the whole of the long flexors of
Fl. hall. Fl. dig.
the second, third, and fourth toes ;
Fig. 8.
while the flexor digitorum longus supplies the remainder of
the flexor tendon of the first toe, and the whole long flexor
tendon of the fifth toe.

ANIMAL MECHANICS.
95
In these animals we may observe a concurrence in action
of the first and fifth toes, like that which takes place between
the first, third, and fourth toes, in the monkeys of the Old
Continent.
The relative strengths of the flexor tendons in the Negro
monkey were found to be-
Lagothrix Humboldtii.
1. Flexor hallucis longus, . . 49 per cent.
2. Flexor digitorum longus, . . 43 ,
3. Accessorius, &c., .... 8 »
Total, .... 100
1. First toe, ......
2. Second toe, . ....
3. Third toe, . . . . .
4. Fourth toe, ..
5. Fifth toe, ·
· · · ·
18.5 per cent.
18.5 »
25.0 »
19.5
18.5 »
0.5
19
Total,.
100.0
In the Hapale or Jacchus, the flexor hallucis supplies the
third and fourth toes only, so that its human name becomes
inappropriate ; and the flexor digitorum supplies the second
and fifth toes, with a small slip to the hallux.
(e.) The Carnivores.-In most of the Carnivores, the hal-
lux is wanting, and in all of them the flexor hallucis, and flexor
digitorum, instead of being distributed separately to the dif-
ferent toes, are united into one tendon (into which also the
accessorius is inserted), which is afterwards distributed to all
the toes, generally four in number, but which may be five or
occasionally six in number.
96
ANIMAL MECHANICS.
'The Carnivores, of the flexor tendons of whose feet I have
kept a record, are the following:-
1. The Lioness (young).
2. The Jaguar (female).
3. The Leopard (male).
4. The Indian Jackall (male).
5. The Bengal Fox (male).
6. The Dingo (female).
7. The Dog (female).
8. The Otter (female).
9. The Virginian Bear (female).
1. In the Lioness I found the following relative strengths
of the long flexor tendons distributed to the four toes :-
Lioness (young), (long flexor tendons of toes).
1. First toe, · · · 27.2 per cent.
2. Second toe, .. 27.0 „
3. Third toe, .. 23.9
4. Fourth toe, .. 21.9 „,
5. Retractor of pad,
100.0
Lioness (flexor tendons of foot).
Weight.
1. Flexor hallucis longus, 31.8 per cti · 0.93 oz. av.
2. Flexor digitorum longus, . 9.2 .0.27 »
3. Flexor accessorius, . . . . . . . . ..
4. Friction, &c., .... 59.0 » ... »
Total, ... 100.0
The angle between the f. hall. and the fl. dig. long, was.
21°
ANIMAL MECHANICS.
97
I was unable to determine satisfactorily the absolute
strengths of the tendon of the accessorius muscle, and of that
which retracts the pad of the foot, but I found that equal
lengths of these tendons had equal weights, so that they
mutually eliminate each other from the
Tables.
2. The tendons of the sole of the foot
in the Jaguar are shown in Fig. 9, which
represents the pad-retractor (a) and accesso-
rius (a') tendons in their natural positions.
The relative strengths of the tendons are
as follows :-

SS
2010
TUL
21
Jaguar (long flexor tendons of toes).
1. First toe, .. 30.9 per cent.
2. Second toe, . 24.2
3. Third toe, . . 23-3 ,
4. Fourth toe,. . 21.6
5. Retractor of pad, ..
Total, . . 100.0
RU
VETTA
.
Fl. hall.
Fl. dig.
FIG. 9.
Jaguar (long flexor tendons of foot).
O
'
1. Flexor hallucis longus, 35.8 per cent. . 0.83 oz.
2. Flexor digitorum longus,. 15.0
0.26 ,
3. Flexor accessorius, . . .
4. Friction, Sc., · · · · 49.2
Total, ...
100.0
The angle between the fl. hall. long. and the f. dig. long.
was 32°.
: H
98
ANIMAL MECHANICS.
3. In the Leopard the relative strengths of the tendons
were found to be as follows:--
Leopard (long flexor tendons of toes).
1. First toe, . i .... 21.9 per cent.
2. Second toe, . . . . 23:6 ,
3. Third toe, ... . 27.0 ,
4. Fourth toe, : ... 27.5 »
5. Retractor of pad, .. .. 1
Total, ...
100.0
Leopard (long flexor tendons of foot).
Weight.
1. Flexor hallucis longus, . 39.2 per cent. • 1.63 oz. av.
2. Flexor digitorum longus, . 15.3 m2 .0.50
3. Flexor accessorius, . . . . .
4. Friction, &c., ·· 45.5 1
Total, ... 100.0
The angle between the A. hall. long. and the fl. dig. long.
was 20°.
4. In the Indian Jackall I found the following relative
strengths of the tendons of the foot :-
Indian Jackall (long flexor tendons of toes).
1. First toe, . . . . . 23.8 per cent.
2. Second toe, .... 28.2
3. Third toe, . ..26.5 2 .
4. Fourth toe, . . . . 21.5 »
Total, ... 100.0
ANIMAL 211
MECHANICS.
Indian Jackall (long flexor tendons of foot).
Weight.
1. Flexor hallucis longus, · 40.3 per cent. · 0.28 oz. av.
2. Flexor digitorum longus, 10.5
· 0.05 1
3. Frictio, &C., .. . · 49.2
Total, ... 100.0
5. In the Bengal Fox, the relative strengths of tendons
were found to be :-
Bengal Fox (long flexor tendons of toes).
1. First toe, . .... 25.0 per cent.
2. Second toe, i ... 27.0 »
3. Third toe, .... 22.5 »
4. Fourth toe, . . . . 25.5
Total, ...
100.0
Bengal Fox (long flexor tendons of foot.)
Weight.
1. Flexor hallucis longus, . 32.9 per cent. · 0.08 oz. av.
2. Flexor digitorum longus, . 9.5 , 0.02
3. Friction, &c., ··57.6
Total, ... 100.0
6. In the Australian Dingo I found the following re-
sults :-
Australian Dingo (long flexor tendons of toes).
1. First toe, . ...24.6 per cent,
2. Second toe, · · · · 28.9 »
3. Third toe, . . . 25.3 ss
4. Fourth toe, . .21.2 ,
Total, ... 100.0
H 2
100
ANIMAL MECHANICS.
Australian Dingo ( long Alexor tendons of foot).
Weight.
1. Flexor hallucis longus, . 53.5 per cent. · 0.59 oz. av.
2. Flexor digitorum longus, 12.7 2 . 0.10
3. Friction, &c. &c., . . 33.8 es
Total, ... 100.0
7. In the Pyrenean Mastiff, two " dew claws" or spurious
toes are found, in addition to the regular four toes characte-
ristic of the digitigrade carnivores. The relative strengths of
the tendons supplying the six toes were found to be-
Pyrenean Mastiff (long flexor tendon of toes).

1. First toe, ..., 4.1 per cent.
2. Second toe, ... 10.7
3. Third toe, ... 20.9
4. Fourth toe, ... 25.0
5. Fifth toe, ...: 22.8
6. Sixth toe, . . . . 16.5
.
www
2
Total,
. . 100.0
Pyrenean Mastiff (long flexor tendons of foot).
Weight.
1. Fl. hall. long, 28.1 per cent. . 1.59 oz. av.
2. Fl. dig. long., 6.5 „ . 0.28
3. Friction, &c., 65.4 "
Total, . 100.0
Fl. dig. Fl. hall.
Fig. 10.
8. The Otter possesses regularly five toes, supplied with
long flexor tendons nearly equal in strength. If the united
ANIMAL MECHANICS.
101
cross section of these tendons be called 100, we find for the
cross sections of the flexor tendons of the foot the following
values :-
Otter (long flexor tendons of foot).
1. Flexor hallucis longus, .,
2. Flexor digitorum longus 80.2 per cent.
3. Friction, &c., .... 19.8 »
Total, .....
100.0
9. The Bear, like other plantigrade Carnivores, possesses
five toes, the tendons of which gave me the following re-
sults, in the Virginian and Japanese Bears :-
Bear (long flexor tendons of toes).
Virginian Bear. Japanese Bear.
1. First toe, .... 11.6 per cent. . 16.3 per cent.
2. Second toe, . .. 24.5 ,, · 22.0 h
3. Third toe, ..., 24.7
· 24.4 "
4. Fourth toe, . . . . 22.4 „ . 22.0
5. Fifth toe, . . . 16.8
· 15.3
Total,
.
. ICO.O
100.0
Virginian Bear. Japanese Bear.
1. Flexor hallucis longus, . 57.5 per cent. . 53.7 per ct.
2. Flexor digitorum longus, 18.6 2 : 14.6
3. Friction, &c.; : ... 23.9 . 31.7 ·
Total, · · 100.0
100.0
102
ANIMAL MECHANICS.
If we compare together the amount of friction experienced
by the long flexor tendons of the foot of the various carni-
vores, we find the following Table :-
TABLE XX.*-Amount of Friction, fc., in the Long Flexor
Tendons of the Toes of Carnivores.

Carnivores.
Friction, &c.
65.4 per cent.
.
1. Mastiff,
2. Lioness,, . .
3. Fox, . . . .
4. Jaguar, . . .
5. Jackall, . . .
6. Leopard, . . .
7. Dingo, · · ·
8. Japanese Bear, .
9. Virginian Bear, .
10. Otter, . . . .
59.0
57.6
49.2
49.2
45.5
.
.
·
.
.
.
33.
Nooroo
31.7
23
* Since writing the preceding Table, I have had an opportunity of examin-
ing the tendons of the foot of a male European Wolf, and of a magnificent Ben-
gal Tiger, with the following results :-
European Wolf (long flexors of toes).
1. First toe, . . . . . . 24.6 per cent.
2. Second toe, . . . . . . 25.8 »
3. Third toe, . . . . . . 25.3 »
4. Fourth toe, . . . . . . 24. 3
Total, ... 100.0
European Wolf (long flexor tendons of foot).
Weight.
1, Flexor hallucis longus,. 13.5 per cent. . 0.30 oz. av.
2. Flexor digitorum longus, . 52.5 2 .0.79 ,
3. Friction, &c., . . . . 34.0 »
Total, ,
.
. 14.40
ANIMAL MECHANICS.
103
It will be seen, on comparing these results with those
already given for Man and the Quadrumans, that the Otter's
foot, used for swimming, resembles that of Man; and that the
feet of the Carnivores differ widely from those of Man and
the Quadrumans, being characterized by an enormous
strengthening of the flexor tendons of the toes. The scratch-
ing and digging actions of the hind claws of these Carnivores
are essentially distinct from the grasping actions of the feet
of the monkeys, as well as from the swimming motions of the
feet of Man and the Otter.
(f.) The Ungulates. Of the even-toed ungulates, I have
examined the flexor tendons of the Ox and Goat; and of
the odd-toed, those of the Rhinoceros.
1. The tendons of the foot of the Ox (Zebu Cow) are
shown in Fig. 11, and their relative cross sections are as fol-
lows:-
Bengal Tiger (long flexors of toes).
1. First toe, . . . . . . 28.3 per cent.
2. Second toe, . . . . . . 24.6
3. Third toe, · · · · 21.8 »
4. Fourth toe, . . . . . . 180 ,
5. Retractor of pad, . .,. . 7.3
Total, .
.
. 100.0
Bengal Tiger (long fesor tendons of foot).
Weight.
1. Flexor digitorum longus, . 12.2 per cent. • 1.57 oz. av.
2. Flexor hallucis longus, . . 35.8 1 6 .52 »
3. Accessorius, . . . . . . 6.0
Friction, &c., . . . . . . 46,0
Total, .
.. 100.0
104
ANIMAL MECHANICS.
fl.dig.
Zebu Cow (long flexors of toes.) Fl.hall.
1. First toe, ... 49.1 per cent.
2. Second toe, ... 50.9

sro
Total, ..100.0
Weight.
1. Fl. dig. long., 74.6 per c. 0.90 oz.av.
2. Fl. hall. long., 18.6 5 4.43 ,
3. Friction, &c., 6.8 ?
Total, . . . 100.0
.
Top
ww
w
2. The flexor tendons of the Goat
(Bombay) resemble those of the Cow very
closely; and gave the following results :-
Fi. hall. Fl. dig.
Fig. 11.
Bombay Goat (long flexors of toes).
1. First toe, ...... 47.6 per cent.
2. Second toe, ... . . . 52.4 1
Total, ... 100.0
1. Flexor digitorum longus, 69.05 per cent.
2. Flexor hallucis longus, 21.43
3. Friction, &c., ... 9.52 ,
Weight.
0.04 02. av.
0.25 1
100.00
3. The arrangement of the flexor tendons of the toes in the
foot of the Rhinoceros (Indian) is somewhat peculiar, as is
shown in Fig. 12.
ANIMAL MECHANICS.
105
.
Bootle
1
..
ap1
.
The plantaris muscle takes an origin from the back of the
outer condyle of the femur, and is inserted into the plantar
fascia that passes over the os calcis, 2
and gives origin to the perforate ten-
dons of the toes. The tendon of
plantaris, flowing freely over the
heel, becomes partly inserted into
the tendon of the long flexor of the
toes, in the centre of the foot, and
partly terminates in the three per-
forate flexors that correspond with
the flexor brevis of other animals.
The long flexor muscles arise from
the entire back of the fibula, and
lower half of the tibia, and unite
into a tendon that represents f. dig.
long. and A. hall. long. united; this
wodacomodadosomihod fl.long
pl.
bendon to Jomel, an abseauyuculincuy Pi. Tendon of plantaris.
in the sole of the foot, by the ten- Fl. long. Joint tendon of fl. hall.
don of the plantaris muscle and flexor
brevis united.
1. 2. 3. perforate flexors of toes.
The relative strengths of the fore- Fig. 12.
going tendons, measured by their cross sections, are as fol-
lows, representing the total cross section of the flexors of
the leg by 100:-

VORW.
nontonio muolond Amnes long. and fi. dig. long.
1, 2, 3, deep flexors of toes.
UCU
Rhinoceros (long flexor tendons of toes).
Weight:
1. Plantaris, . . . . . . 44.3 per cent. 3.5 oz. av.
2. Flexores longi dig. et hall., 55.7 2 21.0 ;,
Total, ... 100.0
106
ANIMAL MECHANICS.
1. Long flexors of toes, . . . 61.1 per cent.
2. Short flexors of toes, .. 18.4
3. Friction, &c., ..... 20.5 2
Total, ... 100.0
tuul
Long Flexors of Toes.
1. First toe, ... 18.4 per cent.
2. Second toe, ... 24.6
3. Third toe, ... 18.1 ,
1
Total, · · · 61.1
U
Perforate Flexors of Toes.
1. First toe, .... 5.6 per cent.
2. Second toe, . ... 6.6 ,
3. Third toe, .... 6.2
Total, ... 18.4
The arrangement of the flexor tendons in the foot of the
Rhinoceros differs from that of any of the animals already de-
scribed for the total strength of the tendons of the toes is
203 per cent. short of that of the tendons of the muscles, in-
stead of exceeding it, as in the former cases.
If we consider, however, only the deep tendons of the
toes, and tendon of the long flexors, as in the former cases,
neglecting the plantaris and short flexor, and perforate ten-
dons of the toes, we find a result similar to those already re-
corded, viz., counting the total cross section of the long flexors
of the toes as 100:-
ANIMAL MECHANICS.
107
Long Flexors of Toes.
1. First toe, · · · · · 30 per cent.
2. Second toe, ... · 40 ·
3. Third toe, . . . . 30 ,
Total, 100
IOO
1. Flexor dig. long., )
2. Flexor hall. long., š· · 91 per cent.
3. Friction, &c., ... 9 »
Total, ... 100
(9.) The Insectivores. I shall take the flexor tendons of the
foot of the common Hedgehog as the type of the Insecti-
vores. They belong to a degraded type, as the fl. hall. long.
and fl. dig. long. are fused into one muscle, and possess a com-
mon tendon distributed by slips to each of the five toes.
Equal lengths of the common tendon and of the five slips
gave the following comparative results, counting the cross
section of the five tendons of the toes as 100:-
Hedgehog (long flexor tendons of toes). .
1. Flexor hall. longus, ,
Weight.
Flomon dis Tommie J . 75 per cent. 0.01 oz. ay.
3. Friction, fc., · ... 25
100
(h.) The Edentates.:-The six-banded Armadillo, and three.
toed Sloth of South America, are the best living types of this
remarkable group of animals.
108
ANIMAL MECHANICS,
1. In the foot of the Armadillo, the A. hall. long. of other
animals constitutes the whole of the deep flexor of the toes,


1
hand
.
foot
Fig. 13.
and is inserted, Fig. 13, into a plantar sesamoid bone, from
which radiate five tendons, distributed to the five toes.
The muscle, which corresponds to the f. dig. long., is
inserted into the near end of the first metatarsal.
The relative strengths of the fl. hall. long. and of the
tendons of the toes are as follows:
Six-banded Armadillo (long flexor tendons of foot.)
Weight.
1. Fl. hall. long., 55.6 per cent. 0.11 oz. av.
2. Friction, &c., · 44.4 2 · 0.02 ,
·
100.0
1. Flexor of first toe, .. 13.2 per cent.
second toe, . . 22.2
third toe, .. 24.4
fourth toe, .. 22.2
5. 2 fifth toe, .. 18.0
Total,
.
. 100.0
2. In the three-toed Sloth, the arrangement of the tendons
ANIMAL MECHANICS.
109
of the foot is as follows: the f. hall. long. and f. dig. long.
unite into a common tendon, Fig. 14; this is joined in the sole
of the foot by the tendon of the large plantaris (pl.), and
the united tendons diverge to form
the tendons of the three toes. The
tendon of the accessorius muscle (acc.)
is inserted into the outer side of the
tendon of the third toe.
Equal lengths of these tendons,
when weighed, gave the following
comparative results :-
Foot of Three-toed Sloth (long flexors).
Weight
1. Accessorius, 14.2 per c. 0.06 oz.av.
2. Plantaris, . 10.0 0.13 »
3. Fl.hall. long., 12.0 ,
4. Fl. dig. long., 21.3 „)
pl.
5. Friction, &C., 42.5 »
flh all.

EP
PS
I
Total, . .
100.0
Fig. 14.
1. First toe, .... 26.0 per cent.
2. Second toe, ... 43.8 ,
3. Third toe, . ... 30.2 »
Total, ... 100.0
(i.) The Rodents. The following observations on the Hare
and Porcupine will suffice to show the general character of
the deep flexor tendons of the feet of this important Order :-
1. In the Hare, the fl. hall. long. and fl. dig. long. unite, as
in the Hedgehog, into a single tendon, before reaching the
heel, and this tendon is afterwards subdivided into four, dis-
tributed to the four toes. Equal lengths, when weighed,
gave the following results:-
110
ANIMAL MECHANICS.
Hare (long flexors of foot).
1. First toe, . . . . . 28.4 per cent.
2. Second toe, .... 26.5'. .
3. Third toe, . . . . . 23.8
4. Fourth toe, . . . . 21:3
100.0
Or per cent.
United tendon of fl. hallo long.
and A. dig. long., ..."
Friction, &c., .......
. 36
100
2.. In the Porcupine, the A. dig. long. and f. hall. long.
pass round separate grooves on the inner ankle, and unite in
the sole of the foot, to give origin to tendons distributed to
the five toes, as shown in Fig. 15.
The relative cross sections of the ten-
dons are as follows, and differ from
those previously found for other ani-
mals, by the flexor tendons of the
muscles being in excess of the flexors
of the toes, as in the case of the Rhi-
noceros.
Porcupine - Deep flexors of Toes.
First toe, ... 8.0 per cent.
Second toe,. .19.2
2
Third toe, ... 21.3 -
Fourth toe, ... 20.0
Fifth toe, . . .
Deficiency, ... 20.0 2
fl.kall.
Wfl.dig.
100.0
Fig. 15.

.
.
e
.
11.5
ANIMAL MECHANICS.
111
Flexor hall. long, ..
Flexor dig. long., .,
Weight.
81.6 per cent.. 0.32 oz. av.
18.4. 2 . 0.10
2.
100.0
AGE
(k.) The Marsupials. The Kangaroos and Phalangers may
be selected as good specimens of Marsupials. In the Kanga-
roos, the plantaris muscle takes an origin from the outer condyle
of the femur, and from the crescent-shaped sesa- 173
moid bone lying beneath it at the top of the
fibula; it is inserted into a tendon that flows
freely over the calcaneum, and becomes ulti-
mately the perforatę tendon of the toes. At the
outer side of the heel, a tendon (a) is given off,
Fig. 16, which runs direct to the outer toe, and
forms half of its perforate tendon, the other
half being formed of a slip given off by the
main tendon near the toes.

WA
Perforate flexor tendons of Wallaby Kangaroo.
Plantaris tendon, . . . . 100.
1. Perforate tendon of third toe, 71 per cent.
2.
fourth toe, 26 »
3. Deficiency, · · · · · 3 ·
't..
.
Total,
...
100
LLLL
Fig. 16.
The deep flexor tendons of the Marsupials are formed of
the single tendon belonging to the f. hall. long. and A. dig.
Ing. muscles conjoined. I have found the following results
to represent their relative strengths :-
1
112
ANIMAL MECHANICS.
Giant Kangaroo (long flexors of toes).
1. First and second toes, ... 7 per cent.
2. Third toe, .. . ... · 75
3. Fourth toe, . . . . . . . 18 ,
Ιοο
Weight.
2.61 oz. av.
Fl. hall. longus, ,
Fl. dig. longus,
conjoinedy 100.
Friction, &c., .. . 0 .
100
TT,
Wallaby Kangaroo (long flexors).
1. First and second toes, ... 7 per cent.
2. Third toe, . . . . . . . 65
3. Fourth toe, · · · · · 27 9
4. Deficiency, . . . . . . . 1 »
100
Fl. hall. long., )
Weight.
{ conjoined, . 100 1.24 oz. av.
Fl. dig. long., ) conj
In the Giant Kangaroo the deep tendons have exactly the
same cross section before reaching the heel, as when distri.
buted to the toes; and in the Wallaby Kangaroo the difference
of cross sections representing friction is only siz or i, per
cent.
In the Phalanger, the fl. dig. long. muscle is fourteen times
the weight of the A. hall. long. muscle, and is distributed,
sometimes to all five toes, and sometimes to the outer four only.
The fl. hall. long. is inserted into the back of the scaphoid
bone.
ANIMAL MECHANICS.
113
Phalanger (long flexors of toes).
Flexor dig. long., i . . 100
1. Second toe, .... 5.9 per cent.
2. Third toe, · · · · · 5.9 »
3. Fourth toe, .. . · 29.4 »
4. Fifth toe, .. .... 29.4
5. Deficiency,. . . . . 29.4
Total, . . 100.0
(1.) The Birds.The arrangement of the long flexor ten-
dons is very similar in all birds, and the tendons are so long,
that the question of the amount of force lost by friction in
their passage round the heel possesses great interest.
The f. dig. long. and A. hall. long. muscles are distinct, and
their tendons generally unite before reaching the heel.
1. Ostrich (long flexor tendons of toes).
1. Outer toe, . .... 7.1 per cent.
2. Inner toe, . . . . . 92.
9 1
100.0
Weight.
1. Flexor dig. long., . 49.7 per cent. . 5.90 02. av.
2. Flexor hall. long., • .- 25.7
. 2.08
3. Friction, . .. ... 24.6 "
Total, . . 100.0
22
In the Ostrich, as in the other Struthionidæ, the tendons
of the f. hall. longus and A. dig. longus unite into a common
tendon, at a point half way down the cannon bone.
114
ANIMAL MECHANICS.
2. Rhea (long flexor tendons of toes).
1. First toe, ... 22.1 per cent.
2. Second toe, ., 45.1 ,
3. Third toe, ... 32 8
Total,
. .
. 100.0
Weight.
1. Flexor hall. long. 17.3 per cent. . 0.75 oz. av.
2. Flexor dig. long., · · 30.3 · · 2.09 »
3. Friction, $c., ... 52.4
Total, ... 100.0
3. Jabiru. In this bird, which is celebrated for its power
of standing on one leg for a long time, the tendons of the A.
hall. long. and fl. dig. long. do not unite until they reach the
sole of the foot; the fl. hall. long. flexes the four toes,
and the f. dig. long. flexes the second, third, and fourth
toes only.
Jabiru (long flexors of toes).
1. First toe, . . . . . 17 per cent.
2. Second toe, ... · 23 ·
3. Third toe, ·
37 )
4. Fourth toe, ...· 23
Total, ... 100
Weight.
1. Flexor hall. long., · 12.3 per cent. · 0.07 oz. av.
2. Flexor dig. long., . 30.9 2 . 0.18 "
3. Friction, foc., .. . 56.8
Total, · · 100.0
ANIMAL MECHANICS.
115
4. Weka Weka Rail.-In this bird, the two long flexor
muscles have their tendons united into a single partially ossi-
fied tendon, which is about an inch in length, and subdivide
into the four long flexors of the toes.
Weka Rail (long flexors of toes).
1. First toe, ..... 7.7 per cent.
2. Second toe, ... · 29.5
3. Third toe, ...33.3
4. Fourth toe, .... 29.5
Total,. . .100.0
Weight.
Flexor hall. long., . 33.3 per cent. . 0.06 oz. av.
Flexor dig. long., .. 19.2 2 . 0.03 15
Friction, &co, ··· 475 1 .
:
Total
: · 100.0
S
5. Silver Pheasant.-- In the pheasant the A. hall. long. is
distributed, by means of an ossified tendon, to the first toe,
and by a tendinous cross band to the ossified tendon of the
second, third, and fourth toes, which is pulled by the f. dig.
longus; and, with the exception of this cross band, the two
tendons are quite distinct.
Silver Pheasant (long flexors of toes).
1. First toe, .... 15.7 per cent.
2. Second toe, .... 26.3 "
3. Third toe, ... . 36.8 »
4. Fourth toe, . . . . 21.2
Total, ... 100.0
1 2
116
ANIMAL MECHANICS.
Flexor dig. long, .. 26.3 per cent.
Flexor hall. long., .. 26 3
Friction, &c., ..47.4
Weight.
0.06 oz. av.
. 0.02
Total,:
.
. 100.0
6. Black Swan. In this bird the tendons of the two long
flexors unite into a common ossified tendon about three-fourths
of an inch in length, which distributes tendons to the second,
third, and fourth toes.
Black Swan (long flexors of toes).
o
1. First toe, . . . . . o per cent.
2. Second toe, .... 23
3: Third toe, · · · 42
4. Fourth toe, ... 35
---
Total, ... 100
Flexor dig. long., , .
Flexor hall. long., ·
Friction, &c., ...
40 per cent. ·
24
36
Weight.
0.06 oz. av.
0.07 1
Total, ...
100
Collecting together the preceding observations on the
flexor tendons of birds, we find that the tendons distributed
to the toes exceed in strength the tendons of the long flexors
above the heel by the following quantities :
ANIMAL MECHANICS.
117
TABLE XXI.-Friction of Long Flexor Tendons of Birds.

Friction, &c.
24.6 per cent.
52.4
1. Ostrich, . .
2. Rhea, . . .
3. Jabiru, . . .
4. Weka Rail,. .
5. Silver Pheasant,
6. Black Swan, ..
56.8
47.5
47.4
36.0
6.
(m.) Reptiles.—The only reptile that I have had an oppor-
tunity of examining carefully is the Alligator of the Missis-
sippi, from the dissection of which I have obtained the fol-
lowing results :-
S
S
Alligator Lucius (long flexors of toes).
1. First toe, ...: 41.0 per cent.
· 2. Second toe, · · · · 29.5
3. Third toe, . . .. 18.0
4. Deficiency, . . ; . 115
Total, ... 109.c
Weight
Flexor hall. long., . 29.5 pe :tent. 0.17 oz. av.
Flexor dig. long., . . 70.5
.0.54
Total, ... 100.0
Il we arrange the animals already described, according to
the nature and amount of friction experienced by the tendons
of the long flexors of their toes, we shall obtain the following
l'esults :-
2
S
118
ANIMAL MECHANICS.
TABLE XXII.--Comparative View of Friction of the Long
Flexors of Toes.
P >P'; or Cross Section of Tendons of Toes greater than
Cross Section of Tendons of Muscles.
Friction.
65.4 per cent.
59.0
57.6
56.8
52.4
49.2
49,2
47.5
.
.
46.0
45.5
44.4
42.5
36.0
1. Pyrenean Mastiff (Canis familiaris), . . . .
2. Lion (Leo barbarus), .......
3. Fox (Vulpes vulgaris), ......
4. Jabiru (Mycteria), . . . . . . .
5. Rhea (Rhea Americana), .....
6. Jackall (Canis aureus),"...
7. Jaguar Leopardus onca), .
8. Weka Weka (New Zealand Rail),
9. Silver Pheasant (Gallophasis nycthemerus),
10. Tiger (Tigris regalis), . . . . . .
11. Leopard (Leopardus varius), . . . . .
12. Armadillo (Dasypus sexcinctus), . . .
13. Sloth (Bradypus tridactylus), . . .
14. Black Swan (Cygnus atratus), ...
15. Hare (Lepus timidus), . .
16. Wolf (Canis lupus), . , , .
17. Dingo (Canis Dingo), .........
18. Japanese Bear (Ursus Japonicus), . . .
19. Virginian Bear (Ursus Americanus),
20. Llama* (Llama yamma), . .. .. ..
21. Hedgehog (Erinaceus Érir opeus), · ...
22. Ostrich (Struthio camelus), . . . . .
23. Otter (Lutra vulgaris), ii,...
24. Man (mean of five subjects), ...
25. Spider Monkeyt (Ateles ater), . , .
26. Goat (Hircus ægagrus), .....
27. Rhinoceros (Rhinoceros unicornis), . .
28. Negro Monkey (Lagothri. Humboldtië), . .
29. Ox (Bos Indicus), . . . . . . . . .
36. Macaque (Macacus nemestrinus), :...
31. Boomer Kangaroo (Macropus major), ..

www
Ovo o

23.9
25.9
25.0
do novico o rio
P>P'; or Cross Section of Tendons of Toes less than Cross
Section of Tendons of Muscles.
Friction.
31. Boomer Kangaroo (Macropus major),
32. Wallaby Kangaroo (Halmaturus Talabatus), .
33. Alligator (Alligator Mississippiensis), ...
34. Porcupine (Hystrix cristata), . . . . . .
35. Phalanger (Phalangista vulpina), .....
0,0 per cent.
1.0
11.5
20.0
29.2
ANIMAL MECHANICS.
119
NOTES ON THE FOREGOING TABLE.
* I found the following results from the dissection of a fine specimen of the
common Llama:-
Llama (long flexors of foot).
1. First toe, ....... 49.5 per cent.
2. Second toe, . . . . . . 50.5
Total, .
... 100.0
1. Flexor hall. long., ..., 39.6 per cent.
2. Flexor dig. long., ... .. 34.5
3. Friction, &c., · · · · · · 25.9
Total, . . . . 100.0
In this animal the cross section of the tendons of the toes of the hind foot
exceeds that of the tendons of the fore foot in the proportion of 444 to 411.
+ The following are the details of the observations on the Spider Monkey:-
1. First toe, ..... 17.5 per cent.
2. Second toe, .... 19.3
3. Third toe, ... 24.7 m
4. Fourth-toe, . . . . 22.7.
5. Fifth toe, .. .. 15.8 »
Total, .... 100.0
Flexor hall. long., :: 47.4 per cent. -0,24 oz. av.
Flexor dig. long., . . 40.3 1 . 0.15 is
Friction, &c., • ... 12.3 m
Total, .... 100.0
The relative strengths of the tendons of the toes are very similar to those
the Negro Monkey (p. 95), and very unlike those of the Macaque (p. 94). The
ft. hall. long. flexes the first, second, third, and fourth toes; while the fl. dig.
long. flexes the first, second, third, fourth, and fifth. This arrangement resem-
bles that of the flexor tendons of the Capuchin and Negro monkeys, in this re-
spect, that the entire flexion of the fifth toe devolves on the f. dig. longus.
120
ANIMAL MECHANICS.
The preceding analysis of the comparative strengths of
the flexor tendons of the f. hall. long and A. dig. long., and of
the toes, proves, that of thirty-five animals examined, the ten-
dons of the toes exceed the tendons of the muscles in thirty
instances, and fall short of them in four instances only. This
is a result that is, à priori, probable ; for the use of the foot
is not, like that of the hand, to grasp an object by means of
the flexion of the digits produced by the muscles of the fore-
arm; but, the toes being applied to the ground as a fulcrum,
the muscles of the calf acting on the flexor tendons of the
toes (which are fixed), resist their own forcible extension by
the weight of the body, by means of the tendons passing round
the heel, and so act as a spring to prevent the foot reaching
the ground with shock. In other words, in the foot, the
force is applied from below, upwards, to the muscles through
the intervention of the tendons; whereas, in a grasping organ,
such as the hand, the force is applied from above, downwards,
from the muscles to the obstacle, or resistance to be overcome.
The difference of strengths, or cross sections in all cases
represents the friction overcome by the tendons; while the
direction in which the force passes is always from the greater
towards the lesser cross section of the tendons.
.
B. Comparative Anatomy of the Flexor Tendons of the Hand.
In the following observations on the comparative anatomy
and mechanism of the flexor tendons of the fore foot, or hand, I
shall confine myself to the deep flexor tendons of the fingers, as
this method will furnish an instructive comparison with the cor-
responding long flexor tendons of the toes, already discussed.
There is one remarkable difference to be noted between the
long flexors of the toes and the deep flexors of the fingers ;
viz., that although both become simplified, as we descend in
the scale of animals, into a common tendon giving off similar
slips to each toe or finger; yet in the leg, the flexor muscles
ANIMAL MECHANICS.
121.
N
do not undergo the same fusion as in the forearm, or as their
tendons undergo. Thus, we always find the A. hall. long. to
be a muscle quite distinct from the fl. dig. long., even when
they act by a united tendon ; and, on the contrary, in the
forearm when the tendons of the A. poll. long, and A. dig. prof.
become united, we generally observe these muscles to become
united also. This difference in the muscles of the forearm
and leg, I believe to be due to a mechanical necessity. In
the propulsion of the body forwards by means of the hind
foot, it is necessary to press the inner toe (hallux) outwards
and backwards against the ground; a motion that is effected
by the separate action of the A. hall. long. muscle, and ma-
terially aided by the action of the peroncus longus muscle,
which lifts upwards and inwards the outer side of the near end
of the foot. Such an action of the fore foot is seldom neces-
sary or desirable, and hence the differentiation of the fl. poll.
long., when it exists, is made to increase the grasping or
hand-like action of the thumb, and not for the purpose of pro-
gression; and is, therefore, found to be less constant in its
occurrence in various animals than the flexor hallucis longus.
(a). Mun. The differentiation of the thumb from the
other fingers is carried farther in man than in any other
animal, because in man the appropriation of the fore foot to
the use of the brain is carried farther. The A. poll. long. is
quite distinct from the f. dig. prof., and in the latter muscle,
the tendon of the index finger is usually distinct from the
tendons of the third, fourth, and fifth fingers, far above the
wrist. It is a curious fact that in the apes, that in other re-
spects most nearly resemble man, the differentiation of the
action of the thumb is most imperfect. Thus, in the Chim-
panzee (Troglodytes Aubryi), Professor Gratiolet found that
the thumb is bent by an oblique slip of the tendon of the
muscle that bends the other fingers ; it is, therefore, influenced
by the common movements of flexion, and consequently not
t
122
ANIMAL MECHANICS.
free. This type, according to Gratiolet, is realized in the
Gorilla as well as in the Chimpanzee : the small tendon which in
these animals flexes the thumb is reduced to a tendinous thread,
which exerts no action, for its origin is lost in the synovial
folds of the tendons which bend the other fingers, and it abuts
on no muscle. The thumb, therefore, in these apes is won-
derfully enfeebled, and in none of them is there a trace of the
large independent muscle (A. poll. long.) which gives move-
ment to the human thumb; and, far from becoming more
strongly developed, the thumb so characteristic of the human
hand seems in the most elevated apes (Orang) to incline to a
complete annihilation. These apes have, therefore, nothing
in the organization of their hand which indicates a passage
into the human form.
I have never had an opportunity of dissecting the hand of
a Gorilla or of an Orang, but I have dissected the hand of
the Chimpanzee (Troglodytes niger). In this animal I found the
flexor tendon of the thumb to be formed by the union of two
small thread-like tendons; one of which, of silken lustre, was
derived from the portion of the fl. dig. sublimis muscle cor-
responding to the index finger, and the other, equally slender,
but wanting the silken lustre, proceeded from the tendon of
the fl. sublimis of the little finger.
The peculiarities of the human ft. poll. longus have always
been insisted upon by anatomists as essentially characteristic
of Man, as distinguished from the Quadrumans; but excep-
tions of the most startling kind are occasionally met with. In
1864, a male subject was brought into the dissecting-room of
Trinity College, one of the hands of which was dissected by
Mr. Finny, Med. Schol., who made the following note at the
time:-
“ Abnormal flexor pollicis longus. The fleshy origin of this
muscle from the bone of the forearm was entirely wanting, and
the tendon of the fl. poll. long. was attached opposite to the
O
U
ANIMAL MECHANICS.
123
base of the third metacarpal bone, to the tendons of the fl. dig.
profundus, on their superficial surface.”
This remarkable arrangement of the flexor tendons of the
hand is identical with that found in the Macaques, and other
monkeys of the Old World, shown in Fig. 17(a). Whether
this man was a Macaque passing upwards into a man, or a man
passing downwards into a Macaque, must be decided by the
reader, according as his sympathies are with Lamarck or
Buffon.
(6). The Quadrumans. The peculiarities of the thumb in
the higher apes have been already alluded to; in the lower
monkeys a remarkable difference is found in the deep
flexors of the fingers, between the monkeys of the Old and
New Continents. This difference is shown in Fig. 17, in


SA
RP
ges
V
U
.
6.
Fig. 17.
which (a) represents the distribution of the flexor tendons in
the hand of the monkeys of the Old World, and (b) their dis-
tribution in the hand of the monkeys of the New World. If
we take the Macaque (Macacus Nemestrinus) (Fig. 17, a), as
the type of the Old World monkeys, we find the following
arrangement. There is no distinct f. poll. long, muscle, but a
tendon branches off to the thumb, starting from the central
124
ANIMAL MECHANICS.
portion of the common tendon of the A. dig. prof., as shown
in the drawing.
If we call the cross section of the common tendon 100, we
find the cross sections of the thumb and four fingers, as fol-
lows:
1. Macaque (deep flexors of fingers).
Flexor dig profundus, ... 100
1. Thumb, ...., 19.3 per cent.
2. Other fingers, ... 40.0 %
3. Deficiency, .... 40.7 .,
Total, ... 100.0
In the hand of the Sooty Mangabey (Cercocebus fuliyinosus)
I obtained the following results :-
2. Sooty Mangabay (deep flexors of fingers).
Flexor dig. prof.,
...
100
1. Five fingers, ... 50.8 per cent.
2. Deficiency, ... 49.2 99
Total, ..100.0
In the monkeys of the New World (Fig. 17 b.), a more
degraded type of thumb is found to exist; for the flexor ten-
don of the thumb proceeds, not from the central and upper
portion of the common flexor tendon of the fingers, but from
the side of that tendon, as shown in the drawing; an arrange-
ment which reduces the thumb, as in animals lower than
monkeys, to a simple finger. In some of these monkeys
also the thumb itself becomes rudimentary, and is not fur-
nished with any flexor tendon whatever. The A teles, or
YO
Y W
ANIMAL NIECHANICS.
125
Spider monkey, owes its scientific name to this circum-
stance.
The following observations show the relative cross sec.
tions of the flexor tendons of the hand in several of the New
World monkeys.
3. Spider Monkey (deep flexors of fingers).
Flexor dig. prof., . .
100
INT
1. Second finger, . .. 18.4 per cent.
2. Third finger, . , 20.4 ,
3. Fourth finger, . 020.4
4. Fifth finger, .. . 14.3
5. Deficiency, · · · · 26.5
Total, . 100.0
4. Negro Monkey (deep flexors of fingers),
Flexor dig. prof.,
. • 100
1. Five fingers,
2. Deficiency, .
.
. 72.6 per cent.
27.4 ?
Total,
... 100.0
5. Capuchin Monkey (deep flexors of fingers).
Flexor dig. prof., . . 100
1. Five fingers, .64.7 per cent.
2. Deficiency, · · · · 35-3 »
Total, . 100.0
126
ANIMAL MECHANICS.
-
(c). The Carnivores. I have examined the deep flexor
tendons of the following Carnivores :-
1. The Tiger.
2: The Wolf.
3. The Pyrenean Mastiff.
4. The Bengal Fox..
5. The Virginian and Japanese Bears.
1. Tiger (deep flexors of fingers).*
Flexor dig. prof., ..
100
1. First finger, · .. . 23.4 per cent.
2. Second finger, ... 15.5
3. Third finger, ... 12.7 1
4. Fourth finger, . . 12.5
5. Fifth finger, ... 11.8
6. Pad refractor, ... 1.4
7. Deficiency, . o 22.7
Total, ... 100.0
The Tiger has no f.poll. long. distinct from the f. dig. long.
muscle, and the thumb is therefore flexed by the same mus-
cular effort as the other fingers; the strength of the tendon
that flexes the thumb, as compared with that of the tendons
of the other fingers, is remarkable, and shows the great im-
portance of this digit in the grasping action of the animal's
paw.
There are some circumstances in the history of this fine
tiger, which throw light upon the details of muscular action,
and which are, in themselves, sufficiently interesting to jus-
tify a slight digression. The tiger and tigress, brother and
.
* These measurements were taken from the left paw
ANIMAL MECHANICS.
127
11
A
sister, arrived in the Dublin Zoological Gardens on the 22nd
March, 1860, being then about two years old ; they were a do-
nation to the Gardens from Major-General Montgomery (Bom-
bay Army), who had shot their mother and carried home the
two cubs, and afterwards kept them in his bungalow for nearly
one year. While under General Montgomery's care they had
become playmates to his children, and it is a remarkable fact,
that when visited by two of these children in the Dublin Gar-
dens, after nearly two years' absence, both animals recognised
their former friends, and expressed their satisfaction by loud
purring and by licking the children's hands gently with the
tips of their tongues, being well aware that the rough surface
of the tongue would cause pain. The claw of the fourth
finger of the right paw of the tiger grew into the finger pad
of the foot early in 1866, and was removed by me in May of
that year; afterwards the claws of the second, third, and fifth
fingers grew into the finger pads, and were cut by me in
January, 1867. On both occasions the tiger showed great
anger at the pain that was caused by the operation, but shortly
afterwards insisted on licking my hands, expressing his desire
to do so by constant purring and by rolling himself upon his
back, in the manner of a kitten at play.
It is worthy of remark that it required the united strength
of eight men, to hold the tiger during these operations, al-
though I have seen a similar operation performed easily upon
a large African lion, with the help of five men.
This tiger died on the 20th February, 1869, of malignant
putrid fever, after forty-eight hours' illness, and on dissecting
his muscles, I paid particular attention to the relative deve-
lopement of the deep flexors of the right and left paws. The
only claw that had again grown into the pads of the foot was
that of the third finger of the right foot.
128
ANIMAL MECHANICS.
Bengal Tiger (deep flexor muscles).
Weight
Right paw (overgrowth of claws), 10.87 oz. av.
Left paw (uninjured), · · · · · 13.30
In all the other muscles of the right and left forearm, there
was a slight preponderance of the muscles of the left (unin-
jured) side over those of the right side, but it was not so
marked in any as in the deep flexors of the fingers, and in
the long palmar muscle, which were directly involved in the
injury occasioned by the overgrowth of the claws. The abso-
lute and relative weights of the muscles of the two forearins
are shown in the following table :-
TABLE XXIII.--Comparison of the Weights of the Muscles of
the two Fore Paws of a Bengal Tiger, in which the Claws of
the Right Foot had grown into the Finger Pads.

Name of Muscle.
Left Paw.
Right Paw.
Patio
96 per cent.
Weight.
oz. av..
3:51
2.12
4.72
5.57
0.44
Weight
OZ. AV.
3.35
2.06
3.07
5.45
0.37
10.87
1. Pron, rad. teres, ..
2. Fl. carp. rad., ...
3. Palm. Iong., ...
4. Fl. carp. uln., . . . o
Fl. dig. subl., , .
Fl. dig. prof., ...
Fl. poil long., ...
Pron. quadr., ...
Sup. rad. long., ...
10. Ext. carp. rad. longr.,
II. Ext. carp. rad. breyr.,
12. Sup. rad. brev.,
13. Ex dig. long., . . .
14. Auricularis, .
15. Ext. carp. uln.,' ..
16. Ext. oss. met. poll., .
17. Indicator, ....
13:30
1.99
3.50
4.20
3.30
1.II
2.60
1.40
maior
ini minni
i mincinnrinio
Owo
ca cola fin con Ow
0.60
It appears from the foregoing Table that the muscles that
• ANIMAL MECHANICS,
129
suffered most from the injury produced by the ulceration of
the foot pads, were--
Palmaris longus, . . 65 per cent.
Flexor dig. longus, · · 81
Ext. oss. met. poll., . . 82 »
2. In the European Wolf, I obtained the following re-
.
sults:
Wolf (deep flexors of fingers).
Flexor dig. prof., . . 100
1. First finger, ... .7.0 per cent.
2. Second finger, ... 19.1 ,
3. Third finger, ... 12.8
4. Fourth finger, ... 12.7
5. Fifth finger, ... 17.
0
6. Deficiency, · · · · 31.4
100.0
3. In the Pyrenean Mastiff, the following results were
found:-
Pyrenean Mastiff (deep flexors of fingers).
Flexor dig. prof.. . 100
1, First finger, .... 8 per cent.
2. Second finger, ... 19
3. Third finger, ...; 24
4. Fourth finger, i. 24
5. Fifth finger, .... 18
6. Deficiency, .... 7
Total, . . . 100
130
ANIMAL MECHANICS.
The flex. poll. long. muscle is not distinct from the fl. dig.
prof. in either the Wolf or Dog, while
in the Fox it constitutes a small but dis-
tinct muscle, one-tenth of the weight of
the deep flexor, having a tendon of its
own, that joins the common tendon of the
deep flexor at the wrist. This arrange-
ment of tendons in the paw of the Fox is
shown in Fig. 18, and constitutes an addi-
tional distinction, separating the Fox from
the Dog and Wolf.
4. The Fox examined by me was a
Bengal Fox, which gave the following
measurements:

fl.dig.long.
.
fla poll. long.
TOA
Fig. 18.
7
Uuuu
Bengal Fox (deep flexors of fingers).
Weight.
Flexor dig. prof., · · 97.5 per cent. : 0.15 oz. av.
Flexor poll. long., . , 2.5 . : 0.015 »
Total, .. 100.0
1. First finger, . . . . 2.5 per cent.
2. Second finger, .... 20.7 »
3. Third finger, .. 19.5 »
4. Fourth finger, ... 18.3 m
5. Fifth finger, .. 18.3
6. Deficiency, . . . . 2017 ,
Total, ,100.0
The tendon of the f. poll. long. acts upon the common
deep flexor tendon in the direction of the prolongation of the
ANIMAL MECHANICS.
131
tendon of the thumb, and must exert, by its oblique pull, a
distinct action upon that digit.*
5. In the Virginian and Japanese Bears I found the fol-
lowing results :-
Virginian Bear (deep flexors of fingers).
Flexor dig. prof., ... 100
1. First finger, .... 9.1 per cent.
2. Second finger, . . . . 15.9 .
3. Third finger, .... 16.9 »
4. Fourth finger, · · ·· 15.3
6. Fifth finger, . ... 7.8 ,
7. Deficiency,. . . . . 35.0 ,
Total, ... 100.0
Japanese Bear (deep flexors of fingers).
Flexor dig. prof., .. 100
G
1. Five fingers, .... 69.4 per cent.
2. Deficiency, . . . . . 30.6 1
Total, · 100.0
(d). The Ungulates.—The Ungulates examined by me, in
reference to the relative strengths of the tendons of the deep
flexors of the fingers above and below the wrist, were the Goat
and the Llama. I examined the wrist tendons of the Rhinoce-
ros also, and found them to resemble those of the foot already
described, both as to arrangement and relative cross sections.
* It has been suggested to me by Dr. Templeton, that the f. poll. long.
enables the Fox to climb, an action of which Dogs and Wolves are incapable.
K 2
132
ANIMAL MECHANICS.
1. In the Bombay Goat I found the following results :-
S
Flexor dig. profundus, .
100
1. First finger, ... 49.2 per cent.
2. Second finger, · · 50.8
Total, ... 100.0
2. In the Llama, the following cross sections were ob-
tained :- .
Flexor dig. prof.,
Friction, foc., ·
. 85.6 per cent.
14.4
Total,
.
100.0
'I. First finger, . . . 49.1 per cent.
2. Second finger, . . 50.9
Total,
... 100.0
0
In the hands of the animals previously described, the
cross section of the deep tendons of the fingers was less than
that of the common tendon above the wrist; which is the es-
sential characteristic of a hand as distinguished from a foot.
In these Ungulates, however, we find that in the Goat, the
cross sections become equal, and that in the Llama, the cross
section of the finger flexors becomes greater than that of the
common tendon ; thus reducing the fore-foot of this animal
to the condition of a true foot, and degrading it from the
dignity of a paw, or hand.
(e). The Rodents. The only rodent, of which I have
ANIMAL MECHANICS.
133
examined the flexor tendons of the hand, is the Porcupine,
which gave me the following results :-
Porcupine (deep flexors of fingers).
Flexor dig. prof., ... 100
1. Second finger, . ... 7.7 per cent.
2. Third finger, . .. 7.6
3. Fourth finger, . ... 7.1 ,
4. Fifth finger, ... 6.6 »
5. Deficiency,. ,... 71.0
Total, ... 100.0
In the Porcupine, no slip from the deep flexor tendon is
given off to the thumb, and there is no A. poll. long. muscle.
It will be remembered (p. 110), that the foot of the porcupine
resembles a hand, in having the cross section of the flexor
tendons greater above the heel than the cross sections of the
tendons of the toes; and it is remarkable, that in the hand of
the same animal, the difference between the similar tendons
is greater (71 per cent.) than in any other hand examined by
me.
(f). The Marsupials.-In the hand of the Wallaby Kan-
garoo, I found the following cross sections :-
•
Five fingers, . . . . .
100
1. Flexor dig. prof., ...
2. Friction, 8c, .. .
98.3 per cent.
1.7 »
Total, ... 100.0
134
ANIMAL MECHANICS.
· If we now collect into one Table, the observations made
upon the flexors of the hand, we shall find the difference be-
tween these and the flexors of the foot to be very remark-
able (vide Table XXII.)
TABLE XXIV.- Comparative View of the Friction of the Deep
Flexors of Fingers.

(A.) P > P'; or Cross Section of Tendons of Fingers greater
than Cross Section of Tendons of Muscles.
Friction.
1. Llama (Llama yamma), . . . . . . . .
2. Wallaby Kangaroo (Halmaturus Valabatus), .
3. Goat (Hircus ægagrus), ........
14.4 per cent.
1.7
0.0

P'; or Cross Section of Tendons of Fingers less
than Cross Section of Tendons of Muscles.
Friction.
0.0 per cent.
7.0
.
3. Goat (Hircus ægagrus), . .
4. Mastiff (Canis familiaris), .
5. Fox (Vulpes vulgaris), . . . . . .
6. Tiger (Tigris regalis), dom' .
7. Spider Monkey (Ateles ater), . .
8. Negro Monkey (Lagothrix Humboldtii), .
9. Japanese Bear (Ursus Japonicus), . . .
10. Wolf (Canis lupus), . . . . . . .
11. Virginian Bear (Ursus Americanus), . .
12. Capuchin Monkey (Cebus Capucinus), . .
13. Macaque (Macacus Nemestrinus), :
14. Sooty Mangabey (Cercocebus fuliginosus), .
15. Porcupine (Hystrix cristata), . . .
31.4
.
40.7
49.2
1.71.0
The preceding Table demonstrates the fact, that in the
hand (or fore foot) of most animals, the cross section of the
tendon of the deep flexor, at the wrist, is greater than the
united cross sections of its branches subdivided to the various
fingers. In fact, the only animal, out of fifteen examined, in
which the reverse condition takes place, is the Llama, whose
fore foot, in its structure and mode of action, may be regarded
as being more truly a foot than a hand,
ANIMAL MECHANICS.
135
a
The following considerations will serve to show more
clearly, the important mechanical differences involved in the
Tables of cross sections of the flexors of the hand and foot.
The mechanism of a true hand, or grasping organ, may be
thus illustrated.
Let OA, AB, and BC, represent three bars turning upon
hinges at A and B, and acted upon by a muscle Om, attached

-
-
22.
C
1
Fig. 19.
to a tendon may, this tendon being inserted at y, and pass-
ing through a loop or pulley at X. When the muscle Om
contracts, the system of bars will take the position OA'B'C',
which is more flexed than the position OABC, or in other
words, the angles at A and B' are smaller than the angles at
A and B. In this movement of flexure, the muscle Om is
shortened, while the tendon mxy remains equal in length to
the tendon m'x'y ; but since the angle at B' is smaller than
the angle at B, the base of the triangle, y'a', is shorter than the
base yx, and therefore the tendon is drawn through the loop
&" in the direction of the arrow, towards the point 0; hence
the friction of the loop x', which is opposite to the direction
of the motion, assists the part of the tendon y'a', which is
situated below the loop. If therefore the force of the muscle
be denoted by P', and the friction of the loop by F'; we shall
136
ANIMAL MECHANICS.
have for the strength of the tendon y'x', the equation (vide
p. 83).
P = F + P.
(15)
The tendon a'm' above the loop must be made sufficiently
strong to resist the force of the muscle P'; while the tendon
y'x below the loop need only have a strength sufficient to re-
sist the force
P-P- F;
or, in other words, the cross section of the tendons of the
fingers of a hand, will fall short of the cross section of the
tendons at the wrist, by a quantity that represents the friction
of these tendons in passing through their various sheaths to
be distributed to the ends of the fingers.
reverse of that of a true hand. When the toes of the foot
reach the ground, they are partly flexed, as at OA'B'C', and
when the point C' touches the ground, the weight of the body
unbends the toes, changing them from the position OA'B'C
to the position OABC. The muscle Om' resists the unbend-
ing of the toes, and is forcibly extended to the length Om,
acting thus as a sort of spring, and preventing the sudden
shock that would otherwise be given to the body each time
that the foot reaches the ground. Hence it is evident, for
the reasons already given, that the tendon y''m' (which is in-
extensible), must be drawn through the loop x, in a direction
opposite to that of the arrow, and that the friction of the loop
acts towards 0, helping the portion of the tendon am situated
above the loop; hence in this case we have
P + F-P;
(15)
from which it appears that the cross section of the tendons of
the toes must exceed the cross section of the flexor tendons
at the heel, by an amount (F) representing the friction of the
sheaths.
ANIMAL MECHANICS.
137
1
7. On the Mechanical Work done by the Human Heart.--The
heart constitutes so peculiar and so important a muscle, that it
is worthy of a distinct consideration. It contracts constantly
by means of a self-acting nerve supply, and works day and
night while life lasts, without ever experiencing a sense of fa-
tigue, or the necessity for rest. Let us examine what is the
mechanical equivalent of the work done by an average human
heart during the course of a single day and night:-
Let Q denote the weight of the quantity of blood that is
expelled from the left ventricle into the aorta at each stroke
of the heart, and let h denote the height of a column of blood
that would measure the fluid pressure inside the ventricle at
the moment of contraction. It follows, from the hydraulic
laws of spouting fluids, that the blood Q, if thrown vertically
upwards, would ascend to the height h, so that the work done
in a single stroke of the left ventricle is represented by the
product
Qxh = Work done.
We have now to estimate the values of the quantities Q.
and h:-
The estimates given by various anatomists of the capacity
of the left ventricle vary from 2 to 5 ounces (water), the
former estimate being that of the empty ventricle, which ad-
mits of considerable distention, especially during life. It is
extremely difficult, if not impossible, to say what is the amount
of distention experienced by the left ventricle before it con-
tracts upon its contents, and expels them. From some ex-
periments that I have made, it seems to me probable that the
distention does not amount to more than fifty per cent., and
that we shall not be far from the truth if we take to be
3 ounces of blood, neglecting the difference of specific gravity
between blood and water.
138
ANIMAL MECHANICS.
Professor Donders* assumes the capacity of the left ven-
tricle to be 188 grammes, which are equivalent to 6.631 oz. av.,
a result which seems to me more like the capacity of both
ventricles, than that of one only.
. In order to find the quantity h, I shall employ the experi-
ments of the Rev. Dr. Hales.f Having opened a large artery,
such as the crural or carotid, in several animals, Dr. Hales in-
serted a brass or glass tube into the opening, and observed the
height at which the blood stood before the animal had lost
any large quantity of blood. The following are some of his
results :-

No.
Artery.
Height in Tube.
.
8.25 feet.
1. Mare, . .
2. Gelding, .
3. Mare, . .
4. Wether, . .
5. Fallow Doe,
6. Dog,. . .
.
.
.
.
Left Crural, ..
Left Carotid, . :
Carotid,
Left Crural,
Crural,
o adivo to
..
.
.
An important fact was ascertained by Dr. Hales during
the course of these experiments, viz., that when some of the
blood was allowed to flow away, the force of the heart's con-
traction diminished, no doubt in consequence of the dimi-
nished resistance offered by the diminished circulation; the
heart, as it were, instinctively measuring the work to be done,
and expending upon it exactly the force required. The height
of blood in the tube, which measured the force of the heart's
contraction, became less and less as the blood was allowed to
flow off, until it reached a minimum height, when the animal
expired. This minimum, carefully measured in the case of
the horses, was found to be-
* Physiologie des Menschen (Leipzig, 1859), p. 109.
+ Statical Essays (London, 1769), vol. ii.
ANIMAL MECHANICS.
139

No.
Maximum height
before loss of blood.
Minimum Height.
Total Blood lost.
8.25 feet.
2.33 feet.
924 cub. in.
I, Mare, . .
2. Gelding,. .
3. Mare, ..
2.83 »
2.42 »
895
1
9.50
1
833
Mean,
.
9.14 feet.
2.53 feet.
884 cub. in.
It appears from the foregoing table that the minimum con-
traction of the horse's heart, corresponding to the resistance
offered by a circulation almost drained of its blood, amounts
to the pressure of a column of blood 2.53 feet in height. Dr.
Hales has also recorded the important fact, that the crural or
carotid artery of the horse, when first cut and allowed to spout
freely, does not throw the blood much higher than 2 feet.
It would appear from this observation that the cutting and
free exposure of a cut artery to the air relieves the heart at
once of the resistance offered by the capillary circulation, for
otherwise it is difficult to understand why the blood should
not spout upwards through 9 feet instead of somewhat over
2 feet.
Experiments on the pressure of the blood inside the human
heart, similar to those of Dr. Hales on animals, have not yet
been made; but it is estimated at 7.5 feet by Dr. Hales from
considerations founded on the comparative bulks of the soft
parts of man and of the horse supplied by the arterial circula-
tion.
The following observation and calculation of the velocity
of blood spouting from a large artery in man, lead, as I be-
lieve, to a more correct estimate, although indirect, of the
pressure of blood inside the left ventricle.
On the 18th of March, 1863, a large fibro-cellular tumour
was removed by Mr. M. H. Colles, in the operating theatre of
the Meath Hospital, from the left groin of a middle-aged,
140
ANIMAL MECHANICS.
large-sized man; in the course of the operation the external
epigastric artery, which appeared enlarged to feed the tumour,
was divided, and before it could be ligatured, strong jets of
blood from it were thrown in various directions about the
room. I noticed, as the poor fellow moved about on the
operating table, that the jets of blood fell short, or enjoyed a
long range, according to the angle of elevation of the orifice
of the bleeding artery, and that there was a certain maxi-
mum range on the floor of the theatre, which was not
exceeded.
I saw immediately that I had before my eyes the solution
of the problem that had puzzled me; as by measuring the
co-ordinates of the maximum range, I could calculate the
velocity with which the blood left the artery.
In the annexed figure, which is drawn to scale, let A re-
present the orifice of the artery, and B the position of the far-

Przy
R 57°7'
65° 46'
FT.
&
8 FT.
OLS FT.
Fig. 20.
thest squirt of blood on the floor; OA was found to be 3 ft.
6 in., and OB was 8 ft. It is well known that AX, the direc-
ANIMAL MECHANICS.
141
tion of projection of the maximum range on the plane AB,
bisects the angle CAB, and that
2-R cos? ¿
(16)
4 cos?
where
h = the unknown height due to the velocity of projection ;
R = the maximum range AB = 8.773 ft.
i = the angle made by AB with the horizon = 24° 14';
e = the angle of elevation, made by AX with the horizon = 32° 53'.
Substituting the preceding values in equation (16), I find
h = 2.586 ft.;
and since the velocity of projection, v, is given by the equa-
tion,
v = V 2gh;
(17)
it follows that
v = 12.905 ft.
The result here found shows that the heart of man, when
experiencing the minimum of resistance, such as that offered
by the circulation when a large artery is divided, or the vessels
nearly empty of blood, contracts with a force measured by
2.58 ft. of a vertical column of blood. The heart of the horse,
under similar circumstances, contracts with a force measured
by a column 2.53 ft. in height.
The following considerations lead us to the conclusion that
it is probable that the maximum hydrostatical force of man's
heart is nearly the same as that of the horse. The experiments
of Dr. Hales, fully corroborated by the recent observations of
Poiseuille, show that the hydrostatical pressure of the blood is
the same in all the arteries of the same animal, that have a
sensible diameter ; from which fact, it is plain that the resist-
ance to the circulation of the blood takes place in the capil.
lary arteries and veins. If therefore we knew the relative
142
ANIMAL MECHANICS.
resistances offered in man and in the horse, to the capillary
circulation, we could estimate the relative forces with which
the blood is propelled in the circulation of these animals.
• The experiments of Poiseuille on the discharge of liquids
through capillary tubes, prove that the resistance offered by
such tubes is directly proportional to the lengths of the tubes,
and inversely proportional to the squares of their cross sec-
tions.
The quantity of liquid discharged by a capillary tube in a
given time is inversely proportional to the resistance, and may
be expressed in general by the following formulam
hdd
Q = AXT
(18)
·
In this expression, Q denotes the quantity of liquid discharged
in a given time; A is a constant; h denotes the charge, or
hydrostatical pressure of the fluid ; and d and I are the diame-
ter and length of the capillary tube.
There is reason to believe that in animals similar in bulk,
the arrangement and structure of the capillaries are such,
that the ratio of the squares of their cross sections to the total
lengths of the capillaries, is practically constant; as may be
shown by a comparison of the sheep and the dog ;-the data
concerning which animals I take from Dr. Hales’ observations
and experiments. The left ventricle of the heart of the sheep
contains 1.85 cubic inches, and its pulse beats 65 times in a
minute; the quantity of blood passing through the capillaries
in a given time being proportional to the product of these two
quantities. The hæmastatical pressure of the blood in the
large arteries was found by Dr. Hales to be 6.46 ft. of blood.
: Equation (18) gives us, for the coefficient depending on
the capillary resistance,
. d Q 1.85 x 65 -
AX=
= 18.6.
A* 7 h
=
6.46
ANIMAL MECHANICS. .
143
The average of the capacities of the left ventricles of the
hearts of six dogs examined by Dr. Hales is 0.954 cub. in..;
and the average statical pressure in the arteries of sixteen
dogs experimented on by him is 4.75 ft. of blood; and the
average pulse of the dog is found to be 97 beats in a minute.
Hence equation (18) will give us, for the coefficient depeni.
ing on the capillary resistance in the dog,
du Q
Ax7-ī
0.954 * 97 – 10.6.
4.71
These results, depending solely on the resistance offered by
the capillary vessels to the flow of blood, show by their
near approach to equality, that in animals, like the dog and
sheep, we may safely assume the resistance offered by the
structure of the capillaries to be nearly the same.
The sheep and dog differ from each other considerably in
size of heart, rate of pulse, and hæmastatical pressure of blood;
yet, notwithstanding these differences, the capillary coefficienc
depending on them all comes out to be nearly the same in
both animals. The capillary coefficient, found by experiments
made on the horse, comes out to be double that of the dog
and sheep, showing that the resistance to the circulation in
the horse is only half that of the smaller animals.
The left ventricle of the horse contains 10 cub. in., and
the rate of pulse is 36 beats in a minute, while the average
hæmastatical pressure of the blood in the arteries is 9.14 ft. of
blood; hence, we obtain for the capillary coefficient in the
horse,
du Q 10 x 36
Axņ= * = 0.13° = 39.3.
This quantity is double that found for the dog, and there-
fore the capillary resistance is one-half.
Dr. Hales did not make any observations on the hæma-
144
ANIMAL MECHANICS.
statical force of the arterial blood in the ox, but there is rea-
son to believe that it is rather greater than in the horse, and
nearer to 10 ft. than 9 ft. If we are at liberty to assume it to
be 9.66 ft., which is the greatest height measured in the horse,
we can calculate the capillary coefficient as follows:- In the
ox, the capacity of the left ventricle was found by Dr. Hales
to be 12.5 cub. in., and the pulse beats 38 times in a minute.
Hence the capillary coefficient in the ox is-
"
Ax =
Q
12.5 x 38
h
0.66
= 39.8.
This coefficient is nearly the same as that of the horse, an
agreement which serves to show that the capillary resistance
depends more on the bulk of the animal than on any other
condition.
If we suppose the coefficient of capillary resistance to be
the same in man as in the horse, we can calculate the hæma-
statical pressure of the blood in the human arteries, as fol-
lows :-The human heart has a capacity in its left ventricle,
when in action, of 3 ounces, or 5.2 cub. in.; and beats 75 times
in a minute. Solving equation (18) for h, we have
Inn
Axi
Substituting for the product of the capacity of the ven-
tricle and rate of pulse, and for the capillary resistance, its
value found for the horse we obtain
h = 5.2 * 75 = 0.923 feet of blood.
39.3
This is the value that I shall assume for the arterial pressure
in man. This estimate of the blood pressure approaches to
that of Donders, * which is 10.527 ft. (3.21 metres).
* Physiologie, p. 109.
ANIMAL MECHANICS.
145
2 3 * 9.923-x 75 x 60 x 2
CT
We are now in a condition to calculate in foot tons, the
daily work done by the left ventricle of the human heart, which
lifts at each stroke 3 ounces through a height of 9.923 ft. Its
value is the following :-
Daily work of left ventriele
24 = 89.706 ft. tons.
16 x 2240
I find, on taking an average of the estimates of Bouillaud,
Laennec, Cruveilhier, Sömmering, and Andral, that the right
ventricle has a thickness, which is to that of the left in the pro-
portion of 5 to 13; and as the cavities are of equal dimensions,
the thicknesses are proportional to their muscular powers ; in
order, therefore, to find the total work done by both ven-
tricles of the heart, we must add to the preceding result ths
of its amount, from which we obtain finally,
Daily work of left ventricle, ... . 89.706 ft. tons.
Daily work of right ventricle, .... 34.502 ?
Total daily work of both ventricles, 124.208 ft. tons.
The total work done by the heart is somewhat greater than
this estimate, for the auricles of the heart do some work, in
propelling the blood into the ventricles; of which work no
account has been taken in the preceding calculation.
The average weight of the human heart is found from the
following Table :-
TABLE XXV.---Table of Average Weight of Human Heart.
2

Ounces.
1. Meckel, ...
2. Cruveilhier, ..
3. Bouillaud, ..
4. Lobstein, ..
5. Boyd [et. 30-40],
6. Boyd [et. 40-50], . .
10.0
7.5
8.375
:0.41
10.36
Mean = 1 :
9.39 02.
-
-
146
1
ANIMAL MECHANICS.
From the weight of the heart and the work done by it in one
day, we can calculate the work done by the heart per ounce,
as follows:-
Work done by the human heart in foot lbs. per oz. per minute,
_124.208 x 2240
9:39 x 24 x 60 – 20.576.
On comparing this result with the work done by other
muscles, given in Tables VII. and VIII. (p. 62), we observe
that the work done by the heart in a given time (20.576) ex-
ceeds considerably the work done by the muscles in a boat
race (15.17). In a boat race, the labour can be maintained
for a few minutes only; while the heart labours at a greater
rate throughout the whole of life.
There is another mode of exhibiting the wonderful energy
of the heart which is worth stating, more especially as it
affords a means of comparing my own result with that of Pro-
fessor Helmholtz, the data of whose calculation I am not ac-
quainted with. Let us suppose that the heart expends its
entire force in lifting its own weight vertically, then the
height through which it could lift itself in one hour is thus
found, by reducing the daily work done in ft. tons to the
hourly work in foot ounces, and dividing by the weight of
the heart in ounces.
Height through which the human heart could raise its own weight in one
hour,
124.208 x 2240 x 16
= 19754 fto
24 x 9:39
Professor Helmholtz* states as the result of his own calcu-
lations, that the heart could raise its own weight through
20250 ft. in one hour. It has been already stated (p. 52),
that an active climber can ascend 9000 ft. in nine hours,
* "Medical Times and Gazette,” May 14, 1864, p. 529.
ANIMAL MECHANICS.
147
which is only at the rate of 1000 ft. per hour, or one-twentieth
part of the energy of the heart.
When the railway was constructed from Trieste to Vienna,
a prize was offered for the locomotive Alp-engine, that could
lift its own weight through the greatest height in one hour.
The prize locomotive was the “Bavaria,” which lifted herself
I through 2700 ft. in one hour; the greatest feat as yet accom-
plished on steep gradients. This result, remarkable as it is, is
only one-eighth part of the energy of the human heart.
Professor Donders* gives an estimate of the force of the
heart, which exceeds considerably the result found by Profesa
sor Helmholtz, and by myself. He assumes the contents of the
left ventricle to be 0.188 kil., or 188 cubic centimeters, or
11.473 cubic inches, a result intermediate between the capa-
cities of the hearts of the ox and horse, measured by Dr. Hales.
Measured in ounces, this capacity is 6.631 ounces, or more
than twice the capacity assumed by me. The blood pressure
is assumed by Donders at 3.21 meters, or 10.527 ft.; and the
force of the right ventricle to be one-third of that of the left
ventricle. The other elements of his calculation are the same
as my own. From these data, the daily labouring force of the
heart comes out equal to
87,400 kilogrammeters = 281.03 ft. tons.
This result, as might be anticipated from the assumed size of
the ventricle, is more than double that found by myself and
Helmholtz, from independent data.
It may be useful to the reader to state here the results of
observers, subsequent to Dr. Hales, as to the pressure of the
blood in the large arteries of several animals. Dr. Hales ob-
served the maximum pressure in the arteries, as measured by
* Physiologie, p. 109.
12
148
ANIMAL MECHANICS.
a vertical column of blood, which was noticed to rise and fall.
Subsequent observers have shown that in the anterior extre-
mity, the blood pressure depends on the respiration, being
greatest during expiration, and least during inspiration; and
that this inequality is less sensible in the arteries distant from
the heart. In order to compare the observations, it is desirable
to reduce Dr. Hales' column of blood to millimeters of mer-
cury. The weight of a cubic inch of blood is 267.7 grs.,
while that of a cubic inch of water is 252.5 grs. The specific
gravity of mercury is 13.596; and 9.14 feet (Dr. Hales' result
for the horse) are equal to 2785 millimeters. Hence we find
Maximum mean pressure of blood in the horse, according to Hales,
- 2705% 207.7. = 217.17 mm. of mercury.
13.596 X 252.5
Poiseuille* directed his attention especially to the deter-
mination of the mean pressure of the blood, independent of
expiration or inspiration ; but has left in his memoir (p. 295),
the data for calculating, in the case of the horse, the maximum
pressure corresponding to that observed by Dr. Hales. From
the experiment there recorded, on the carotid artery of a
horse, I have calculated the maximum mean pressure of the
blood during expiration, and find
Maximum mean pressure in the carotid artery of the horse, according to
Poiseuille,
= 229.75 mm. of mercury..
The minimum mean pressure, during inspiration, in the
same artery, may be calculated from the data given, at 63.61
mm. of mercury. The mean of all the maxima and minima
being 146.68 mm. of mercury. The maximum pressure found
by Poiseuille is a remarkable confirmation of the accuracy of
the experiments made by Dr. Hales.
S
* « Journal de Physiologie par Magendie,” tom. viii., p. 272.
ANIMAL MECHANICS.
149
Poiseuille (p. 301) gives the following as the mean of all
the pressures found in the horse and dog :-

No. 1.
Horse.
Dog.
146.68 mm.
147.00.99
157.25 »
154.33
182.05 »
fin fino roon
148.88 mm.
147.36 ,
141.45 13
157.39. »
145.75 »
166.60
179.04
141.40 %
171114 »
Mean,
157.46 mm,
155.44 mm.
Spengler* repeated the experiments of Poiseuille, and ob-
tained, among others, the following results from the arteries
of the horse. The mean maximum pressures during expira-
tion were,
1. First horse, . ..
229.31 mm. of mercury.
229.31 in
2. Second horse, .. 200.67
3. Third horse, ... 171.34
4. Fourth horse,. .. 170.00
5. Fifth horse, .. 220 00
6. Sixth horse, ... · 230.00
Mean maximum,
. 203.55 mm. of mercury.
This result agrees well with those of Hales and Poiseuille.
S
U
* « Archiv für Anatomie,” &c., von Müller (1844), p. 49.
150
ANIMAL MECHANICS.
The mean of all the pressures, including maxima and
minima, were found by Spengler (pp. 58, 9), to be,

Horse.
Dog.
Goat.
170 man.
185 mm.
140 mm.
Morf
175 29
160 19
165 1
135
150
165
1
12
Mean,
1675 mm.
158.75 mm.
140 mm.
The rate at which the blood circulates through the body
may be estimated from the size of the left ventricle, as com-
pared with the total quantity of blood in the body. The total
quantity of blood in the body has been found by weighing,
before and after death, criminals executed by decapitation.
From such observations made by Lehmann, Weber, and
Bischoff, the total quantity was found tô range from 9 lbs. to
10 lbs. If we assume the latter quantity, and take the capa-
city of the left ventricle at 3 ounces, it is plain that a quantity
of blood equal to that contained in all the vessels, will have
passed through the heart at the end of 53 strokes, which, if
the pulse be assumed at 75 to the minute, will correspond to
42 seconds. This represents the time occupied by the entire
circulation of the blood. The time occupied by absorption,
circulation, and secretion, combined, may be set down at less
than 4 minutes, as appears from the following interesting
experiment, devised by Professor Macnamara, of the Royal
College of Surgeons in Ireland, at which I had an opportunity
of assisting. An old soldier, a patient of the Meath Hospital,
suffering from hydrocele, was trained by us to urinate at the
word of command, at intervals of one minute, into a succes-
sion of test tubes. When his training had been completed by
ANIMAL MECHANICS,
151
S
11
a few days' instruction, the sac of the hydrocele was emptied.
and injected with two drachms of tincture of iodine. The pa-
tient then urinated at intervals of one minute into the test
tubes previously prepared, commencing at one minute after
the injection. The fourth test tube responded faintly to the
starch test for iodine; and the fifth and subsequent tubes
showed its presence very distinctly. From this experiment
it is evident that the entire process, viz., absorption by the
tunica vaginalis ; transmission by the lymphatics and capillary
veins to the heart; retransmission by the renal arteries to the
kidneys; secretion of urine, and its transmission to the bladder
by the ureters-occupied less than four minutes; or about
five times the entire time required for the circulation of the
blood.
8. On the Muscular Forces employed in Parturition.-
In the first stage of natural labour, the involuntary muscles of
the uterus contract upon the fluid contents of this organ, and
possess sufficient force to dilate the mouth of the womb, and
generally to rupture the membranes. I shall endeavour to
show, from the principles of muscular action already laid down,
that the uterine muscles are sufficient, and not much more
than sufficient to complete the first stage of labour, and that
they do not possess an amount of force adequate to rupture,
in any case, the uterine wall itself. In the second stage of
labour, the irritation of the fetal head upon the wall of the
vagina provokes the reflex action of the voluntary abdominal
muscles, which aid powerfully the uterine muscles to complete
the second stage, by expelling the fætus. The amount of
available additional force given out by the abdominal muscles
admits of calculation, and will be found much greater than the
force produced by the involuntary contractions of the womb
itself.
The mechanical problem to be solved for both cases is one
152
ANIMAL MECHANICS.
of much interest and some difficulty, as it is the celebrated
problem of the equilibrium of a flexible membrane subjected
to the action of given forces. It has been solved by La-
grange (Mecanique Analytique, p. 147), in all its generality.
In the most general case of the problem, the following beau-
tiful theorem can be demonstrated :--Let T denote the tensile
strain acting in the tangential plane of the membrane, applied
to rupture a band of the membrane one inch broad; let P
denote the pressure resulting from all the forces in action, per-
pendicular to the surface of the membrane, and acting on a
surface of one square inch; and let pı and pz denote the two
radii of principal curvature of the membrane, at the point
considered. Then we have the following equation :
U
yli 1
P=Txt-
(19)
\PI. P2)
If the surface, or a portion of it, become spherical, the two
principal curvatures become equal, and equation (19) be-
comes
(20)
P 27
In the case of the uterus and its membranes, the force P arises
from hydrostatical pressure only, and is therefore easily mea-
sured, and the supposition of spherical curvature is approxi-
mately admissible.
The natural position of the gravid uterus is shown in
Fig. 21, in which OP is the axis of the uterus, and AB a ver-
tical line drawn through G, the centre of gravity of the foetus;
this vertical line must pass through F, the centre of floatation
of the fætus, or centre of gravity of the liquor amnii displaced
by the foetus.
In the first stage of labour, the contraction of the muscu-
lar walls of the uterus compresses its liquid contents, and the
ANIMAL MECHANICS.
153
membranes are pushed through the mouth of the womb,
bulging out into the segment of a sphere, and are in most

ht
IN
SMS
SY
W
AAR
.
.
TW
bre
w
SM
AUS
S
ho
www
Fig. 21.
cases ruptured by the pressure of the liquor amnii contained
in them.
It is evident from the form of the gravid uterus, shown in
Fig. 21, that its curvature is greatest near its mouth, and equa-
tion (20) shows that for a given hydrostatical pressure, the
tensile strain is proportional to the radius of curvature; hence
this strain will be greatest at the fundus of the uterus. If we
assume the shape of the uterus to be that of a prolate ellip-
soid, whose longer diameter is 12 inches, and shorter diame-
ter 8 inches; its mean curvature will be that of a sphere whose
diameter is 9.158 inches.
The volume of the gravid uterus is found from the expres-
·sion-
Volume = 4 7 ab?;
(21)
TT
in which a and 6 are the semiaxes, and 7 is the ratio of the
154
ANIMAL MECHANICS.
circumference of a circle to its diameter; substituting for a
and b their numerical values, we find the contents of the
uterus to be 402.13 cubic inches.
The surface of the gravid uterus may be found from the
equation-
Surface = 2.770 (sin e + V1 -%); (22)
in which e is the eccentricity of the generating ellipse. If
the numerical values be substituted in this expression, it will
be found that the surface of the uterus is 270.66 square
inches.*
Some highly interesting conclusions may be drawn from
the preceding calculations, combined with the weight of the
total muscular tissue of the uterus. Heschl estimates the
weight of the uterine muscles at from 1 lb. to 1.5 lb. Mont-
gomery found the muscles of the gravid uterus to weigh 1.5 lb.;
and Levret estimates them at 51 cubic inches, which, with a
specific gravity of 1.052 (vide p. 84), I find to be equivalent
to 1.93 lb. Taking the mean of these estimates, we have-
1
Weight of Muscular Fibres of Gravid Womb.
Heschl, . . . . . 1.25 lb.
Montgomery, .. . 1.50 »
Levret, · · · · · 1.93 »
Mean, .. 1.56 lb.
If we now suppose this quantity of muscle to be spread over
the entire surface of the uterus, we find,
* Levret estimates the contents of the gravid uterus at 478 cubic inches,
and its surface at 339 square inches.
Poppel estimates the contents at 300 cubic inches, and the surface at 2in
square inches.
ANIMAL MECHANICS.
155
Mean thickness of muscular wall of uterus,
- 1.56 X 7000 x 1000.
1000 = 0.1519 inch.
252.5 X 270.66 x 1052
If we suppose a ribbon, one inch in width, to be formed
from the wall of the uterus, its thickness will be 0.1519 inch;
and as each square inch of cross section of muscular fibre
(p.69) is capable of lifting 102.55 lbs., we find for the greatest
tensile force producible by the contraction of the uterine
muscles.
Tensile strain of uterine wall per inch = 102.55 * 0.1519 = 15.577 lbs.
Substituting this value of T, in the equation (20)
P = 21
and for q, its mean value 9.158 in., we obtain the maximum
hydrostatical pressure inside thé gravid uterus, that can be
produced by the contraction of its muscular fibres.
Maximum hydrostatical pressure produced by uterine contraction
2 x 15.577 = 7.402 lbs.
9.158
One hundred experiments were made by Duncan and
Tait, upon the hydrostatical pressure necessary to rupture the
membranes which contain the liquor amnii, which are re-
corded in Dr. Duncan's book* (pp. 306-311). The greatest
pressure observed was 3.10 lbs., and the least was 0.26 lb. ;
and I find that the mean rupturing pressure of all their ex-
periments was 1.2048 lb.
Combining this experimental result with the calculation
already given, of the amount of pressure producible by the
muscular tissue of the womb, we may conclude that the uterine
muscles are capable of rupturing the membranes in every
I
* “Researches in Obstetrics." Edinburgh : 1868,
156
ANIMAL MECHANICS,
O
S
case, and possess, in general, nearly three times the amount of
force requisite for this purpose.
The weight of muscle in the virgin uterus is about one
and a half oụnces, and this muscle is increased to one and a
half pounds during pregnancy; that is, it is increased about
sixteen fold. If ever there was a muscular system produced
to effect a specific object, the uterine muscle may be regarded
as such ; and if the principle of least action in nature be valid,
we should expect such a muscle to be developed only to the
extent required, and not beyond it. Considered from this
point of view, the preceding calculation must be regarded as
interesting; for it shows that the uterine muscle possesses a
force of 3.4 lbs., intended to overcome a maximum resistance
of 3.1 lbs. It would be a waste of power to endow the uterus
with more force than I have shown it to possess, for it is not
necessary that the uterus should complete the second stage of
labour, as the abdominal muscles are available for this pur-
pose; so that by using them, and not giving the uterus more
force than is absolutely necessary for the first stage of labour,
an admirable economy of muscular power is effected.
Viewed in connexion with the doctrine of Final Causes, the
uterine muscle presents a subject well worthy of contemplation.
It is plainly necessary that the first stage in the expulsion of
the foetus should not be entrusted to a voluntary muscle, and
hence an involuntary muscle is gradually provided, which
takes the initiative, and commences the process of parturition,
the completion of which is then accomplished by the aid of
voluntary muscles, to the employment of which, at this stage,
no moral objection can be raised. It is also necessary (if the
Contriver be all-wise, or if the principle of Least Action in
nature be true), that the involuntary muscle, so produced,
should not possess more or less force than is requisite for its
purpose. The uterine muscle does not grow, to meet a grow-
ANIMAL MECHANICS,
157
112
11
ing resistance (as happens frequently in other cases), ang its
precise degree of strength cannot be produced by a tentative
process ; for, in healthy gestation, the uterine muscle never
tries its force against the membranes it is called upon to rup-
ture, until the actual period of parturition has arrived. We
witness, therefore, the instructive spectacle of a growing force
intended to overcome, at a definite future period, a definite
resistance, against which it never measures its strength until
the actual moment of conflict has arrived.
The mind must be singularly constituted, which fails to
see in such an arrangement the clear evidence of a Divine
Contriver, adapting means to an end, foreseen by Himself.
The extreme force of uterine contraction produces a pres-
sure of 3.402 lbs. per square inch, which is equivalent to a
pressure of 54.106 lbs. acting upon a circle of four and a
half inches in diameter, which is assumed as the average area
of the pelvic canal.
The maximum force used to expel the foetus, by both ute-
rine and abdominal muscles combined, is estimated by Joulin, *
from forceps experiments made on the dead body, at
110.23 lbs., a result which is regarded by Dr. Duncant as too
large. Dr. Duncan himself considers 80 lbs. as the maximum
force ever employed in difficult cases. This would correspond
with an hydrostatical pressure inside the uterus of 5.03 lbs.
per square inch, which is greater than the uterine muscles,
unaided, are capable of producing.
Dr. Duncan's experiments on the strength of the uterine
wall itself show that an hydrostatical pressure of from 17 to 25
YI
conclusively demonstrates, when taken in conjunction with
S
* " Traité complet des Accouchements," p. 447.
op " Researches in Obstetrics," p. 323,
158
.
ANIMAL MECHANICSY
IU.
nT
(D
1
the preceding calculations, that the uterus can never destroy
itself by its own contraction. It would, indeed be, à priori,
ridiculous to suppose that Nature would produce an involun-
tary muscle capable of destroying itself, and so causing the
death of its owner, by its own blind force.
In concluding my remarks on the uterine muscles, I may
observe that the arrangement of these muscles on various
parts of the organ affords a confirmation of the accuracy of
Lagrange's principle as applied to them. It is well known
that these muscles attain their greatest thickness at the fundus
of the womb, where they are fully double the thickness of the
muscular layer at the neck of the womb. Now, so long as
the contents of the womb are fluid, or semi-fluid, a constant
pressure is exerted perpendicularly to all portions of its walls,
requiring by Lagrange's theorem (20), a tensile muscular
force varying as the diameter of curvature at each point.
From observations on the curvature of the gravid uterus, it
may be inferred that the diameters of curvature at the fundus
and neck of the womb are nearly in the proportion of 29
to 17. When the womb is emptied of its contents, and the
uniform hydrostatical pressure inside is removed, the uterine
walls contract with degrees of force proportional to the dia-
meters of curvature they possessed in their former condition;
the fundus contracting with twice the force of the parts near
the neck—an action eminently calculated to promote the safe
and speedy reduction of the organ to its diminished bulk,
without risk of inversion.
It is worth while to consider, for a moment, the explana-
tion offered by the Lamarckian Theory of Natural Selection,
of the relation between the strength of the uterine muscles
and the resistance of the membranes they are required to
rupture. According to this theory, an animal that possessed
such a proportion as I have shown to exist between the force
ANIMAL MECHANICS.
159
and resistance to be overcome, must be the descendant from
ancestors who attained this proportion accidentally, and trans-
mitted it to their posterity, who thus acquired an advantage
over their neighbours in the competition for life. Thousands
and tens of thousands of those who have preceded us must
have perished in the race, from not having had the precise
balance between uterine force and resistance, which we hap-
pily possess ; and we may regard ourselves as the survivors of
a contest in which the slain greatly outnumber the victors.
According to this School of Philosophy, an economy of two
per cent. of wax will make a bee; and a saving of a quarter
ounce of uterine tissue may, possibly, constitute the sole dif-
ference between a Celtic and a Teutonic woman.
Happy is the man whose mother possessed the hereditary
tendency to form no more uterine muscle than was absolutely
necessary to push him into the world ! Happy is the bee who
can form her cell with a saving of two grains of wax in the
hundred, for she will distance in the race of life her compe-
titors who have not inherited this precious art!
In the second stage of labour, the voluntary action of the
abdominal muscles is called into play, to aid the expulsive
efforts of the uterine muscles. I have attempted to calculate
the force available from the contraction of these muscles as
follows:-
The abdominal muscles are four in number, viz. rectus
abdominis, obliquus externus, obliquus internus, and transver.
salis. The last three muscles form curved sheets, acting
upon the corresponding muscles of the opposite side by means
of tendinous aponeuroses which meet in the linea alba, and
form the sheath of the vertical rectus abdominis muscle. From
the arrangement of all four, it is plain that the tensile force of
muscular contraction in the curved wall of the belly from the
xiphoid cartilage to the symphysis pubis, is to be measured
.160
VIM
ANIMAL MECHANICS,
by the sum of the united forces of all the muscular sheets. If
we knew the force of each muscle, and the principal curva-
tures of the belly in the middle line, we could calculate, by
Lagrange's theorem, the hydrostatical pressure inside the ab-
dominal cavity, and available to expel fæces, urine, or a
fætus.
In order to ascertain the force of the muscles, I measured
carefully their average thicknesses in three subjects, of whom
one was a young woman who had borne children, and the
others were men of ordinary size and appearance. The re-
sults obtained were the following:
y
TABLE XXVI.-Thickness of Abdominal Muscles.

No. 1.- Male.
No. 2.-Female.
No. 3.—Male.
o 29 in.
0.34 in.
Rectus abdominis, ..
Obliquus externus, ..
Obliquus internus, . .
Transversalis, ...
0.275 in.
0.200 ,
0.235 39
0,127 1
0.1999
0.25 »
0.17 »
0.1519
0.24 ,
0.1419
.
Totals, ..!
0.837 in.
0.86 in.
0.91 in.
The average total thickness of the muscular walls is 0.869
inch, which is nearly identical with the measurements ob-
tained from the female subject. It has been already explained
(p. 46), that we must add 50 per cent. to the weight of mus-
cles in the dead subject, in order to bring them to the living
weights. This correction gives us 1.3053 inches, for the mean
thickness of the muscles causing tension in the central line of
the belly, where the forces of all the muscles come into play
together. Multiplying this thickness by 102.55 lbs., or coeffi-
cient of muscular contraction, we find,
T = 1.3035 * 102.55 = 133.67 lbs.
This is the tensile strain producible by the contraction of the
S
ANIMAL MECHANICS.
161
abdominal muscles along the curved central line of the
belly.
There remain now to be ascertained the principal curva-
tures of the abdominal surface, and to use the equation, (19)
P-Tt3
lei 22
so as to determine P, the hydrostatical pressure per square
inch inside the cavity of the belly, and available, either in
whole or in part, for the expulsion of the foetus during the
second stage of labour.
In order to ascertain the curvature of the belly, I made
experiments upon three young men, placed lying upon their
backs upon the floor, and made them depress and raise the
abdominal wall as much as possible. The result was as fol-
lows:- Taking a straight line from the upper part of the sym-
physis pubis to the xiphoid cartilage as the fixed line of com-
parison, it was found possible to depress the navel one inch
below this fixed line, and to raise it two inches above it.
When the belly was distended to the utmost, by the action of
the abdominal muscles, I measured the longitudinal and trans-
verse curvatures, by measuring the sagittas corresponding to
a given length of tangent, with the following results :-

No.
Diameter of Longitudinal Diameter of Transverse
Curvature.
Curvature.
22.93 inches.
12.30 inches.
J. G. H., ..
H.O., ...
S. H., . ..
22.73
22
12.80
12.80
22.52
2
Mean, ..
22-727 inches.
12-633 inches.
MEJLPS
M
YT
162
ANIMAL MECHANICS.
VIVA
IM
The curvature of the distended belly, at the navel, is found
to be, from the foregoing measurements,
- III
I
i O2 11.3635 6.3166 4.0596
Multiplying this curvature into the tension of the abdomi-
nal muscles at the navel, already found, viz. 133.67 lbs. per
inch, we obtain, finally,
P_133.67
* = 32.926 lbs. per sq. inch.
0.6 lhe
P= 4.0596
This amount of expulsive force, per square inch, is avail-
able, although not usually employed to assist the uterus in
completing the second stage of labour. If we suppose it ap-
plied to a surface of a circle 41 inches in diameter, the usual
width of the pelvic canal, we find that it is equivalent to
523.65 lbs. pressure.
Adding together the combined forces of the voluntary and
involuntary muscles, we find,
Involuntary muscles = 54.106 lbs.
Voluntary muscles, = 523.65 1
Total, . . 577-75 lbs. av.
Thus, we see that, on an emergency, somewhat more than
a quarter of a ton pressure can be brought to bear upon a re-
fractory child that refuses to come into the world in the usual
manner. *
In order to determine, by actual experiment, the expulsive
* The preceding result will, no doubt, remind the curious and well-informed
reader, of the statement made by Mr. Shandy, on the authority of Lithopædus
Senonensis de partu difficili, that the force of the woman's efforts, in strong labour
pains, is equal, upon an average, to the weight of 470 lbs, avoirdupois, acting
dervendicularly upon the vertex of the head of the child.
ANIMAL MECHANICS.
163
P=
113
2.937
1
force of the abdominal muscles, I placed two men, of 48 and
21 years of age, respectively, lying on a table upon their backs,
and put a disk measuring 1.87 in. diameter just over the
navel ; weights were placed upon this disk, and gradually in-
creased until the extreme limit of weight that could be lifted
with safety was reached ; this limit was found to be, in both
cases, 113 lbs. As the circle whose diameter is 1.87 in. has
an area of 2.937 square inches, the pressure perpendicular to
the abdominal wall produced by the action of the abdominal
muscles is,
3. = 38.47 lbs. per sq. inch.
The force is somewhat greater than that calculated from this
curvature of the belly, and the coefficient of muscular con-
traction, already determined; viz. 32.93 lbs. per square inch.
It should be remembered, however, that in my experiments
on the abdominal muscles, I pushed the trial of strength al-
most to the verge of producing slight peritonitis, as both the
subjects of my experiment complained of sharp shooting
pains in the abdominal walls for 24 hours after the experi-
ments were made.
The foregoing theory of the muscular forces employed in
parturition leads us naturally to be cautious in the use of
chloroform, and other anæsthetics, which produce their effects,
first upon the voluntary, and afterwards upon the involuntary
muscles. When chloroform is used, in parturition, beyond
the stage at which it produces simple drunkenness, and indif-
ference to pain, it is positively injurious; for it destroys the
action of the voluntary abdominal muscles, which constitute
the chief part of the force employed in difficult labours.
Hence, the muscular tissue of the uterus is left to overcome
an obstacle beyond its strength, and without the aid that na-
ture intended to be given by the abdominal muscles. The
11 2
164
ANIMAL MECHANICS.
result is, that labour ceases until consciousness returns, and
enables the ill-treated woman to avail herself of the apparatus
of voluntary abdominal muscles provided by nature for her
use.
In these remarks I confine myself altogether to the phy-
sical aspects of the use of chloroform in labour, and forbear
to express an opinion as to the moral effects likely to be pro-
duced by the exhibition of an intoxicating agent, administered
merely for the luxurious purpose of deadening the pain of
the exercise of a purely natural physiological function. .
9. On the Classification of Muscles and their Mode of
Action. It has been already shown that muscles consist of
linear fibres, which may be variously arranged; and as lines
form surfaces, and combinations of surfaces form solids, it is
evident that a natural classification of muscles must be essen-
tially geometrical, and that the mechanical action of muscles
must depend upon their geometrical configuration.
The following classification of muscles is proposed as a
simple and natural arrangement:
I MUSCULAR FIBRES LYING IN THE SAME PLANE.
1. Fibres parallel and rectilinear.
(a) Direct Prismatic.
(6) Rhomboidal.
(c) Penniform.
2.. Fibres intersecting and rectilinear.
(a) Triangular.
(6) Deltoidal.
(c) Quadrilateral.
3. Fibres curvilinear and parallel.
(a) Sphincters.
ANIMAL MECHANICS.
165
II. MUSCULAR FIBRES FORMING CURVED SURFACES.
1. Fibres rectilinear.
(a) Skew Surfaces.
2. Fibres curvilinear.
(a) Ellipsoidal Surfaces.
*Fibres in same plane, parallel to each other, and rectilinear.
-This natural group of muscles has been described by Bo-
relli,* who has named their several species, Direct Prismatic,
Rhomboidal, and Penniform.
Direct Prismatic Muscles. If we suppose two bars, AA
and BB, joined by a number of parallel elastic strings, to be
drawn asunder, as re-
presented in Fig. 22,
the elasticity of the
strings will draw back A
the bars into their origi-
nal position, by the same
kind of action as the con-
tractile fibres of a Direct
Prismatic muscle. If f
denote the contractile
force of each fibre, and
in their number, the re-
sultant force of the mus-
cle will be
F = nf, (23)
and will act parallel to
the fibres, in the di-
rection ab. If the pa- B.
Fig. 22.
rallel sheets of mus-
cular fibres be united, so as to form a solid muscle, the re-
* De Motu Animalium, vol. 1., p. 8.

V
|
pool a
.
th
A
166
ANIMAL MECHANICS.
sultant will pass through the centres of gravity of the areas
marked on the bones by the origin and insertion of the fibres.
Good examples of the Direct Prismatic muscle may be found
in the sartorius and omo-atlantic muscles of various animals.
m 11
Il

Fig. 23.
Rhomboidal Muscles.--If the two bars, AA and BB, be
held, as in Fig. 23, so that the fibres joining them are oblique
to the bars, we have the Rhomboidal muscle of Borelli. In
this case the resultant force is also
F=nf;
but as it acts in the right line ab, which is oblique to the bars
AA and BB, it may be resolved into two forces at each bar,
viz., 67, 6V, and aH', aV'; of these forces, 6 H and aH'act
parallel to the bars, and tend to bring them into the position
of the prismatic muscle; the other forces 6 V and a V' tend to
ANIMAL MECHANICS.
167
draw the bars together. If o denote the angle ab H, the ho-
rizontal and vertical forces exerted by the muscle will be
H = F cos o
(24)
V-F sino
NI
/
.
/
/
TO
T
J
ples of the Rhomboidal type of muscle.
Penniform
Muscles. — The
structure ofpen
niform muscles
is illustrated by
Fig. 24, in which
two bars, AA,
BB, parallel to
each other, are
connected by
two sets of par-
allel fibres, with
a third bar, CC,
parallel to the
other bars. It
is a combination
of two rhom-
boidal muscles,
which give a
vertical result-
ant in the di-
rection CV.
If F denote,
as before, the
resultant force
of each rhom-
boidal muscle,
Fig. 24.

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168
ANIMAL MECHANICS.
and o denote the angle ab V, made by its fibres with the par-
allel bars, the resultant force of the penniform muscle will be
R = 2F cos .
(25)
Many muscles, when dissected, are found to contain parallel
sheets of fascia, resembling the bars AA, BB, and the mus-
cular fibres take origin from these sheets of fascia, forming
true penniform muscles, which, taken together, make up the
entire complex muscle. Thus, the deltoid muscle is consi-
dered by Albinus to be composed of seven distinct pennate
muscles. In the biceps humeri of the Ungulates I have found,
also, a penniform arrangement of the fibres with reference to
fascial sheets that run along the whole course of the muscle.
Let AX and BX (Fig. 25)
represent two fibres of a pen-
nate muscle, and let l be the A
length of each; during their
contraction, they are each
shortened by a length 8l, and
the point X is moved to the
point x; draw up perpendi-
cular to BX, then Xp is equal
to dl, and the distance through
which X is moved is
Xx = dl sec .
Fig. 25.

Multiplying this distance by the resultant force (25), in the
direction XY, we have
Work done by the contraction of a pennate muscle
= 2 F cos o sec pôl = 2 FOI.
(26)
And, since 2F8l is the work inherent in the muscular fibres,
if arranged in a prismatic manner, we conclude that there is
S
ANIMAL MECHANICS.
169
neither gain nor loss of work due to the penniform arrange-
ment.
In the Nylghau, the muscular fibres of the biceps humeri
are 1.84 inches in length, and make an angle of 50° with the
central line. These penniform fibres will therefore act like
the prismatic fibres of an ordinary biceps muscle, whose
length is
L = 1.84 * sec 50° = 2.85 inches.
02
The actual length of the biceps in the Nylghau is 6 inches,
and yet the total amount of its contraction is only that due to
a muscle of 2.85 inches in length. The intention of this ar-
rangement of penniform fibres is to produce a powerful pull
through a small space. In the brachiæus muscle of the same
animal, the fibres run parallel together through the entire
length of the muscle, which is prismatic, and wound spirally
round the humerus. The length of the fibres is 8 inches, and
from their parallel, and not penniform arrangement, this
muscle is capable of a contraction nearly three times greater
than that of the penniform biceps.
In the Ass, the length of the penniform fibres of the
biceps humeri is 1.06 inches, and they form with the tendinous
axis an angle of about 24°. Hence the action of this penni-
form biceps will be similar to that of a prismatic muscle
whose length is
1 = 1.06 x sec 24° = 1.15 inches.
The actual length of the biceps muscle in the Ass is 6.6 inches.
The fibres of the brachicus muscle in the Ass are parallel, and
run along its entire length, which is 9 inches. Hence the
brachiæus is capable of a contraction nearly eight times
greater than that of the biceps muscle.
Let it be required to find a prismatic muscle, whose action
170
ANIMAL MECHANICS,
shall be exactly the same as that of a given penniform muscle.
In order that the action of the two muscles shall be, in all
respects, the same, it is necessary that they shall contract
through the same space and with the same force, the product
of the space and force being the work done by the muscle in
a single contraction. If I denote the length of the penniform
fibres ab (Fig. 24), and the angle ab V ; then the contrac-
tion due to the penniform muscle will be equal to that of a
prismatic muscle whose length is
L = 1 sec o.
(27)
Let ada bBb (Fig. 26) represent a penniform muscle, whose
length AB is , the angle ABC being . If perpendiculars
AC and AD be drawn from A to the fibres Bb and Bb pro-
duced, it is plain that the lines AC and AD will represent
the total number of fibres employed in the muscle; hence, if
f be the force of each fibre, the total force employed at each
side will be
F = nf = f * AD = f, sin d;
R = 2F cos p.
Hence R = 2f sin cos = f) sin 20 (28)
This is equivalent to the force of a prismatic muscle, whose
cross section is , sin 20, or equal to the line CD.
but
For,
1 = AB.
, sin p = AB sin g = AD.
sin cos 0 = AD cos = DE.
22 sin o cos = 2DE = CD.
the actual width of the penniform muscle is
w = 2l sin p.
ANIMAL MECHANICS.
171
Hence the width of the equivalent prismatic muscle may be
found :-let this width be called W, and we have
W CD = 20 sing coso
w * B6 21 sin

cos
and, finally,
W-wcosp. (29)
These formulæ, applied
to the biceps muscle of
the Nylghau and Ass,
give the following equi-
valentprismatic muscles.
In the Nylghau, the
biceps muscle has the
following measurements:
1=1.84 in., p= 50°,
X = 6 in.
Hence,
L=l sec o = 2.85 in.
W= w^<0$P = 2.09 w.
Fig. 26.
In the Ass we have
1 = 1.06 in., o = 24°, 1 = 6.6 in.
Hence,
L = 1 sec o = 1.15 in.
W=w #cos = 5.688w.
It is interesting to observe that the short thick prismatic
muscles, to which the penniform biceps of the Nylghau and of
172
ANIMAL MECHANICS.
the Ass are equivalent, are totally different from the long thin
brachiæus muscles, associated with them in the flexion of the
forearm.
If we assume the specific gravity of muscle to be 1.050,
the weight of a cubic inch of muscle, expressed in avoirdu-
pois ounces, will be-
Cubic inch of muscle = ---
1.050 x 252.5 x 16
= 0.606 oz.
7000
In the Nylghau, the weight of the biceps muscle is 3.44 OZ.,
and its length 6 inches; hence, its cross section, w, is
.
W = -3.44
___3•44_=0.946 sq. in.
26x0.606
but
W = 2.99 W = 1.977 sq. in. Equivalent Prismatic biceps of
L
= 2.85 in. ) Nylghau.
The weight of the brachiæus is 3.43 oz., and its length is 8
inches ; hence its cross section is-
W =
3:43
W = 8 x 0.606
Brachiæus of Nylghau. .
1 =
8 in.
These two muscles are of nearly equal weight, and therefore
contain nearly equal amounts of inherent work, and we can
easily verify the statement already made (26), that they give
out equal quantities of work, although one has its fibres pris-
matic and the other penniform,
Work done by biceps (penniform)
= 1.977 x 2.85 = 5.63.
Work done by brachiæus (prismatic)
= 0.707 x 8 = 5.65.
In the Ass, the weight of the Biceps is 6.83 oz., and its
length is 6.6 inches; hence its cross section, w, is
w = a
6.83
6.6 x 0.606 = 1.71 sq. in.
ANIMAL MECHANICS.
173
Hence,
W = 5.688 w = 9.726 sq. in. | Equivalent prismatic biceps
1 =
1.15 in. I of Ass.
The weight of the brachicus is 3.23 oz., and its length is
9 inches; hence its cross section is-
W =
3.23
9 x 0.606 = 0.592 sq. in. l ,
Brachiæus of Ass.
1 =
9 in.
In a fine Sambur Stag examined by me, the biceps humeri
muscle was found to be pennate in the arrangement of its
fibres; their length being 2.56 in., the angle made by
them with the fascia of insertion being 34°. The weight of
the muscle was 5.91 Oz. In this case, the length of the
biceps muscle was 10 inches, from which we find its mean
cross section
W =
-5.91
10 0.606 = 0.975 sq.in.
From this we find, as before, the prismatic muscle equivalent
to the pennate biceps-
L = 1 sec 0 = 3.09 in.
To i cos 0
W = W Y = 3.16 sq. in.
The brachiæus muscle in the Sambur is 4.95 oz. in weight
and 9.5 inches in length, and it is a prismatic muscle; its
cross section is,
4.95
606 = 0.860 sq. in.
W-
It is interesting to contrast together the two muscles which
flex the forearm in the three animals described, in order to
see, at a glance, the very different uses which these muscles
are intended for.
174
ANIMAL MECHANICS
TABLE XXVII.— Ungulate Biceps and Brachiæus.

BIChfs.
BRACHLÆUS.
Ratio of Force
of Bice is t)
Brachiæus.
Ratio of Con-
truction of Bra-
ichiæus to Biceps.
W.
L.
w.
Sq. In.
1.98
In
2.85
sq. In.
0.71
2.30
2.81
Nylghau, .
Ass, ...
Sambur,
1.15
0.59
16.47
7.83
9.72
3.16
3.09
10.86
3.67
3.08
2
This Table shows clearly the very different actions of the
biceps and brachiæus muscles in the Ungulates. The biceps
muscle contracts with a force much greater than that of the
brachicus, while the space through which it acts is much
less than that of the brachiaus. The two muscles differ but
little in external appearance, but by means of the penniform
arrangement of the fibres of the biceps, a rope-like muscle is
enabled to do the duty of a flat short muscle, which it would
be impossible to place directly upon the bones of the arm.
The rapid lift of the fore foot, which is so characteristic of
the swift Ungulates, and by none more beautifully shown
than by the Deer tribe, is mainly due to the powerful action
of the biceps through a short space; while the slower and
longer flexions of the forearm are due to the distinctive action
of the brachiæus muscle.
In a Brahmin Cow examined by me, the arrangement of
the fibres of the biceps was penniform, though I omitted to
measure their length, and the angle made by them with the
central fascial sheet of their origin. The measurements of
the biceps and brachiæus muscles in this animal were as fol-
lows. The biceps weighed 3.88 oz., and had a total length of
6 inches. The brachiæus weighed 2.66 oz., and its fibres,
which were arranged as in a prismatic muscle, measured
20
PI
n
ANIMAL MECHANICS.
175
8.15 inches in length. These measurements correspond with
the following mean cross sections :-
Biceps, ..... 1.067 sq. in.
Brachiæus, . ... 0.539 .
The angular velocity with which the forearm turns on the
elbow joint is not proportional merely to the force of the
muscle, but to the product of that force into the distance from
the point of insertion to the axis of the elbow joint. I mea-
sured these distances carefully, for the purpose of obtaining
a comparison between the angular velocities produced by the
biceps and brachiæus, muscles respectively. The following
Table contains the measurements and results; and I have
added corresponding measurements taken from the arm of a
very fine Lioness.
In the Lioness the biceps muscle weighed 6.70 oz., and had
a length of 8 inches; and the brachicus muscle weighed
2.60 oz., and had a length of 8.75 inches.
TABLE XXVIII.—Angular Velocity of Forearm produced by
Biceps and Brachiæus Muscles.

BICEPS.
BRACHIÆUS.
Ratio of An-
gular Veloci -
ties produced
Distance of
Distance of by Biceps and i
Cross Section. Insertion from Cross Section. Insertion from Brachicus.
| Axis of Joint.
Axis of Joint.
In.
Sq. In,
1.98
In.
1.02
Sq. In.
0.71
2.26
1.26
9.72
1.24
0.59
2.00
10.21
Nylghau, 1
Ass, . ..
Sambur,
Lioness,
3.16
1.72
2.44
2.59
1.38
2.29
0.49
2.16
3.00
The total angular movement of the forearm produced by
each muscle will be proportional to the length of the muscle
176
ANIMAL MECHANICS.
divided by the distance of its point of insertion from the axis
of the joint; the ratios of the angular motions caused by the
two muscles is shown in the following Table:-
TABLE XXIX.--Angular Motion produced by the Biceps and
Brachiæus Muscles.

BICEPS.
BRACHIÆUS.
Ratio of Angu-
lar Movement
produced by
Biceps and
Brachiæus.
Length.
Distance of
Insertion.
Length.
Distance of
Insertion.
In.
2.85
In.
1.02
In.
2.26
1.15
1.24
2.00
Nylghau, .
Ass, ...
Sambur,
Lioness,
0.789
0.206
0.461
0.862
3.09
1.72
9.5
8.00
2.29
8.75
Bringing together all the preceding results, we obtain the
following Table :-
TABLE XXX.—Comparison of the Action of the Biceps and
Brachiaus Muscles.

Ratio of Forces.
Ratio of Velocities.
Ratio of Angular
Motions.
1.26
10.21
Nylghau, ..
Ass, ..
Sambur, ..
Lioness, .
Man, * ..
2.80
16.47
3.67
2.82
0.789
0.206
0.461
0.862
2.59
3.00
1.50
1.82
1.159
The Ass greatly exceeds the other animals in the relative
* The data from which the comparison of the Biceps and Brachiæus muscles
in Man is made are given in pp. 63, 64.
ANIMAL MECHANICS.
177
force of the biceps, and the velocity of motion produced by it,
as compared with the brachiæus ; and the angle through
which it bends the forearm is only one-fifth the angle of
flexion due to the brachiæus; while in the Lioness these
angles are nearly equal, and in Man the angular motion
caused by the biceps is greater than that produced by the
brachiaus.
These peculiarities are common to the Ass and Horse
in which latter animal the “gig action” of the fore foot, so,
much admired, is altogether due to the same cause as in the
Ass, viz., the rapid flexion of the elbow joint caused by the
pennate arrangement of the fibres of the biceps muscle, pro-
ducing a powerful contraction through a short space.
The mylohyoid muscle, which forms the floor of the mouth
(diaphragnia oris), is another good example of the penniform
arrangement of muscular fibres. This muscle, as it exists in
Man, is shown in Fig. 27, where H denotes the hyoid bone,

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and S the symphysis. The posterior fibres of the muscle are
attached directly to the tongue bone, while the anterior fibres
N
178
ANIMAL MECHANICS.
meet in a central raphe, HS, which is the direction of the re-
sultant motion produced by the simultaneous contraction of
all the penniform fibres. The lengths of the fibres measured
from behind forwards increase from 0.95 in. to 1.50 in., and
then decrease gradually down to zero. The angle ø, made by
them with the central raphe, is 45°, when the top of the
tongue bone is brought into the same plane as the floor of the
mouth. If I denote the length of any fibre, aná 8l the amount
of its contraction, it follows that any point x, lying in the
central raphe, will be drawn upwards and forwards through
a space represented by slv 2, where l is the length of the
line xa, which is proportional to the distance of x from the
symphysis of the jaw; hence the point S will have no motion
at all, and the velocity will increase from S to the centre of
H, where the fibres are of the greatest length.
Another example of penniform muscle may be seen in the
accelerator urine, in which the fibres form angles of 23º
with the central raphe.
nyo
Muscular Fibres rectilinear, lying in the sume Plane, and not
parallel to each other.
This natural group of muscles may be divided, like the
former, into
three distinct
types, which
may be nam-
ed, respec-
tively, Tri-
angular, Del-
toidal, and
Quadrilateral
Fig. 28.
muscles.
Triangular Muscles. In these muscles, the origin is a

ANIMAL MECHANICS.
179
t
point, and the insertion of the fibres is a right line Let 0
(Fig. 28) denote the origin, and AB the line of insertion. The
muscular fibres diverge from the point 0, lying between the
lines 0 A and OB, and each fibre contracts with the same
force, whatever be its length. Draw the line 0X bisecting
the angle AOB. If the muscular fibres be taken in pairs,
making equal angles at opposite sides of 0X, the resultant of
each pair of fibres will lie along the right line OX, which will
therefore be the direction of the resultant of the whole mus-
cle; and since the bisector of the angle AOB divides the base
AB into segments AX, XB, proportional to the sides AO
and OB, it is easy in all cases to find the point X, in the line
of insertion, through which the whole force of the contracting
muscle is directed.
One of the best examples of triangular muscle, with which
I am acquainted, is to be found in the biceps femoris of the
large Felide. In a large Lion examined by me, the biceps
weighed 27.75 oz. It forms a massive triangular sheet, hav-
ing for its origin the highest point of the tuber ischii, and is
inserted by strong fascia, continuously, from the middle of
the femur, covering the vastus externus, down the entire length
of the outer side of the leg, as far as the tendo Achillis, round
which it is fastened.
The total length of the fibres from the origin to the lowest
point of insertion was 22.25 inches, and the length of the
fibres passing to the highest point of insertion was 8.75 inches.
The base of the triangle, when the leg is extended, was 17.25
inches. If we divide this length in the proportion of 0 A to
OB, or 22.25 to 8.75, we shall find
AX = 12.38 inches.
BX = 4.87 ,
This point corresponds exactly with the top of the fibula, so
11
2
2
N 2
180
1
ANIMAL Il
.
MECHANICS2
that the effect of the whole triangular muscle is the same as
that of a single prismatic muscle passing from the tuber ischii
at the top of the fibula.
Another remarkable example of triangular muscle is found
in the anterior fin muscle of the Angel Shark; this muscle
. consists of a number of distinct muscular bundles, each be-
longing to a single ray of the fin, and forming altogether a
triangular muscle, whose sides measured 5 in. and 21 in. re-
spectively, the base measuring in a straight line 7 inches;
the muscle itself is not terminated by this base, but has a
curved base, as shown in the Fig. 29. In this case, as in the

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Fig. 29.
last, the bisector 0X, of the angle AOB, will be the direc-
tion of the resultant of the radiating fibres, whatever be the
shape of the curve that forms the base of insertion of the
muscular fibres. In the Angel Shark I found 20 distinct
bundles lying between OB and OX, and 30 bundles lying
between 0 A and OX; but the first bundles were 50 per cent.
thicker than the second, so that the muscular force was dis-
tributed equally through equal angles round the point 0.
Let us suppose the radiating fibres of a triangular muscle
to be so arranged that equal angles will contain equal num-
bers of muscular fibres; and let it be required to calculate
the work done by the contraction of such a muscle, and
2
ANIMAL 14
181
MECHANICS,
1
compare it with the inherent or potential work of the muscu-
lar fibres contracting separately.
We may imagine a circle described round the point 0
with a radius assumed equal to unity, and such that the fibres
passing to their insertion completely fill the circumference of
the arc of this circle. If 0 denote (Fig. 28) the angle XOX,
made by any muscular fibre with the bisector OX, and f de-
note the force of contraction of the muscular fibres applied
perpendicularly to a unit length; then fdo will be the force
acting in the line Ox, and the resultant of all these forces,
estimated in the direction OX, will be
R=f
cos 0d0 = 2f sin 0.
1
When the whole muscle AOB contracts, each fibre is short-
ened so as to allow the base AB to be drawn towards 0, in
the direction OX, remaining parallel to itself. If b denote
the bisector Ox, and 86 be the amount of contraction, the
total work done by the triangular muscle will be-
Work of triangular Muscle = R86 = 2f 86 sin 0 = 2f&q, (30)
where q denotes the perpendicular (XY) let fall from the
point X upon the side of the triangle.
If we assume, that the cross section of a triangular muscle
is everywhere the same, this cross section may be conveniently
used as the unit to which f is applied, and we obtain the fol-
lowing important theorem :-
IV
The Work done by the contraction of any triangular muscie
is equal to the Work done by a prismatic muscle haring the same
cross section and a length equal to twice the perpendicular drop-
ped on the side, from the foot of the bisector of the vertical angle
of the triangular muscle.
182
.
TT TNT
ANIMAL MECHANICSThe total amount of work inherent in the muscle may be
found as follows: if I denote the length of any fibre Ox, then
ldo will be proportional to the work of its contraction; and
the total inherent work will be-
Inherent Work of Triangular Muscle = f 81d0
-0
If we call ß the angle BXO (Fig. 28), we have
6 sin ß
sin ( - )
to
Inherent Work = f8l sin ß
do
sin (ß - 0)
=f86 sin B log cot ž (B - 0)
cot 1 (Broſ
(31)
This result admits of a geometrical illustration (Fig. 30).
In the triangle AOB draw the bisector of the vertical angle

.
.
.
.
A
BÉ --
* U
Fig. 30.
Ox, and from the point X let fall the perpendicular X Y =q.
Produce the sides AO and AB, and draw the exscribed circle,
ANIMAL MECHANICS.
183
touching AB at the point Z, and let fall the perpendicular
OU = p, from the vertex upon the base of the triangle.
It is easy to see that
OAZ = ß-0
OBZ = ß+ 0
cot }(2 - 0), AZ
cot ] ( + 0) BZ
OU = p = b sin ß.
Therefore
The work inherent in a triangular muscle is proportional
to the perpendicular let fall from the vertex upon the base,
multiplied by the Napierian Logarithm of the ratio of the seg.
ments of the base made by the point of contact of the exscribed
circle ;
on
and
of the bietor of the ven the side of
The work done by the same muscle will be proportional to
double the perpendicular let fall upon the side of the triangle
from the foot of the bisector of the vertical angle.
In the case of the biceps femoris of the Lion, I made the
following measurements :-
p = 3.28
AZ = 9.50
BZ = 2.72
9 = 1.85.
From these data we find-
p loge 12 = 3.28 x 1.25 = 4.10,
and
29 = 3.60.
184
ANIMAL MECHANICS.
Hence, the inherent work of the muscle is to the work done by
it as 41 to 36; or the work done is 88 per cent. of the inhe-
rent work, and the loss of work amounts to 12 per cent., in
consequence of the triangular arrangement of the fibres.
In the preceding investigation, I have assumed that the
muscular fibres proceeding from AB to () are distinct from
each other, and pass continuously from origin to insertion
without interfering or blending with each other. On this
supposition, the fibres are supposed to be arranged, not in
one plane, but like the radiating blades of a fan, in such a
manner that the cross section of the muscle made by a sphere
described with O as centre and with any radius, shall be
always ronstan'
In order to ascertain the law of distribution of fibres in a
triangular muscle, I made the following measurements in the
great pectoral muscle of the wing of the common Buzzard.
The weight of the muscle was 0.705 oz. av., and the lengti
of its excreme fibres was 6 inches. I took the breadth and
thickness of the cross section of the muscle at 6, 4, and 2 inches,
respectively, from its humeral insertion, and calculated the
area of the cross section from these measurements, with the
following results :
Pectoral Muscle of Buzzard.

Distance of Section
from lusertion.
Length of Section.
Mean Thickness of
Section.
Area of Cross
Section.
2 inches.
. 0.22 inches.
1.92 inches.
2.26 22
0.422 sq. in.
C.418 »
4
11
0.185
»
6.00
0.07
0.420
This Table demonstrates the accuracy of the supposition
ANIMAL MECHANICS.
185
.
OSCO11
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made as to the distribution and arrangement of the fibres,
and I believe that the same arrangement exists in every
muscle, however complicated in form; or, in other words,
that each muscular fibre passes from origin to insertion, re-
taining a constant thickness or cross section.
In the common Heron, I obtained results quite conform-
able with those found from examination of the pectoral muscle
of the Buzzard. In Fig.
31 is shown the pectoral
muscle of the Heron
OP is the insertion on
the ridge of the arm .
bone; XYZ is the curve
formed by the origin of
the muscular fibres, which was
radiate in a fan-shaped
manner from their in-
sertion, OP, to their ori-
gin, XYZ. I measured
the length of the line
XYZ, and took the thick-
ness of the muscle at
various points along this
line ; and also along a
Fig. 31.
curved line of section, xyz, made so as to divide all the radi-
ating fibres. The cross sections along XYZ, and myz are
shown in Fig. 32.
The section XYZ had a mean thickness of 0.22 in.
through two-thirds of its length, and of 0.335 in. through
one-third, giving a mean thickness of 0.258 in. The section
xyz had a mean thickness of 0.56 in. in the middle third, and
of 0.50 in, and 0.38 in. in the outer thirds, giving a total

186
ANIMAL MECHANICS
mean of 0.48 in. The length of the line XYZ was 7.42 in.,
and that of xyz, 4.27 in.* Hence we find the following :--
Pectoral Muscle of Heron.

Section.
Length.
Thickness.
Arca.
XYZ, ...
7.42 inches.
0.258 inches.
1.914 sq. in.
XYZ
..
4.27
-
0.48
99
2.0499
22
This result shows that the cross
section of the muscle is constant;
but, in order to verify the result,
I cut out two templets in zinc, of
exactly the shape of the cross sec-
tions in Fig. 32, and weighed
them, with the following re-
sult:--

Weight of (a) = 63.4 grs.
Weight of (b) = 65.6 ,
When we consider the difficulty
of measuring exactly the cross
section of a muscle, the foregoing
results must be regarded as prov-
ing that its cross section is, prac-
Fig. 32.
tically, constant.
Among the advantages that arise from this arrangement
of fibres in complex muscles, one of the most obvious is the
equal strength which it gives to all parts of the muscle, which
would not be secured by any web-like arrangement, allowing
* The total weight of the muscle was 2.68 oz. av.
ANIMAL MECHANICS.
187
of ene blending of the fibres with each other. Thus, in a tri-
angular muscle formed from a uniform sheet of muscle of equal
thickness throughout, the strength of the section near the
vertex would be much less than of that near the base, and the
muscle might be torn by the application of external violence;
whereas, in a muscle whose fibres are arranged in a fan-
shaped form, the cross section being the same everywhere,
the muscle is of equal strength, and the tendency to tear
across is lessened.
The following measurements, made on a male human sub-
ject, confirm the results obtained from the pectoral muscle in
Birds :-
Pectoral Muscle in Man.

Length of Section
Mean Thickness.
Cross Section.
2.20 inches.
0.31 inches,
0.682 sq. in.
4.25
1
0.16
10
0.680
22
»
11.50
,
0.06
0.690
There is always a loss of Work done in a triangular muscle,
which is due to the peculiar arrangement of its fibres in a fan-
like form; and, in this respect, muscles with radiating fibres
differ remarkably from the penniform muscles, in which, as I
have shown, the Work done is always equal to the inherent
Work. This may be proved as follows: let P, P, P", &c.,
be any forces, and let p, p, p", &c., be the displacements
(estimated in the directions of the forces) undergone by their
points of application ; and let R, r, be the resultant of the
forces, and the displacement of its point of application, esti-
mated in the direction of the resultant itself;—then we have,
by Lagrange's principle of virtual velocities,
Rr = Pp + P'p + P"p" + &c.
(32)
188
ANIMAL MECHANICS.
hand be too, Fita
Let AOB, Fig. 33, be a triangular fan-shaped muscle, and
let OY be the bisector of the vertical angle, and its length
2

!
Fig. 33.
equal b. Since the line of insertion, AB, remains constant in
length, when the muscle contracts, the line AB will move,
parallel to itself in the direction YO, into a position A'B',
and the fibre O Y will contract to its full amount of inherent
work, through a space AA' = BB' = 86; if we take any other
fibre, Ox = l, and draw xx' parallel to OY, and with O as
centre describe the circle x'p, then xp will be the space
through which the fibre l contracts; but this space is less
than xa = 8l, which bears to l the same ratio that AA' = 86
bears to b. Hence the fibre l, in contracting, only gives out
the work
O
1
fd0 x xp = f8b. cos Odo,
instead of the work
faldo,
which is inherent in it.
If we suppose (Fig. 34) AB and A' B' of indefinite length,
and draw ay parallel to OY in such a manner that Ox = Oy; it
is easy to see that all the fibres lying to the left of xy will be
shortened by the transference of AB to A'B', while all the fibres
.
ANIMAL 189
11
III
.
MECHANICSlying to the right of xy will be lengthened. No Work at all
is done by the fibre Ox, or those to the right; and of the

-
BY
Fig. 34.
fibres to the left of ay, all do some Work; but the Work done
by them, with the single exception of the fibre OY, is less
than their inherent Work. Taking the sum of all the Work
done by each fibre, we have-
+0
Work done = feb cos 0d0 = 2f86 sin 0,
a result that coincides with the Work done, found directly
from the resultant of all the forces (30).
When a fan-shaped muscle, instead of being triangular in
shape, has a curved base, as in the pectoral fin of the Angel
Shark, Fig. 29, the Work done by the muscle will still be re-
presented by double the perpendicular let fall from the foot
of the bisector of the vertical angle upon the side; and the
work inherent in the muscle will depend upon the integral
of ldo, taken all round the curve.
If this curve be a portion of a circle whose centre coin-
cides with the vertex of the radiating fan-shaped fibres, the
comparison of the inherent Work and Work done becomes
very simple. This case is nearly realized in the great pectoral
190
ANIMAL MECHANICS.
muscle of man, Fig. 35. Let 20 be the angle contained
between the extreme radiating fibres. Then, if I be the length

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Fig. 35.
of each libre,
Inherent Work = 2f8l x 0.
Work done = 2f8l sin 0.
and
Work done sin 0
Inherent Works
(33)
In the great pectoral muscle of man, Fig. 35, a circle PQ,
described with a point 0 on the humerus as centre, coincides
appromixately with the origin of the fibres, ABCD; and
since the angle 20, contained between the extreme radiating
fibres, is 90°; equation (33) becomes for the pectoral of man,
Work done sin 0
Inherent Work o
2V 2
x
= 0.9003.
ANIMAL MECHANICS.
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Hence there is 10 per cent. of Work lost, in the pectoral
muscle, by means of the fan-shaped arrangement of its
fibres.
The temporal
muscle in man,
shown in Fig. 36,
is a remarkable
example of fan-
shaped muscle;
and there must be
a considerable loss
of work occasioned
by its shape; this
loss of work would
be very difficult,
Fig. 36.
however, to calculate, in consequence of the elliptical form
of the line of origin of the fibres.
The latissimus dorsi in man may be considered as a muscle
whose fibres take origin from an arc of a circle, whose centre
is the point of insertion in the arm bone; the mean angle
contained by the extreme fibres is 350_hence we find, from
equation (33),

SN
Ally
sin 0
Work done
Inherent Work
- 300=0.08.
3007
3054
This shows a loss of only 2 per cent., caused by the fan-shaped
arrangement of the fibres, whereas the pectoral muscle loses
10 per cent., in consequence of the greater angle contained
between its extreme fibres.
The radiating fibres of the muscle of the iris give us an
example of the maximum loss of Work caused by fan-shaped
arrangement. We have seen that when the angle between
the extreme fibres increases from 350 to 90°, the loss of Work
192
4
V
done increases from 2 to 10 per cent. When the Gbres
radiate through a semicircle, we have
Work done sin 0 2
Inherent Work 7 - = 0.636.
This corresponds to a loss of nearly 36 per cent.
Deltoidal Muscles. The deltoid figure of geometers is a
quadrilateral figure, formed by constructing two isosceles tri-
angles, with unequal vertical angles upon the opposite sides of
the same base.
· Deltoidal muscles are found in nature, formed by the union
of two triangular muscles.
An excellent example is to be found in the trapezius
muscle of man, shown in Fig. 37.

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Fig.37.
The whole figure XAYB is a deltoid, and is formed by
the union of the two triangular muscles XAY and XBY; the
Parent
ANIMAL MECHANICS.
193
1
spines of the scapulæ are shown at AA and at BB' ; the line
AB joining the vertices of the two triangular muscles is
necessarily perpendicular to the spinal column XY; and the
bisectors of the vertical angles (AO and BO) are found to fall
behind and below the line AB; so that when the whole
trapezius is in action, the resultant forces draw the scapulæ
in the directions A and BO, shown by the arrows. This
action of the trapezius muscle is usefully employed, for ex-
panding the chest, in the kalisthenic exercises used by young
ladies.
Using the same notation as in the case of the biceps fe-
moris muscle of the lion, I have found the following results
from measurements made on the trapezius muscle of man:
XY = 16 inches.
X1 = 9 »
YA = 12
B = 83º = âOX.
0 = 47°30' = 0 AX.
p = 7 in.
9 = 5:16 in.
From these data we find, from equations (30) and (31),
Work done
Inherent work
cot } (B-05
ploge. cot (B + 0)
10.32
- 34
7 loge
cot. 170 451
cot. 65° 15')
=
0.7
The work done by the trapezius muscle is, therefore, only
77 per cent. of its inherent work, and there is a loss of 23 per
cent. due to the fan-shaped arrangement of its fibres.
The pectoral muscles of many animals, especially the
rodents, form, by union of the muscles of the right and left
sides, admirable examples of Deltoidal muscles.
In fig. 38 I have drawn the pectoral muscles of the com-
mon squirrel, 0, 0 being the humeral insertions of the two
194
TI
II
ANIMAL MECHANICS.
muscles, and AB the sternum; 0X is the bisector of the
angle BOA.

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Fig. 38.
Using the same notation as in equation (34), we have
B = 80°, p = 27,
0 = 30°, I = 10,
and, finally,
= 0.66.
cot (250)
ploge cot (559)
This shows a loss of 34 per cent., arising from the fan-shaped
arrangement of the fibres.
Quadrilateral Muscles.-It frequently happens that the
fibres composing a triangular muscle do not run on com-
pletely to a narrow insertion, but are inserted into a bone or
fascia, lying in the same plane with the line of origin. Such
a muscle may be called a quadrilateral muscle, and its origin
and insertion, like the base of the triangular muscle, may be-
come curved lines instead of right lines. The typical qua-
drilateral muscle is shown in fig. 39, in which AB, A'B' are
the origin and insertion of the muscular fibres. Produce the
extreme fibres AA', BB, to meet at 0, and draw OX'X
ANIMAL MECHANICS.
195
bisecting the angle A OB; and from
X'Y', perpendicular to O A.
X, X', let fall XY,

VA
-----
Fig. 39.
The quadrilateral muscle may be regarded as a triangular
muscle, A OB, from which a portion A'OB' has been cut
off, and the properties of the quadrilateral muscle may be
readily deduced from those of triangular muscles.
I have shown (equation 30) that the work done by the
triangular muscle AOB, is proportional to the perpen-
dicular X Y; but the work done by the part cut off, A'OB',
is proportional to the perpendicular X'Y'; hence the total
work done by a quadrilateral muscle is proportional to the
difference of X Y and X'Y'. Hence, whatever be the shape
of two quadrilateral muscles, if the difference between X Y
and X'Y' be the same in both, the two muscles will do the
same work in contracting: also, if the inclination, AOB
between the extreme fibres of two quadrilateral muscles, be
the same, the work done by each will be proportional, simply,
to the length of the right line XX'; for
XY = X 0. sin 6
X'Y' = X'O. sin e
Q 2
196
ANIMAL MECHANICS.
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and, therefore
XI-XV" = (XO - X'O). sin A = XX'. sino
Hence, when the angle 0 is given, the difference between X Y
and X'Y', bears a constant ratio to the right line X X'.
An example of the Quadri-
lateral muscle, with curved origin. Is.
and insertion, may be found in
the muscle, fig. 40, which re-
presents in the Nemestrine Ma-
caque the threefold muscle, ten-
sor vaginæ femoris, glutæus maxi-
mus, and agitator caudæ ; the line
Is, Il, represents the bony origin
of this threefold muscle, and the
lower curved line represents its
insertion into the fascia of the
thigh; its fibres converge towards
a certain point O, the angle be-
tween the extreme fibres being
Fig. 40.
40°; and the resultant force of all the fibres lies in the line O X,
which bisects the angle between the extreme fibres.
Quadrilateral muscles, whose lines of origin and insertion.
lie, strictly, in the same place, are very rare in nature, and
the whole group of muscles, called quadrilateral, tends to
merge itself in the more complex form of muscle, presently
to be described, in which the fibres, although rectilinear,
form, when taken together, ruled or skew surfaces.

Sphincter Muscles, whose Fibres are curvilinear and parallel,
lying in the same Plane.
In a sphincter muscle, each fibre forms a closed curve,
circular, or elliptical in shape; and the collection of such
ANIMAL MECHANICS.
197
fibres forms the entire muscle. Sphincter muscles are usually
placed round an orifice, which is closed by the contraction of
the fibres. These two systems of fibres counteract each other,
and by means of their alternate action the orifice may be
opened or closed to any required extent.
We shall first consider the theory of a single circular fibre,
but before doing so, it is necessary to prove an elementary
theorem in Geometry, of which we shall make use, not only
in the theory of sphincter muscles, but also in the theory of
muscles forming curved surfaces.
If from any
point 0, outside a
circle, Fig. 41, two
tangents be drawn
and the chord join-
ing their points of
contact, and the
diameter passing
through O be also
drawn:
Fig. 41.
Then let
go = radius,
t = tangent,
a = half the chord,
x = intercept between circle and point 0,
y = intercept between circle and chord :-
I say that when the point ( approaches indefinitely near
the circle, the intercepts x and y will become equal to each
other.
For
t2 = (2r + x) x (Euc. iii. 36).
az = (214 - Y) y (Euc. iii. 35).
Subtracting these equations from each other, we have
2r (ix - y) + (2x2 + y) = {2 - a”.

198
ANIMAL MECHANICS.
But t and a become equal to each other, and to half the arc
contained between the tangents when 0 indefinitely approaches
the circle ; hence the foregoing equation becomes
21 (x - y) + (ac2 + y2) = 0;
and, as x and y are indefinitely small, the term x2 + y2 disap-
pears, being of the second order, and the equation becomes,
finally,
2r (x - y) = 0
Or
x = y .
Q. E. D.
Let xyz, Fig. 42,
represent a circular
fibre of a sphincter
muscle, and let any
point 0, indefinitely
near the fibre, be taken,
and from this point
imagine two tangents
drawn to the circle.
When the whole fibre
contracts there will be
produced a tangential
zl
strain at each point of
Fig. 42.
the circle ; and it is
necessary for equilibrium that forces perpendicular to the
circle shall act at each point. We are required to find the
relation between the system of tangential and perpendicular
forces.
It must be remembered that the sphincter fibres can pro-
duce tangential strains only, for a muscle can only contract;
and that the perpendicular forces, which equilibrate the tan-
gential strains must be produced by a distinct set of radiating
fibres.

ANIMAL MECHANICS.
199
At the points of contact x, y, the tangential strains act in
the directions. Ox and Oy, and they must be equilibrated by
a perpendicular force acting in the direction Co, passing
through the centre. Let T represent the tangential strain at
each point x and y; the resultant of these two forces is
· R = 2T cos TOC.
Since the arc xy is small, we have the intercepts between 0
and the circle, and between the circle and chord xy, equal to
each other; let the tangent, arc, or semichord be called a, and
the intercept c; then we have
cos TOC =
a 20
a
=
a
e'
and therefore
R=270
If P represent the perpendicular force acting on each unit
length of the circle, the resultant R just found must equili-
brate the force P applied to the whole arc intercepted between
x and y; and hence we have
R = 2 Pa..
Equating these two values of R, we obtain,
2 Pa = 27
and, finally,
(35)
Hence, if the tangential strain be given, the perpendicular
force will vary inversely as the radits of the circle, or directly
as its curvature ; and if the perpendicular force be given, the
tangential strain will vary directly as the radius of the circle.
S
200
ANINAL NIECHANICS.
Fig. 43.
The first of these cases is one of common occurrence, as the
sphincter is often called upon to contract upon a resisting
body; in this case, the cross section of the sphincter ring
measures the total amount of the tangential contraction possible,
and the body filling the orifice will be compressed by a force
which increases as the aperture of the orifice diminishes.
The best example of sphincter
and radiating fibres to be found in
nature occurs in the muscles of
the iris, which close and open the
aperture of the pupil of the eye.
These are shown in Fig. 43.
The sphincter muscle forms a
flat narrow band on the posterior
surface of the inner rim of the iris
close to the pupil, and is about
oth of an inch wide.
The radiating fibres, which are antagonistic to the sphincter
fibres, commence at the ciliary, or outer margin of the iris,
and are directed inwards, in bundles, towards the pupil; at
the pupil itself, they blend with the sphincter, some of the
radiating fibres reaching as far as its inner margin.
Under the influence of these opposing muscles, the dia-
meter of the pupil ranges from rd to both of an inch. At any
given diameter of the pupil, equilibrium exists between the
forces T and P, developed by the sphincter and radiating
fibres; the ratio between these forces, however, does not re-
main constant, but ranges from 3 to 20, according to the dia-
meter of the pupil.
The sphincter muscle of the iris derives its motor supply
from the third pair of nerves, while the dilator muscle is sup-
plied from the sympathetic; thus, the opposing forces of these
antagonist muscles are instructed to act from distinct nerve
centres,

ANIMAL MECHANICS.
201
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Fig. 44.
-
.
The relations be-
tween the fibres of a
these muscles (as re-
vealed by the micro-
scope) are shown in
Fig. 44, which repre-
sents a portion of
these fibres, in the
iris of the Albino-
rabbit, (Kölliker)
magnified 350 times.
In this figure ab
represents the sphinc-
ter, and cccc the
radiating fibres.
The arrange-
ment of these
fibres shows in a
clear manner how
admirably they
are constructed to
react upon each
other in the man-
ner already de-
scribed.
The sphincter
muscle of the eye
forms, also, an
excellentexample
of this class of
muscles.
This muscle
00 (orbicularis
palpebrarum) is
shown in Fig. 45;


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ANIMAL MECHANICS.
2
it forms a thin elliptical sheet, surrounding the fissure be-
tween the eyelids, and covering their surface, and spreading
for some distance outwards, on the temple, upwards on the
forehead, and downwards on the cheek. The fibres may be
regarded as fixed at 1, the inner angle of the eye, where
they terminate either in the tendo palpebrarum or in the
superior maxilla or frontal bone.
The muscle is often considered by anatomists as composed
of an external and internal ring, according as the fibres lie
upon the eyelids (ciliaris) or lie outside them. The external
portion is bounded above by the frontal muscle, FF, with
whose fibres the outer fibres of the sphincter are blended ;
and the lower portion of the external sphincter sends off cross
slips, by which it blends itself with the malar muscles, MM,
The frontal and malar
muscles are the antagonists
of the external portion of
the sphincter; but the an-
tagonists of the inner por-
tion (ciliaris) are the radi-
ating fibres shown in Fig.
46, named levator palpebra-
rum, uu, levator palpebræe
superioris, ll, a portion of
the orbicularis muscle.
The outer fibres of the
Fig. 46.
external sphincter are
nearly circular, and have a radius of curvature equal to 0.89
in.; the fibres of the inner sphincter are elliptical, and their
outer portion corresponds to an ellipse, whose axes are in
the proportion of 9 to 5-the radii of curvature at the ex-
tremities of the axes being 9.25 in. and 0.45 in. respectively.
When the whole orbicularis muscle contracts, its outer

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ANIMAL MECHANICS.
203
fibres, being circular, contract symetrically towards the centre;
but the elliptical fibres, being attached to the tendo palpebra-
rum on the inner side, on contracting draw with them the
entire orbicular muscle towards the nose; the force with which
this is done varying as the curvature of the fibres at the
outer side of the eye.
The motor nerves of the orbiculàris muscle are derived
from the temporal branch of the facial nerve (7th pair). These
same nerves furnish the motor supply of the frontalis muscle.
The other antagonist of the orbicularis muscle, viz., levator
palpebræ, derives its motor supply from the 3rd pair of nerves
(motores oculorum). There would thus seem to be a more
complete antagonism between the levatores palpebrarum and
the orbicularis internus than between the frontalis and orbicu-
laris externus.
The mouth furnishes us with an extremely complex but
most beautiful example of sphincter and radiating muscles,
whose varied action produces an infinite variety in the shape
and expression of the lips.
The orbicular muscle of the mouth, like that of the eye, is
elliptical, and constitutes the only force available for closing
the lips. The radiate muscles of the mouth are eighteen in
number, nine at each side, placed symmetrically in relation to
those of the opposite side. The nine muscles at each side of
the mouth are arranged in two sheets, an outer and an
inner; the outer sheet consists of six muscles, and the inner
sheet consists of three muscles, which overlap the muscles of
the outer sheet at the angles of the mouth, where the curva-
ture of the orbicular muscle is greatest.
These muscles are-
204
ANIMAL MECHANICS.
OUTER LAYER.
1. Quadratus labii superioris (levator labii superioris).
2. Orbicularis malaris.
3. Zygomatici (major et minor).
4. Risorius.
5. Subcutaneus colli (platysma).
6. Triangularis oris (depressor anguli oris).
INNER LAYER.
7. Quadratus labii inferioris (quadratus menti).
8. Buccinator.
9. Caninus (levator anguli oris).
In fig. 47 I have drawn the directions of the foregoing
muscles in the relative position which they occupy with
respect to the sphincter muscle.
ER.
qis
915

om
gts
21
ami
---
--------8_be
1
n
860
Fig. 47
The latter muscle is represented by the ellipse, whose foci
are () and O'; on the left-hand side of the ellipse are shown
the directions of the outer layer of radiating muscles, and
ANIMAL MECHANICS.
205
(to avoid confusion) the deep layer of radiating fibres is shown
on the right-hand side of the diagram.
1. I. I. Quadratus labii superioris. The fibres of this sheet
of muscle act perpendicularly to the orbicularis oris, from the
extremity of the minor axis to a line passing through the
focus, and making an angle of 40° with that axis. This sheet
of muscle acts upon three-fourths of the elliptic quadrant, and
is the only radiate muscle that counteracts the orbicularis in
this part of its extent.
2. Orbicularis malaris. This little muscular slip is a portion
of the orbicularis oculi, on the outer side, and passes into the
upper lip, with the fibres of the quadratus labii superioris.
Its mechanical action is feeble, but identical with that of the
fibres of the latter muscle. Its chief use is not mechanical
but expressional, for it correlates the action of the sphincter
of the eye with that of the lifter of the upper lip ; as in the
highly natural gesture of winking with one eye, and lifting
at the same time the corner of the corresponding lip. It was
fully described by Santorini, who calls it perpetuus lacertulus
ab imo orbiculari.
3. The Zygomatic muscular slips occupy the position
shown in the diagram, and require no further description, as
they are well known.
4. 4. Risorius. This muscle forms a thin sheet of radiating
fibres, extending from 35° above the axis major to 40° below
the same; in conjunction with the deeper seated buccinator, it,
is intended to control the action of the orbicularis at the
extremity of its major axis, where its curvature is greatest,
and consequently the perpendicular force also required to be
S
greatest.
5. The Platysma of the neck (subcutaneus colli) sends a
sheet of fibres to the angle of the mouth, which act in a di-
rection 30° below the major axis.
206
I
TY
ANIMAL MECHANICS.
6. 6. The triangularis oris (depressor anguli oris) forms a
sheet, radiating from 68° below the major axis to a line,
OQ, very nearly at right angles to the same axis.
7.7. Quadratus labii inferioris (quadratus menti) forms a
radiating sheet whose fibres are perpendicular to those of the
orbicularis, through four-fifths of the lower elliptic quadrant,
from the minor axis to a direction 22° below the major axis,
near the corner of the mouth.
To understand exactly the relative positions of the deep
and superficial layers, we should imagine the ellipse folded
on its minor axis, so as to bring all the radiating lines on the
same side of the mouth.
This muscle acts alone on the orbicularis, from B to Q,
where the curvature of the sphincter is least, and is assisted
by other muscles, as it approaches the angle of the mouth
where the curvature is greater. It corresponds precisely
in its action and position with the quadratus labii superioris
on the upper side of the ellipse.
8. 8. 8. 8. Buccinator. This important muscle consists of
a flat, thin, strong set of fibres, in contact with the mucous
membrane, and forming a considerable portion of the wall of
the mouth ; its fibres range from 34° above to 34° below the
major axis, the upper fibres passing into the upper lip, the
lower fibres into the lower lip, and the central fibres decus-
sating, so as to pass, as shown in the figure, into the opposite
lip.
This whole arrangement gives a maximum of perpendicular
force along the major axis, at the extremity of which the
curvature is greatest. Its upperfibres O'P' or OP, mark the
boundary of overlapping of the deep and superficial muscles on
the upper side of the mouth. Thus the whole ellipse may be
divided into regions as follows :-
PAP' and QBQ counteract orbicularis by one sheet of muscles only.
PQ and P'Q' counteract orbicularis by two sheets of muscle.
S
D
ANIMAL MECHANICS
207
9.9. The canine muscle (levator anguli oris) acts in a
manner different from all the preceding; its fibres are tan-
gential to the ellipse in the neighbourhood of the extremity
of the axis major, and therefore aid the orbicularis rather
than counteract it. It may be regarded as a force in reserve
to aid the orbicularis where its curvature is greatest, and
where the most powerful apparatus of radiate muscles is
placed.
If we examine the outer sheet of muscles, we find that
they form a continuous radiating muscle through APQ, with
one break between the lower border of the risorius (4) and
the triangularis oris (6), below the angle of the mouth; this
break is marked xy on the figure, and is feebly supplied
by a few fibres of the platysma and triangularis. If we now
examine the inner sheet of muscles, we find them to form a
continuous sheet through BQ'P (or BQP), with the ex-
ception of a space, a'y', through which the buccinator and
quadratus labii inferioris overlap each other, forming a double
sheet.
It is interesting to observe that this space, w'y', overlaps
the space xy, when the figure is folded on its minor axis, so
that the overlapping of the sheets of inner fibres compensates
for the deficiency of the outer sheet.
The nineteen muscles of the mouth are all supplied with
motor power from the seventh pair, and their constantly varied
action produces the wonderful play of the lips, required
in articulation; as the curvature of each portion of the or-
bicularis changes, corresponding changes must take place'
in the action of the radiating fibres that counteract it, the
result being an endless variety in the expression of the most
expressive feature of the face.
This consideration furnishes an argument against the use
of the beard, which not only takes away from its wearer the
i
208
ANIMAL MECHANICS.
power of showing his feelings by the play of his lips, but also
confers upon him the dishonest advantage of concealing their
expression by means of a barricade of red, black, or grey
bristles.
The sphincter muscles already described-viz., of the iris,
eye, and mouth, have a corresponding set of antagonist radi-
ating muscles, intended to open the apertures round which
the sphincters are placed ; but there are other sphincters sur-
rounding apertures, which it is not necessary to open by
muscular agency. Such sphincters are the anal and vaginal
sphincters, which close their respective apertures, and are
not interfered with by any antagonist muscles.
The anal sphincter consists of two distinct parts; an inter-
nal part, composed of circular fibres of involuntary muscle,
and an external part, composed of elliptical fibres, which are
under the control of the will. The fibres of the elliptical
sphincter take origin from behind, in the os coccygis, and pass-
ing round the anus, unite in front of it, at the central point,
(tendo perinci); from this point again, in the female, the fibres
of the vaginal sphincter proceed, opening out to surround the
vaginal orifice and vestibule, and are finally inserted on the
corpora cavernosa clitoridis. The two sphincters thus form a
muscular sheet, resembling the figure 8, and may act together
or separately.
The levator ani, which forms the pelvic diaphragm, does
not act as an antagonist to the anal sphincter; on the contrary
its action aids the sphincter, for those of its fibres which blend
with the sphincter fibres are tangential to the curve of the
latter (like the fibres of the canine muscle in the mouth), and
therefore assist instead of opposing the constricting action of
the sphincter muscle.
During parturition, the levator ani and sphincter ani co-
operate to close the aperture of the anus, so as to allow the
ANIJAL MECHANICS.
209
abdominal muscles to exert their full force in expelling the
contents of the uterus and vagina.
Muscular Fibres forming curved Surfaces.
In all the preceding cases, we have supposed the sheet
formed by the muscular fibres to form a plane, but this
condition is rarely fulfilled with mathematical exactness.
The muscular sheets, in general, form surfaces more or less
curved, and it becomes necessary to take into consideration
the various curvatures of the several surfaces formed by the
sheets of the muscular fibres.
Surfaces are divisible into two classes--viz., those whose
curvatures at any point are in the same direction, and those
whose curvatures at any point have opposite directions. The
first class of surfaces may be called convex, or ellipsoidal sur-
faces; the second class may be called skew surfaces.*
If we imagine a convex surface, like an egg, uterus, or
bladder, and at any point taken on the surface draw a line per-
pendicular to it, and imagine a plane passing through this
line to turn round upon it, this plane in each of its positions
will intersect the surface in a certain curve. In the convex
surface, the curvatures of all these curves at the point in ques-
tion will be all convex or all concave together, in the same
direction.
If we imagine a skew surface, like a saddle, or dice box,
and draw a perpendicular to it at any point, the plane re-
volving round this perpendicular will intersect the surface in
certain curves, some of which are convex, and some concave;
and the convex curves are separated from the concave curves
by two lines intersecting upon the surface at the given point;
* This term is here used in a sense slightly different from its meaning when
employed by geometers.
210
ANIMAL MECHANICS,
so that the surface possesses a convex curvature in one of
the regions formed by those intersecting lines, and possesses
a concave curvature in the other region. In the direction of
the intersecting lines themselves, the surface has no curvature
at all, for these lines divide the convex from the concave
portion of the surface.
The lines drawn upon skew surfaces, along which the
surface is not curved, may become right lines, and, as the
surface itself is the aggregate of all the lines composing it,
we may have a skew curved surface composed altogether of
rectilinear fibres. This case frequently occurs in animal me-
chanics, the rectilinear generators being the actual fibres of
the muscles.
As the ellipsoidal muscles are more easily understood than
the skew muscles (notwithstanding that they are composed
altogether of curved and not rectilinear fibres), I shall com-
mence my description of curved muscular surfaces, with that
of ellipsoidal muscles, and proceed afterwards to show the
properties of skew muscles.
Ellipsoidal Muscles.-If a tangent plane be drawn at any
point of a convex or ellipsoidal surface, it will touch the sur-
face in one point only; and if a plane be drawn parallel to the
tangent plane and very close to it, so as to intersect the sur-
face, this plane will cut the surface in an ellipse, or circle,
according as the principal curvatures at the point in question
are unequal or equal. If tangent planes be now drawn to the
surface along the elliptic curve of intersection, they will form
a tangent cone to the surface, and this cone will be a circular
or elliptic cone, according as the curve of intersection is a
circle or ellipse.*
S
* The curve of intersection with a surface made by a plane very near and
parallel to a tangent plane is called by geometers the indicatrix curve.
ANIMAL MECHANICS.
211
In the simplest case of ellipsoidal muscles, that in which
the curvature at a given point coincides with that of a certain
sphere, let O be the vertex of the tangent cone, Fig. 48, OT,
OT being sides of the cone touching the surface at t, t.
Let C be the centre of the circular indicatrix, twyt, and
join OC. This joining line pierces the surface at the point S,
at which point, a tangent plane would be parallel to the plane

-
--
--
-
-
-
--
w
"
-
-
-
-
------
Fig. 48.
of the indicatrix ; and it can be shown, as in the case of the
circle p. 198, that the line OC is bisected at the point S, be-
cause O is indefinitely near the surface.
Let the surface now be supposed to be composed of mus-
cular fibres, and to contain a fluid reacting perpendicularly
to the surface, it is required to determine the conditions of
equilibrium. Let P denote the force acting perpendicularly
on the unit surface, and let T denote the tangential strain
caused by the muscular contraction, acting along the side of the
tangent cone, upon the unit of length of the indicatrix. It is
necessary and sufficient for equilibrium, that the perpendicular
pressure upon any element xy C, of the indicatrix, shall equi-
librate the corresponding muscular strain acting in the element
P2
212
ANIMAL MECHANICS.
of the tangent cone xy0. Let > be the centre of the circle
Sx or Sy, and o its radius. Since O is very near the surface,
Oy and Cy are to be considered equal ; let each of them be
called a.
The perpendicular pressure acting on the element Cxy is
Px za do
where do is the angle «Cy; and the tangential strain in the
element of the cone is
T x ado.
This must be resolved along the line OR, and equated to the
perpendicular pressure. This gives us
Txado x cos (YOR),
or
Tx adoxa.
Equating this expression to the perpendicular pressure, we
find that a’do goes out, and leaves us the equation
P-27'
This is the well known equation which is used by architects
in the problem of the equilibrium of a spherical dome.
When the surface has a spherical curvature at any point,
as in the preceding case, the tangential strain T, is the same
in every azimuth at each point; but when the curvature,
becomes ellipsoidal, and the indicatrix is an ellipse, the tan-
gential strain T' is different in different azimuths, „while the
perpendicular pressure P remains constant.
Let us now examine the general case.
Let S (Fig. 49) be the point of the ellipsoidal surface in
question ; and let täyt be the indicatrix ellipse formed by a
1
07
ANIMAL MECHANICS.
213
plane parallel to the tangent plane at S; and let Otæyt be the
tangent cone drawn to the surface along the indicatrix. It
m
V
.

11
Fig. 49.
can be shown as before, that OS is equal to C. Let the
following notation be used.
P = perpendicular pressure acting on the unit of surface.
T = tangential strain acting on the unit of length.
= Ct = semiaxis major of indicatrix.
b = Ct = semiaxis minor of indicatrix.
C = CS = so.
p = CX, Cy, = radius vector of indicatrix at any point.
Let xy be any element of the indicatrix, and produce the
tangent xyz to meet a perpendicular Oz, let fall upon it from
the vertex of the cone; and draw Cz. The lines Oz and Cz
are both perpendicular to the tangent to the indicatrix, and
are to be considered equal to each other, because the point o
is very near the surface. Let the element xy be called ds,
and let Oz or Cz be called p. It is necessary and sufficient
for equilibrium that the pressure P acting upon the elemen-
tary triangle Cay, shall be equilibrated by the vertical com-
ponent of the tangential strain acting on the element ay.
214
ANIMAL MECHANICS.
The perpendicular pressure acting on the elementary
triangle Cay is
Px 1 pds.
The tangential strain on the element xy, is perpendicular
to this element, and parallel to Oz, and its vertical component
Tds x cos (CO2);
but
cos (CO2) =
where p is the radius of curvature of the section of the surface
made by the plane OzC.
Hence, finally,
P x š pds = Tds x2
P=21
(36)
In using this equa-
tion it must be con-
stantly borne in mind
that is the ra-
dius of curvature of
the section passing
through the perpen-
dicular let fall on
the tangent to the
indicatrix, and not
that of the section
passing through the
radius vector.
Let åt Sty, Fig.
$0, represent this

Fig. 50.
S
ANIMAL MÉCHANICS.
215
section, and wz Stw the circle of curvature at the point S,
having its centre at R; then
C = OS = SC
p= 20 = %C
p = Rz = RC.
In the right angled triangle Rz0, we have
(20) = RO ~ CO;
p2 = 200.
(37)
Since 2c, the height of the vertex of the tangent cone, is the
same for all points of the indicatrix, we see that
Or
oc pa,
and since (because P is constant), by equation (36) T' varies
as p, it must vary as pa. Hence, we have the following
elegant theorem.

Fig. 51.
Let Fig. 51 represent the indicatrix ellipse at any point
of a convex surface; let C be its centre, and CR, CP, the
radius vector and perpendicular on tangent at any point. The
surface forms an elliptic dome, of very small height, standing
216
ANIMAL MECHANICS.
upon the indicatrix. It follows from the foregoing, that the
tension of the surface in any azimuth CR, varies directly as
the square of CP.
Hence the greatest and least tensions will be at the ex-
tremities of the major and minor axes of the ellipse, and will
be proportional to the squares of those axes.
The preceding problem, as is evident, contains the so-
lution of the conditions necessary in architecture to construct
an ellipsoidal dome.
It is sometimes convenient in problems in Animal
Mechanics to use the mean tension of the muscular wall,
without regard to its variation as the azimuth varies; and for
this purpose, we must take the mean of all the tensions in
every possible azimuth.
The tensile strain at each point of the indicatrix is Tds,
acting parallel to the line Oz, Fig. 49, and perpendicular to
xy. The component of this tension, in the direction OC, is
S
Tds cos (20C) = Tds x ? ;
but, by equation (37)
p_20
p
p'
and hence, the tensile force acting on the element ay, in the
vertical direction OC, becomes
Ids x 20
P
We must now regard T as a constant, having the mean
value of the tensions taken in every azimuth, and integrate
the foregoing expression all round the ellipse.
ANIMAL MECHANICS,
217
To do this wo zaust find the value of
in an ellipse.
Assume
x = a sin o
y = b cos ,
then
ds = do ✓a cos 2p + Z2 sin 20
ab
P- Jacos 20+ b2 sin "P
ds
(a' cos 20 + b² sin ap) do
ab
P
The integral of this expression taken all round the ellipse
is
a² +6²
fils
JP
-
TT
-
and
20
120
= ab x T 10
207 milis
2cT
*
200
or finally
2ct
207 10ls
mi I
= hab x T1-t
\P P2)
where pı and p: are the radii of curvature of the surface
j along the axes of the ellipse.
The vertical component of all the elementary tensions,
just found, must be now equated to the sum of all the perpen-
dicular pressures acting on the indicatrix, which is
Pxfipes = P x tab.
218
ANIMAL MECHANICS.
P = Tt
Hence we obtain
I I
Px zab = gab x T = + =
lei P2,
or
TI
la pel
which is Lagrange's theorem, already employed at page 152.
The first example of ellipsoidal muscles that I shall dis-
cuss is the case of the abdominal muscles, already considered
in a general way in treating (p. 151) of the muscular force
employed in parturition. The principal curvatures of the
abdomen at the navel (p. 161) were shown to correspond to
circles of curvature of 22.7 in. and 12.6 in. diameter respec-
tively; and the tensile strain in the walls of the abdomen is
produced by the simultaneous action of the eight abdominal
sheets of muscle. According to the theory just developed,
the tensile strains produced by all the muscles in the vertical
and transverse directions at the navel, should be in the pro- ·
portion of the diameters of curvature in the same directions ;
and the axes of the indicatrix ellipse at the navel should be in
the proportion of the square roots of those diameters.
A severe test may thus be applied to the theory, and to
the measurements of the forces produced by the abdominal
muscles. For the purpose of applying this test I shall make
use of the muscles of subject No. 2 (female), p. 160, the
measurements of whom corresponded nearly with the mean of
all.
The four muscles at each side of the central line have the
following mean thickness :-
1. Rectus abdominis, .... 0.29 inches.
2. Obliquus externus, · · · · 0.25 »
3. Obliquus internus, . 0.17
4. Transversalis, .... 0.15 »
ANIMAL MECHANICS1
219
.
ELBA
NIINIA
.
Will
Of these muscles, the rectus abdominis produces a tensile
strain altogether in the vertical direction, and the transversa-
lis causes a tension in the horizontal direction only. These
two tensions are nearly in the proportion of 2 to 1, which ap-
proximates the ratio of the diameters of curvature. If there-
fore these muscles, alone, were in action, the theory would
be fully vindicated. The oblique muscles, however, produce,
each of them, a vertical and horizontal component strain at
the navel, which must be added to the former, to determine
the total strain acting in the walls of the abdomen in those
directions. Let us now proceed to calculate these compo-
nents, and test the entire the ory.
Let 0, E (Fig. 52), re-
present the obliquus externus
muscle, and SNP the linea
alba, S denoting the end of
the sternum, N the navel,
and P the pubes ;—11. being
the crest of the ilium. The
line AB represents the inner
limit of the muscular fibres,
where they become inserted
into the fascial sheet enclos-
ing the rectus abdominis.
The fibres of the external
oblique muscle are approx-
imately parallel to each
other, and constitute a pris-
matic muscle spread in a
sheet over a convex sur-
face,--and the prolonga-
tions of its fibres to the
Fig. 52.
linea alba make angles of 35º with that line. Each of the

Ic
AN
220
ANIMAL MECHANICS.
muscular fibres, when combined with the corresponding fibre
of the other side, produces a vertical strain acting from P to-
wards S, and a horizontal strain which is equilibrated by the
equal and opposite strain on the other side of the abdomen ;-
and there is a remarkable difference between these two tensile
strains, for the transverse strain is constant along the entire
linea alba, while the vertical strain is greatest at P, and di-
minishes from P to S. Take, for example, the fibres NX,
passing through the navel; the muscular fibres between NX
and B, produce along the line PN vertical components acting
towards N, and consequently they produce no strain whatever
at the point N; on the contrary, the fibres lying between
NX and A, all combine to produce a vertical strain at N,
acting towards the point 8. It is therefore evident that the
vertical strain on the linea alba caused by the external oblique
muscle, increases from S to P, and varies in the proportion
of the distance of the point considered from 8. The compo-
nents of the external oblique muscle, acting upon a linear inch
at the navel, may now be calculated.
Draw the line NX, and let fall the perpendicular AK
upon it. If t be the thickness of the muscle, and the angle
XNS, we have for the vertical component,
7
V = AKx + x 2 cos ;
and for the horizontal component,
H = AK x + x sin o.
In order to find the vertical tensile strain, on a linear inch,
we must take account of the width of the recti abdominis
muscles, for as the fibres terminate in the line AB, at the outer
border of the recti muscles, their vertical component is to be re-
garded as distributed along the entire breadth of those muscles.
ANIMAL MECHANICS.
221
Hence, if w denote the width of the recti muscles, and p the
length of AK, the vertical component per linear inch will be
V = pxt x 2 cos 0
.
(38)
W
The horizontal strain corresponding to a linear inch of the
· linea alba will not, of course, be the strain corresponding to a
width (AK) of muscle, but is produced by a width equal to
sin 0; substituting this width for AK, we find
H = t x sin po.
.
(39)
I find, by careful measurements, the following values to
be substituted in the foregoing expressions-
AK = 4.25 in.
t = 0.25 »
0 = 35°
w = 5,41 in.
Hence we obtain, finally,
V = 4.25 x 0.25 x 2 cos 35º
35 = 0.321
5.41
H = 0.25 * sin? 35º = 0.082.
S
The internal oblique muscle is shown in Fig. (53), where
P, N, 11., denote the pubes, navel, and crest of ilium, and
AB the insertion of the muscle into the outer border of the
rectus sheath. The internal oblique muscle is not prismatic,
but quadrilateral, and its action is more complicated than that
of the external oblique. The upper fibres give a vertical com-
ponent which draws the linea alba, downwards, towards the
222
ANIMAL MECHANICS.
А
-
*
navel, and the lower fibres give a vertical component which
draws the linea alba, upwards, towards the navel; hence,
neither sets of fibres
can produce any tensile
strain at the navel itself.
The only portion of the
muscle that causes a
vertical tension at the
navel, is the portion
contained between the
horizontal fibres XY
and the line NX, which
is the direction of the
Y fibres passing through
the navel itself. This
portion of the muscle,
and the corresponding
portion of the similar
muscle, at the other
Fig. 53.
side, will behave as a tri-
angular muscle, whose
resultant ZX bisects the
angle NXY, and is equal to (p. 181)

-
Z
-
X
::
.
W
2 f sin 0,
where 0 denotes half the angle NXY, and f denotes the cross
section of the muscle. Hence the vertical resultant of the in-
ternal oblique muscle at both sides is
2 f sin 0 x 2 cos o,
where o denotes the angle made with the linea alba by the
line ZX, and is the complement of 0. This resultant must be
divided by the total width of the recti muscles, in order to find
ANIMAL MECHANICS.
223
the strain per linear inch, as in the case of the external oblique
Hence we find, finally,
V 2f sin 0 x 2 cos o 2f versin 20
(40)
20
w
In this equation, f is equal to the product of the cross section
of the muscle, multiplied by its width enclosed between the
lines NX and XY.
Cross section of internal oblique = 0.17 in.
Width of do., . . . . . . = 2.97 »
O = 90° - 0.....= 13°
Width of recti muscles, ... = 5.41 in.
Hence we have
V_ 2 0.17 x 2.97 x versin 26°
- = 0.019.
5.41
The transverse strain produced at the navel by the inter-
nal oblique muscle, depends on the direction of the fibres in
its neighbourhood only, and is found by the same formula as
the transverse strain of the external oblique muscle, viz.- .
H = t x sin ,
1 (41)
which gives us, for the transverse strain, per linear inch at
the navel,
H = 0.17 x sin 64° = 0.137.
Adding together the vertical and transverse tensile strains
produced by all the muscles at the navel, we find-
Vertical Tension due to Abdominal Muscles.
Rectus abdominis, ...... 0.290
Obliquus externus, . . . ... 0.321
Obliquus internus, ...... 0.019
Transversalis, .... ... 0.000
0.630
224
ANIMAL MECHANICS.
Transverse Tension due to Abdominal Muscles.
VU U
LL
U
V
Rectus abdominis, ... ... 0.000
Obliquus externus, : ..... 0.082
Obliquus internus, ...... 0.137
Transversalis, . . . . . . . 0.150
4
.
0.369
From the preceding results, it follows that the transverse
strain at the navel is 58.57 per cent. of the longitudinal
strain-
Transverse strain - 369 x 100 = -8.64 per cent.
Longitudinal strain 630
At p. 161, I have given the measurements of the abdomi-
nal curvatures of three young men, when the abdomen was
distended to its utmost extent. From the mean of all three
we obtain, in the case of young men, the result that the dia-
meter of the transverse circle of curvature is 55.58 per cent.
of the diameter of the longitudinal circle of curvature-
Transverse diameter 12633 X 100
Longitudinal diameter 22727
-- = 55.57 per cent.
As it was desirable to ascertain the amount of curvature
of the abdomen, at the time when its muscles are used in the
act of parturition, I obtained the following measurements of
five healthy women, taken at the commencement of labour.
I instructed the head midwife of Sir Patrick Dun's Hospital
Maternity to measure for me, with tape, the following quan-
tities :-
1. Height of woman.
2. Girth transversely round navel.
3. Length of curved arc from the end of the sternum to
the top of the symphysis pubis.
TTTT
ANIMAL MECHANICS.
225
In well formed skeletons the distance from the top of the
symphysis to the end of the sternum is 17ths of the height of
the body, so that from the measurement of the height, I
could find the length of the chord of the longitudinal arc,
whose length was measured directly.
If 0 denote half the arc subtended by this chord, we have
the following equations, to determine p, and 92, the radii of
longitudinal and transverse curvatures :
chord = 52 x height
07
arc
chord
0
sin 2
chord
sin 0 = 20
girth -
= 202
The measurements made were the following:-
Parturient Women.

Height.
Longitudinal
Arc.
Transverse
Girth.
16.8 in.
39.3 in.
60 in.
63.6 ,
31.2
No. 1.
No. 2.
No. 3.
No. 4.
No. 5.
61.5 »
19.019
16.8 ,
15.7 »
21.8 ,,
59.8 1
69.9 19
From these data, the following table was calculated by means
of the foregoing equations :-
226
ANIMAL MECHANICS.
Parturient Women.

Diameter of
longitudinal
Curvature
Chord.
Semiangle
Diameter of Ratio of Curva-
transverse tures
| =100 x Pa
=2p1
= 22
Pi.
1.
43°42
56.8
| 15.22 in.
16.13 ,
15.60 ,
22.03 in.
19.50
55°48'
37°36'
25.51.»
12.51 in.
9.93 »
14.10 „
14.71 ,
12.63 »
50.9
55.3
57.6
63.1
15.17 »
26°00'
34.60 ,
20.01 »
17.73 »
The mean of the ratios in the last column gives us, for
parturient women,
Transverse diameter
Longitudinal diameter = 50.74 per cent.
This ratio agrees very well with the 55.57 per cent., found
from the measurement of the abdominal curvatures in young
men, and they both agree with the corresponding proportion
of transverse and longitudinal tensile strains (58.57 per cent.),
deduced by calculation, from the dissection of a healthy young
'woman, who had borne several children.
If we compare together the mean curvatures of the several
cases, we shall find, as we might expect, a considerable differ-
ence depending on height and general configuration of body.
But it is worthy of remark, that the mean value of all the
curvatures differs very little from the mean maximum curva-
ture of the abdomen, voluntarily distended :-
ANIMAL MECHANICS:
227
Mean Curvature of Abdomen in Parturient Women.

Mean Curva-
w Longitudinal Transverse Dia-
NO. Diameter = 2pil meter = 202
ture =
t
-
P2
P₂
22.03 in.
12.51 in.
3.987
2. | 19.50 »
9.93 »
_1
3.289
14.10 )
4 540
25.51 »
34.60 »
20.01 »
schot
14.71 »
12.63 ,
3872
Mean
4.1738
This mean curvature differs little from the mean curvature
of the voluntarily distended abdomen of young men—viz. :
so given in p. 162.
can
ansverse
We can calculate, from the vertical and transverse ten-
sions of the abdominal muscles at the navel, given in p 223-4,
and from the curvatures just given, the perpendicular pres-
sure per square inch, caused by the action of those mus-
cles, in each of the preceding cases. For, by equation (36),
we have
P-21;
which becomes, on adding 50 per cent. to the measurement
of the dead muscles,
P-4 * 0.945
2pi
for the longitudinal strains; and
P-4 * 0:553
2p2
Q 2
228
ANIMAL MECHANICS.
for the transverse strains. Substituting in succession, the
values of żpı and 2p2, we obtain-
Pressure producible by Abdominal Muscles acting at navel,
in Parturient Women.

Deduced from Longi-
tudinal Curvature.
Deduced from Trans-
verse Cuvatuie.
| 17.59 lbs. P sq. in. | 18.13 lbs. P sq. in.
19.88 1
22.84 1
| 15.19 » 16.09 »
15.42 „ „
| 19.37 . . . 17.96 m
| 11.20
Mean
16.646 lbs.
18.088 lbs.
These pressures are produced by the abdominal muscles, act-
ing at the navel, under the conditions of curvature, proper to
each case; and are only about half as great as the pressure at
navel, obtained by direct experiment (page 163); but they
do not represent the maximum pressures producible by the
abdominal muscles. In order to show this, let us calculate
the tensile strains produced at each point of the linea alba.
The longitudinal and transverse strains, produced by the
recti and transversalis muscles, are constant at each point of
the linea alba; but this is not the case with the oblique mus-
cles. The vertical and horizontal strains of the external oblique
are, as shown by equations (38) and (39),
v 2t cos 0
COs xp
V
=
20
H = t sin 8.
In these equations, everything is constant except p, the per-
pendicular thickness of the portion of the external oblique
ANIMAL MECHANICS.
229
above the point of the linea alba under consideration. The
value of p, at the navel, is 4.25 in. (p. 221), and it increases
as we go downwards towards the pubes, where it becomes
8.11 inches. While, therefore, the horizontal component of
the external oblique is everywhere the same, its vertical com-
ponent reaches a maximum just above the pubes.
This maximum is represented by
8.11 X 0.25 x 2 cos 35 -0.614
5.41
The vertical and transverse strains produced by the inter-
nal oblique, are both variable, as shown by equations (40)
and (41)
V- 2f versin 20
w
H = t sin .
Ir these equations, f, 0, and p are all variable, and in the fol-
lowing manner :-Let the point be taken on the linea alba
(about half way between the navel and the pubes), where the
fibres of the muscle become horizontal. At this point there is
no vertical strain, and the horizontal strain is a maximum, be-
cause o becomes a right angle; the transverse strain at other
points varies as the square of sin o, but is always less than at
· the point named. With respect to the vertical strain at any
point, it is thus found: 20 = 90° - p is the angle made with
the horizontal line, by the fibres passing towards the point,
and f varies, as the base of the triangular muscle formed by
these fibres and the horizontal fibres. Now, it is evident that
the versine of 20 and the base of the triangular muscle both
increase as we pass along the linea alba, in either direction,
above or below the central point between the navel and pubes,
where the whole action of the internal oblique is transverse ;
so that the vertical strain produced by this muscle has two
maxima, one near the pubes, and the other near the ster-
(1
2
230
ANIMAL MECHANICS.
num. The measurements for these two points are,
At pubes, 20 = 65°:f = 3.12 * 0.17,
At sternum, 20 = 45° f = 7.16 * 0.17.
Hence, we find the vertical strain producible at the sternum,
by the internal oblique muscle, to be
2 * 7.16 * 0.17 x versin 450
- = 0.135.
5.41
and the vertical strain, producible at the pubes, by the same
muscle,
2 * 3.12 X 0.17 x versin 65º .
5.41
Let us now add together the vertical strains at the point
in the linea alba, just above the pubes, and we shall find,
Rectus abdominis, · · · · · · 0.290
Obliquus externus, ..... 0.614
Obliquus internus, · · · · · · 0.113
Transversalis, . . . . . . . 0.000
- = 0.113
1.017
We have seen that the vertical tension, producible by the
muscles acting at the navel, was represented by 0.630 only; so
that in order to calculate the maximum hydrostatical pressure,
producible inside the abdomen by the abdominal muscles
acting to the greatest advantage, we should increase the num-
bers given in the table, page 228, in the proportion of 1017
to 630.
If we take the largest number in the table, corresponding
to No. 2, we have
maximum pressure
inside abdomen = 22.84 * 656 = 36.87 lbs. per sq. in.
This result agrees well with that given, from direct experi-
ment (p. 163), as the maximum expulsive force of the ab-
cominal muscles-viz, 38.47 lbs.
1ΟΙ
ANIMAL MECHANICS.
231
It is interesting to observe that the tension in the vertical
direction produced by the external oblique muscle, varies as
the distance of the point of the linea alba under consideration
from the sternum, which is precisely the law according to
which the pressure of the contents of the abdomen, regarded
as semi-fluid, would vary ;. viz., as the depth of the fluid
measured from its surface. .
The Diaphragm, which forms the upper surface of the
abdominal cavity, furnishes an example of a convex ellip
soidal muscle ; it is used continually in respiration, and
occasionally in the expulsion of the contents of the abdomen.
Its force is considerable, although not equal to the maximum
effort of the abdominal muscles; and being an involuntary
muscle not under our control, when it provokes the abdominal
muscles to take on a violent reflex action, as in coughing, its
effects are sometimes dangerous, as, for example, in cases of
hernia. It is, therefore, important, not merely in a scientific
but also in a practical point of view, to know the amount of
hydrostatical pressure it is capable of producing upon the
contents of the abdomen.
From measurements made by me, I infer that the mean
curvatures of the diaphragm, when most convex, correspond
with circles of 5.601 inches, and 5.109 inches in diameter
at the two sides of the body, and that the mean thickness of
its muscular walls is 0.18 in. Adding 50 per cent to the
latter measurement as before, and taking the mean diameter
of curvature, we obtain from equation (36)
2T 4T
P = =-
P 2p ²
and from the coefficient of muscular force (p. 69),
K = 102.55 lbs.
p
.4 x 0.27 X 102.55
98 = 20.68 lbs. per sq. in,
5.355
232
ANIMAL MECHANICS.
This result is comparable with the expulsive force of the ab-
dominal muscles acting at the navel, although inferior to
their maximum force exerted near the pubes.
Muscular Skew Surfaces.
The last and most complex muscular surfaces to be con-
sidered are those corresponding with skew surfaces in
geometry. The simplest form of skew muscle occurring in
nature is that shown in Fig. 54, where AB and A' B' are two

.
-
.
...-
LAM
IIIIIIIIIIIIIIIIIIIII
11
Fig. 54
bones not lying in the same plane, and each divided into an
equal number of parts, such that the anharmonic ratio of any
four points on one bone is equal to the anharmonic ratio of the
ANIMAL MECHANICS.
233
four corresponding points on the other bone; and the corre-
sponding points on the two bones are supposed to be joined
by straight muscular fibres capable of contraction.
The surface formed by these muscular fibres is a skew
surface, of a simple kind, called the Hyperboloid of one sheet.
It may be regarded as formed from the Quadrilateral muscle,
p. 195, by placing the origin and insertion of the muscular
fibres AB and A'B', in different planes, so as to form a skew
quadrilateral instead of a plane quadrilateral.
A skew surface, of which the Hyperboloid of one sheet is
one of the simplest examples, may be readily imagined from
the shape of a saddle; and at each point it possesses curvatures
which are not all in one direction, as in the convex or ellip-
soidal surface. If a tangent plane be drawn to the skew
surface at any point, this plane, instead of touching the sur-
face at a single point, will cut the surface along two inter-
secting right lines, which divide the surface at the point of
contact into two regions, in one of which the curvatures are
convex, and in the other concave, while along the right line
themselves, as is evident, there is no curvature at all.
If a plane be drawn parallel to the tangent plane, and
very near it, it will inter- A
sect the surface in a hy.
perbola and not an ellipse.
Let this hyperbola, or
indicatrix, be represented
in Fig. 55, its asymptotes
XY and AB, are parallel
to the two right lines, 6
along which the tangent
Fig. 55.
plane intersects the surface, and the indicatrix plane meets the
surface inside the angles X0A and YOB, but does not meet
the surface inside the angles XOB and YOA. If, however,

-
Y
234
ANIMAL MECHANICS.
Y
a second plane : be drawn very near the tangent plane, and
at the other side of it, this plane will intersect the surface as
shown in Fig. 56, not A
meeting the surface
within the angles
X0A and YOB, but
cutting it along the
hyperbola contained
in the angles XOB
and YOA. : As be-
fore, XY and AB are x'
TB
asymptotes to the hy-
Fig. 56.
perbola, and parallel to the right lines along which the tan-
gent plane intersects the surface. The curvatures of the sur-
face, in the two angular regions marked out in Figs. 55, 56,
&c., are in opposite directions, and, therefore, the perpen-
dicular pressures acting on the muscular sheets in these two


Fig. 57.
regions are also in opposite directions, while along the lines
ANIMAL MECHANICS.
235
1
S
X Y and AB, there is no curvature and no force perpendiculer
to the muscular fibres.
In the simplest case of muscular skew surface that occurs
in nature, let AB and A'B', Fig. 57, represent straight bones
which are the origin and insertion of the rectilinear muscular
fibres joining them. The curved surface formed by the mus-
cular fibres (vide Fig. 54) is a surface of the second order,
known as the Hyperboloid of one sheet, the direction of whose
centre may be found readily by means of the following simple
construction.
Draw the diagonals AB' and A'B of the skew quadrilateral
ABBA, and bisect each diagonal in x and y, and join x and
y, producing indefinitely both directions; the right line xy
passes through the centre of the hyperboloid.
This construction may be thus proved—Fig. 58. Since
B

Fig. 58.
the bone AB, and the muscular fibre AA' both lie upon the
surface, the plane BA A' is a tangent plane at the point A ;
and in like manner, the plane A'B'ß is the tangent plane at
the point B'; and similarly the tangent plane at A' is the
236
ANIMAL MECHANICS,
plane of the bone A'B', and of the muscular fibre A'A; and
the tangent plane at B is the plane ABB'. Draw any number
of chords ab', ab', &c., parallel to the diagonal AB'; since
these are all chords of the common surface, their points of
. bisection all lie upon a diametral plane By A' of that surface,
and passing through y, the point of bisection of the diagonal
AB. In like manner, if a number of chords a'b, a'b, &c., be
drawn parallel to the diameter A'B, their points of bisection
will also lie upon another diametral plane AxB', and passing
through X, the point of bisection of the diameter A'B. The
common line of intersection xy of these two diametral planes
By A' and AxB', will, therefore, pass through the centre of
the skew surface, or hyperboloid.
We are now in a position to determine the mechanical
conditions of equilibrium A
of such a muscle. Let
XY and AB, Fig. 59, de.
note the directions of the
two generating lines at
any point 0, and let pa-
rallel planes be drawn
at equal small distances
above and below the tan- x
gent plane; and let their.
Fig. 59.
indicatrix hyperbola be projected, as in the figure upon the
tangent plane. Then, if a and b denote the lines Oa and Ob,
the equations of the indicatrix hyperbolas, which are con-
jugate to each other will be, if referred to the axes Oa and

1
ob,
edhe
Z2 = $1.
(42)
All that has been previously stated respecting the tangent
cone and indicatrix ellipse, in convex muscular surfaces, holds
ANIMAL MECHANICSU
237
.
true in the skew surface for its tangent cone and indicatrix
hyperbola. Thus, the tensile strain in the tangent plane at
0, will vary in any azimuth as the square of the perpendicular
curvature of the section of the surface passing through that
perpendicular. Let us trace the variation in the tensile strain
round the point 0. In the direction Oa the strain is propor-
tional to a?, and diminishes from Oa to OY, varying always as
the square of the perpendicular let fall from O upon the tan-
gent to the hyperbola YaB; in the direction OY, this per-
pendicular vanishes because OY is the asymptote, and the
tensile strain and curvature vanish with it. When we pass
OY, the tensile strain increases from zero to b?, in going from
OY to OB; varying now according to the square of the per-
pendicular let fall upon the tangent to the hyperbola YbA,
from the centre 0, and so on. Thus the tensile strain has
two maxima, a, and b, corresponding to the axes Oa and Ob,
and vanishes between these maxima, in the directions XY
and AB, the asymptotes of the indicatrix hyperbolas. In
the convex muscular surface, on the contrary, the tensile
strain ranged from a maximum aż to a minimum b, and never
vanished at all. The perpendicular pressures, produced by
the tensile strains in the skew surface, are in opposite direc-
tions inside the angular spaces containing the two hyperbolas,
because the curvatures of the surface have opposite signs in
these two regions.
The joint effect of the tensile muscular forces in a closed
convex surface is to compress the contents enclosed within
it; but in the muscular skew surface, which is an open surface
and cannot be made to enclose a space, the joint effect of a
contraction of all its fibres results in an effort made by the
surface to destroy its own curvature and return into the con-
dition of a plane surface. Thus, if one of the bones AB be
238
ANIMAL MECHANICS.
fixed, and A'B' free to move (Fig. 54), the contraction of the
fibres will bring A'B' into the same plane with AB, making
ABA'B' a plane quadrilateral; this reduction of the skew
quadrilateral to a plane quadrilateral being accompanied by a
peculiar twisting motion of the entire sheet of muscle varying
with the nature of its curvatures at each point. .
Let us now consider, in detail, the action of a quadrilateral
skew muscle, in which one bone, AB, or A'B' is fixed, and
the other is either altogether free, or subject to various con-
ditions in its motion, the most usual of which is, that it is
compelled to turn always round some fixed socket or joint,
whose position, as well as that of the fixed bone, is known
and given. As the muscle may be supposed to perform its :
work by two distinct motions, one of which reduces the skew
quadrilateral to a plane, and the other is confined altogether
to that plane, it will be convenient to divide the propositions
relative to skew quadrilateral muscles into two parts, viz.-
1. Action of a quadrilateral muscle in a plane. .
2. Reduction of a skew quadrilateral muscle to the corre-
sponding plane muscle.
The following postulate is necessary in order to solve the
problems that present themselves, and will be admitted by
some readers as an a priori truth, and by others as an induc-
tive truth demonstrable by instances.
POSTULATE.
The Framer of the Universe (Δημιουργός του κοσμού) has
the maximum of Work possible for it under the given external
conditions. . .
From this postulate, the following results appear:-
PROPOSITION A. (Problem.)
Given a fixed bone. A'B', and a perfectly free bone AB,
lying in the same plane, forming the origin and insertion af a
ANIMAL MECHANICS.
239
quadrilateral muscle; it is required to find the motion of AB,
when the muscle contracts.
The most general possible movement of AB may be re-
presented by a movement of translation parallel to itself, and
a movement of rotation round a point.
Produce AA' and BB' to meet at 0, and draw OX, the
bisector of the vertical angle. I say that no work is done by a

.
Fig. 61,
rotation of the bone AB round an axis passing through X
the foot of the bisector, and perpendicular to the plane of the
muscle, and, therefore, that the muscle by its own contraction
can never produce such a rotation. For, let sy be the new
position of the bone AB, when turned through a small angle
of rotation, «XA = yXB = W.
Let aa'o be the original direction of any muscular fibre,
and let ab be the path described by the point a during the
rotation; draw bp perpendicular to ad' and join d'b. Let
Ou = 1
Oa' = 1
aOX = 0
BX0 = B
0X = 6.
240
ANIMAL MECHANICS.
It is plain that ap denotes the change in length of the fibre
aa' during the rotation, and the work done by it is
oldo - ap x do,
but
ap = ab x sin ( - 0)
sin 0
ab = w + a X = w * OXX
sin (8-0)
Therefore,
ap = w * OX sin ,
and
8ᏓdᎾ = ob sin ᎾdᎾ .
Integrating this element from A to BO, we find
p+ 0
Work done =
wb sin 0d0 = 0.
Hence, no work is done by the muscle in any rotation of the
bone AB round an axis passing through the foot of the bi-
sector 0 X, and perpendicular to the plane of the muscle.

ID
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
&
Fig. 61.
Let us now suppose the bone AB, Fig. 61, to be moved
parallel to itself into the position xy, and let
* AB = 0;
ANIMAL MECHANICS.
241
it is required to find the conditions necessary to produce the
maximum amount of work done by such a movement.
Let aa' be any fibre, and let ab be the line traversed by
the point a; and draw bp perpendicular to aa'. Then
&l= ad' - a/b = ap = ab x cos (pab) but pab = B - 0-0.
Therefore,
old0 - ab cos (B-0-0) do.
Let ab, which is the same for all points of the bone be called
A, then we have
Work done - Oldo = A cos (B-0-0) do,
= A{sin (ß-0+0) - sin (ß-0-0)}.
or
Work done = 2A cos (3 - 0) sin 0. (43)
This expression is a maximum when ß = 0, or when the bone
is moved parallel to itself, in a direction parallel to OX, the
bisector of the vertical angle of the triangle formed by pro-
ducing A 4' and BB' the two extremo fibres of the Quadrila-
teral muscle. Therefore,
If there be given a fixed bone A'B, and a perfectly free bone
AB lying in the same plane, forming the origin and insertion of
a Quadrilateral muscle ; when that muscle contracts, the foot of
the bisector of the vertical angle OX must move towards O, but
the bone AB may have any position whatever, subject to the fore-
going condition.
From the preceding investigation it appears that the mo-
tion of the bone AB is indeterminate, but subject to a cer-
tain condition, viz., that the point X of the bone shall move
towards the point 0); a condition which agrees with what
has been already stated, pp. 181, 195. In nature, however,
nothing is indeterminate, and conditions are always added,
which completely fix the actual motion in every case. One
of the inost common additional conditions in nature is the
LLC
242
ANIMAL MECHANICS.
following restriction on the free motion of the bone AB. Let
A'B (Fig. 62) be the fixed bone, and AB the moveable bone,
which is subject to the condition of always turning round a

---

-------
Wis
-
Fig. 62.
joint, or socket, placed at the point S. Hence, we have the
following problem.
Proposition B. (Problem.)
Given a fixed bone A'B, and a moveable bone AB lying in
the same plane, subject to the condition of always turning round
an axis passing through a certain point S, and perpendicular to
the plane of the bones AB and A'B'; it is required to find the
locus of the joint S, so that the work done by the muscle, in con-
tracting, shall be a maximum.
Let aa' be any fibre, and let the bone AB receive a small
ANIMAL MECHANICS.
243
rotation dw, round the socket S; and let the point a move to
the point b; and let OX be the bisector of the vertical angle
AOB. Also, let
O Sa = w,
SOX = , OS = k,
XOa = 0, Sa = A.
Draw a'b, and let fall bp perpendicular upon aa'. We now
have
Š (1-1) = ap= ab x cos (baa') = ab sin ( Sao),
but
ab = Adw,
and
sin (SaO) = sin (w +0+0);
therefore,
8(1 - 1) = Adw sin (w +$ + 0);
sin ( + 0)
sin (w + 0 + 0)
Therefore, finally,
0(1-1) = ków sin (0 + 0).
The work done by the muscle during the rotation through dw,
is, therefore,
but
A = k
= kow
sin (0+0) do.
The integration of this expression gives us for the work done
18 (2-1) do = zkèw sin o sin 0 (44)
This result will be a maximum when p = 90°. Hence we ob-
tain the following solution of the problem proposed:
Produce the extreme fibres AA' and BB”, to meet in the point
0, and draw OX, the bisector of the vertical angle of the triangle
R 2
244
ANIMAL MECHANICS.
so formed; it is necessary, in order to produce the maximum
work due to a rotation round the socket S, that this socket shall
be placed somewhere upon a right line drawn through the certex
of the triangle, and perpendicular to the internal bisector of that
angle.

Fig 63.
The same result may be thus obtained without integra-
tion, from the principles laid down in p. 181. Let AB, A'B',
Fig. 63, be the quadrilateral muscle, and let OX be the bi-
sector of the vertical angle A OB.
OS = kg
SOX = 0,
AOX = 6.
The resultant of all the forces of the muscular fibres lies in
the right line Ox, and its moment, in turning the bone
AB round the socket S, will be a maximum when the per-
pendicular SP, let fall from the socket upon OX the direction
of the force, is a maximum. The resultant force (p. 181), is
2f sin 0;
and from the figure,
SP = k sin o.
ANIMAL MECHANICS.
245
0
Hence, the moment of the force round the socket will be
Moment = 2kf sin 0 sin o,
which agrees with equation (44).
If we suppose the angle 0 to be given, the work done will
be a maximum when k sin p = SP is a maximum; and if k,
the distance of the socket from the vertex, be also given,
then p = 90°, or OS perpendicular to OX, will be the ar-
rangement capable of producing the maximum effect.
Illustrations of Prop. B.-On examining, in nature, the
actual relative positions of the socket of the joint, and vertex
of the triangular or quadrilateral muscle, we find them always
placed in accordance with the result just found; so that, at
the moment at which the muscle makes its most vigorous
contraction, the direction of its resultant is perpendicular to
the line so, and, therefore, produces the maximum effect in
turning the bone round the socket of the joint.
(a). Latissimus dorsi in Man.-In Fig. 64, let S represent
the socket of the shoulder joint, and the insertion of the
latissimus dorsi, whose u
s
extreme fibres make
with each other an
angle of 35° (p. 191).
The bisector OX of
the vertical angle of
the muscle passes, very
nearly, through the
spinous process of the
first lumbar vertebra.
Draw SH horizontal,
and place the arm 80
in such a position that X
Fig. 64.
the angle soX shall
be a right angle. The following measurements are readily
made :-

IT
246
ANIMAL MECHANICS.
HS = 8 inches.
HX - 10
SO = 2.16 12
From which we find
SX = 12.8 inches.
HSX = 51° 20'.
XS0 = 80° 17.
Adding together the two angles, we find
HSO = 131° 37';
or, that the arm is depressed 48° 23' below the horizontal line,
when SO is perpendicular to OX. In giving a back sword
cut, or back-handed stroke to a racket-ball, this is the angle
of depression at which the moment of the force of the latissi-
mus dorsi round the socket S is a maximum.
The angular velocity produced by the contraction of the -
muscle is thus found. Let
X SO = 0.
X0 = d.
XOS = D.
SX = a.
w = angular velocity.
1 = moment of inertia of arm
F' = force of muscle.
SO
We have, from the principles of Dynamics,
(any - w*= 24F|sin do
sin 6 = sin 6
x= Vær + kis – jak cos 0.
but
ANIMAL MECHANICS.
247
1
247
Ilence, we have
F 2ak sin Odo
I va? + ke2 – 2ak cos 0
27
w?= Iva? + £2 – zak cos 0 + const.
62
we = 27 de
w
?
(45)
where ex denotes the total change of length of 0 X in
passing from the angle 0, to the angle 02. Hence the
angular velocity of the arm produced by the contraction of
the latissimus dorsi varies as the square root of that contrac-
tion; and it is important, being supposed given the amount
of the contraction, to find the conditions necessary to produce
the resulting angular velocity with a minimum angle of
rotation., This may be thus determined,
m2 = Q? + k2 – 2ak cos 0.
Hence,
xox = ak sin 080,
хох Sr
30 = ak sin 0 k sin o
Hence, 80 will be a minimum when sin o is a maximum, or
when 0 = 90°, which coincides with our previous result.
When the arm is depressed 48° 23' below the horizontal
line, so as to make the angle XOS a right angle, the length
of OX is 12.62 inches; and if the total contraction of the
muscle be one-eighth part of its length, we have for the
minimum angle of rotation, before and after passing through
the position of the arm, where XOS is a right angle,
80 = 2 = 79 = 210.
248
ANIMAL MECHANICS.
Whenever, therefore, the arm is depressed by the latissimus
dorsi, from 27° 23' to 69° 23', passing through an arc of 42°,
a greater angular velocity will be attained, in passing through
a given angle, than is possible for any other motion of the
arm. Racket players, cavalry soldiers, and others who
require to give a swift back stroke with the arm, instinctively
learn the proper range of depression, and the relations of
position of the muscle, socket, and arm, are all arranged in
strict conformity with Proposition B.
In the action here described, the muscles which extend
the forearm upon the arm act in unison with those which
depress the humerus, and it is interesting to compare their
relative forces. I determined the weights of the two groups
of muscles in a male and female subject, with the following
results :-
Depressors of Arm.
Male. Female.
Latissimus dorsi, . . . . 7.55 oz. 4.58 oz.
Teres major, . . . . . 4.09 oz. 1.67 oz.
11.64 oz.
6.25 oz.
Female.
Extensors of Forearm.
Male.
Triceps longus,
internus, \ ... 13.10 oz.
99 externus, )
6.59 02.
Rutio of Depressors of Arm to Extensors of Forearm.
1164
Male, . .
.
.
.
.
.
0.89
1310
Female, ......
625
659
0.95
Vean, ... 0.92
ANIMAL MECHANICS.
249
From this it appears that the two groups of muscles are
nearly of equal weights, and therefore contain nearly equal
quantities of work stored up; so that their simultaneous
action is highly advantageous. It may be noticed that in the
case of a ginglymoid joint, like the elbow, where the motion
of the bone is circular, the condition required by Proposition B.
is necessarily fulfilled in every position of the bone.
(6). Latissimus dorsi of the Tiger and Lion. The position
and relations of the latissimus dorsi in the Tiger are shown
in Fig. 65, where S is the centre of the glenoid cavity, C is

au
SS
TKK
Fig. 65.
the centre of motion of the elbow joint, and Q is the position
of the centre of the tendon of the latissimus dorsi, inserted on
the inner side of the humerus. The line, QA, drawn to the
fifth dorsal vertebra, represents the anterior fibres of the
muscle, and QB, drawn to the third lumbar vertebra, repre-
sents its posterior fibres. The line, QX, bisecting the angle,
250
ANIMAL MECHANICS.
AQB, passes through the ninth dorsal vertebra, and is the
line of force acting upon the humerus.
In the position of the Tiger's arm, represented in the
figure, the angle, SQX, is 93°, and the force of the latissimus
position of maximum advantage.
It is important to observe that, when the angle SQX
becomes a right angle, the teres major (whose origin is repre-
sented at the angle of the scapula, and which is inserted with
the latissimus dorsi) acts with its entire force along the
bisector QX, and at right angles to SQC; so, that when the
latissimus dorsi is acting to the greatest advantage, producing
a maximum velocity with a given angle of rotation; at the
same time, its auxiliary muscle, teres major, acts also to the
greatest advantage, and exactly in the line of force of the
latissimus dorsi. The weights of these muscles in several
Felidæ are as follows:--
Depressors of Arm in the Felidæ.

Tiger,
Lion.
Jaguar
(female).
Leopard.
39.75 oz.
25.50 oz.
4.19 oz.
8.40 oz.
Latissimus dorsi,
Teres major, .
11.45 ,
9.50 »
1.68 »
2.65 ,
Total force,
51.20 oz.
35-00 oz.
5.87 OZ
11.05 oz.
The latissimus dorsi and teres major act upon the humerus,
tending to turn it round an axis perpendicular to the plane
of the triangular sheet of muscle; but another important
muscle assists in this rotation. This muscle is the triceps
longus, which takes its origin on the line marked on the pos-
terior edge of the scapula, and is inserted, in conjunction with
ANIMAL MECHANICS,
251
the other muscles that act upon the elbow joint, into the
olecranon process. The right line 01 represents the di-
rection in which the force of this muscle is applied, and
it will produce its maximum mechanical effect when the
angle SOI becomes a right angle, in conformity with Pro-
position B.
In the position of the arm shown in the figure, the angles
are as follows:
SQX = 93° (latiss. dorsi and teres major),
SOI = 95° (triceps).
The arrangement of both groups of muscles is so contrived,
that these angles pass through 90° together, thus enabling
both sets of muscles to act at the same moment with maxi-
mum effect upon the humerus.
The triceps and its associated muscles in the Felidæ have
the following weight:-
Extensors of Forearm in the Felide.

Tiger.
Lion).
Jaguar
(female).
Leopard
4.20 oz. 1
7.27 oz.
Triceps longus, · 30.65 oz.
Triceps internus, .!. 7.10 »
1.48 »
-
-
-
+
+
1.54
-
-
Triceps externus,
Triceps accessorius, . /
1.79 ,
3.45 »
6.67 1
0.60 ,
14.05 »
2.95 ,
3.05
-
0.20 ,
Anconæus, ...
0.33 .
Total
57.80 oz. | 43.50 oz 7.75 oz. 13.78 oz.
If we compare these weights, which represent the work done
252
ANIMAL MECHANICS.
in each contraction of the group of muscles, with those already
given, we find-
Ratio of Depressors of Arm to Extensors of Forearm in
the Felidæ.
Tiger. Lion. Jaguar (female). Leopard.
0.88 0.80 - 0.76
0.80.
With respect to the action of the triceps and its assistants
upon the elbow joint, the following considerations are of in-
terest. The triceps will produce, in conformity with Pro-
position B. its maximum effect upon the forearm, turning
round the centre C (Fig. 66), when the quadrilateral OSCI
comes into the position in which the line of force of the triceps
O1 is at right angles to the line CI. In this position of the
quadrilateral, the right line Cl, drawn from the point C to
meet the circle I'II", becomes a tangent to that circle.
It is evident from the figure, that the angle OIC becomes
a right angle at the same time as the angles SQX and Sol;
and therefore, that in the position of maximum effect of all
the muscles acting on the shoulder and elbow joints, the lines
SO and CI become parallel to each other, and are both at
right angles to the direction of the triceps 01.
From the preceding facts, it is plain that the triceps is
intended to co-operate with the latissimus dorsi and teres major;
muscles, when both they and it are placed in their relative
positions of maximum effect upon the rotations, round both
the shoulder and elbow joints.
I have repeatedly watched the back stroke of the arm in
all the large Felidæ, and am unable to perceive the slightest
difference in the time of action of the elbow and shoulder
joints in the Tiger, Jaguar, or Panther; in the Lion it appears
ANIMAL MECHANICS.
253
to me that the extension of the forearm precedes the backward
motion of the arm by a very short interval.*
In Fig. 66, I have drawn to scale the relative positions of
the various sockets, and origin and insertion of muscles in the
X

Fig. 66
arm of the Bengal Tiger. In this figure, S is the centre of the
shoulder joint, O is the centre of origin of triceps longus, M is
the centre of origin of teres major, X is the extremity of the
bisector of vertical angle of latissimus dorsi, C is the centre
of the elbow joint, I is the centre of insertion of the triceps
longus.
The measurements of the fixed lines in the Tiger are as
follows:
SO = 5.62 inches. SM = 9.50 inches.
SC = 10.50
SX = 16.12 12
CI = 3.00 u
SQ = 2.89 »
* In the action of a Horse, walking or trotting, the forearm always moves
before the arn, by a perceptible interval.
25+
ANIMAL MECHANICS.
The point of insertion, Q, of the latissimus dorsi and teres major
describes a circle, of which S is the centre, and the fibres of
these muscles must be contrived of such a length, that XQ
and MQ shall be equal to their respective lengths, when half
contracted; and, supposing the whole contraction to be one-
eighth of the length, the lengths of the muscles, before and
after contraction, must be
Before contraction. After contraction.
Latissimus dorsi, ... *** QX
Teres major, ....* QM
16* QX
16XQM
This corresponds to a range of rotation round S, amount-
ing to 21°, at each side of the line SC, which is the line of
maximum effect.
While the latissimus dorsi is engaged in shortening itself
by one-sixteenth, rotating the arm through 21°, into the position
SC, the triceps longus also shortens itself by one-sixteenth,
and causes the forearm to rotate round C, through an angle
of 9° 30', into the position of maximum effect, in which OI
and CI are at right angles; and in the entire complex down-
ward and backward stroke of the Tiger's paw, the arm passes
through 42°, while the forearm passes through 19º.
It is important to observe that the condition fulfilled by
the Tiger's arm, by which the angles SQX, SO1, and CIO
become right angles at the same time, and so allow all the
muscles of the shoulder and elbow to co-operate together,
each acting to the greatest advantage, is not a random con-
dition attained by chance; but must have been the result of
deliberate forethought and contrivance. This can be proved
by showing that the condition involves rigorous geometrical
consequences, which might or might not have been fulfilled
ANIMAL 255
VINE
MECHANICS.
by bones and muscles selected at random. Let us suppose
that S,O,Q,M, and X, or everything relating to the shoulder
joint, be given; it is required to contrive for the Tiger an
olecranon and triceps, which shall enable all the muscles of
shoulder and elbow to act simultaneously, and to the greatest
advantage.
Let
SO = a CI = x (unknown olecranon).
SC = b OI = y (unknown triceps).
OSC = 0
The angle o is given, because it is the angle whose cosine is
found from the relation
SQ 289
cos o = SX
1612
0 = 79° 40',
because XQ is a tangent to the circle described by Q, round
the centre S.
It is easy to see that the olecranon and triceps are given
by the equations
x = a - b cos o
y = b sin .
If the values of a, b, and q; above given, be substituted in
these equations, they will determine the lengths of olecranon
and triceps, which (and which only) will enable all the muscles
of the shoulder and elbow in the Tiger to act simultaneously
to the greatest advantage.
Whenever, as in the case of the triceps longus, a muscle
passes over two joints, a considerable amount of rotation
round both joints is possible, without requiring the fibres of
the muscles to be lengthened or shortened. Thus, in the
quadrilateral figure OSCI, the sides OS, SC, and Cl, are
absolutely constant; and it may be shown that the arm may
Tin
256
ANIMAL MECHANICS
be rotated through a considerable angle, without altering the
length of the fourth side, Ol; and, consequently, without
lengthening or shortening the triceps longus.
In the forward direction, the arm SC may be moved into
the position, SC", such that OI plus IC shall be equal to the
right line, Ol! C'"; and in the backward direction, the arm
may be moved into the position, SC'I', such that SC plus Ci
shall be equal to the right line, SC'I'. While the arm is
moved from SC to SC", through an angle of 65°, the fibres of
the triceps muscle, of, are neither lengthened nor shortened ;
if the arm be brought farther forward than SC", its fibres will
be stretched, and the muscle provoked to contract itself; and
the arm cannot be brought farther backward than the position
SC", unless the fibres of the triceps be shortened.
(c). Latissimus dorsi (Albatross).

-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
N
.
www
-
-
-
-
-
-
-
-
-
-
-
-
.
-
The latissimus dorsi of
the wing of the Alba-
tross, (one half natural
size) is shown in Fig. 67,
in which S, the socket
of the humerus, is 2.20
inches distant from 0,
the point of insertion of
the muscle, which is
triangular in shape, its
extreme fibres making
an angle of 51° with
each other. In the
mean position of the
wing, in the dead bird,
Ý X
the angle sox, con-
Fig. 67.
tained between OS and
0X, the bisector of the musular fibres is 86º. During life,
-
-
-
www
-
.
.
d
u
o
_
.
· ANIMAL MECHANICS.
257
the contraction of the muscle always swings the wing through
the position corresponding with p = SOX =90°, so as to pro-
duce, in striking the air, a given angular velocity, with a
minimum angle of rotation of the wing. Assuming, as before,
the contraction of the muscular fibres to be one-eighth of
their length, we readily obtain the following result :-Since
0X is equal to 4.62 inches-
So = Rsin P
- dx
R sin
462
16 220
Hence the wing of the Albatross, when acted upon by the
latissimus dorsi, will sweep through an angle of 15, making
the angle soX a right angle in the course of the rotation,
and producing a maximum velocity of stroke, for a given
angle of rotation.
TY
(d). Hanistring Muscles of Tiger and Lion.
The next example of Proposition B that I shall give is
taken from the hamstring muscles of the great Felidæ, which
constitute one of the most wonderful instances of well-devised
mechanism to be found in the entire range of the Animal
Kingdom.
In Fig. 68 are shown the pelvis and leg of the Bengal Tiger,
in which is the tuber ischii and origin of the biceps femoris
muscle, whose extreme fibres are 0 A, to the middle of femoral
fascia, and OB wound round the tendo Achillis, and directed
towards the calcaneum. The angle AOB is 50°, and the
bisector OX passes through the top of the fibula, and is the
direction of the resultant force of all the fibres of the biceps
This great muscle, instead of being inserted into a single point,
is spread out, so as to act upon several joints at the same
time; its upper fibres act on the hip joint only; its middle
fibres act upon both the hip and knee joints, and its lower
S
258
ANIMAL MECHANICS.
fibres influence, in addition, the ankle joint. We shall, how-
ever, regard it as a whole, represented by its resultant force
acting in the direction OX. The other hamstring muscles,

ON
WA
sm.st.
WE
l
nu
2
17
Ilus
U
Fig. 68.
semi-tendinosus and semimembranosus, take origin from the
tuber ischii, immediately beside the biceps, and are inserted
into the tibia, in a position corresponding with the position
ANIMAL NIECHANICS.
259
of the line OX; so that our conclusions, with respect to the
biceps, will apply to all the hamstring muscles.
Let S be the centre of the acetabulum. ( the tuber ischii,
( the centre of motion of the knee joint, and I the top of the
fibula I obtained the following measurements from the male
Tiger, already mentioned at p. 127:-
OS = 5.49 in.
SC = 11.75 »
C1 = 2.63 in.
01 = 14.00 »
17
The distances OS, SC, anc C/ are alsсlutely fixed; but the
distance 01 may be shortenec by the action of the biceps and
other hamstring muscles. I have taken the length of O1 at 1.4.
inches, before the contraction of the muscles.
Let us first inquire what range of motion the leg can
have, before the line OI is shortened or extended. This
question gives us the quadrilateral figure OSCI, whose four
sides are given; and we are required to move the quadrilateral
about OS regarded as fixea, sc as to find the extreme limits
possible for its construction. W'ith C as centre, describe the
circle l'Il"; and with S as centre, describe the circle C'CC".
The leg of the Tiger may be moved forward, or flexed
upon the body, until the lines Ol and IC form one continued
right line, in the position OI"C", where I"C" is equal to
IC; the leg cannot be moved forward beyond this position
without lengthening the muscle Oix, and thus provoking it
to contract. The leg of the Tiger mạy be moved back-
wards, or extended upon the body, until the lines SC and CI
form one continued right line, in the position SC'I', where
C'I' is equal to Cl; and the leg cannot be moved farther
backwards without a shortening of the muscle OIX. Pro-
duce the line OS to O', and measure the angles O'SC" ann
O'SI', which represent the extreme limits of the position of
the leg, compatible with the supposition, that the muscle
s 2
260
ANIMAL MECHANICS.
01 X has been neither lengthened nor shortened. We find-
O'SC" = 31°
O'SC'' = 1050
showing a range of 74° of motion round the socket S, through
which the biceps and other hamstrings are neither lengthened
nor shortened.
When the leg attains the position C'I', the angle sol' is
83", and shortly afterwards passes through 90°, when the ham-
string muscles contract. If we examine the angle CIO, at
the point l', when the angle SOI' is approaching a right
angle, and when the hamstring muscles are producing their
maximum effect with respect to the hip joint S, we shall see
that C'I'O becomes acute, and that it is far removed from the
position in which its maximum effect would be produced
upon the knee joint C. In fact, the position of the leg, in
which the hamstring muscles would produce their maximum
effect upon the knee joint, corresponds with the point c, from
which a tangent ci, to the circle I'II", would be equal to CI.
In this position of the leg, the angle ciO is 90°, while the
angle SOi, corresponding to it, is only 64º. From this it fol-
lows, that the knee joint is not related to the hip joint, in the
same way that the elbow joint is related to the shoulder joint;
and that it is not intended that the muscles acting upon each
should attain their positions of maximum effect at the same
time.
In the Tiger's leg, it is the ankle joint that bears to the hip
joint, the same relation that the elbow bears the shoulder
joint, in the arm ; for, let y be the centre of the ankle joint,
and join YB to the calcaneum; and let G denote the origin
of the external gastrocnemius muscle, which acts from G to B.
This powerful muscle, and the corresponding internal gastro-
cnemius, will produce their maximum effect upon the ankle
joint, when the angle YBG becomes a right angle, in con-
ANIMAL MECHANICS.
261
formity with Proposition B; and it is easy to see from the
figure, that the angles SOI and GBY will pass through right
angles at nearly the same time. The gastrocneme muscles, in
the hind leg, therefore, stand in the same relation (mechani-
cally) to the hamstring muscles, that the triceps longus and
its allies bear to the latissimus dorsi and teres major, in the
fore leg.
I have found the following weights of the hamstring and
gastrocneme muscles in the Felidæ :-
Hamstring Muscles in the Felida.

Tiger.
Lion.
Jaguar
(female).
Leopard.
27.75 oz. | 3.77 02. | 9.66
oz.
Biceps femoris,.. 47.60 oz.
Bicipi:i accessorius | 1.28 ,,
Semitendinosus, . 13.70
Semimembranosus, . | 23.40 ,
Gracilis, .... 13.35 »
10.00 »
26.25 1
9.00 »
1
2.61
0.92
Total
99.33 oz.
73.00 oz. | 9.02 02.
20.89 oz.
Gastrocneme Muscles in the Felide.

Tiger.
Lion.
Jaguar (female).
Lcopard.
4.93 oz.
Gastrocnemius externus,
Gastrocnemius internus,
Plantaris, ...::|
9.80 oz.
9.50 ,
7.35 ,
2.38 21
Total
26.65 oz
2,84 oz.
| 7.31 oz.
It will be observed, that, while the hamstring muscles are
attaining their position of maximum effect upon the hip joint,
262
4111
ANIJAL JECHANICS.
1
i
uy
S
O
by the angle SOI becoming a right angle, and the gastro-
cneme muscles are acquiring their maximum effect upon the
ankle joint, by the angle GBY becoming a right angle, that
both these muscles act at a considerable disadvantage upon
the knee joint; for the angles c10 and CGB become very
acute, when the angles sol and GB Y pass through right
angles. Still, both hamstring and gastrocneme muscles tend
to flex the knee joint, in every position of the legs; an effect,
which would be disastrous to the combined movement of the
muscles of the hip and ankle, when the Tiger walks, runs, or
springs upon its prey. Therefore, the flexing effect of these
muscles upon the knee, is always counteracted by the extend-
ing effect of the rectus femoris, and its allies, upon the same
joint; and these antagonist muscles are called into action, to
an extent necessary to keep the knee joint from bending, or,
in some motions of the leg, to an extent that actually extends
the leg upon the thigh.
It is very interesting, therefore, to observe that the rectus
femoris, which is the natural ally of the hamstring and gas-
trocneme muscles, acting in the line R.rf., is at right angles
to the lever cR, at the same time that the angles SOI and
GBY, of the hamstring and gastrocneme muscles, pass
through 90°.
In Propositions A and B, we have considered the case of
two bones, lying in the same plane, and united by a quadri-
lateral sheet of muscular fibres; one bone being supposed
fixed, and the other bone compelled to turn round an axis of
rotation perpendicular to the plane. We nave now to ex-
amine the more complex case in which the moveable bone
turns round some axis lying in the plane of the two bones;
an action, by which the plane quadrilateral becomes a gauche
quadrilateral, and the muscular fibres form the surface known
as an hyperboloid of one sheet.
I shall first suppose the axis of rotation to be perpendi.
ANIMAL MECHANICS.
263
cular to the bisector of the muscular fibres, when lying in the
same plane, and investigate the relations between the bones
which are necessary to produce a maximum effect, when the
muscle contracts.
In Fig. 69, AB and A'B' are the two bones, A' B' being
supposed fixed, and AB moveable, round an axis SK, pass-
ing through the fixed socket S, and perpendicular to OX

A
Fig. 69.
S
the bisector of the muscular fibres. Let aá be any fibre
which takes the position a'b, when a small rotation occurs
round the axis SK, moving the point a to b, which lies out-
side the plane ABA'B'.
In the triangle aa's, let
aa' = a (-a'= 1 - 1' = p.
aS = A a'S = A' asa' = d.
264
ANIMAL MECHANICS.
02
Hence we have
p? = A + A'? - 2AA cos X.
In the triangle bSa', let
ba' = p 68 = A (by hypothesis) á'S = A' bsa' = y.
Hence we have
p2 = A + A'? – 2A Acos y.
If we imagine a sphere described round S, intersecting
the several lines, we shall have (Fig. 70), the line SK, which
is the axis of the cone, described by the line Sa. meeting the
sphere in the centre of a small
circle, in which the cone pier-
ces it.
The spherical radius of this
circle will be the angle
KSa = 0.
Let S, a, á be the original
positions of all the lines, and
let b denote the new position
taken up by a, after a rotation
Fig. 70,
through an angle
asb = w

Draw the arc ba', which will be equal to y, while ac will be
equal to a.
In the spherical triangle ba's, we have
6S = 0, ba' = y, Sa' = 0 -- X, 6 Sa' = w.
Therefore
cos y = cos w sin o sin (o – X) + cos o cos (o – X),
and when w is a moderately small angle
COSW = 1 ano
COS WI-
ANIMAL MECHANICS.
265
which reduces the preceding equation to
cos y = {sin o sin (o - X) + cos o cos (o - x)}
-. sin o sin (o – x);
or, finally,
.
cos y = cos x - ,* sin o sin (6 — «).
Substituting this value of cos y in the preceding equations,
we find
p'? - p2 + w?.AA' sin o sin (o - x)
(46)
In Fig. (69), the angle o is KSa; and if we draw a perpen-
dicular a X from a, upon the bisector YX, it is easy to see
that
YX YO + OX
sin o = as = -as
Let
OS = a, YOS = P, a0X = 0),
and we obtain
sin o =
a cos o + 1 cos 0
and, in like manner, since o - X = K Sá
sin (o - 2) =
a cos 0 + l' cos 0
A'
Hence, equation (46) becomes
p2 = px + w2 (a cos 0 + l cos 0) (a cos 0 + l' cos 0),
or,
(a cos 0 + l cos 0) (a cos o + 1' cos O)
no
rt?! Vos q + cos olloc
22
266
NIR
ANIMAL MECHANICS.
expanding the root, and remembering that w is moderately
small, we obtain
cos o + 1 cos 0) (a cos 0 + l' cos O) \
and finally, substituting for p, its value I - 1', we have
Di-D = opw* (a cos p+l cos 0) (a cos p + l’cos ).
lo (47)
This expression denotes the shortening or lengthening of a
single fibre ad', produced by the rotation (w) round the axis
KS; and the work done by the entire muscle, in a single
contraction, will be
Work done = lspd6 = wf(a cosø+lcos8) (a cos $ + lº cos ) de
1-1'
and we are required to investigate the conditions, which shall
render this work a maximum.
Using the same notation, as in Prop. B, we have
; b sin ß b'sin ß!
sin (ß-6) *sin (@–0).
Hence,
ſöpde =
Wal{a cos o sin ( - 0) + b sin ß cos 0} {a cos o sin (B'-0) + B' sin ß' cos 0} de
b sin B. sin (B'-0) – b' sin ß'.sin (ß-0)
D
A = b sin ß sin (' - 0) – b'sin ß' sin (ß- 0);
we shall find, by expanding the numerator,
(48)
ſopde =
,* sin B sin B' (a cos $ + b) (a cos 4+23 Jc082 Odo
ANIJIAL MECHANICS.
267
(sin 2 000
+ " , * acosa o cos ß cos Blomme
- a cos px
{(a cos $ + b) sin ß cos ß' + (a cos + b) sin ß cos B} *
fsin 0 cos 0 do
O
Previous to integrating the general expression for the
work done by the skew quadrilateral muscle (48); it will be
interesting to discuss the simpler case of a triangular muscle,
which is deduced from the general expression by reducing
the bone A'B' to a point coinciding with the vertex 0. This
may be best done by combining the general expression for
the work done, viz.,
wap(a cos o+lcos O) (a cos 0 + 1' cos ) do
1-1
with the condition, 1 = 0.
Hence, we obtain,
w? Sa cos o (a cos p+l cos 0) (0
opdo =
2
which becomes, by substituting for I, its value
; b sin
sin (ß - 0)
after a few reductions
1-
| epdo = a cos y (qco$ 4 + b)] cos odd. (49)
that a cos tip cot eſ sin oan.
.
Integrating this expression from + @ to - 0, the second
term will disappear, and we find for the work done,
Work donc = { $pdf = wa sin 0 . a cos p (a cos p + b) (50)
S
268
ANIMAL MECHANICS.
This work will be a maximum, when p = 0, when it becomes
| Spdo = wăsin 6 a(a + b).
Therefore, when a the distance OS (Fig. 69), of the socket
joint from the vertex of the triangle is given, the maximum
effect will be produced when the socket lies on the bisector
of the vertical angle of the triangular muscle, that is, on a line
perpendicular to the locus of socket for maximum effect, for
axis perpendicular to the plane of the muscle considered in
Prop. B. Hence, we see that the conditions for maximum
effect are essentially different for the two axes of rotation, and
that it is not possible for a given muscle to produce the maxi-
mum effect, by its rotation of the bone AB round both axes

.
Fig. 71.
This is a remarkable result, and will lead us to important con-
sequences.
ANIMÁL 269
NTRS
JECHANICS.
The following geometrical construction for the work done
for any position of the axis SY, Fig. 71, is easily remembered,
and applies to every case. Describe the circle XTO, whose
diameter is 0X = b, the bisector of the angle made by the
muscular fibres. From Y, the point where the axis of rotation
intersects the bisector YOX, draw YT a tangent to the circle
XTO; I say, that the work done by the muscle, during a
small rotation of the bone AB round the axis SY, is propor-
tional to the square of the tangent YT. For the work done
is proportional to
a cos o (a cos 0 + b),
and,
YO = a coso,
YX = a cos o + b,
also,
(YT)2 = YO ~ YX. Euc. iii. 36.
Therefore,
Work done, varies as (YT)2.
Q. E. D.
If the axis of rotation be supposed to move parallel to
itself, the work done will be greater, the farther off the point
Y is from the circle, either beyond 0, or beyond X. When
it passes through either 0 or X, the work done becomes zero;
and for positions like S'Y' lying between 0 and X, the work
done becomes negative, and is proportional to the square of
the ordinate Y'T"; and the negative work done will be a
maximum, when the axis of rotation passes through the centre
of the circle XTO; or through the point of bisection of the
bisector of the vertical angle A OB.
The mechanical interpretation of the preceding facts, is-
1°. That if the axis of rotation, round which the bone AB
is compelled to turn, be placed beyond 0 or X in either
direction, no amount of contraction of the muscle can alter
its position, for any change of position would be equivalent
to a total lengthening of the fibres, which is impossible.
YT
2
S
270
ANIMAL MECHANICS,
2
2°. That if the axis of rotation pass through either 0 or X
(the vertex of triangular muscle or foot of bisector of vertical
angle), the bone AB will remain, indifferently, in any position
in which it is placed, for no work is done in altering its
position,
30. That if the axis of rotation lies between 0 and X, a
contraction of the muscular fibres will immediately cause a
motion of the bone round the given axis, for any change of
position will be accomplished by a total shortening of the fibres,
which is possible.
4º. The triangular muscle will exert a maximum effect
in displacing the bone AB, round the axis passing through
the socket joint, when that axis of rotation bisects the line
OX, because in this case the total shortening of the fibres will
be a maximum.
The positions of the axis of rotation, beyond ( or X,
which correspond to positive values of the work done, denote
positions of stable equilibrium, and the work done by the
muscle consists in forcing back the bone into the plane con-
taining itself and the axis of rotation.
The positions of the axis of rotation, passing through
O or X, correspond with positions of neutral equilibrium,
when the bone and muscle are indifferent to the position in
which they are placed, and no work is done by the contrac-
tion of the muscle, tending to rotate the bone round the
axis.
The positions of the axis of rotation, lying inside 0 and X,
which correspond to negative values of the work done, denote
positions of unstable equilibrium, and the work done by the
muscle consists in pulling the bone out of the plane contain-
ing itself and the axis of rotation.
All the preceding results may be verified, geometrically,
by drawing the cone described by the bone AB round the
axis of rotation, and considering, in detail, the lengthening or
S
ANIMAL MECHANICSL2
271
.
shortening of the muscular fibres joining the bone AB with
the vertex 0.
The preceding investigation leads to the following :-
PROPOSITION C.
If, in a triangular muscle. AOB, the moveable bone AB ro-
tate round an axis, perpendicular to OX, the bisector of the
angle AOB-
. 1º. The contraction of the muscle will tend to restore the
bone to its original position, if the axis of rotation intersect the
bise:tor Ox, beyond either 0 or X; and its restoring power
will increase with the distance of the intersection of the axis from
O or X.
2°. If the axis of rotation pass through O or X, the contrac-
tion of the muscle can have no effect in moving the bone.
3º. The contraction of the muscle will tend further to dis-
place the bone, if the axis of rotation intersect the bisector OX,
between () and X; and its displacing power will be a maximum
when the point of intersection of the axis bisects the bisector
OX.
In addition to the construction shown in Fig. 71, for the
work done by the muscle, whether positive or negative, the
following construction may be used, and is interesting, be-
cause it is a particular case of the construction belonging to
the general equation (48).
Let us suppose the work done to be represented by the
square of the ordinate of a curve, whose abscissa is a cos o
= %. Then, by equation (50), we have, for the triangular
muscle,
y2 = x!b + x).
(51)
When yis positive, this is the equation of an equilateral hy-
272
ANIMAI. MECHANICS.
perbola, whose axis major is OX, the bisector of the angle
made by the muscular fibres.
Let this hyperbola
be aOb, cXd, Fig. 72 ;
then the square of any
chord ab, cd, will be
proportional to the work
done, by the contraction
ofthe muscle, in rotating
the bone round the chord
considered as an axis.
When y’ is negative,
Fig. 72.
equation (51) denotes a
circle described on the
bisector OX as diameter; and the square of the chord ef is
proportional to the work done, by the contraction of the
muscle, in rotating the bone round the chord ef.
The denominator A, used in equation (48), is
A = b sin ß sin (ß' - 0) – b' sin B sin (B - 0);
À = A cos 0 + B sin 0,
where
A = (b -b') sin ß sin ß.
B=0 sin ß cos B'- b' sin ß' cos ß.
If B = 0, the equation (48) becomes much simplified, and is
reduced to the following :-
ſ&pdo = sin ß sin B' (a cos 9 + b) (a cos 0 + 6') score
+0.7.12 cos? . cos ß cos 6 (Sin due (52)

or
Nle
S
The third term of equation (48) disappears altogether, for
it consists of pairs of elements, of the following form :-
sin cos odo sin cos 0.10
A cos 0 + B sin 9 A cos 0 - B sin 0
ANIMAL AT
273
MECHANICS.
or,
- 2B sin cos 0d0
A cosa 0 – B2 sina 6
and each such pair of elements is zero, when B = 0.
The geometrical condition involved in the equation
B=0 .
is shown in Fig. 73. Let the bones AB and A'B' be produced
to meet in the point Q.

-
O
Fig. 73.
If the condition B = o be granted, then the point of inter-
section, Q, must lie somewhere on the line OQ drawn through
the vertex 0, perpendicular to OX, the bisector of the angle
made by the muscular fibres. For
OQ = 0X tan ß = 0X'tan B';
or,
'
6 tan 3 - 8 tan 3 = 0,
which is equivalent to
O
B = b sin ß cosß' – b' sin ß'cos ß = 0.
When this condition is fulfilled, equation (52) becomes,
writing,
274
.
ANIMAL MECHANICSK
.
v ſcos 20d0 v
X= cos Y =
sin 20d0
cos 0
P = sin ß sin B (6+ a cos )(b' + a cos'),
Q = cos ß cos ß". acosp?,
fòpdo = (PX + QY)
'12 (6 - 6') sin ß sin p')
If, as before, we write
3* = PX +QY,
x = a cos de
we obtain the equation of the curve the square of whose ordi-
nate represents the work done by the contraction of the
muscle, the axis of rotation corresponding to the ordinate in
question. This curve becomes
ya = PX + QY,
or,
· y' = X sin ß sin B' (b + x)(8 + x) + Y cos ßcos ' x2. (53)
This is the equation of a central conic, whose major axis lies
on the bisector OX, and whose centre is situated at a distance
from 0 towards X represented by
(6 + 6') X sin ß sin 8
2' X sin ß sin ß + Y cos B'cos i
Since X and Y are essentially positive, and since the con-
dition B = o requires ß and B' to be both acute, & will be
essentially negative, and the centre of the conic will lie inside
the vertex of the triangle AOB.
The positions of the axis on the bisector of AOB, which
correspond to neutral equilibrium, are found by equating (5:3)
ANIMAL MECHANICS.
275
to zero, which gives the following quadratic, whose roots de-
termine the positions of neutral equilibrium :-
X sin ß sin B'(b + x) (b' + x) + Y cos ß cos Bix2 = 0
The condition for real roots in this quadratic is, that
X sin ß sin ß'[(6 – 6')?X sin ß sin ß'- 466' Y cosß cosß!} (55)
shall be positive.
If this condition be fulfilled, and the points corresponding
to neutral equilibrium be taken on the bisector of AOB; for
all axes lying outside those limits the equilibrium will be
stable, and the work done positive, and represented by the
square of the ordinate of an hyperbola constructed with the
intercept between the points of neutral equilibrium as its
transverse axis; and for all axes lying inside the points of
neutral equilibrium, the equilibrium will be unstable, and the
work done negative, and represented by the square of the
ordinate of an ellipse constructed with the intercept between
the points of neutral equilibrium as its major axis.
Hence a geometrical construction, similar to that shown in
Fig. 72, may be made to represent the work done; pro-
vided we replace the circle and equilateral hyperbola by an
ellipse and hyperbola with unequal axes.
If the equation of the central conic (53) be referred to its
centre, as origin of co-ordinates, it will have the form
+
na
2 - 1;
(56)
where
e X sin ß sin ß' {(6 – b)2X sin ß sin ß' – 466'Y cos ß cos B'}
4 {X sin ß sin ß' + Y cos ß cos ß'}
n? =
X sin ß sin B'{(6-6)*X sin ß sin Bʻ– 466' Y cos B cos B'}
ma
4 {X sin ß sin ß' + Y cos ßcos B'12 (57)
These values for m2 and 1 are essentially positive when the
T 2
276
ANIMAL MECHANICS
t
quantity (55) is positive, which is the condition necessary for
the existence of real positions of unstable equilibrium.
When the equilibrium is stable, yz is positive, and the
equation becomes
x2
42
m² n²
=
I
which represents an hyperbola, whose axes are m and n, the
real axis being m, and situated on the bisector of the angle
АВ.
When the equilibrium is unstable, ya is negative, and
the equation becomes
m2 + m2 = 1;
which represents an ellipse, whose axes are m and n; the axis
m being situated on the bisector of the angle AOB; and
the maximum value of the negative work, corresponding with
the position of maximum instability of the muscle, determined
by equation (54), will be represented by nº.
From the preceding equations it appears that .
22
= X sin ß sin ß' + Y cos ßcos ß';
m?
S
or, that the conjugate axis will be greater or less than the
transverse axis, according as this function is greater or less
than unity.
There are three distinct cases to be considered according
as the expression (55) is positive, zero, or negative.
1°. If the expression (55)
(6 - 6') ?X sin ß sin ß' – 466'Y cos B cos ß'
be positive, then ma and n’, in the equations (56, 57), are both
ANIMAL MECHANICS,
277
positive, and these equations will represent, for all values of y?,
positive or negative, an ellipse and hyperbola, having the same
axes.

Fig 74.
Let C be the common centre of the ellipse and hyperbola,
Fig. (74), found by the equation (54), by measuring off
OC = d. Take CA = m and CB = n, and construct the el-
lipse and hyperbola corresponding to these lines as semi-axes,
the line 0 X being the real axis of the hyperbola. Then, as
before, the work done by the quadrilateral muscle in rotating
the moveable bone round any axis ab, cd, or ef, will be pro-
portional to the square of these chords ; and if the chord of
rotation intersect the hyperbola, the muscular contraction will
restore the skew quadrilateral to the plane; but if the chord of
rotation intersect the ellipse, the contraction of the muscle
will convert the plane quadrilateral into a skew muscle; and
finally, the contraction of the muscle will produce no effect
at all, if the axis of rotation pass through the extremity of
the axis of the ellipse and hyperbola.
2°. If the expression (55)
(6 – 6) ²X sin ß sin ß'- 466'Y cos ß cos B'
be equal to zero; the semi-axes m and n will both become
278
ANIMAL MECHANICS.
1
zero also; and the hyperbola will be reduced to its asymptotes,
while the ellipse disappears altogether. Hence, in this case,
the equilibrium of the plane quadrilateral muscle will be
always stable for every axis of rotation, except that passing
through the intersection of the asymptotes; for which axis of
rotation the equilibrium will be neutral ; and, finally, there is
no position of axis possible for which the equilibrium becomes
unstable, so that the contraction of the muscle would convert
the plane quadrilateral into a skew muscle.
3°. If the expression (55)
(6 - 6') ?X sin ß sin ß' – 466' Y cos B cos B'
be negative, m’and na will both be negative; and there is no
possible position of the axis which will render y either zero
or negative. The construction for this case becomes the
following (Fig. 75):-

-ZA
Fig. 75.
conjugate hyperbola, whose real axis is CB = n, on the line
CB, perpendicular to OX, the bisector of the angle made by
the muscular fibres. If a rotation take place round any chord
ab, cd, the work done by the muscle will be proportional to
the square of the chord of rotation, and its contraction will
always restore the skew muscle to the plane, and the equi-
· librium will never become neutral or unstable.
ANIMAL MECHANICS.
279
In the three cases now considered, we have :
1°. The axis of rotation passing through the centre of the
conic (56, 57), is the axis of maximum instability. .
20. The axis of rotation passing through the centre of the
conic (56, 57), is the axis of neutral equilibrium.
3º. The axis of rotation passing through the centre of the
conic (56, 57), is the axis of minimum stability.
The condition contemplated in the second of the foregoing
cases, viz., that the expression (55)
(6 – b') X sin ß sin B' – 466' Y cos ß cos ß' = 0
may be readily found. For, since
b tan 3 = 0 tan 3,
we have
tan 8' = , tan B,
which reduces (55) to the following:
tun 22. -
tan *ß = (6-)
482
ON
tan ß = 1ïv X
/
tan 6 -
21
(58)
S
When the angle ß exceeds the value determined by
equation (58), the expression (55) will be positive, and a
position of axis of unstable equilibrium will be possible ; but
when ß is less than the value assigned by equation (58), the
equilibrium will, in all cases, be stable, and there is no position
of axis of rotation which will render it unstable.
We shall now consider the general expression (48) for
the work done by a quadrilateral muscle revolving round an
axis in its plane, and perpendicular to the bisector of the
angle AOB, contained between its extreme fibres.
280
ANIMAL-MECHANICS.
If we write
poos 2000
( sin udo
2000,
sin
cos 0d0
X
=
CZ-
If
equation (48) becomes, writing as before, x = a cos o,
2&pd6 = X sin B sin B(6 + x) (B' + x)
+ Y cos ß cos B x2
- 20 (sin ß cos B' (6+) + sin B'cos B (6+®).
gj = (dpdo
we obtain, for the geometrical representation of the work
done, the following conic:
y = {X sin ß sin ß' + Y cos ß cos ? - Z sin ( + B')) x2
+ {(6+ 69 Xsin ß sin ß-Z(bsin ßcos B' + b'sin ß'cos B) } x
+ 66' X sin ß sin B'.
(59)
The centre of the conic,
y2 = Lx2 + Mx + N,
is situated at a distance & from the origin, determined by the
equation
2Lx + M = 0,
or
8=-
(6+b) Xsin ß sin B'-Z(bsin ßcos B' + b'sin B'cos B)
X sin ß sin B' + Y cos ß cos ß' - Zsin (B + B')
(60)
Conclusions, similar to those obtained from equation (53),
may be drawn from equation (59).
1°. The equilibrium of the plane quadrilateral muscle will
be stable when yº is positive.
2°. The equilibrium will be neutral when y: = 0.
ANIMAL MECHANICS.
281
30. The equilibrium of the plane muscle will be unstable
when y’ is negative.
4º. The stability of the equilibrium increases indefinitely
as the axis of rotation becomes more and more remote
from 0...
5º. The instability of the muscle reaches a maximum at
a position of the axis of rotation passing through the centre
of the conic determined by equation (60).
The definite integrals, X, Y, Z, are readily found as
follows. We have
cos 2009
J A cos 0 + B sin '
ſ sin 2020
J A cos @ + B sin A'
z
4
f_sin 0 cos de
J A cos 6 + B sing
where
A = (6 - 6') sin ß sin-B'
B = b sin ß cos ß' - b' sin ß' cos B
Assume
R sin a = A,
R cos a = B;
then we have
R = VA + B',
and
A
Iſ cos 2000
Ā Jsin (a + 0)
iſ sin 2000
Y = 7
sin (a t 05
"
1 (sin cos 0do
R) sin (a + 0
282
ANIMAL MECHANICS.
Integrating these expressions between + 0 and - 0, we obtain,
after some reductions, the following values, which may be
readily computed, for given values of A and B :-
X =
2sin a sin 0 + cos Ⓡa loge.
Tootan } (a + A))
Pigeo tan ļ (a - 0))
Y =
tan ļ (a + 0))
- zsin a sin 6 + sin la loge.
geo tan į la – 01
(1)
(61)
tan (a + 8))
Z= 52cos a sin 0 - sin a cos a loge. Tan ini
The best examples, in Nature, of the rotation, by means
of a quadrilateral muscle, of limbs round an axis passing
through a fixed socket, and perpendicular to the bisector of
the angle made by the extreme fibres, are to be found in the
wings of birds, which are depressed by the contraction of the
great pectoral muscle, in the manner here considered. I shall
give a few examples, to illustrate the general principles just
given.
(a). Wing of the Albatross.-In Fig. 76, the bone AB
represents the origin of the pectoral muscle, from the fur-
culum, A, and sternum, B, and A'B' represents the insertion
of the muscle into the humerus ; both bones being placed in
the same plane, when the wing is extended, previous to the
contraction of the muscle. The centre of the shoulder-joint
is shown at S, and the bone A'B' is rotated, by the action of
the pectoral muscle, round the axis ST, perpendicular to OX,
the bisector of the angle A OB.
We are required to calculate, from the preceding theory
the position of the axis of rotation, corresponding with the
maximum work done, when the system is in a position of
unstable equilibrium ; or, in other words, we are required to
find the centre of the conic (59).
ANIMAL MECHANICS.
283
The theory just laid down, supposes the bones AB and
A'B' to be straight bones; and before applying it to the wing
of the Albatross, it is necessary to calculate the magnitude

es
20
map
A!<
1311
Fig. 76.
and position of a straight bone A"B", which would be me-
chanically equivalent to the curved bone AB. Let A B be
284
ANIMAL MECHANICS.
cu
n-
0
any curved origin (Fig. 77) of a muscle, whose fibres con-
verge to the point 0. The work done by this muscle is
represented by ſido, where l is the length of each fibre-and
this integral is readily found by construction, by drawing a
number of radiating lines from the point 0, and drawing per-
pendiculars, pq, pq, from the points of intersection with the

Fig. 77.
curve AB, upon the consecutive radiating lines. The sum
of all these perpendiculars, added together, will be equal to
the integral ſido, when the radiating lines are taken sufficiently
near to each other.
Let us draw the line AB (Fig. 76), joining the extremities
of the bone AB; and seek to find a right line A"B", parallel
to AB, and included between the extreme fibres, 0A, OB,
whose mechanical effect shall be the same as that of the
curved bone.
The work inherent in a triangular muscle, with a right
line for base, is, by equation (31), proportional to
tan ] ( + 0).
b sin ß loge Fan I (B3 - 0)
where b is the bisector of the vertical angle, and ß is the
angle made by it, with the base.
ANIMAL MECHANICS.
285
285
Hence we have.
tani (B + 0)
Sido = b sin ß loge
loge tani (B - 0)
(62)
In the case of the pectoral muscle of the Albatross, I found
by measurement
Fid0 = 7.17 inches.
20 = 47°.
3 = 1 29°.
Introducing these values into equation (62), and solving for b,
we find
b =
7.17
60
= 8.1317 inches.
sin (1299) loge tan 52° 45'
The right line A"B" is drawn, in Fig. 76, parallel to AB, and
intersecting the bisector 0X, at the distance just found, from
the vertex 0.
A straight bone, A"B", acting as the origin of the quadrila-
teral pectoral muscle, would produce upon the wing bone,
A'B', a mechanical effect similar to that of the curved bone,
AB; but it would, obviously, cause much inconvenience by
its awkward shape and length. Hence we see the reason for
the curved outlines of many bony origins of muscles, which
frequently occur in Nature. The curved bone fits into a
smaller space, gives rounded outlines to the structure, and at
the same time produces all the mechanical effects of the
straight bones, which mathematicians use in their calculations;
and we are obliged, in order to interpret Nature, first to
translate the expressive language of her beautiful and sym-
metrical curves, into the less perfect language of the rigid
lines and bars used by geometers.
886
ANIMAL MECHANICS.
Collecting together all the measurements of the wing of
the Albatross, we have
b = 8.13 inches. B = 129°
b' = 2.67 2 B' = 680 20 = 47°.
From these we readily obtain
A = (b – b') sin ß sin ß' = 3.934
B = b sin ß cos ß' – b' sin ß'cos B = 3.925
R=NA? + B2 = 5.55
a = tan -- = 45°
tan (a + )
= 1.2770
tan ] (a - 0)
v
1.2024 = + 0.2166
5.55
y
0.0740 – + 0.0138
5.55
2 =- Y = - 0.0138.
The centre of the conic (59) is found from equation (60),
which is equivalent to
(6 +84) & - 2 (6 cot B' + b cot B)
2 X + Y cot B cot ß' - Z (cot ß + cot B')'
We have, by the preceding values,
(b + b') X = 10.80 x 0.2166 = + 2.3393
- 2 (b cotß' + b'cot B) = 0.0138 x 1.12 = + 0.0154
Numerator, ........ .. + 2.3547
X = . . . . . . . . . . . . + 0.2166
Y cot fcot ß' = 0.0138 X -0.327 = - 0.0045
- 2.(cot ß + cot B'} = 0.0138 - 0 406 = -0.0056
Denominator, ......... + 0.2065
X L. I2 = to
ANIMAL MECHANICS.
-287
Therefore, finally
-- } 0.2065 = - 5.70 inches.
The negative sign of 8 denotes, that the axis of rotation
of maximum work lies at the near side of the vertex 0.
Draw PQ perpendicular to OX, so that
OC = 5.70 inches,
then C is the centre of the conic (59), and its minor axis,
PQ, is the axis of rotation, which would produce the maxi
mum work in the motion of the wing.
In order to construct the conic, let us divide equation (59)
by sin ß sin B', and we obtain
sin ”sin * = Lx2 + Mx + N;
(63)
where
L = X + Y cot ß cot ß' - Z (cot B + cot B');
M = (b + b') X - Z (6 cot B' + b'cot B);
N = 56 x.
Substituting, in these expressions, the values already giver
we obtain
sin ß sin
= 0.206 x? + 2.355
This equation gives us, when y = 0,
21)
y = - 5.700 + 3.138 inches,
2.,)
288
ANIMAL MECHANICS.
from which it appears, that the semi-axis major of the ellipse,
whose centre is C (Fig. 76), is 3.138 inches.
Equation (63) denotes an ellipse or hyperbola, according
as ya is negative or positive ;—that is, according as the equili-
brium of the bones is unstable or stable.. Making ya negative,
we have
y = - sin ß sin ß' (Lx? + Mx + N),
and this expression will become a maximum, and equal to the
square of the semi-axis minor, when
M
= = -
= -5.70 inches,
which corresponds with C, the centre of the conic.
Writing the equation (63)
- y
=
5
; = Lx? + Mx + N,
sin B sin B
we have, for the semi-axis minor of the ellipse formed by
(', x),
y = V - (L8? + M8 + N) = 1.424 inches,
The ellipse (y', .) drawn to scale is shown in Fig. 76; and
ts equation, referred to C as origin of coordinates, is
( x 12
13.138)
-
+
. y 12
(1.424)
=
I.
The work done by the muscle, causing the bone A'B' to
rotate round any two chords PQ and ST, is proportional to
the squares of those chords; and we may, therefore, compare
the actual work done round the chord, ST, with the calculated
maximum round the chord PQ.
ANIMAL 1
289
11
.
MECHANICSThe point S is the observed centre of the glenoid cavity of
the shoulder joint, and ST is the actual axis round which the
wing rotates. This axis of rotation is placed at a distance of
4.83 inches from the vertex 0; and, as we have just seen,
the axis PQ, corresponding to the imaginary straight bone
A"B", lies at a distance from O, equal to 5.70 inches. The
distance, therefore, between the actual axis, ST, and the
calculated axis, PQ, is 0.87 inch. By substituting in equa-
tion (64), for x, the intercept 0.87 between ST and PQ, we
ſind
ly
(1.424) = 0.9232
or
ya = 1.872;
but
m2 = 2.028.
Therefore
n2 - y2
222
156
2028
I
$3'
or, the work done by the rotation of the wing of the Albatross
round its observed socket differs by 7}th part from the cal-
culated possible maximum derivable from the imaginary
straight bone A"B".
In Proposition C, we have seen that, in a triangular
muscle, the position of maximum effect of the axis of rotation
corresponds with the point of bisection of the bisector of the
vertical angle; and it may be seen, from the preceding cal-
culation, that, in a quadrilateral muscle with straight bones,
the axis of rotation of maximum effect nearly corresponds to
the point of bisection of the portion of the bisector included
within the quadrilateral. For the distance from 0 to this
point of bisection is
S
b + B' 8.13 + 2.67 = 5.40 inches ;
22
290
ANIMAL MECIIANICS.
which may be regarded as a useful approximation to the true
position of the maximum axis, which is 5.70 inches.
The method of finding a quadrilateral muscle, with straight
bones, equivalent to a given muscle with curved bones which
has just been given, leads to difficult calculations; and,
moreover, does not seem to bring the calculated axis PQ,
sufficiently near to the actual axis $T, to satisfy the require-
ments of exact science; it may, therefore, be useful to give
another mode of computation, somewhat less laborious, and
which may be made to approximate as near as we please to
the conditions imposed by the actual curves formed by the
bones of origin and insertion.
The general expression (47),
2
J
fein WP (a cos o + I cos 0.) (a cos o + ' cos 0)
f&pdo = * [la cos
i-
for the work done during a rotation (w) round an axis per-
pendicular to the bisector may be thus written, making
a cos p = x :
[(x + I cos ) (x + 1' cos 0)
..
.ſåpdø = [(* * cos 07-23
or
do = y:= Lx? + Mx + N,
(65)
where
1
ſ do
v
lll + l) cos Odo
N. PIl cos 20:10
DI - ľ
177 M = 1
Now, if the bones of the wing be carefully laid down to scale,
and a number of radii drawn through the point 0, the definite
integrals L, M, N, may be found, by construction, to any
degree of approximation, as shown by the following example
of the Albatross, in which the measurements are made in
twenty-fourths of an inch :-
ANIMAL UL
291
MECHANICS.
Wing of Albatross.

Angle. I l-1.
I cos 0.
I cos 0.
me
109
162
190
O co ao
204
213
186
239
195
245
198
194
231
From these measurements, the following values are cal-
culated :-
Wing of Albatross.

.
(+1') cos al
Angle.
ll' cos 20
7-ľ
0.03225
0.01136
0.00833
0.00735
6,1219
2.7160
2.1917
2.0074
1.8851
1.7164
1.5968
1.5506
1.5202
1.4742
284 80
141.75
115.58
103.50
94.98
82.83
74.53
0.00675
0.00588
0.00538
0.00512
0.00505
0.00515
71.76
69.29
65.49
Sum,
0.0962
22.7803
1104.51
Hence we obtain, approximately,
PLA
I do
y = 0.0962.
U 2
292
ANIMAL MECHANICS.
Po
111 =
|(l + l') cos Odo
T-1 = 22.7803.
Pto
11' cos 2000
T TT = 1104.51
J-O
The centre of the conic,
ya = Lx? + Mx + N,
(66)
is found by the equation,
• - - * - - 1:22.7803 - - 118.4
= - 4.93 inches.
The actual distance of the axis passing through the socket S,
from the vertex 0, is 4.83 inches; and the distance calculated
from our former approximation was 5.70 inches. The result
of the present calculation is 4.93 inches.
If m and n denote the semi-axis major and minor of the
ellipse (66) we have
M – 4LN
m =
(67)
n = m V L = 15.62.
When expressed in inches, these values are
m = 2.099 inches ;
n = 0.651
The line LM, Fig. 76, is the minor axis of the conic (66), and
it is distant from the actual axis ST, by only one-tenth of an
inch.
If we substitute one-tenth of an inch for , in the equation
"
N
21
= 50.37;
12
12.099)
/ Y
10.651)
ANIMAL MECHANICS.
293
ly } .
we find
gol = 0.9773;
10.651)
y = 0.4142.
This represents the amount of work done round the axis ST';
and the work done round the axis LM is
m = 0.4238.
n² - y² 96 I
Hence we find
od m? * 4238 – 44.
or the work done by the rotation of the wing of the Albatross
round its observed socket differs by only anth part from the
calculated possible maximum work. This degree of agreement
between calculation and observation is sufficient to satisfy all
reasonable expectations.
(6.) Wing of the Wood Pigeon.— The next example that I
shall select is the wing of the Wood Pigeon, shown in Fig.
78. The origin of the pectoral muscle is shown at AB, and its
insertion at A'B'. The angle AOB is 23°, and the socket of
the wing is placed at S; the line ST being the axis round
which the wing-bone revolves.
Dividing the angle AOB into six equal parts, I found
the following measurements, made in thirty-seconds of an
inch :-
Wing of Wood Pigeon.

Angle,
I 1-
l I cos 0. ' l' cos 0.
I!
5.20
09.00
10.40
16.00
21.20
26.10
32.00
I 15
129
173
156
From these measurements the following values are calcu-
lated :
294
ANIMAL MECHANICS.
Wing of Wood Pigeon.

Angle.
(1+1') cos 0
TV
09.00'
0,09091
0.01317
0.01041
0.00869
0.00775
0.00699
0.00041
9.8182
2.1843
1.8437
1.6609
1.5319
1.3567
1.3462
32.00
Sum,
0,14433
19.7449
Il cos 20
TT
262.82
71.64
56.06
50.55
44.22
37.15
34.61
557.00

TOMMITTIM
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
Fig. 78.
ANIMAL MECHANICS.
295
Hence, we have, for the centre of the ellipse (66)
8--
--
19.7449 = 68.4.
2
0.14433
The distance of the actual axis of the wing, ST, from O is 64,
while the distance of the calculated axis of maximum work
is 68.4.
(c.) Wing of the Common Heron.-The wing of the Heron
is shown in Fig. 79, where AB is the origin, and A'B' the

CA
i
Fig. 19.
insertion of the pectoral muscle ; S is the socket of the wing-
bone, and ST perpendicular to 0 X, the bisector of the angle
AOB, is the axis round which the bone A'B” rotates. The
angle A OB is 45°, and, when divided into ninc equal parts
296
ANIMAL MECHANICS..
:
of 5° each, gives the following measurements in thirty-seconds.
of an inch:--
Wing of Heron.

Angle.
I-1'. ! i cos 0.
I cos 0.
113
100
III
121
! 26
124
130
132
123
122
From these measurements, I obtained the following results,
by calculation :
Wing of Heron.

(2+1') cos 0
Angle.
Il' cos?
TT
ehit tengo ono
0.03571
0.02632
0.02083
0.01539
0.01205
0.01000
0,00901
0.00826
0.00791
0.00814
2.4649
1.9474
1.7292
1.5077
1.3855
1.2700
1.2342
1.1736
1.1349
1.0901
36.93
27.50
24.37
21.18
19.08
15.82
17.52
12.00
11.52
10.08
Sum,
0.15370
14.9375
193.90
Hence we find, for the centre of the conic (66),
14.9375
48.60.
0.1537
ANIMAL MECHANICS.
297
The axis PQ is drawn, in Fig. 79, at this distance from the
point 0; while the measured distance of the actual axis ST
from the point O is 44.
(d.) Wing of the Macaw.—The wing of the Macaw is
shown in Fig. 80, where AB is the origin of the pectoral

S
WNIK
Fig. 80.
muscle, and A'B' is its insertion. S is the centre of the
glenoid cavity, and ST' is the axis round which the wing-bone
revolves. PQ is the calculated axis, corresponding to the
minor axis of the ellipse (66). The angle AOB is 90°, which
I divided into parts of 10° each, with the following results,
measured in thirty-seconds of an inch :
298
ANIMAL MECHANICS.
Wing of Macaw.

Angle.
1-1.
I cos 0.
I cos 0.
58
106
114
127
143
112
77
106
115
120
126
127
91
From these measurements, I obtained the following re-
sults, by calculation :-
Wing of Macaw.

Angle.
1 (1 + l') cos e
T-
ľ L
li cos 20
-
loll
0.02381
0.02174
0.01852
0.01415
0.01041
0.00943
0.00877
0.00787
0.00699
0.00893
1.1191
1.2174
1.2407
1.2429
1. 2084
1.1793
1,1491
1.0787
0.9650
0.9196
Sum,
0.13062
11.3202
101.17
ilence we find, for the distance of the centre of the Conic
(66), from the vertex 0,
•=-5
1 11 3202 = - 43.333.
1206
The actual distance of the axis ST, passing through the
socket S, is 45.
ANIMAL MECHANICS.
299
.
(e.) Wing of the Pheasant. In the wing of the Phea-
sant, Fig (81), the angle AOB is 45°, which I divided into

III
11
17
Fig. 81.
nine equal parts, and made the following measurements,
in thirty-seconds of an inch :-
Wing of Pheasant.

Angle.
1-1.
I cos 0.
V' cos 0.
100
I 20
I 18
135
177
و ب با ما
184
202
211
300
ANIMAL MECHANICS.
From these measurements, the following results were ob-
tained by calculation :-
IVing of Pheasant.
.

Angle.
1
T-7
(1 + 1') cos e
Int'
Il cos 20
T-ľ
0.07692
0.02500
0,01515
0.01351
0.01000
0.00833
0.00614
0.00547
0.00503
0.00463
29.00
6.231
2.750
1.515
1.6892
1.3500
1.2500
1,1717
1.0765
1.0704
1.0231
153.00
45.90
28.38
20.06
15 20
13.07
11.16
9.77
Sum,
0.17018
19.5805
342.43
IIence, the axis PQ, passing through the centre of the Conic
(66), lies at a distance from the vertex 0, equal to-
8=-
19.5805
0.17018 * - 57.53.
i
The actual distance of the axis ST is şi.
In equation (66),
ya - Læ* + Mx + N
the axis O X will not intersect the conic in real points unless
MP - 4LN
be positive. This, therefore, becomes the condition for the
existence of axes of unstable equilibrium, and for the existence
of the ellipse of unstable equilibrium. We are, therefore, bound
to ascertain that this condition is fulfilled in the wings of the
birds already investigated. The following Table shows that
they all satisfy this test :-
S
ANIMAL MECHANICS.
301
Condition of Unstable Equilibrium in the Wings of Birds.

Bird.
I
M
12 - 4 IN
22.7833
1104.51
557.05
+ 93.91
+ 68.27
0,14433
19.7449
Albatross, .. 0.0962
Woodpigeon, il
Heron, ... 0.15370
Macaw, ...
0.13062
Pheasant,. ..
14.9375
193.90
+103.90
11.3202
101.17
+
75.27
+ 150.29
0.17018
19.5805
342.42
The positive values in the last column of this Table show,
that in every case there exists a real axis of maximum insta-
bility, or tendency in the plane muscle to become skew.
If we compare together the preceding examples, and bear
in mind the necessary errors of observation, and also that we
have been compelled to employ summation instead of integra-
tion, so that our results are only approximations; there can
be no doubt that, in the wing of the bird, the socket of the
humerus is so placed, in relation to the origin and insertion
of the pectoral muscle, as to produce the maximum amount
of work possible by the contraction of that muscle.
entitled to lay down the following:-
PROPOSITION D.
When a quadrilateral muscle unites any two bones whatever,
and motion takes place round an axis lying in their plane, and
perpendicular to the bisector of the angle made by their extreme
fibres ; two points may be found on that bisector, such that-
1°. If the axis of rotation lie outside these points, the con-
traction of the muscle will tend to replace the bones in the same
plane, and their equilibrium will be stable.
2º. If the axis of rotation pass through either of these points,
302
ANIMAL MECHANICS.
the contraction of the muscle will produce no effect, and the
equilibrium of the bones will be neutral.
3°. If the axis of rotation lie inside these points, the contrac-
tion of the muscle will tend to convert the plane quadrilateral
into a skew muscle; and this effect will be a maximum, when the
axis passes through the point midway between these points; and
the equilibrium of the bones will be unstable.
The Divine Architect of the Universe has placed the axis
of rotation of the wings of birds in the position of maximum
work, or so that a given amount of work shall be performed
by the minimum amount of muscle; which is in conformity
with the “ Principle of Least Action” observable in all de-
partments of Nature. Hence we may regard the Postulate
of page 238 as proved for this class of muscles, and as thus
having additional probabilities in its favour.
In Propositions A and B, we have considered the case of
two bones, lying in the same plane, and united by a quadri-
lateral sheet of muscle, one bone being supposed fixed, and
the other compelled to turn round an axis of rotation perpen-
dicular to the plane.
In Propositions C and D, we have considered the same
case, when the axis of rotation lies in the original plane of the
quadrilateral muscle, and is perpendicular to the bisector of
the muscular fibres.
In order to complete the consideration of the action of the
muscle, when the bone rotates round any axis whatever, we
shall now consider its action when the bone rotates round an
axis perpendicular to both the former axes of rotation; that
is to say, when the rotation of the moveable bone takes place
round an axis parallel to the bisector of the muscular fibres.
Let AB and A'B' (Fig. 82) be the bones, and let the
same notation be used as before. Let S be the socket of the
joint, and SZ the axis of rotation, parallel to OX, the bisector
of the angle made by the fibres. Draw SY perpendicular to
YOX, and let
ANIMAL TRA
303
MECHANICS.
OS = a
YOS = 0
ao x = 0
Oa = 1
Oa' = 1

SITWASSWR
Fig. 82.
Let the point a be moved to the point b, by a rotation (w)
round the axis Sz. Then, if
Sa = A
b Sa' = Y
Su = A' aSZ = o.
aSa' = x
we have, since A and A' are constant,
p = aa' = ✓ A’ + A'? – 24 Acos x;
p = ba' = V A² + 42 – 2A A' cos y.
If we describe a sphere
round S as centre, we shall
have (Fig. 83),

Sa = $l = 0
aa' =
(ht = 3
U Su = w.
Fig. 83.
304
ANIMAL MECHANICS.
From the spherical triangle Sba', we find
cos y = cos w sin ó sin (o + x) + cos o cos o + x;
or, when w is a moderately small angle,
cos y = cos « -.. sin o sin (c + a).
Substituting this value in the preceding equations, we find
p'-p
w? AA
, sin o sin (o + x).
2
P
Now, it is easy to see, from Fig. 82, that
A sin o = a sin p + 1 sin ();
A sin (o + x) = a sino + lsin 0);
p=1-l
Hence we obtain
we (a sin ø + sin () (a sin 0 + l' sin (): (68)
:p put iny
and, finally, for the work done, by a rotation round SZ,
through the small angle we
de_w? [(a sin 0 + 1 sin () (a sin 0 + l' sin 6) do
1-1
". (69)
If we substitute, in this equation, the values of land l' given
in page 266, and follow the method there employed, we obtain,
finally,
rcos 2000
= aż sin ?p sin ß sin
ſopdo = v* sin sin B sin ß* fcos 2010 (70)
in (1sin B – a sin go cos B) (bʻsin BP – a sin o cos B) sin dan
al
psin
cos 6.0.
asin p{(6 + b) sin ßsin ß'-asin º sin (ß +ß".
2
ANIMAL MECHANICS7
305
.
2
This expression might be integrated in terms of the
definite integrals, X, Y, Z (p. 282), and the whole theory of
work done by plane quadrilateral muscles deduced from it ;
but we shall obtain results more in accordance with the actual
circumstances of nature, by using the method of differences,
as employed in the discussion of the wings of birds. Previous
to adopting this method, however, it is worth while to apply
equation (69) to the simple case of a triangular muscle, which
often occurs in nature.
Making l = 0, and substituting for l its value,
6 sin ß
sin (ß - 0)
we find
S
t
+0
famio - a sin ni sing = { sin o) a
E
a
a sin octo
lía sin o sin ß cos0 + (6 sin B - a sing cos B) sin 0j do
b sin BJU
= 2a sin? o sin
Or,
9
0% a sina o sin 0
(71)
S
From this expression for the work done by a triangular
muscle, when rotated round an axis parallel to the bisector of
its vertical angle, we obtain the following conclusions:
1°. If the vertical angle and bisector of the muscle be
given (0,b), the work done will be independent of the position
of the bone.
2º. The work done varies as the square of the distance of
the axis of rotation from the bisector.
306
11.
ANIMAL NIT
MECHANICS.
3º. For the bisector itself, taken as axis of rotation, the
equilibrium of the muscle is neutral, and for all other positions
it is stable, and never becomes unstable.
If we compare these results with the corresponding results
for rotation round axes perpendicular to the bisector of the
angle of the muscular fibres (pp. 269, 270), we see very im-
portant consequences, which may be thus stated :-
PROPOSITION E.
No triangular muscle, by the contraction of its fibres, can
compel the bone that forms its base to revolve round an axis of
rotation lying in the plane of the muscle, unless the socket of the
joint lie between the perpendiculars drawn at each extremity of
the bisector of the vertical angle of the muscle.
For, if the rotation be resolved round two rectangular axes,
parallel and perpendicular to the bisector, the equilibrium of
the muscle, with respect to the first component axis of rota-
tion, is always either stable or neutral; and, with respect to
the second component axis of rotation, the equilibrium is also
either stable or neutral, unless the axis lie between the per-
pendiculars drawn at each extremity of the bisector.
I have already shown, in the lutissimus dorsi of Man, of the
Tiger, and of the Albatross (pp. 246, 249, 256), and in the
hamstring muscles of the Tiger (p. 258), that the socket of
the joint is placed upon the perpendicular at the vertex to the
bisector of the muscle, and that the object of that arrangement
is to produce the maximum amount of Work possible by a
rotation round an axis perpendicular to the plane of the muscle.
It follows from Prop. E, that these muscles can produce no
rotation whatever round any axis lying in the plane of the
muscle.
These muscles are, therefore, intended to cause rotation
round one axis only, and to do so with a maximum amount
of useful work.
ANIMAL MECHANICS.
307
This consideration affords us a glimpse of the reason why
so many muscles are made to act in various ways round the
shoulder and hip joints of animals ; each muscle is intended
to produce a rotation round some axis or definite group of
axes ; and, in order to enable it to do so with maximum effect,
its power to cause rotations round other axes is taken away.*
We shall now consider, in detail, the case of rotation round
an axis parallel to the bisector, with respect to the five birds
already discussed.
If we expand equation (69), writing
11
x2 =
$pdo,
y = a sin o,
we find
q? - Py+ Qy + R;
(72)
where
Polen Q-1° +2° sin 0; R = 1W sint
t
"
0
1-ľ
Equation (72) represents a central conic, whose centre lies
upon the axis of y, on the perpendicular drawn at 0, to the
bisector 0 X, and at a distance from 0, determined by the
equation
2Py + Q = 0.
When the equation (72) is referred to axes of co-ordinates
passing through the centre, it becomes
(73)
+ m2 = m2
(74)
* May not the remarkable similarity in the bones and muscles of various
animals, in corresponding parts of their structure, be the result of some such
geometrical and mechanical necessity, foreseen by the Divine Contriver, instead
of being the result of common descent from a remote ancestor possessing a similar
arrangement of bones and muscles ?
X2
308
ANIMAL MECHANICS.
where
ma
QP - 4PR
;
4P
(75)
m2 Q? – 4PR
4P2
When 2 is positive, equation (74) represents an hyperbola,
and the equilibrium of the muscle is stable ; and when x2 is
negative, equation (74) represents an ellipse, and the equili-
brium of the muscle is unstable.
If P, Q, R, be positive, it is evident from (75) that
m2 and 12 will have the same sign as
Q? – 4PR;
and if this quantity become negative, the ellipse represented
will be imaginary; or, in other words, there will be no axis
of unstable equilibrium possible.
(a.) Wing of the Albatross. - The following measurements,
taken from the wing of the Albatross, are made, as before, in
twenty-fourths of an inch:
Wing of Albatross.

Angle.
1-
1
1
sin 0
1
l sin 0
our o
f + + + +
00
IIIII ++ to to t
O
to whether
III
194
ANIMAL MECHANICS.
309
From these values we can calculate the following Table :-
Wing of Albatross.

angle.
(1 + 1') sin 01 Il' sin²0
1-3
Tam
l.
N
+ 2.6774
+0.9091
+0.5250
0.2941
0.01136
0.00833
0.00735
0,00675
0.00588
0.00538
0.00512
0.00505
0.00515
+ to fo tt1 1 1 1
öö ó
1081
-0.0471
-O 1882
-0.3282
-0.4596
0.6598
onö omimo
6.35
13.27
Sum,
0.0962
2.8308
103.37
The sums here given are the quantities P, Q, R; and it will
be observed that the elements of Q have opposite signs at
each side of the bisector, so that its value is the difference of
two groups of elements, and not the sum, as in the corre-
sponding quantity-
M=1
rrl+1') cos 0
IIence we are not surprised to find that
Q:- 4PR =-31.763,
from which we obtain, by equation (75),
n =- (29.29),
m = -(0.08)
As these values are both negative, equation (72) will
represent an hyperbola, whose transverse axis is parallel to
the axis Ox, and whose centre lies upon ( Y, as shown in
Fig. 84. The centre of this hyperbola is found from equa-
tion (73),
2P'y + R = 0;
T
!
11
ANIMAL MECHANICS
dz.
2.8308
0.0062 * 14.71.
Tu
bo
The actual distance of the socket S, from the line O Y, is 37ths
of an inch.

Vị P.
IN
hohet
th
Fig. 840
in Fig. 84, I have drawn carefully to scale, for the wing
ANIMAL MECHANICS.
311
of the Albatross, the two conics represented by the equa-
tions,
y? - Læ? + Mx + N;
22 - Py2 + Qy + R,
S
1
referred to OX and OY as co-ordinate axes.
The ellipse whose centre is C may be called the ellipse of
instability, and the axis of rotation, LM, passing through its
centre, is the axis of maximum instability. The hyperbola
whose centre is D may be called the hyperbola of stability,
and the axis of rotation, PQ, passing through its centre, is the
axis of minimum stability. The intersection of these two axes,
2, determines the exact position of the centre of the socket
of the shoulder joint, which fulfils the conditions imposed by
the principle of " least action ;” because if the wing rotate
round any axis passing through this point, and lying in the
plane of the muscle, the contraction of the muscle will produce
a maximum amount of work done, by the component rotation
parallel to OX; and, although the contraction of the muscle
cannot produce any rotation round an axis. parallel to OY,
yet, if any such rotation take place from external causes, the
contraction of the muscle will restore the bones to their
original positions, with the minimum amount of work lost.
The near approach of the calculated socket 2 to the
observed socket S is a proof of the truth of the Postulate
(p. 238); and we shall see that the wings of the remaining
birds confirm the conclusion here drawn.
(6.) Wing of the Wood Pigeon.-- The following measure-
ments on the Wood Pigeon and other birds are made in 32nds
of an inch:
312
ANIMAL MECHANICS.
Wing of Wood Pigeon.

Angle.
1-2'.
I sin 6
l l' sin e.
-
-
II
+
--
t
of a
to
IIS
129
143
1
oon
180
1
From these values the following Table is calculated :-
D
Wing of Wood Pigeon.

Angle.
(1+7') sin 0
T
Il sin e
Ti
27.64
09.00'
5.20
10.40
16.00
21.20
0.00091
0.01317
0.01041
0.00869
0.00775
0.00699
0400641
+ 3.1818
+0.1080
+0.1770
0.0000
-0,1393
- 0.2517
-0.3166
nooo win
20:40
32.00
Sum, 0-14433
3.0592
34.21
From these values, we can readily calculate,
2
U
Q2 - 4PR=- 10.391.
Since this quantity is negative, the equilibrium of the wing
is always stable, with respect to axes of rotation parallel to
OY. .
The position of the axis of minimum stability is found as
before,
0 = 1 3.0592 = 10.60.
0.14433
ANIMAL MECHANICS.
313
The actual distance of the socket S from the line O Y is
nds of an inch.
(c.) Wing of the Heron.-- The following measurements wer
made upon the wing of the Heron, in 32nds of an inch.
,
Wing of Heron.

Angle.
I!.
I sin 0.
Il sin e.
to
OC Oo oo
ot
+ + tot
go
to
+
1
100
111
121
1 1
1
1 1
126
I 22
1
1
From these measurements, the following Table is calcu.
lated :-
Wing of Heron.

1
Angle.
ta
(1+1') sin 0
Tall
li' sip20
Tn

6.75
how was sono
0.03571
0.02632
0.02083
0.01539
0.01205
0.01000
0.00901
0.00826
0.00794
o 00819
+1.0714
+0.6316
+0.4166
+0.2 54
+0.0723
-0.0500
-0,1622
-0.2562
-0.3651
-0.4508
2.84
1.33
0.51
0,06
0.04
0.29
0 70
1.33
2.05
Sum, 0-15370
1-1230
15.90
From these values, we find,
Q2 - 4PR = -8.514,
314
ANIMAL MECHANICS.
showing that the equilibrium is always stable, and
8 = 1.1230 = 3.65.
0.1537
The actual distance of the socket from OY is zands of an
inch.
(d.) Wing of the Macaw.—The following measurements on
the wing of the Macaw are made in 32nds of an inch :-
Wing of Macaw.
Angle.
1-1
|
I sin
l' sin 0
4
2
+
ano
+
46
+
+
+
106
+ to +t+IIIII
1
1
114
127
143
© 20 AW N- N
1
1
112
1
From these measurements, the following Table is calcu-
lated:
Wing of Macaw.

Angle.
(2+2') sin 0
2-1
Il' sin 20
1.71
0.02381
0.02174
0.01852
0.01415
O 01041
0.00943
0.00877
0.00787
0.00699
0.00893
+1.1428
+ 0.8480
+0.5741
+0.3286
+0.1042
-0. 0132
-0.3070
-0.4960
- o 6783
-0.9018
x = c o c o – 5x
Sum, 0-13062
0:5014
31:36
ANIMAL MECHANICS.
315
From these values, we find
Q2 - 4PR =- 16.134,
showing that the equilibrium is always stable ; and.
8 =
.5014
,
70.13062
* = 1.92.
The actual distance of the socket from the axis O Y is kinds
of an inch.
(e.) Wing of the Pheasant. ---The following measurements
were made on the wing of the Pheasant, in 32nds of an inch :-


Angle.
1-!.
I sin 0.
i sin 0'.
III!!+++++
ha a ir at om
che
+ + + + + III
AWNN- warna
From these measurements the following Table is con-
structed:
Wing of Pheasant.

Angle.
(2+2') sin 0
1-1
Il sin? 0
1-7
0.07692
0.02500
0.01515
0.01351
O 01000
0.00833
0.00614
0.00547
0.00503
0.00463
+ 2.9231
+0.7750
+0.3788
+0,2300
+ 0.0700
-0.0500
-0.1533
0.2404
.-0.3316
-0.4120
moooooo
0.49
0.95
1.59
Sum, 0.17018
3.1896
37.51
316
INVIU
L
From these values we find
Q2-4PR=-15-361,
showing that the equilibrium is always stable, and
3.1896
. 0.17618 = 9.37.
The actual distance of the socket from the line O Y is nds
of an inch.
Collecting together into one Table the co-efficients of
equation (72), we obtain-

Bird.
62 - PR
Albatross, ..
Wood Pigeon,
Heron, . . .
Macaw,. . .
Pheasant,
0.09620
0.14433
0.15370
0.13062
0.17018
2.8308
3.0592
1.1230
103.37
34.21
15.90
31.36
37.51
- 31.763
– 10.391
- 8.514
0.5014
- 16.134
3.1896
- 15.361
From the foregoing investigation, it appears that Prop. E
may be extended to the curved quadrilateral muscles which
form the great pectoral in the wings of birds, and that the
contraction of the muscle cannot produce a rotation of the
wing round any axis lying in its plane, unless the socket o"
the joint be placed somewhere in that plane lying betweer.
the tangents drawn to the ellipse of instability at the extremities
of its diameter on the line OX, which bisects the angle formed
by the fibres of the muscle.
The exact position of this socket, required to fulfil the
conditions of maximum work done and minimum work lost, is
found by the intersection of the axes of maximum instability
and minimum stability.
The difference between the calculated and observed posi-
tions of these two axes is shown in the following Table :-
21
VIIULU
TY
ANIMAL MECHANICS.
317
IT
TV
Ylili1lL.
.
Difference between Calculation and Observation in the position
of the Socket in the Wings of Birds.

Bird.
Axis of maximum
instability.
Axis of minimum
stability.
.
0.30 inch.
Albatross, .
Wood Pigeon, .
Heron, . .
Macaw, , , ,
Pheasant, . .
0.10 inch.
0.14 »
0.14 »
0.05 )
0,20 19
0.27
0,10 22
0.41 22
0.15 99
Mean Error,
0.126 inch. 1
0.246 inch.
The differences between calculation and theory shown in
the foregoing Table (one-eighth and one-quarter of an inch)
are not greater than should be expected in such a class of
observations. It may be noted, also, that the mean error in .
the calculation of the axis of maximum instability (which is the
more important of the two axes) is only half the mean error
in the calculation of the axis of minimum stability.
We have seen by Prop. E, applied to the shoulder and
hip joints of various animals, that muscles intended to produce
the maximum work by rotation round axes perpendicular to
their plane are incapable of producing a rotation round any
axis lying in their plane. Let us now inquire how muscles
like those of the pectoral muscles of birds, intended to produce
the maximum work round an axis lying in their plane, behave
with regard to the axis of rotation perpendicular to their plane
drawn through the socket of the shoulder joint.
It will be readily seen, by reference to the proof given of
equation (44), Prop. B, that it is independent of the shape of
the bones, whether straight or curved ; so that if k, 0, 0, be
given, there are an infinite number of muscles capable of pro-
ducing the same work. This equation-
Work done = 2kow sin q sin 0,
IL
y
1
2
318
ANIMAL MECHANICS.
may therefore be applied directly to calculate the work done
by the pectoral muscles of birds, round the axis passing
through the socket of the shoulder joint, and perpendicular
to the plane of the muscle. The maximum work possible
would be obtained by placing the socket upon the perpendi-
cular to the bisector drawn at the vertex 0, and corresponds to
Maximum work = 2ków sin .
Hence we have,
Work done
I = sin d.
Maximum work po
(76)
By direct measurement of the angle o, we obtain the follow-
ing Table :
Ratio of Work done by the Pectoral Muscle of Birds round the
observed Axis perpendicular to its Plane, to the maximum
possible Work.
WULUI

Bird.
sno
II°.00'
I .55
Albatross, . . .
Wood pigeon, ..
Heron, . . • •
Macaw, . .
Pheasant .
9.10
0.190
0.033
0.159
0.333
0.087
19:30
5.00
Mean ratio, ... 0-1604
If we take for the socket of the shoulder the calculated
instead of the observed positions, we find by measurement of
the angle o the following Table :-
ANIMAL MECHANICS.
319
Ratio of Work done by the Pectoral Muscle of Birds, round the
calculated Axis perpendicular to its Plane, to the maximum
possible Work

Bird.
sin o
60.55
9.35
Albatross, . . .
Wood pigeon, . .
Heron, . . . .
Macaw, . . .
Pheasant, .
9.20
O 120
0.166
0.162
0.035
0.161
2.00
9.15
Mean ratio, ... 0·1288
From these results it appears that the rotation of the wing
round the observed axis, perpendicular to the plane of the
muscle, gives only 4th part of the maximum work possible ;
and that a similar rotation round the calculated axis gives
only Ith of the maximum work possible.
This conclusion confirms generally that already arrived at
in p. 307, and shows that each muscle is constructed mainly
for the performance of a certain kind of work, and that it
performs any other kind of work under great disadvantages.
The condition necessary, in order that a given quadrila-
teral muscle should be absolutely incapable of producing any
rotation round an axis perpendicular to its plane is readily
found. For, the expression (44) for the work done must
vanish, which gives us sin q = 0; or that the socket must be
placed upon the bisector 0X (Fig. 84). Hence the centre Dof
the hyperbola must coincide with O, and in equation (72) the
term involving y must disappear; hence we have,
fo
12 + l') sin
1 1-7
O,
(77)
J-0
as the condition required, in order that rotation round an
axis perpendicular to the muscle shall be impossible.
320
ANIMAL MECHANICS.
The quantity Q, as I showed before, consists of the dif-
ference of two sums, and there is no difficulty in supposing
a case in which it should vanish altogether. For the birds
already discussed, its values are-
Albatross, . . . . . 2.83
Wood Pigeon, ...3.06
Heron, . . . . . . 1.12
Macaw, · · · · · · 0.50
Pheasant, . . . . . . 3.19
Quadrilateral and skew muscles, in which the socket of the
joint does not lie in the plane containing the origin and insertion
of the muscle.
In page 262, I commenced the discussion of Quadrilateral
muscles, in which the plane of the origin and insertion of the
muscle includes the socket of the joint, and fully illustrated
the action of this important class of muscles by means of the
great pectoral of the wings of birds. We have now to
examine the class of muscles in which the length of the fibres
is such as not to permit the origin and insertion to come into
the same plane as that which contains the socket of the joint.
In this class of muscles, the maximum extension of the
fibres possible still leaves the plane passing through the socket
and insertion distinct from the plane passing through the socket
and origin of the muscle; and we are to suppose the two bones,
each lying in its own plane, placed in the position of maximum
extension of fibres, the work to be done being to compel one
bone to turn round the other bone, the intersection of the
planes containing each being the axis of rotation.
It is easy to show, as a preliminary condition, independent
of any special arrangement of the muscular fibres connecting
the two bones, that, in order to produce the maximum moment
to turn the planes round their mutual intersection, regarded as
an axis of rotation, that it is necessary that the planes passing
ANIMAL MECHANICS.
321
B
through the socket, and the origin, and insertion, respectively,
must be at right angles to each other.
With the socket of the joint as centre, describe a sphere,
and let AB and A'B' (Fig. 85) be the spherical arcs, which
are the projections of the bones upon the surface of the
sphere. Let I be the
intersection of the
planes containing
the two bones, and
let A A' be the pro-
jection of any mus-
cular fibre; let IA = ,
a, IA' = d', and let
the angle between -
the planes be I. If
the plane I A'B' be
made to revolve
Fig 85
round the intersection of the twc planes through a small
angle w, so that A and B assume the positions A" and B”;
then let
A'IA" = W,
A A' = x,
AA" = y.

B
If p be the original length of any fibre, and A, A', the
distances of its extremities from the socket; and also, if«p' be
the length of the contracted fibre corresponding with the
rotation through the angle w, then we have
· pa = A + A'2 - 2A A' cos X ;
p'? = A2 + A" - 2A A' cosy;
but
cos x = cos I sin a sin a'+cos a cos a';
cos y=cos (I - w) sin a sin a' + cos a cos a'.
322
ANIMAL MECHANICS,
Hence (since w is a small angle),
pe - p = 2 AA (cos y - cos x)
(p+p) (-p) = 2 A A'w sin I. sin a sin a';
or,
.
pop= AA'w . sin I. sin a sin a'.
Hence we obtain, for the shortening of a single fibre, by a
rotation W,
| AA sin a sin a'
op=w sin 1.1
If, from the ends of the fibre p, we let fall two perpen-
diculars, and T', upon the common intersection of the planes,
it is easy to see that
7 = A sin a,
t' = A' sin a' ;
from which we find
op=w sin I. *
and, summing together all the shortenings of all the fibres, we
obtain
(TT
(dp)=w sin I
(““-
IP
(78)
It is evident that the work done by the muscle will be a
maximum when
ma
(dp) = maximum ;
and this quantity will be a maximum independent of the
sum, (**), (which depends on the special arrangement of
the muscular fibres), when I = 90°.
ANIMAL MECHANICS.
323
Hence we obtain the following preliminary condition :-
CONDITION.
Whenever two bones, joined by muscular fibres, tend to turn
each other round an axis of rotation formed by the intersection
of two planes passing through the bones, the work done by the
contraction of the fibres will be a maximum, when the axis of
rotation is formed by the intersection of planes at right angles to
each other.
Let us now consider the simple case of a single muscular
fibre, of a given length p, the ends of which are required to
remain in two planes at right angles to each other, in con-
formity with the foregoing preliminary condition; it is required
to find the position of the fibre, so that the moment of the
force causing one bone to rotate round the line of intersection
of the planes shall be a maximum.
Let OI (Fig. 86) be the line of intersection of the two
S
o
a

Fig. 86.
rectangular planes, and let P, P', be the ends of the fibre,
whose length, PP = p, is given. Draw Pa, P'd', each in its
own plane, perpendicular to OI; and let
PP' = 1,
Pa= 7,
P'd' = ',
aa' = 4.
Y 2
324
ANIMAL MECHANICS.
Since the planes are at right angles, we have
p2 = 7pi+7? + up,
.
and, by equation (78),
TT TI' = maximum,
Differentiating these two equations, we find
7d7 + 'di' + udu= 0,
TUT' + 'de = 0.
Eliminating di', we obtain
(77% – 72) dm + qudu = 0.
This equation of condition is satisfied by making
= 7T',
u= 0;
md
or, the points a and a' must coincide, and the perpendiculars
77, 77', must be equal to each other. In other words,
The fibre PP' must lie in a plane perpendicular to both the
intersecting planes, and its extremities must be equidistant from
their intersection.
By means of this proposition, we are enabled to solve a
number of problems, suggested by the adductor muscles of
many animals.
If the muscle be triangular, then the resultant of all its
fibres lies in the bisector of the vertical angle, which bisector
is of a given length, and may be regarded as a single fibre,
which is to be placed between the intersecting planes in con-
formity with the foregoing proposition.
If the muscle be a plane quadrilateral, we are then given
two bones, AB and A'B', which lie in two planes intersecting
D
2
ANIMAL MECHANICS.
325
at right angles, their extremities beiny joined by fibres, p,p',
of given length, and we are required so to place the bones,
that the work done by the contraction of the fibres in rotating
one plane upon the other shall be a maxiinuin.

Umuman
TITTATIITTIMET
B'
b
1
Z
ZZZZZZZZZZZATUITAM
ZZZZZ
Fig. 87
Let OI (Fig. 87), as before, be the intersection of the
planes, and let AB and A'B' be the given bones lying in the
two planes, respectively. Produce the plane ABA'B' to meet
OI in the point b; and imagine the fibres AA' and BB' to be
produced to meet, and that X'X is the bisector of the angle
under which they meet. The resultant of all the fibres joining
A B and A'B', lies in the line XX', which may be regarded
as the only fibre with which we have to deal. The line XX'
must, therefore, be placed in a plane perpendicular to the
planes containing AB and A'B', and the perpendiculars
Xa= 7 and X'a=7', must be equal to each other. Now, in
the triangles, Xab and X'ab, we have Xab = X'ab = 90°; Xa =
X'a; and ab common to both. Hence we have X = X'l,
and therefore 6XX'=bXX.
326
MECHANICS.
ANIMAL ANTIN
Let Fig. 88 represent the plane of the quadrilateral
ABA'B', in which YXX' is the bisector of the angle AYB.
B'

MUHIMUNITIMITIMIT
X
B
-
Fig. 88.
Hence, we have
BXX' = BYX+ XBY
B®X®X = A'YX'+ X'A'Y;
but BYX= A'YX'; therefore
XBY= X'A'Y;
or, in the quadrilateral A BA'B', the external angle is equal
to the opposite internal angle; or the sum of two opposite
angles is equal to two right angles; hence the quadrilateral
ABA'B' is inscribable in a circle.
Hence,
If two bones, joined by muscular fibres forming a plane quad-
rilateral, lie in two planes at right angles to each other, the
maximum work done in rotating the planes round their axis of
LED
.
vuu
ANIMAL MECHANICS.
327
intersection, will occur when the bones are so placed as to form a
quadrilateral inscribable in a circle.
This remarkable proposition enables us to determine rea-
dily the truth or falsehood of the Postulate, p. 238, in several
cases. For, if the muscle be really inscribable in a circle,
the rectangle under its diagonals must be equal to the sum of
the rectangles under its opposite sides, by Ptolemy's Theorem;
and the sides and diagonals may be measured on the fresh
subject, by means of a compass, without any serious risk of
any great error.
The Adductor muscles of all animals, as a rule, form skew
sheets of surface, and not plane sheets; but several of the
adductors, in the position of extreme abduction, form plane
muscles in the limiting position, and thus enable us to test
the truth of our Postulate.
I shall number the adductor muscles in the following
order for convenience of reference:-
TYT
Adductor Muscles.
Human Equivalents.
1. Adductor primus. 1. The condyloid portion
of Adductor magnus.
2. Adductor secundus. 2. The remaining portion
of Adductor magnus.
a. Ischiadic sheet.
(a.) (.)
B. Pubic sheet.
3. Adductor tertius. 3. Adductor brevis.
4. Adductor quartus. 4. Adductor longus.
5. Adductor quintus. 5. Pectinæus.
The fourth and fifth of these adductors offer examples of
plane quadrilateral muscles, whose origin and insertion lie in
rectangular planes.
1. Lion. In the Lion the fourth and fifth adductor
muscles (longus and pectinæus) have origins and insertions of
the same length, and lying beside each other on the bones.
328
ANIMAL MECHANICS.
The following measurements were made from a large adult
lion :-
Pectinæus Muscle (Lion).
1. Length of origin, · ·... 1.70 in. = a.
2. Length of insertion, .... 2.95 ,
3. Length of anterior fibres, .4.52 , = c.
4. Length of posterior fibres, : : 7.61 „ =
5. Length of first diagonal, .. 5.77 , = e.
6. Length of second diagonal, . 6.76 , = f.
Now, since a, b are opposite sides, and c, d are also oppo-
site, and e, f are diagonals, we have, by Ptolomy's Theorem,
ab + cd = ef ;
(79)
but
ab = 1.70 x 2.95 = 5.01
cd = 4.52 x 7.61 = 34.39
ab + cd = 39.40
and also,
ef = 5.77 x 6.76 = 39.00.
These results are so close as to prove that, in the Lion, the
adductor quartus and the adductor quintus are quadrilaterals,
inscribable in a circle.
2. Man.-I made the following measurements, on a well
developed male subject :-.
Pectinæus Muscle (Man).
1. Length of origin, . . . . . 1.82
2. Length of insertion, ..:1.52
3. Length of anterior fibres, .. 4.70
4. Length of posterior fibres, · · 6.30 ,
5. Length of first diagonal, . . 5.52 ,
6. Length of second diagonal, . 5.66 ,
ö ö og om
=
ANIMAL MECHANICS.
329
Hence we have
ab = 2.76, od = 22.61
ab + cd = 32.37
ef = 31.24.
This result is sufficient to show that, in Man, the Pectincus
muscle is nearly inscribable in a circle.
3. Llama.- In the Llama, I obtained the following :-
܂
Pectinæus Muscle (Lluma).
1. Length of oriyin, ...
1.91 in. =
2. Length of insertion,.. 3.50 » = b.
3. Length of anterior fibres, . • 3.05 , = c.
4. Length of posterior fibres, . . 6.91 , = d.
5. Length of first diagonal, . . . 4.09 , =
6. Length of second diagonal, . . 6.77 .,
ܢܽܘ
Hence, we find
ab = 6.68, cd = 21.07
ab + cd = 27.75
ef = 27.69.
It appears, to my mind, impossible to evade the conclusion,
that the particular form of muscle employed in the preceding
cases, was the result of design and foresight; for the chances,
a priori, would be many millions to one against the occurrence
of the particular quadrilateral which can be inscribed in a
circle, if the formation of the muscle depended on slow ran-
dom variations of external conditions, only.
I shall now proceed to consider the general case of a skew
muscle, formed by fibres arranged in any manner joining two
bones, AB, and A'B', not lying in the same plane. Let IA,
and I'A' be the directions in which the bones AB and A'B'are
placed; I and I' being the points corresponding to II', which
330
ANIMAL MECHANICS.
is the shortest line between the directions, and perpendicular
to both. Draw I' X parallel to IA, and let
II' = 20
A'I'X = 20.
Take C, the centre of the line II', and draw Cx parallel
to the bisector of XI'A', and draw Cy perpendicular to Cx

B
ONZALEZ
.
KULLILULATIVITATISTIILIT
Fig. 89.
and to II'; and take Cl for the axis of s. Let Cx, Cy, C:
be the axes of co-ordinates, then the equations of the bones
AB and A'B' will be
y – mx = 0 2 – C = 0
y + mx = 0 + c = 0.
(80)
where m = tan p.
Hence, the equations of any two planes passing through
AB and A'B', respectively, will be
(y - ma) +(2 - c) = 0
(y + mx) + X' (2+ c) = 0,
(81)
where , X are arbitrary parameters.
ANIMAL MECHANICSA IV
331
.
I have already proved that the work done in rotating A'B'
round the intersection of these planes will be a maximum,
when the angle between the planes is a right angle. Hence,
in order to obtain the maximum effect, we must make the
two planes (81) intersect at right angles. The angle 0,
between the planes is found by the equation
cos 6 =
1 – m2 +
VAP + mia + I VAN? + m2 +
(82)
when () = 90°, cos 0 = 0, hence we have the condition
XX'= m
– 1.
(83)
If we now eliminate the parameters , X', from the equa-
tions (81) and (83), we obtain the following locus of intersec-
tion of planes at right angles to each other, passing through
the bones AB and A'B',
(g” – mºzº) + (1 - m*)(zº - c°) = 0. (84)
This represents an Hyperboloid of one sheet, having its
Ellipse de Gorge in the plane of y, %; and Il' one of its axes ; :
and the rectilinear generators of this hyperboloid are the
intersections of the rectangular planes passing through the
bones AB and A'B'.
In order, therefore, that the maximum work shall be done
by the muscle AB A'B, it is necessary that the axis of rota-
tion shall coincide with some one or other of the generators
of the hyperboloid (84); but this is not sufficient, for my
Postulate (p. 238) requires, in addition, that of all the
generators of the hyperboloid, that particular generator shall
be chosen as axis of rotation, which shall give us the maximum
maximorum of work done. We must, therefore, in the first
place, discover which, of all the generators, possesses this
property, and then, by measurements on the dead subject,
332
ANIMAL MECHANICS.
test whether or not the actual axis of rotation in the limb of
the animal coincides with the mix, marimorum axis. .
Let PP' = p, be any fibre joining two points, (x, y, z) and
(x', y', z) on the bones AB and A'B'. It has been already
proved that the moment of this fibre, to turn A'B' round the
intersection of the planes passing through the two bones, is
moment =
where f, f', denote the lengths of the perpendiculars let fall
from each point upon the plane passing through the other
point. If we express f, f', in terms of the coordinates, we
obtain
fif"
4
(mx + c\') (mx' + ca)
V1? + m + I V12 +212 + I
P
ff"
P
4
P
mc (x) + x'1') + K
712 + m² + I V3.2 + m + 1
where
K = murix' + cm (m? – I).
(87)
Differentiating (86) with respect to , and X', and remember.
ing the condition
1dx' + 'di = 0.
we obtain the following equation of condition to determine
the position of the generator which gives us the maximum or
minimum amount of work,
cm (12 + m2 + 1) (1? + m2 + 1) (x) – a'2') +(88)
- (m? + 1) (mc (2) + X'X') + k} (- ) = 0
This is the condition required and expressed for a single
fibre. The equation of condition for the entire muscle, com-
ANIMAL I
333
11
.
NECHANICSposed of any number of fibres, anyhow placed, may be thus
deduced.
If we write
8-) x=2(%)
K = m** (*) +®® (m2 – 1) 2(1)
(89)
then we find
Work =
mc (X1 + X'X) + K
*V12 + m2 + I VX2 + m² + I
(90)
and the equation of condition becomes .
(91)
cm (1? + m2 + I) (^2 + m² + I) X) - XX?
= (m? + 1) {mc ( X1 + X'X') + K} (12 - 12).
This equation admits of a solution on inspection, viz.,
dell'
X = X',
which implies that if the fibres were so arranged as to make
X = X', then the generator X = X', which makes equal angles
in space with the two bones AB and A'B', is the required
maximum. I have not, as yet, found any skew muscle whose
fibres give X = X', and therefore we must set aside the most
obvious particular solution of our equation of condition, and
seek its general solution. If we eliminate Xby means of the
relation
W = m? - I,
S
and expand (91), arranging by powers of ), we find, after a
334
ANIMAL MECHANICS.
few reductions, the following bi-quadratic equation of condition
to determine the position of the axis of maximum work-
- K (m2 + I) 14
+ 20 {m X (m+ + I) – mX':(m+ - 1)} 13
+ 20 (m2 – 1) {mX (mt – 1) - mX' (m* + 1)} a
+ (m? – 1) (m" - 1) K = 0.
(92)
)}1
I am now required, in order to justify my Postulate, wher.
applied to the complex case of skew muscles, to find the root
of equation (92) for various muscles, and to show that the
actual axis of rotation corresponds with that root. I select
for examination the adductor secundus of various animals.
This muscle, in all animals that I have dissected, forms a
double sheet folded at the centre on itself, in this manner :
the posterior fibres proceeding from the ischium are inserted
the highest upon the femur, and gradually increase in length
until we reach the most dependent point of the pubo-ischium ;
this portion of the muscle I shall call adductor secundus (a);
in like, but reversed order, the anterior fibres proceeding
from the pubes are inserted highest up on the femur, and
gradually increase in length until they join the long fibres of(a),
with which they coincide; this portion of the muscle I shall
call adductor secundus (). Thus, the two sheets of muscle,
(a) and (), are placed side by side, as if folded against each
other on an axis corresponding to the longest and middle
fibres of the muscle; and, although it is evident that the
two sheets generally may act together, yet, mechanically
considered, they are to be regarded as quite distinct
muscles.
(1) Adductor secundus (B)--Llama.--Having dissected
down upon the adductor muscles of a fine male Llama, after
previously ascertaining the limits of abduction of the femur,
before removing the skin and gracilis muscle; and having
ANIMAL MECHANICS.
335
carefully placed the thigh in the position of extreme abduction,
I made the following measurements :
Llama.
Lengths of equidistant fibres.
P = 4.27 in.
Pi = 6.70 ,
P2 = 8.46 ,
pz = 10.80 ,
Lengths of origin and insertion.
a = 1.96 in. (origin).
a' = 6.00 ,, insertion.
Constants of Hyperboloid.
20 = 2.38 in.
20 = 49°15'
2
The constants of the Hyperboloid were measured by
fastening straight steel wires along the directions of origin and
insertion, and then measuring the shortest distance between
them, and the angle made by the two lines.
The positions of the points 1 and I', on the bones, may
either be measured directly, like 2c and 20, or may be calcu-
lated from the other measurements. In order to check the
whole calculation, I measured their position, and found
a = + 1.96 in.
a' = + 4.40 ,, ;
where a signifies the distance from I to the (anterior) fibre p,
measured on the bone AB; and a signifies the distance from
I'to the same (posterior) fibre p, measured on the bone A'B'.
It is to be noted that the fibre p is anterior on the pubes, and
posterior on the femur; so that all the fibres cross each other,
forming a well marked skew surface. The values of a and a'
may be calculated as follows, from the measured lengths of
any two fibres (p, p:)
p2 = a + a'2 – 2aa' cos 20 + 4c?,
ps2 = (a - a) + (a' + a')? – 2(a – a) (á + a') cos 20 + 4c? ;
336
ANIMAL MECHANICS.
from which we obtain-
(93)
pga – pa = -2(a + a' cos 20 a
+2{á + a cos 20} a'
+ {a? + a"? + 2aa' cos 20}
Introducing the numerical values, I find
14.550' = 11.72a + 43.28.
obtain from each pair a similar relation between a and a', in
all of which it is easy to see that the co-efficients of a and a
the several measurements. I shall write down the results of
the six possible combinations :-
Absolute Term.
(p - P3) . . . . . . 43.28
'(p - p2) .43.23
(p - pu) . . . . . . 61.65
(P1 - P3) ·.·.·· 33.14
(p1 - P2) .....25.29
(P2 - Ps) · · · · · · 43.63
Vean, ... 4170
The value found from the fibres (P, P3) is nearly the mean
value, which is what we should expect; for the errors of
when the differences of the fibres and the lengths of the
intercepts between the points of application are greatest.
Taking the mean value, we have
14.55 a' = 11.72 a + 41.70,
a' = 0.805 a + 2.866.
(94)
ANIMAL MECHANICS.
337
Introducing this value into the equation,
pa = a + a'– 2aa' cos 20 + 40%,
we obtain
0.602 a² + 0.888a? – 4.356 = 0; (95)
and, finally, remembering (94), we reach the following pairs
of real values for a and a':-
S+ 205 in. (observed = 1.96 in.)
1-3.53 ,
{+ 4.51 in. (observed = 4.40 in.)
1 -0.025 ,
From the preceding agreement of calculation and observation,
we may regard the true positions* of I and I' as having been
accurately ascertained.
We are now in a condition to determine the quantities,
X, X', K, which occur in the coefficients of the equation of
condition (92); for
U
mX - sin $ 2 ) mx'= sin q = 0)
(96)
aa'
K = sin? 0
(44) + c2 (m? – I
Using the measurements made in the freshly-dead muscle,
we obtain the following:-
+
6.51 »
Llama, Adductor secundus (B).
a = + 2.05 in.
= + 4.51 in.
aj = + 1.40 ,
a2 = + 0.75 »
a'= + 8.51 ,
az = + 0.09 »
a's = + 10.51 »
* The second values of a and a', refer to the position of the bones, in which
they make together an angle of 20 reversed; and they have no relation to the
present problem.
338
.
ANIMAL MECHANICS.
s!
= + 0.480
= + 1.056
dia 31
51
= + 0.209
= + 0.972
42 = + 0.088
= + 1.006
SIM 1 2
43 = + 0.009
= + 0.973
-
= + 0.786
= + 4.007
aa'
= + 2.165
0.234
Telefone e + 1.360
aia's
? = + 1.360
-0.149
0.149
Pi
020
+
= 0. I 18
aza 3
P3
= + 0.088
ava' s = + 0.058
( ) = + 4.367
-0.092
(1) - 0.593
aa')
a = + 4.367
p
From these enumerations, we find, from equations (06),
m X = + 0.330
m X' = + 1.683
K = + 0.096.
Introducing these values into equation (92), we obtain
- 0.116 /4 + 3.375 13 + 4.903 1 + 0.042 = 0. (07)
ANIMAL MECHANICS.
339
It is evident, from inspection, that equation (97) has a
very small root and a very large root, both real ; because the
coefficients of the first and last terms are small, as compared
with the intermediate coefficients ; and the smallness of these
coefficients depends on the small value of K, which is a factor
of the first and last coefficients.
These small and large roots of equation (92) would
become
1 = 0,
(98)
Tabs
in one or other of two cases, viz. :-
m = 1,
K = 0.*
(99)
Having determined the value of 1 from equation (97), we
must introduce this value into equation (92), in order to
calculate the actual value of the maximum work or moment
of the muscle. This equation may be thus written, by
elimination of X:-
cmX1? + K + cm X' (ma – I)
112 + m2 + IV (ma + I) 1? + (m2 – 1)2
Since the denominator of this fraction does not change sign
with the change of sign in 1, it is plain that the numerator
equated to zero determines the position of the two neutral
generators of the hyperboloid, with respect to which the skew
muscle does no work, and that these generators divide all the
generators of the hyperboloid into two groups, for one of
* It may be interesting to note that the geometrical meaning of K = o is,
that the line joining X and X' is a tangent to the hyperboloid conjugate to the
locus hyperboloid (84).
Z 2
340
ANIMAL MECHANICS.
which the work done is negative and for the other positive.
The equation to determine the neutral axes is
cm X . \ + K
+ cm X' (m? – 1) = 0.
(101)
- This equation becomes, in the case of the Llama,
0.393 12 + 0.096 X - 1.58 = 0.
The roots of this equation are
5 +1.89
1 - 2.13
If we now solve the equation (07), we find two real
roots, viz. :-
= - 0.0085,
1 = + 29.3.
The remaining roots of the equation are imaginary.
The small value of corresponds to the maximum negative
work done by the muscle, as may be seen from the following
Table, formed by introducing successive values of 1 into
equation (100) :-
Llama-Adductor secundus (B).

+ 0.1000
0.0000
-0.0085
- 0.1000
- 5.6568
-7.2828
- 7.2848
-7.2088
The large value of a corresponds with the maximum
positive work of the skew muscle.
ANIMAL MECHANICS.
341
Hence we see, by a process of reasoning similar to that
employed in reference to the great pectoral muscle of the
wing of the bird, that the axis of maximum instability corre-
sponds to 1 = -0.0085, and that the axis of maximum sta-
bility corresponds to 1 = + 29.3.
rre-
K.

Fig. 9o.
The value of corresponding to the axis of maximum
instability is so nearly zero, that we may assume i = 0, without
sensible error. This value of a corresponds with X' = -, 50 ;
that the equations (81) of the planes passing through the
bones are reduced to
y – Mx = 0,
+ C = 0.
(102)
The corresponding axis of rotation is shown in Fig. 90.
342
ANIMAL MECHANICS.
Let IA and I'A' represent the directions of the bones
AB and A'B', and let II be the shortest line joining these
directions. Through the point I draw I'X parallel to IA ;
then the axis IX represents equations (102).
It follows from this construction, that if the Postulate
(p. 238) be true, the socket S' of the hip joint must lie some-
where on I'X, the axis of maximum instability
The following measurements were made on the pelvis and
femur of the Llama. Draw from the centre of the acetabulum,
S, two perpendiculars, SK and SK", upon the line of pubic
origin and femoral insertion of the adductor secundus (B); the
lengths of these perpendiculars and of the line SI were then
measured, as follows:-
SI = p = 3.21 in.
SK = p = 2.77 ,
SK' = p' = 1.02 »
If it be true that the point S lies on the axis I'X (since
II = 20, and A'T'X = 20), we must have, within the limits of
errors of observation,
a
On
p = 2C,
5 = sin 20.
(103)
Vp? - p2
Introducing the measured values, we obtain
P = 2.77 in.
20 = 50° 58'.
These values differ from the constants of the hyperboloid-
* 2c by 0.39 of an inch,
20 , 1° 43'.
These differences are within the limits of errors of observation,
and we may consider the Postulate to be proved true, when
applied to the skew adductor of the Llama.
ANIMAL MECHANICS.
343
(2) Adductor secundus (B)-Greyhound. The next ex-
. ample of a skew muscle that I shall give is the adductor
secundus (B) in the Greyhound. The subject of my dissection
was the celebrated Irish Greyhound, “ Master Magrath,” who
is justly considered as the best and fastest dog that ever ran
a course. The following brief notice of this wonderful dog,
taken from “ The Belfast Newsletter," may be of interest to
the reader:-
“Master Magrath' (named after an orphan boy who reared him) was
by · Dervock,' out of Lady Sarah,' and made his first appearance at the
Lurgan meeting in October, 1867, when he was about one year and eight
months old. At that meeting, although little more than a week in train-
ing, he won the “Visitors' Cup' (32 dogs), beating, amongst others, ‘S.S.'
and · What's the Tip.' His second appearance was at the Creagh meeting
the week following Lurgan, when he divided the Moneyglass Purse'
(32 dogs) with his kennel companion, "Master Nathaniel.' His next
appearance was at Altcar, in February, 1868, when he won the Waterloo
Cup.' His first course was with 'Belle of Scotland,' whom he outpaced
both in the no-go and final spin, beating her very cleverly, indeed. His
second trial with · Kalista' might have ended unfortunately for him, as he
got an ugly fall in the run up; but his great cleverness saved him, as he
was on his legs again, and had puss in his jaws, before Mr. Kay's bitch had
a chance of putting him out. His third course with 'Marionette' was such
a hollow, as well as clever, victory, that there was immediately a rush to
get out by those who had taken a liberty with the nomination. His
coming against 'Brigade' in the next round was quite the event of the
meeting, and, although the trial was not A1, the speed and cleverness
exhibited by Lord Lurgan's dog justly entitled him to the award. His
next course with Lobelia' he made short by his extreme cleverness; and
when, finally, he met 'Cock Robin,' it was, bar accidents, ' Lombard-street
to a China orange,' and 4 and 5 to i was offered in vain. The deciding
course was run undergreat difficulties, and, had not. Master Magrath'
been one of the cleverest greyhounds that ever went to slips, he, in all
probability, as the course was run, would have been put out. At the
Lurgan meeting of 1868 he divided the · Brownlow Cup Stake' with Mr.
Stooker's Sir William,' again beating .S.S. and three others. At the
Waterloo meeting of 1869, he met and beat 'Borealis,' Hard Lines
344
ANIMAL MECHANICS.
* Charming May, Randolph, Lobelia,' and Bab-at-the-Bowster;' the
style in which he beat the last two gaining for him, from good judges, the
opinion that a faster or cleverer greyhound never was slipped. In 1870
he sustained his only defeat, which was from Mr. Trevor's 'Lady Lyon,' the
course creating quite a consternation in all sporting circles ; but, whatever
the cause, it was not Magrath’s' style of running. At the Lurgan meet-
ing of 1870, he won the · Brownlow Cup,' beating, in the last two courses,
* Smuggler' and 'Fritz.' His last appearance in public was at Altcar, in
February, 1871, when he defeated • Wharfinger," "Eyes of Fire' (drawn
after an undecided course), “Rocketer,' "Letter T,' • Black Knight,' and
• Pretender.' It will thus be observed that out of the thirty-seven courses
which Master Magrath' ran in public, his colours were only lowered
once."
“ Master Magrath."
VY
Lengths of Equidistant
Fibres.
p = 4.81 in.
po = 6.11 ,
P2 = 8.20
Lengths of Origin and
. Insertion.
a = 1.20 in.
a' = 3.16 ,
Constants of Hyperboloid.
20 = 1.68 irs,
20 = 69ºoo'.
The positions of the points I and I', on the pelvis and
femur were carefully ascertained by the same method as that
used in the case of the Llama, and the following measure-
ments and calculations made: -
- Master Magrath” --Adductor secundus (B).
a = 1.20 in. a' = 4.14 in.
aj = 0.60 ,
a's = 5.72 ,
a2 = 0.00 22
a'. = 7.30 ,
ANIMAL MECHANICS.
345
Big
= 0.249
&
= 0.100
ola ble size is
* = 0.860
*= 0.936
* = 0.890
2C) = 2.686
$
= 0.000
0.000
(9) -0,349
I
---
aa' – 1.031
= = 0.208
= 0.561
- = 0.163
Pi
aya's = 0.567
o n = 0.000
2 (0) = 1.592
aza'z
-= 0.000
P2
-0.163
= 0.122
%(= 0.493
= 0.122
From these data we find
m X = 0.197
mX' = 1.520
K = 0.326;
and finally, equation (92) becomes
- 0.479 14 + 2.395 13 + 2.128 +0.135 = 0.
This equation, as before, has a small real root, viz.,
X = -0.063,
which leads to the construction (Fig. 90).
346
ANIMAL MECHANICS.
On measuring the lines, I found
SK = p = 1,65 in.
SK' = p = 0.80 ,
SI = p = 1.86 ,
From which we find
p = 1.65 in.
sin 20 = 1
Vpi-p2
20 = 68° 28'.
These values differ from the constants of the hyperboloid-
2c, by 0.03 of an inch;
20, by o° 28'.
(3.) Adductor secundus (3) Woman.--The next example
of skew muscles that I shall give is the case of a woman,
on whom the following measurements were made :-
Woman-- Adductor secundus (B).
Lengths of Origin and
Lengths of Fibres.
Insertion. :
p = 5.67 in. a = 2.55 in.
Pa = 8.00,
a' = 7.27 ,
P2 = 11.57 ,
Constants of Hyperboloid.
20 = 3.60
20 = 70°.
The positions of the points I and I' were determined, as
in the case of the Llama, by taking the fibres in pairs, and
ANIMAL LUL
347
MECHANICS.
using the mean of all three. In this way I obtained (93) the
following equation:-
14.54 a' = 5.10 a + 42.38
a' = 0.35 a + 2.91.
Substituting this value of a' in the equation,
pa = a² + a'2 – 2aa' cos 20 + 4c",
we find
1.12 a² + 2.03 a – 10.71 = 0;
and finally,
a = 1 + 2.32
a = 1-4.13
d=$+3.72
1+ 1.47
Taking the first pair of values, which correspond to the
problem in hand, we find-
Woman-Adductor secundus (B).
= + 2.32 in.
+ 1.04 »
= -0.24
a' = 3.72 in.
a's = 7.36 ,
a'. = 11.00 ,
- = + 0.409
0.656
+
+
&515 Si
+0.130
+0.920
- 0,02 I
= +0.951
2 (9) + 0.518
() = + 2.527
I
am
+
+
2.
348
US
ANIMAL MECHANICS.
.
aa
= + 1.522
= 0,176
.
:
+
O
=
0.
Pi
· ad = + 1.522
anda = +0957
agos - 0.228
(*) - +2.251
= 0.176
= 0.125
-0.086
>() - 0.387
aza'2
= - 0.228
- = 0.086
M
From these data we find
K = +0 103
m X = + 0.297
mX' = + 1.448
and, finally, the equation of condition (92) becomes
- 0.153.14 + 5.288 13 + 3.710 1 + 0.040 = 0.
The root of this equation, which gives the axis of maximum
work, is
X = - 0.011.
(4.) Adductor secundus (B)-- African Leopard.—The fol-
lowing measurements of the skew adductor muscle were made
upon a fine male Leopard, from the west coast of Africa :-
UU
African Leopard.—Adductor secundus (B).
Lengths of Origin and
Lengths of Fibres.
Insertion.
p = 2.42 in. a = 2.76 in.
P1 = 3.27 ,
a' = 5.40 ,
P2 = 6.42 99
ANIMAL MECHANICS.
349
Constants of Hyperboloid.
20 = 2.05
20 = 60° 30'
The points I and I' were found by combining the fibres in
pairs, the mean result giving the following relation between
ů and a':-
13.56 a' = 10.90 a – 16.31
a' = 0.804 - 1.203.
From which I obtained, as before,
0.84 a? – 0.73 a - 0.20 = 0
J-0.34
1- 1.38
+
Taking the first pair of values, as before, I find—
1
o
African Leopard—Adductor secundus (B).
a = + 1.09 in.
ai = – 0.29 »
a't = + 2.36 ,
az = - 1.67 »
a'z = + 5.06 ,
34 in.
81
"= + 0.450
=-0.145
ö .
Elő
= - 0.090
= + 0.722
+
o
42 = -0.260
2 = + 0.788
| P2
-
>()- +0.00
) = + 1.363
350
ANIMAL MECHANICS.
av = -0.153
-
-0.413
dia'i
Pi
= -0.209
= 0.306
S
(') --- 1.678
°(E)=0,874
From these data we find,
K = - 1.024
mX = + 0.050
m X' = + 0.681 ;
and, finally, the equation of condition (92) becomes
+1.372 14 + 1.343 13 + 1.091 ) - 0.077 = 0.
.
The root of this equation, corresponding to the axis of maxi-
mum work, is
= + 0.07.
In the preceding cases, and in others which I have cal-
culated, too numerous to quote, we find that the generator of
the hyperboloid, which constitutes the axis of maximum work,
corresponds very closely with the value,
1 = 0, '
the geometrical construction of which is shown in Fig. 90.
It is not difficult to show that, in this case, the entire skew
muscle may be mechanically replaced by a single fibre, sup-
posed to be endowed with the united force of all the fibres ;
ANIMAL MECHANICS.
351
and by constructing this fibre, we may obtain a clear and
direct view of the action of the skew muscle, and, as I believe,
discover the reason for the invention of such muscles, and, at
the same time, test directly and satisfactorily the truth or
falsehood of the Postulate (p. 238),
Making 1 = 0, in equation (100), which gives the work
done by the skew muscle for any generator, we find for the
work done by the generator I'X (Fig. 90), the following
expression :-
(%) - omX", = 2e sin 24 2(); (104)
O!
V m
+ 1
because mX' = sin 0 36)
ause
Let us now assume
(105)
where n denotes the number of fibres, P the length of the
equivalent fibre, and L' its distance from the point I'. Hence
we find
20 x L' sin 20
=
n
-
(106)
which represents the work done by a single fibre, n times as
strong as any single fibre, attached to the bone A'B' at the
point 0', corresponding to L', and having a length, P.
The only assumption in the foregoing is (105), that
=)
which means that the ratio of L'to P is given. Hence, there
are an infinite number of single fibres, any one of which
352
ANIMAL MECHANICS.
would be the mechanical equivalent of the skew muscle, pro.
vided it be so placed between the bones as to make
7 - 30%)
I am, therefore, at liberty to make a second assumption, for
the purpose of fixing the other end of the fibre, and so render
the Problem of drawing an equivalent fibre definite.
I choose a point 0, on the bone AB, determined on the
same principle as the point Oʻ, viz. :-
Lila
Pn
(107)
and if from this point I draw a fibre to A'B', satisfying the
condition (105), the fibre joining O and O', and having the
united force of all the fibres, will be mechanically equivalent
to the entire skew muscle.
But the single equivalent fibre 00', so found, will not
satisfy the Postulate (p. 238), unless its extremities lie in a
plane perpendicular to the axis of rotation, and at equal dis-
tances from that axis (vide p. 324); from which it readily
follows that, if the Postulate be true, the points 0 and 0 must
lie at equal distances from the socket of the hip joint.
Let us proceed to inquire whether this be so or not.
Determination of the single Fibre equivalent to a Skew
Muscle.
1. Llama-Adductor secundus (B).-In the case of the
Llama, we have, for four fibres,
>() = +0.786
=
+
20) – +4.007
=
T
ANIMAL MECHANICS.
353
Hence, equations (105) and (107) give
L'
= + 0.196,
= + 1.002;
but
P2 = L? + L'– 2LL' cos 20 + 4c?,
from which we find, by eliminating L and L', and using the
constants of the hyperboloid,
20 = 49° 15',
20 = 2.38,
P2 – 5.6644,
0.213
P = 5.157 in.,
and
L = + 1.010 in.,
L' = + 5.167 in.
On laying down the points 0 and O', just found, upon the
pelvis and femur, and measuring their distances from the
centre of the socket of the hip joint, I found-
Distance from 0 to centre of socket = 2.88 in.
Distance from O'to centre of socket = 3.14 ,,
Difference, ..
0.26 in.
The points 0 and O'both lie within the origin and inser-
tion ; but it is to be noticed that the point O' is brought down
to the near end of the insertion on the femur, within 0.56 in.
of its extremity. We may also remark that the equivalent
fibre, 00', does not coincide in direction with any of the real
fibres of the muscle, none of which intersect each other in
space.
S
2 A
354
ANIMAL MECHANICS.
2. “ Master Magrath” --Adductor secundus (B).- In the
case of the celebrated Irish Greyhound we have
20 = 1.68 in., 20 = 60° 00',
= + 0.349,
= + 2.686,
corresponding to three fibres : from these data we find, as
before,
= 0.116, :
= 0.895,
2.8224
L = 0.38 in.,
0.26
Lin.
P = 3.29 in., L' = 2.95 in.
On laying down these points upon the pelvis and femur, I
found-
Distance of O from centre of socket = 1.95 in.
Distance of Ofrom centre of socket = 2.28 ,
Difference,
,
.
0.23 in.
Here the point O lies inside the origin of the muscle, but the
point o'lies completely outside the insertion of the muscle,
at a distance of 1.19 in. nearer to the hip joint than the upper
extremity of the line of insertion ;-and the equivalent fibre
00' is shorter than any real fibre of the skew muscle.
3. Woman-Adductor secundus (B).-In this case we
have
20 = 3.60 in., 20 = 70°.00',
2(%) - 0.518,
26%) = +2.527,
corresponding to three fibres : from these data, we find, as
before,
ANIMAL MECHANICS.
355
L'
= +0.173,
P
= +0.842,
P2 - 12.96
P= 5.99 in.,
* 0.361'
L = +1.04 in., L' =+5.04 in.
On laying down these points on the pelvis and femur, I
found-
Distance from 0 to centre of socket = 3.84 in.
Distance from Oʻ to centre of socket = 3.64 »
Difference,
.
.
.
0.20 in.
The points 0 and O' are both inside the origin and inser-
tion.
4. African Leopard-Adductor secundus (B).-In this case
we have,
20 = 2.05 in.,
20 = 60° 30',
2 (9) - +0.100,
=
+
0.1
2C)- + 1.363
corresponding to three fibres : from these data we find, as
before,
5 = +0.033,
+0.454
+
P2 = 44025
P = 0.808
P = 2.28 in.,
L = + 0.07 in., L' = + 1.03 in.
On marking off the points O and O' on the bones, I found
.
2 A 2
356
ANIMAL MECHANICS.
Distance of O from centre of socket = 1.82 in.
Distance of Ofrom centre of socket = 2.32 ,
Difference, ... 0.50 in.
The points 0 and O' are both inside the origin and insertion,
but the point O' is only 0.41 in. inside the posterior end of
the insertion, showing, as in all the former cases, that the
equivalent fibre is inserted very high up on the femur.
5. African Leopard--Adductor secundus (a).— I shall se-
lect, as my last example of skew muscles, the ischiadic sheet
of the second adductor of the African Leopard. In this case,
the shortest fibres are the most posterior fibres on the ischium,
and are inserted highest up on the femur: this difference of
arrangement will require a change of sign in some of the
quantities used in my calculation, which will be readily un-
derstood from the formulæ employed
African Leopard-Adductor secundus (a).
Lengths of Origin and
Lengths of Fibres.
Insertion.
P = 3.55 in.
a = 1.66 in.
P1 = 4.24 ,
a' = 5.40 ,
P2 = 6.42 ,
Constants of Hyperboloid.
20 = 2.26 in.
20 = 74° 40'.
The positions of the points I and I were found by combining
the lengths of the fibres, in pairs, as in the following
equation :-
po2 – p2 = 21a - a' cos 20} a
+2{a' - a cos 20} a'
+ {ax + a'2 - 2aa' cos 20).
ANIMAL MECHANICS..
357
The resulting relation between a and a', which is the mean of
· the three different combinations, is
.
3.04 a + 9.92 a' + 2.43 = 0.
From this relation we find, as before,
1.25 a* -- 0.02 a - 7.43 = 0,
(+ 2.44
Il-0.99
a' =
1 - 2.44
1 +0.50
Adopting the second pair of values, which correspond
with the question in hand, we find :-
African Leopard- Adductor secundus (a).
a = - 2.44 in.
a, = - 1.61 ,
= – 0.78 ,
a' = +0.50 in.
ai' = + 3.20 ,
az' = + 5.90 ,
a=-0.687
= + 0.141
4= -0.380
= + 0.755
= - O. I 22
m =- 0.122
>C)-- 1.189
= +0.919
26 ) - + 1.815
=
Hence, as there are three fibres employed, we find
=-0.396,
= +0.605,
P2 – 5.1076
P = 3.82 in.,
L = - 1.51 in.,
L' = + 2.31 in.
0.35
358
ANIMAL MECHANICS.
On marking off the points 0 and O' on the pelvis and
femur, I found
· Distance of O from centre of socket = 2.63 in.
Distance of O from centre of socket = 2.58 ,,
Difference, ..
0.05 in.
UL
The preceding facts demonstrate in a remarkable manner
the truth of the Postulate, p. 238, as applied to the case of
skew muscles. The postulate requires the distances of the
points 0 and 0 from any point situated on the common in-
tersection of the two planes containing the bones AB and
A'B' to be equal; and, from the conditions of the joint, the
centre of the socket is supposed to lie on that intersection.
Bringing together all the results, we find
Proof of the Postulate for Skew Muscles.

Muscle.
Distance
from 0 to
Socket.
Distance
from O't®
Socket.
Difference.
Angle.
2.88 in.
0.26 in.
WEN
Adductor secundus (B).
1. Llama, . .
2. “ Master Magrath," .
3. Woman, . . . .
4. Leopard, ....
Adductor secundus (a),
5. Leopard, . . . .
0.23 »
3.14 in.
2.28
64 99
470 281
79 28
ہ تب یہ
0.20 ,
1.8279
32 99
0.50 ,
43 29
51 53
2.63 »
2.58 , 1 0.05 , 1 44 27
NA
Mean, . | 47° 21'
The last column of this Table shows the angle made by
the fibre 00' with the plane containing the pelvic origin AB.
This angle, according to the Postulate, ought to be 45°; its
mean value is 47° 21', which is a very remarkable approxima-
tion in observations of this kind.
ANIMAL MECHANICS.
359
Let P denote the length of the equivalent fibre joining
Cand situated in the two rectangular planes. In its
position of maximum effect, the perpendicular let fall upon
the line 00' from the intersection of the planes will be a
maximum, and will measure the work done; wethus have
Maximum Work = ?
For any other position of the fibre, making an angle 0
with one of the planes, it is easy to show that the perpen-
dicular, or
Work done = P sin 6 cos 0 = sin 20.
Hence we have
Work done
7 = sin 20.
Maximum Work
=
S
(108)
We can thus calculate a Table showing the percentage of
the maximum theoretical work possible, which is actually
done by the observed muscle :-
Lulu
Ratio of Obserred Work done by Skew Muscles to the Maximum
Possible Work.

Muscle.
Work done.
Adductor secundus (B).
1. Llama, .
99.63 per cent.
2. “Master. Magrath,” .. 98.78
3. Woman, . . . . .
99.86
4. Leopard, . . . . . 97.13 19
Adductor secundus (a).
5. Leopard, . . . . .1 99.98
Mean, ...
99.076 per cent.
360
ANIMAL MECHANICS.
The results obtained from the calculation of the equivalent
fibre 00', furnish us with more than a proof of the Postulate;
the invention and use of skew muscles. In order to illustrate
my meaning, I shall place, in the form of a Table, the dis-
tances of the points of insertion on the femur, of the longest
and shortest fibres of the muscle, together with the distances
of the point of insertion of the imagined) equivalent
fibre 00.
Position of Equivalent Fibre.

Distance from Distance from | Distance from
Socket to upper Socket to lower Socket to
end of inser- end of inser insertion
tion.
tion.
of 00'.
Adductor secundus (B).
1. Llama, . . .
2. “Master Magrath,” .
3. Woman, . . .
4. Leopard, ... .
Adductor secundus (a).
5. Leopard, ...
2.52 in.
3.10 ,
3.32 97
1.47 »
8.53 in.
6.16 ,
13.10 ,
بہ تب ہ
1.47 ,
6.70 10
2.58,
.
This Table shows clearly the tendency of the skew form
of muscle to bring the resultant line of force higher up on the
femur, and nearer to the hip joint; but it may be still better
seen from the next Table, which gives the actual transfer on
the femur from the point of bisection of the line of insertion
along the linea aspera,
ANIMAL MECHANICS.
361
Transfer of Force caused by Skew Muscle.

Muscle.
Distance from
O, to centre of
insertion,
Length
of
Femur.
Proportionate
Transfer.
2. 34 in.
12.81 in.
Adductor secundus (B).
1. Llama,.
2. “Master Magrath,"
3. Woman,
4. Leopard, : : : :
2.77
2.3259
18.3 per cent.
32.3 99
14. I
15.1 99
1.33 ,
၀ဝ ဝ ။
Adductor secundus (a).
5. Leopard, . . .
0.89 ,
8.80 ,
IO. I
10. Theory of the Hip and Shoulder Joints-Hip Joint.--
We are now in a condition to attempt the solution of a most
interesting problem, which may be thus stated :-Being given
all the muscles that surround the hip joint, let it be proposed
to ascertain the position of the socket of the joint which will
enable all the muscles to perform the maximum amount of
work. The following classification of the hip joint muscles
may be employed :-
Muscles of Abduction.
1. Agitator caudæ.
2. Glutæus primus, ...(Gl. maximus.)
3. Tensor vaginæ femoris.
4. Glutæus secundus (a), · · (Pyriformis.)
5. Glutæus secundus (B), .. (Gl. medius.)
6. Glutæus tertius, .... (Gl, minimus.)
7. Glutæus quartus.
8. Glutæus quintus.
g. Obturator internus.
362
II11
ANIMAL NI
MECHANICS,
Muscles of Flexion.
1. Rectus femoris.
2. Sartorius.
3. Iliacus.
4. Psoas.
Muscles of Extension,
1. Biceps femoris.
2. Semitendinosus.
3. Semimembranosus
4. Gracilis.
5. Quadratus femoris.
Muscles of Adduction.
I. Adductor primus, ...)
2. Adductor secundus (a), . {(Ad. magnus.)
3. Adductor secundus (ß!, . )
4. Adductor tertius, ... (Ad. brevis.)
5. Adductor quartus, :. (Ad. longus.)
6. Adductor quintus, . : (Pectinæus.)
7. Obturator externus.
These muscles are not all present together in any animal,
but the greater number of them are to be found in all ani-
mals, and their constancy of occurrence and similarity of ac-
tion have justly attracted the attention of anatomists, and
several theories have been proposed to explain their simila-
rity, among which the most remarkable is the theory which
accounts for this similarity by the hypothesis of the descent
of the several animals from a supposed common ancestor.
I shall choose the Lion as the animal, in the case of which
I shall attempt to find the position of the socket of the hip
ANIMAL MECHANICS.
363
joint which shall produce the maximum effect of all the
muscles.
Muscles used in Abduction. The whole group of muscles
(1 to 8) whose action is abduction of the femur, either
wholly or in part, may be regarded as examples of muscles
of the same class as the great pectoral in the bird's wing;
that is, in the position of maximum extension of the fibres,
the origin and insertion may be regarded as lying in the
same plane, which plane contains also the centre of the soc-
ket of the hip joint. Hence the position of this centre, for
maximum work, must lie somewhere on the axis of maximum
instability
1. Agitator caudo.—This muscle takes its origin, for a
length of 3.14 in. from the ist, 2nd, and part of the 3rd cau-
dal vertebræ, and is inserted by means of a long tendon into
the outer side of the patella. Its form is shown in Fig. 91,
in which AB is the
origin and A'B' the
insertion. S is the ac-
tual socket of the hip
joint, and ST the axis
of rotation. The mus-
cle is so nearly trian-
gular in shape that we
may assume as a close į
approximation (pp.
271, 289), that the
axis LM, which bi-
sects the intercept of
OX between the bones,
is the axis of maxi-
mum work. On mea-
suring the space be-
tween LM and ST, I
Fig. 91.
found

A
A
B
364
ANIMAL MECHANICS.
Distance between calculated and observed axes = 0.23 in.
n11
2. Gluteus primus.—This muscle takes its origin from the
sacral vertebræ, and is inserted into the outer edge of the fe-
mur, below the great trochanter, for a length of 2.90 inches.
Its form is shown in Fig. 92, and its fibres are parallel to each
other.
The coefficients L, M, N, p. 292, are-
SA
to

t&
.
W
20minuta
.
NUTUOLIT
SYULUIX
IL
LUTATOTTUNUL
OTT
+0
1 +11) cos 009
M
=
to
.
N
(1 + l') cos2 0d0
I-V
C
In the case of pa-. L"
rallel fibres, these be-
come,
L=3(1-)
HTUM
TIMTIAZANIE
N=3 (474)
Fig. 92.
And, since the point O has moved off to infinity, 1+ l' may
be regarded as equal to twice the distance of the point of
ANIMAL MECHANICS.
365
bisection of the middle fibre from the point O: let this dis-
tance be represented by K, and we have,

UZMATA
+
OTIECAS
C
L=3(27)
M = 2.52 (+7)
N = 2K 8 (2-1),
.
.
.
and the centre of the M
conic is found from
the equation
2Lx + M = 0,
which becomes
2 (0+ K ) = 0,
"
i
n
iril..!!
ALA!ITO Shillim
ilill in!
and shows that the
axis of maximum work
is the axis which bi-
sects the middle fibre.
In Fig. 92, AB
is the origin, and A'B'
the insertion of the
muscle, and LM re-
presents the calcu-
lated axis, and ST,
passing through the
socket S, the actual
axis of rotation.
Fig. 93.
Distance between calculated and observed axes = 0.40 in.
-
No
366
ANIMAL MECHANICS.
3. Tensor vaginæ femoris.- This muscle takes origin from
a length of 2.76 in, along the lower margin of the anterior
portion of the ilium, and is inserted by a broad fascia into.
the outer side of the thigh.
The origin and insertion are shown in Fig. 93, at AB and
A'B'.
Producing the fibres to meet in 0, and dividing the angle
AOB into three equal parts, I found-
Lion ( Tensor vaginc femoris).

Angle.
I-1
i cos
e
l
cos e
7.85 in.
6.34 in.
7.75
9.20 ,
14.10 in.
14.80 ,
15.60 ,
7.06.9
äŐ
6.70 ,
5.91 ,
10.40 ,
16.25 »
From these measurements we obtain

11+ l'
Angle.
11-7") cose
oºoo'
0.1577
3.4621
2.8207
5.15
0.1290
10.30
0.1087
2.4239
10.45
0.0961
2.1308
Sum,
0.4915
10-8375
Hence we obtain, by the equation to find the centre of the
conic (66),
2 Læ ·+ M = 0
10.8375 = 11.02 in.
0.4915
ANIMAL MECHANICS.
367
In Fig. 93, the axis LM is drawn at this distance from
the vertex 0, and the axis ST is drawn parallel to it, passing
through the socket S.
Distance between calculated and observed axes = 0.25 in.
4. Glutæus secundus (a).—This muscle (pyriformis) takes
its origin from the under surface of the sacral vertebræ, for a
length of 2.60 in., and is inserted into the inner surface of
the top of the great trochanter. Its form is shown in Fig. 94,
AX

mon
LINMINIMUM
MULLIAUTINTI
ITID
Fig. 94.
where AB and A'B' are its origin and insertion. It is so
nearly triangular in shape, that we may safely assume LM,
which bisects the intercept of OX between the bones, to be
the axis of maximum effect. ST, as usual, is the axis passa
ing through the observed socket S.
· Distance between calculated and observed axes = 0.16 in.
368
ANIMAL MECHANICS.
5. Glutæus secundus (B).-This muscle takes its origin
from the upper two-thirds of the entire surface of the outer
side of the ilium, and is inserted into the whole of the top
and outer surface of the great trochanter. It is shown in

-
Salud
Fig. 95.
Fig. 95, where AB is the curved middle line of its origin,
and A' B' the curved middle line of its insertion. Dividing
the angle AOB into three parts, we obtain-
ANIMAL MECHANICS.
369
Lion-Glutæus secundus (B)
UU

Angle.
It'
I cos @
ľcos e
in
6.78 in.
3.09 in.
5.54 »
8 26 ,,
2.74 »
oo
10.95 »
2.72 ,
oo
11.72 »
2.41 ,
From these measurements we obtain--

Angle.
1
2.5243
0.2558
0.1805
0.1228
1.9856
1.6793
0.1015
1.5359
Sum,
0.6606
7.7251
and lastly,
=
-
7.7251
0.6606 = 5.85 in.
The axis LM is drawn at this distance from the vertex 0,
and ST' is drawn through the observed socket.
Distance between calculated and observed axes = 0.21 in.
6. Glutæus tertius.- This muscle takes its origin from the
whole of the lower third of the outer surface of the ilium, and
2 B
370
ANIMAL MECHANICS
is inserted into the top and outer side of the great trochanter,
in front of the Glutvus secundus (B). Its form is shown in

VIIMIIMMTWA
MI
VA
Fig. 96.
Fig. 96, where AB is the origin, and A'B' the insertion of
the muscle. By dividing the angle AOB into five equal
parts, I found the following measurements :-
Lion-Glutæus tertius.
U

Angle.
11-11
I cos
I'cos e
1 02 in.
3.30 in.
3.50 ,
1. I219
4.02 in
4.47 »
5.15 »
5.81 ,
6.74 ,
8.30 ,
5.77 »
1.20 ,
7.75 ,
1.20 ,
ANIMAL MECHANICSUN
371
.
From these measurements we obtain-

Angle.
Angle. io (4+4).05 6
COS O
0.3030
0.2857
0,2475
0.2137
0.1733
0.1261
1.5273
1.5857
1.5520
1.4872
1.3761
1,2258
Sum,
1-3493
8.7541
Hence we find

X = -1 8.7541
1.3492 = 3.24 in.
The axis LM, in Fig. 96, is
drawn at this distance from
the vertex 0, and the axis
ST is drawn parallel to it
through the socket S.
Distance between calculated
and observed axes = 0.18
in.
7. Glutæus quartus. -
This muscle takes origin
from a horizontal line on
the ilium, 2 in. long, below
the Glutous tertius, and in
front of the acetabulum, and
is inserted on the inner front
of the great trochanter, in-
side the insertion of Glutæus
Fig. 97.
2 B 2
372
ANIMAL MECHANICS.
tertius. Its origin AB, and insertion A'B', are shown in
Fig. 97. On dividing the angle AOB into three equal parts,
I found the following results :-
Lion-Glutõus tertius.

Angle.
1-1
1. I cos
e
l
cos
| 3.17 in,
4.47 in.
1.37 in.
7.201
4.7015
ب
ب
4 95 »
| 5.21 ,
22.00
3.73
1.65 ,
From these measurements we obtain

Angle.
a l
icos
oºoo!
1.8423
1.9375
7.20
14.40
22,00
0.3155
0.3125
0.3003
0.2681
1.9730
1.8391
Sum,
1.1964
7.5919
Hence we find
X = -1 7.5919
9 = 3.17 in.
1.1964
Laying down this line LM, and drawing ST through the
socket S, we find
Distance between calculated and observed axes = 0.19 in.
ANIMAL MECHANICS.
373
1
8. Glutæus quintus. -This muscle is not found in the
Lion.
9. Obturator internus. The tendon of this muscle, as in
all animals, passes over a pulley formed by the posterior edge
of the ischium, and is inserted into the upper portion of the
pit of the great trochanter. If we measure the distances
from the centre of the socket to the insertion of the muscle
and to the centre of the tendon where it passes over the pul-
ley, we find-
Distance from insertion to centre of socket = 2.40 in.
Distance from pulley to centre of socket = 2.30 ,
From these measurements we readily find
0 = 46°.13
Work done
Maximum Work = 99.91 per cent,
sin 20 = Movimum Work
;
We have now seen that in the case of all the muscles used
in abduction, the centre of the acetabulum is nearly pierced
by the axis of maximum work, proper to each one of the
seven muscles ; and it is to be remembered that the head
of the femur does not fit the acetabulum exactly, so that there
is a play permitted to the axis of rotation, which does not ne-
cessarily pass accurately through the centre of the socket in
every motion. The diameter of the acetabulum is 1.42 in.,
and hence the rotation may take place, in some instances,
round a tangent to the acetabulum sphere, at a distance of
0.71 in. from the centre.
Collecting together the seven muscles of abduction, we
find :-
374
ANIMAL MECHANICS..
Abduction Muscles of Lion's Hip-Difference between Theory
and Observation.

Muscle.
Weight
of
Muscle.
Distance
between cal-
culated and
Observed axes.
.
6.50 oz. av.
0.23 in.
000,
1. Agitator caudæ, , . .
2. Glutæus primus, . . .
3. Tensor yaginæ femoris, .
4. Glutæus secundus (a), .
5. Glutæus secundus (B),
6. Glutæus tertius, . . .
7. Glutæus quartus, . .
8. Obturator internus, . .
ܝܲܢ 6 ; ; ܩܗܿ ܩܵ
οοοοοοοο
is
4.00
40,53 oz. av.)
0.215 in.
The mean difference between calculation and observation
amounts to only 0.215 of an inch, being less than one-third of
the total range (0.71 in.) permitted to the axis of rotation by
the arrangements of the joint.
The foregoing mean error is calculated as the arithmetical
mean of the error in each muscle, or, in other words, all the
muscles are treated as of equal importance. The work
actually done by each muscle is, however, proportional to its
weight, and we may calculate the mean error, taking into
account the relative importance of each muscle, as measured
by its work actually done. In order to find this mean, we
multiply the weight of each muscle by its error, and divide
the sum of all the products by the total weight of all the
muscles. We thus find
.
ANIMAL MECHANICS.
375
Abduction Muscles of Lion's Hip-- Difference between Theory
and Observation, tuking into account the relative importance
of each muscle.
-
Weight
Muscle,
Weight
x
Error.
.
.. Agitator caudæ, . . .
2. Glutæus primus, . .
3. Tensor vaginæ femoris, .
4. Glutæus secundus (a),
5. Glutæus secundus (B), .
6. Glutæus tertius. . . .
7. Glutæus quartus, . .
8. Obturator internus, . .
6.50 oz. av.
3.5. .
6.50 19
2.50 18
14.50 12
2.50 19
0.53 19
4.00
1.3950
1.4000
1,6250
0.4000
3.0450
0.4500
0.1007
0.4000
Total, .
.
40.53 oz. av.
8.8157
1232
Dividing this last sum by the total weight of the muscles, we
find the mean error to be 0.217 of an inch, which agrees with
the former result. This agreement shows that, in the plan of
the Hip joint, the perfect action of each muscle, small and
great, is equally provided for; just as in a well-ordered
republic, the happiness of the humblest citizen is of as much
importance to the State as the happiness of the citizen of
highest rank.
Before leaving the muscles used in abduction, it may be
useful to record the actual directions of the axes round which
they cause the femur to rotate.
We may take the longest axis of the pelvis joining the .
tuber ischii with the anterior inferior spine of the ilium, as
our reference line, measuring the angles made with this line
towards the ischium, positive when above this line, and
negative when below it.
On laying down the several axes, I found the following :-
376
ANIMAL MECHANICS
Abduction Muscles of Lion's Hip-Position of Rotation axes.
M

Muscle.
Angle made
with long axis
of pelvis.
– 15°45'
ooo
+ 12 00
1. Agitator caudæ, . . .
2. Obturator internus, . .
3. Glutæus secundus (a), .
4. Glutæus primus, . . .
5. Glutæus quartus, . .
6. Glutæus tertius, . . .
7. Tensor vaginæ femoris, .
8. Glutæus secundus (), .
.
.
.
.
.
.
+ + + + + +
.
Muscles used in Adduction. In discussing the muscles
used in abduction, we have gone half way round the hip
joint; in the muscles used in adduction, we shall complete
the remaining half of the circle, in the reverse order, as shown
in the Table of muscles (p. 362).
1. Adductor primus.- This muscle takes its origin from
a surface 2.61 inches in length, extending nearly from the
tuber ischii to the posterior end of the symphysis pubis ; and it
is inserted for a length of 4.40 inches into the far end of the
linea aspera. It forms a skew muscle, although not so
markedly as the other adductors; and we shall investigate the
conditions of its maximum work, by the method laid down for
skew muscles.
Lion-Adductor primus.
Lengths of extreme Lengths of Origin and
Fibres.
Insertion
· P = . 8.00 in.
a = 2.61 in.
Pi = 11.60 ,
a' = 4.40 ,
Constants of Hyperboloid.
20 = 4.30 in.
20 = 90°
ANIMAL MECHANICS.
377
It will be unnecessary to investigate the biquadratic (92),
for the constant 20 = 90° gives us m= 1, and therefore i = 0,
is a root of the equation, as may be seen on inspection.
To determine the points I and I', we have
pa = a² + a^2 + 40,
pz= (a + a)2 + (a' + a')2 + 4c",
from which we obtain
Pi - p2 = 21a + 2a'a' + a² + a'? ;
or, introducing the numerical values,
5.22 a + 8.80 a' - 44 39 = 0,
a' = – 0.6 a + 5.04.
Introducing this expression into the first equation, we find
1.36 a– 6.05 a - 9.12 = 0,
and finally,
1+5.64
1-1.15
1 + 5.73
An inspection of the bones shows that the last pair of
Talues are those to be employed, and thus we find
a =) T1.00
a = - 1.15 in.
ai = + 1.46 ,
a' = + 5.73 in.
ai' = + 10.13 »
0.144
= +0.716
It
-
-
=(--0018
) = -0.018
20)- + 1.524
--
= + 1.524
D
378
ANIMAL MECHANICS.
From these results we can calculate the equivalent fibre
00', as follows:-
P =-0.009, = +0.762,
P2 = I² + L'? + 4c?,
P = 6.64 in.,
0.414
L = - 0.06 in. L' = + 5.06 in.
Laying down the points 0 and O' on the bones, I found-
Distance from 0 to centre of socket = 5.16 in.
Distance from 0 to centre of socket = 6.40 ,
P2
18.40
Difference,
.
.
.
1.24 in.
C
These measurements correspond to an angle of 51° 7'
made by the fibre 00' with the plane passing through the
bone AB, or by equation (108).
Work done
= sin (20) = 97.7 per cent.
Maximum work
2. Adductor secundus (a).—This muscle takes its origin
along a surface 2.68 inches long, from the tuber ischii to the
posterior end of the symphysis pubis, and is inserted in continua-
tion of the insertion of adductor primus, higher up on the linea
aspera, for a length of 3.45 inches. It is more skew than
adductor primus, but less so than adductor secundus (B). The
following measurements were made upon this muscle :-
Lion-Adductor secundus (a).
Lengths of equidistant Lengths of Origin and
Fibres.
Insertion.
p = 5.34 in. a = 2.68 in.
Pa = 6.40 ,
a' = 3.45 ,
Pa = 7.60 ».
ANIMAL MECHANICS.
379
Constants of Hyperboloid.
20 = 4.06 in.
20 = 91°15'.
The angle between the bones is so nearly 90° that we may
assume, as before, that I = c is a root of equation (92).
To determine I and I', we have
pz2 - p2 = 2aa + 2a'a' + (a+ a'?),
which gives us
5.36 a + 6.90 a' – 10.17 = 0,
a' = -6.78 a + 1.47.
Substituting this value of a' in the expression for p?, we
obtain :
1.61 a’ – 2.29 a - 9.87 = 0,
and finally,
1 + 5.30 in.
1 - 1.10 in.
1 - 1.87 ,
On inspection of the bones it is evident that the latter pair
of roots are those required in the problem. Hence we find --
a
=
a
t
2.93 »
a = - 1.87 in.
Q1 = – 0.53 »
+0.81 ,
a' = + 2.93 in.
az' = + 4.66 ,
az' = + 6.39 »
- = -0.350
-
= +0.549
== -0.083
= +0.728
42 = + 0.107
= + 0.841
>() -- 0.326
= -0.326
26 + 2.18
- l = + 2.118
380
AN
ANIMAL MECHANICS.
7
In order to find O and Oʻ, we have
--0.109,
= +0.706,
P2 = L+ L'? + 40%,
P = 16.4836,
P = 5.802 in.
L = - 0.632 in.,
L' = + 4.096 in.
On laying down the points 0 and 0', thus found, on the
bones, I obtained-
Distance of O from centre of socket = 4.57 in. .
Distance of O from centre of socket = 5:31 ,
Difference, ... 0.74 in.
0 = 49° 17'.
Work done
Maximum work
= sin 20 = 98.88 per cent.
3. Adductor secundus (B).- This muscle takes its origin
for a length of 3.72 inches along the symphysis pubis, and is
inserted, side by side, with Adductor secundus (a), into 3.45
inches of the upper portion of the linea aspera.
Lion-Adductor secundus (B).
Lengths of Equidistant Lengths of Origin and
Fibres.
Insertion.
P = 5.86 in.
a = 3.72 in.
Pa = 8.03 ,
a' = 3.45 ,
P2 = 10.16 ,
Constants of Hyperboloid.
20 = 3.20 in.
20 = 49° oo'.
ANIMAL MECHANICS,
381
In order to calculate the positions of the points 1 and I',
we have (taking into account the precautions as to sign already
mentioned), vide equation (93),
pe? – p2 = – 2 (a + a' cos 20) a + 2 (á + a cos o) a',
+ (a? + a'+ 2aa' cos 20).
From this we find
11.78 a' = 11.96 a + 26.30,
a' = a + 2.23.
Substituting this value in the equation
p2 = a + a 2 – 2aa' cos 20 + 402,
we find
0.69 a² + 1.53 a – 19.13 = 0,
and finally,
1 + 4.27
(+6.50
“ 1-6.49
- 4.26
The first pair of values are those required by our present
problem. Using them, we obtain
Lion- Adductor secundus (B).
a = + 4.27
a' = + 6.50
a, = + 2.41
av = + 8.23
= + 0.55
az' = + 9.96
- = + 0.728
+
-
= + 1.109
42 = + 0.900
= + 1.025
+
R 310,888
42 = + 0.054
= + 0 980
+
>) - +1.082
= + 1.082
%) = + 3.114
= + 3.114
382
ANIMAL MECHANICS.
aa
0.171
—
p
= + 4.736
a ai
= 0.1 24
= + 2.470
pa
0202
= 0.098
0.539
P2
aa'
C) = 0.393
= + 7.745
p
Hence we obtain, using equations (96),
K = + 0.535,
mX = + 0.449,
mX' = + 1.292.
If we now employ these values to calculate the coefficients
of (92), we find for the biquadratic which determines the
position of the generator of maximum work, ..
- 0.647 14 + 5.456 18 + 4.495 1 + 0.405 = 0.
One of the real roots of this equation is
d=-0.09
Hence the construction (Fig. 90) applies, and we can find the
positions of 0 and 0, and the length of the equivalent fibre,
as follows:
L
L'
P= + 0.36I,
D=+ 1.038.
Using these values in the equation,
P% = + L'? - 2LL' cos 20 + 40%,
we find
P = 6.026 in.
P2 = 10:24
- 0.282
L = + 2.175 in.
L' = + 6.255 in.
ANIMAL MECHANICS .
383
Laying down these points, as before, on the bones, I found
Distance of O from centre of socket = 3.63 in.
Distance of O' from centre of socket = 4.46 ,
Difference, ., . .0.83 in.
0 = 50° 51.
Work done
Maximum work
ve = sin 20 = 97.92 per cent.
4. Adductor tertius (brevis). This muscle is wanting in
the Lion.
5. Adductor quartus (longus). These muscles take
6. Adductor quintus (pectineus). S origin, side by side,
from the anterior margin of the pubes, for a length of
Å x B

CRIMITILAT
Fig. 98.
1.70 in., and are inserted in like manner into a line 2.95 in.
in length, lying beside the insertion of Adductor secundus.
384
ANIMAL MECHANICS,
and to its inner side. When the thigh is abducted, these
muscles form plane quadrilaterals, and I have already shown
(p. 328) that they are nearly inscribable in circles, which is
the geometrical condition necessary to produce the maximum
work.
In Fig. 98 I have drawn these muscles to scale, from the
measurements given in p. 328. The pelvic origin is AB,
and the femoral insertion is A'B'. If the quadrilateral
ABB'A' be inscribable in a circle, then if A'A and B'B be
produced to meet in 0, and OYX be drawn bisecting the
angle AOB, the triangle CXY should be isosceles, and the
side CX equal to the side CY. I found the following mea-
surements, on the scale of nature:-
CX = 4.29 in
CY = 4:30 ,
From this result we can calculate the ratio which the work
actually done by the muscle bears to the maximum work pos-
U

Fig. 99.
sible. Let CXY (Fig. 99) be the same triangle as in Fig. 98,
and let SC be the axis of rotation passing through the
socket S. Draw through XY the plane YPX perpendicular
ANIMAL MECHANICS
385
(or nearly so) to the axis SC, and let us use the following
notation :-
CY = 6, CX = a,
XCY = $, YP = y, XP = x.
It is easy to see that we have the following relations:-
ya – x2 = 62 – a?,
y2 + x2 = 22 + a² - 2ab cos ,
from which we obtain
y 16 (6 - a cos o)
* Va (a- b cos )
= tan 6,
Maximum work
Work done
= sin 20.
Substituting, in the foregoing, the following observed
values,
a = 4.29 in., b = 4.30 in., 0 = 9° 30',
we obtain
= tan (49° 45'),
Work done
Maximum work = 98.03 per cent.
sin 20 = Movimum work
7. Obturator externus.--This muscle has the usual origin,
and is inserted into the deepest portion of the pit of the great
trochanter; it may be regarded as a triangular muscle, and the
proportion of the work done by it to the maximum work
possible may be found by measuring the distances from the
insertion and central point of origin from the centre of the
socket. We thus find-
Distance from insertion to centre of socket = 2.33 in.
From centre of Obturator foramen to centre of socket = 2.27 ,,
2 C
386
ANIMAL MECHANICS.
Hence we find
0 = 45° 45',
Work done
sin 20 = Vari
- Maximum work = 99.96 per cent.
Collecting the preceding results into one Table, we find
Adduction Muscles of Lion's Hip-Difference between Theory
and Observation.

Muscle.
Work done.
Difference between
work done and
maximum work,
97.70 per cent.
2.30 per cent.
1, 12
2.08
97.92
1. Adductor primus,
2. Adductor secundus (a),
3. Adductor secundus (B),
4. Adductor tertius, .
5. Adductor quartus, . .
6. Adductor quintus, . .
7. Obturator externus, . .
1.37
.
.
.
98.63
2
1
1. 37
99.96
0.04
Mean, ...
98.62 per cent.
1:38 per cent.
Nothing can be more complete than the proof afforded by
the foregoing Table, that the adduction muscles, like the
abduction muscles, are all placed in reference to the socket
of the hip joint, in such a manner as to produce the maximum
amount of work.
It is unnecessary to discuss the case of the flexion and
extension muscles, for, in p. 258, et seq., I have fully proved
that they are placed (not merely in relation to the hip joint,
but also in relation to the knee and heel) so as to produce the
maximum of work possible.
It would be tedious to make similar calculations for the
muscles of other animals; and it is sufficient to state that I
have completed many such calculations, for the hip, shoulder,
ANIMAL MECHANICS.
387
and other joints of many animals, and that I have never met
with any exception to the following Propositions, which may
be regarded as summing up my results :-
1°. Each muscle is constructed in relation to its joint, in such
a manner as to perform one kind of work, only; and it performs
that work to maximun advantage.
2°. The number of muscles employed is determined by the
number of distinct actions required from the limb.
3º. The shape and form of the bones employed are the
necessary consequence of the shape and power of the muscles in
action.
4°. The smallest muscle in the combination is as carefully
adapted to its conditions of maximum work as the largest muscle.
my
IUVI
From these Propositions, supposed to be extended to the
action of every muscle and joint, it appears to me to follow
as necessary consequences :-
1°. That a foreseeing Mind planned the type of the limb,
and of its actions.
2º. The idea of the limb and of its necessary actions being
given; the number, shape, and arrangement of the necessary
muscles can be calculated and predicted with as much certainty
as an astronomer can predict an eclipse.
3º. That the shape and arrangement of the bones follow of
necessity, from the necessary arrangement of the muscles.
4°. That any alteration, however slight, in any part of the
combination of bones, muscles, and joints, would entail a loss of
work, and lead to a less perfect mechanism.
5º. Hence, the permanence and stability of each species (so
far as relates to bones, muscles, and joints) is absolutely secured,
on the principles so admirably laid down by Mr. Darwin.
6. The profound study of the mechanism of joints lends no
support to the postulate, that the similarities found to exist in the
VI
I
2 c 2
388
.
ANIMAL 1
MECHANICS.
2
bones, muscles, and joints of animals may be explained by common
descent from a supposed common ancestor.
Shoulder Joint. The shoulder girdle differs from the
pelvic, in being moveable, so that the centre of the socket
round which the arm revolves is not fixed in one definite
position, but may be fixed for the moment by the action of its
own muscles, or it may be itself in motion during the motion
of the arm. Notwithstanding its greater complexity of action,
the same principles may be applied to the shoulder joint,
that were used in the discussion of the hip joint; and they
lead us to the same general conclusions, as to the absolute
perfection of the mechanism of the shoulder joint in all
animals.
If we compare together the muscles of the shoulder and
hip, setting aside those of the shoulder which are peculiar to
it, from the mobility of the shoulder girdle, we find the fol-
lowing Table of comparative muscles, corresponding to each
other in action :-
Shoulder Joint.
Hip Joint.
Muscles of Abduction.
Muscles of Abduction.
1. Trapezius inferior, . . . . Agitator caudæ.
2. Trapezius superior, 1
. . Glutæus primus.
3. Deltoideus,
4. Teres major,. . . . . . Tensor vaginæ femoris.
5. Supraspinatus, ... Glutæus secundus.
6. Infraspinatus, ..... Glutæus tertius.
7. Teres minor, . ...... Glutæus quartus.
8. Infraspinatus secundus, . . Glutæus quintus.
9. Subclavius, . . . . . Obturator internus.
Muscles of Flexion.
Muscles of Flexion.
1. Triceps longus, . . . . . Rectus femoris.
2. Tricipiti accessorius, ... Sartorius,
3. Subscapularis, .... Iliacus.
4. Latissimus dorsi, ... Psoas magnus.
N
ANIMAL MECHANICS.
389
Muscles of Extension.
Muscles of Extension.
1. Biceps coracoradialis, . . . Biceps femoris.
2. Biceps glenoideus, ... Semitendinosus.
3. Brachiæus, . . . . . . Semimembranosus.
4. Coracobrachialis capsularis, . Quadratus femoris.
Muscles of Adduction.
Muscles of Adduction.
1. Coracobrachialis, ... Adductor primus.
Adductor secundus.
2. Pectoralis major, ...
Adductor tertius.
Adductor quartus.
LAdductor quintus.
3. Pectoralis minor, .... Obturator externus.
4. Pectoralis quartus, . . . . Gracilis.
S
The similarity in the muscles of the shoulder and hip
is a necessary consequence of the idea which is common to
both joints; viz., a single long bone revolving round a centre,
and acted on by muscles terminating in that bone, and taking
their origin from a girdle, either absolutely or relatively at
rest. There are some twenty or twenty-five actions necessary
in such a system, and, therefore, a muscle is “ told off” to
perform to maximum advantage each proper action. In the
order of nature, the following method is observed in all the
mechanisms of the body, of every animal :-
1°. A contriving Mind.
2°. The Idea of the mechanism and its actions.
3º. A perfect muscle to perform each action.
4º. Bones of form, magnitude, and position, which can be
predicted from the condition of maximum work imposed on
each muscle employed.
Here we obtain a solid and permanent foundation for the
Science of Comparative Osteology, founded on Geometry and
390
ANIMAL MECHANICS. .
O
Mechanics, combined with the Postulate of maximum work;
and starting from this foundation, in the inverse order to that
employed in the case of the Lion's hip, we might proceed to
re-clothe a given skeleton with its proper muscles, and cal-
culate with accuracy the amount of the forces employed in
every action of the body of the living animal.
II. Muscular Types.--I now propose to develop, by
means of a few examples, the theory of Muscular Types, of
various animals, based upon the preceding proofs of the
individual importance of each muscle ; and shall choose as
my first example the well-marked Type of the Felidæ.
(a.) Muscular Type of the Felidæ.-- In discussing the
theory of muscular types, I shall confine my attention, chiefly,
to the hip and shoulder muscles. Let the weight of each
muscle be taken, as measuring the work done by it, and let
the total weight of all the muscles acting on the joint be
represented by 100, so that the weight of each muscle is
given as a percentage; we may thus construct Tables for the
several animals in which all the figures are comparable. In
order to avoid unnecessary columns of figures, I do not
reproduce the actual weight of each muscle, but give the
total actual weight of all, so that any reader can calculate
back from the Table the original weight of any muscle. The
animals selected for the present purpose are-
I. Adult African Lion (magnificent specimen).
2. Adult African Lioness (not related by blood to Lion).
3. Bengal Tiger (magnificent specimen).
4. Indian Leopard (male).
5. African Leopard (male).
ANIMAL MECHANICS.
391
Muscular Type of Larger Felida-Hip Joint.
Muscle.
Lion.
1
Lioness. /
Tiger.
Mean.
14.10
3.76
3.91
Abductors.
1. Agitator caudæ, . . .
2. Glutæus primus, .
3. Tensor vaginæ femoris, .
4. Glutæus secundus (a), , .
5. Glutæus secundus (B), , .
6. Glutæus tertius, . . . .
7. Glutæus quartus, . . .
8. Obturator internus, . . .
Per cent. Per cent. Per cent. Per cent.
3.57 1 3.12
3.60
1.93 2.07 1.60 1.87
3.57 4.39
7.21
I.19
1.13
1,24
I, 19
0.16
0.25
2. 24
1.95
OFyw
N
Flexors.
1. Rectus femoris, . .
2. Sartorius, . . .
3. Psoadiliacus,
6.05
5.74
.
.
.
.
.
6.45
5.39
5.62
7.90
4.75
6.58
5.91
5.45
15.25
17.50
ܝ ܠ
Extensors.
1. Biceps femoris, .. .,
2. Bicipiti accessorius, . .
3. Semitendinosus, . . . .
4. Semimembranosus, . . .
5. Gracilis, . . :
6. Quadratus femoris, . . .
5.50
o isso to
ܨ ܨܢ ܨܢ ܨܢ
owoc do
.
.
Adductors.
1. Adductor primus,
2. Adductor secundus, \
3. Adductor quartus,
4. Adductor quintus,
5. Obturator externus, .
20.24 | 21.02
1.92 1.72
7.88 |
21.07
14.071
1.42 1.69
1.16 1 1.39
.
.
.
1.43
1.58

100
100
100 | 100
Oz, av.
Oz, av.
Oz. av.
Total weights,.
181-96
143.86
276.09
The total weights of the muscles in the Tiger and Lion
show the great superiority of the Tiger, and this superiority
is further confirmed by the weights of the muscles of the
shoulder joint, which will be given presently. In fact, if we
take the Tiger as our standard, and compare with him the
2
S
392
ANIMAL MECHANICS.
Lion and Lioness, it is easy, from the weights of muscles
given in the Tables, to calculate the following:-
Comparative Strength of Tiger and Lion.

Tiger.
Lion.
Lioness.
100
65.90
Hip joint, ..
Shoulder joint, ..
52.11
| 51.55
IOO
69.93
Y
From this Table it appears that the strength of the Lion is
about two-thirds of that of the Tiger, and that the strength
of the Lioness is about one-half that of the Tiger. From the
greater development of the fore-quarter of the Lion, as com-
pared with that of the Lioness, we find that the shoulder
muscles of the Lion bear a somewhat greater proportion to
those of the Tiger, than the hip muscles do; while in the
Lioness the proportions of the muscles of the shoulder and
hip continue to have the same relation to the corresponding
muscles of the Tiger.
In the time of the Emperor Titus, Tigers and Lions were
forced to fight in the Amphitheatre at Rome, and it is
recorded by Martial that the Tiger was able to kill the Lion,
a fact which has been confirmed by accidental contests be-
tween these animals occurring in modern menageries.
" Lambere securi dextram consueta magistri
Tigris ab Hyrcano gloria rara jugo
Sæva ferum rabidâ laceravit dente leonem :
· Res nova, non ullis cognita temporibus.
Ausa est tale nihil, sylvis dum vixit in altis,
Postquam inter nos est, plus feritatis habet."'*
* Martial. De Spectaculis Ep. X.
ANIMAL MECHANICS11
393
,
Muscular Type of Lesser Felida
Hip Joint.

Muscle.
Indian | African
Leopard. | Leopard.
Mean.
Per cent.
4.44
Per cent.
3.57
Per cent.
4.01
1.82
1,36
3.93
Abductors.
1. Agitator caudæ, . , .
2, Glutæus primus, .
Tensor vaginæ femoris, .
Glutæus secundus (a), .
5. Glutæus secundus (B), .
6. Glutæus tertius, . . .
7. Glutæus quartus, . .
8. Obturator internus, . .
5.89
0.99
imo o mio mi
1.31
0.89
1.20
.
.
.
0.20
0.31
1.72
0.25
2.01
1.87
6.20
6.69
6.82
5.64
6.44
6.45
6.63
5.84
17.07
N
Flexors.
1. Rectus femoris, . . . .
2. Sartorius, . . . . .
3. Psoadiliacus, . . . . .
Extensors.
1. Biceps femoris, . . . .
2. Bicipiti accessorius, . . .
3. Semitendinosus, .. .
4. Semimembranosus, . . .
5. Gracilis,
6. Quadratus femoris, . . .
Adductors.
1. Adductor primus, . . .
2. Adductor secundus, . . .
3. Adductor quartus, I
4. Adductor quintus, . .
5. Obturator externus, . . .
9.19
4.14
9.76
14.II
0.63
0.71
7.61
7.67
13.83
I. 14
13:38
1.39
1.14
7.64
13.61
1.26
1.17
1.15
100
100
100
Oz, ay.
Total weights, . .
58-33
Oz. av.
48:33
These results show a general close agreement with those
obtained from the larger Felidæ; and, if we compare the
muscles in groups, according to their general action, we
obtain the following comparative Table :-
394
ANIMAL MECHANICS.
Muscular Type of Felidae—Hip Joint.

Abductors.
Flexors.
Extensors. Adductors.
21.46
36.62
Lion and Tiger, .
Leopard, ...
17.77
18.82
24.15
23.66
19.72
37.80
We shall now proceed to examine the muscular type of the
shoulder joint in the Felidæ, using the same animals as before:-
Muscular Type of Larger Felidoe-Shoulder Joint.

Muscle.
Lion.
Lioness.
Tiger.
Mean.
Per cent.
i
- 3.7
Per cent.
4.53
Per cent.
4.04
1.38
1.20
1.15
Per cent.
3.85
1.80
1.48
1.30
8.81
3.50
1.27
imico
mirisio +
nimico +
Scapular.
1. Trapezius clavicularis, ..
2. Cleidomastoideus, . . . .
3. Trapezius inferior, . . . .
4. Trapezius superior, ...
5. Omo-Atlanticus, . . . .
6. Serratus magnus, . . . .
7. Rhomboideus, .....
Abductors.
1. Deltoideus, . . . . . .
2. Teres major, . . . . . .
3. Supraspinatus, . . . . .
4. Infraspinatus, . . . . .
5. Teres minor, . . . . . .
Flexors.
1. Triceps longus, . . . . .
2. Tricipiti accessorius, ...
3. Subscapularis, .
4. Latissimus dorsi, ....
Extensors.
1. Biceps humeri, · · · · ·
2. Brachiæus, · · · ·
Adductors.
1. Coracobrachialis, . . . .
2. Pectoralis major, . . . .
3. Pectoralis quartus, . . . .
ito no
innego
is noso
inovo sobo
Novo
9. 10
0.87
sovo
5.48
11.79
3.86
3.68
3.64
1.71
1.49
:
1.63
0.12
17.81
4.04
0.07 0.06
19.06 u
4.381
0.08
22.29
100
100
100
| 100
Total weights, .
Oz. av. Oz. av. ] Oz. ay.
235.67 | 173.72 337.00
ANIMAL MECHANICS.
395
Muscular Type of Lesser Felida-Shoulder Joint.

Muscle.
Indian | African
Leopard. | Leopard.
Mean.
Per cent.
Scapular.
1. Trapezius clavicularis,..
2. Cleidomastoideus, , ...
3. Trapezius inferior,. ...
4. Trapezius superior,
5. Omo-Atlanticus, . . .
6. Serratus magnus, . . .
7. Rhomboideus, . . . .
Per cent. Per cent.
3.91 3.83
I13 1.95
1.74 1.34
3.87
1.54
1.54
1.46
.
1.46
1.09
9.11
1.03
9.22
4.22
1.06
9.17
5.43
4.82
5:39
Abductors.
1. Deltoideus,
2. Teres major, . ...
3. Supraspinatus, ...
4. Infraspinatus, ..
5. Teres minor, : ...
3.8%
ouwen
ouwer
0.27
Flexors.
1. Triceps longus, . . . .
2. Tricipiti accessorius, . ...
Subscapularis, . . . . .
4. Latissimus dorsi, ....
9.72
10.03 19.40
0.92 / 0.87
6.616.10
11.58 | 14.43
0.80
6.35
13.01
Extensors.
mi
1. Biceps humeri, . . . . .
3.34
2. Brachiæus, ......
1.70
Adductors.
1. Coracobrachialis, ... 1 0.11 0.03 | 0.07
2. Pectoralis major, 1
3. Pectoralis quartus, } .'
: | 18.33 | 19.32 | 19.08
100
100
Oz. av.
Total weights, ..
Oz. av.
72-50
58-93
If we compare the muscles of the shoulder joint in groups,
according to their general action, we obtain the following
results :-
396
ANIMAL MECHANICS.
Muscular Type of Felida-Shoulder Joint.

Scapular.
Abductors.
Flexors.
Extensors. | Adductors.
21.02
28.41
22.37
Lion and Tiger, . 1
Leopard, ...
22.93
23.46
5.27
5.34
22.08
29.97
19.15
t
(6.) Muscular Type of the Canidæ.-The muscular type of
the Canidæ is quite distinct from that of the Felidæ, and is not
so constant in the members of the entire family, which differ
from each other more considerably than the members of the
Felidæ do.
The Canidæ differ from the Felidæ in the hip and shoulder
joints in the following particulars :-
First, negatively-
The Canidæ do not possess the following muscles :-
1°. Agitator caudæ, ut:
2º. Glutæus quartus, I
3º. Pectoralis quartus-Shoulder.
Secondly, positively-
The Canidæ possess always a double Sartorius, and
occasionally a double Tensor vaginæ femoris.
S
I shall take the Greyhound as the type of the family, and
give the percentages of this animal in detail, adding a summary
of the percentages of groups of muscles taken from other
members of the family.
In the following Table, I give “ Master Magrath’s” muscles
in one column, and place beside it the average muscles found
from other Greyhounds :-
ANIMAL MECHANICS.
397
Muscular Type of Greyhound-Hip Joint.

Muscle.
"Master
Magrath."
Average
Grey:
hound.
Per-
centage.
2,03
4.98
Per-
centage.
1.90
4.70
Abductors.
1. Glutæus primus, . . . .
2. Tensor vaginæ femoris, . .
Glutæus secundus (a), 1
3. Glutæus secundus (B), ) .
4. Glutæus tertius, . . . ,
5. Obturator internus, ...
10.13
0.91
1.40
äori
Flexors.
1. Rectus femoris,. ...
2. Sartorius (double), ...
3. Psoadiliacus, . ·
4.69
4.26
nerer
5.18
Extensors.
1. Biceps femoris, . ...
2. Bicipiti accessorius, ..
3 Semitendinosus, ...
4. Semimembranosus, ..
5. Gracilis, ..
6. Quadratus femoris, ..
ம்மம்
og önnino
9.42
Adductors.
1. Adductor primus, . . . .
2. Adductor secundus (a), ,
3. Adductor secundus (B), ) .
4. Adductor quartus, . . . .
5. Adductor quintus, ...
6. Obturator externus, ..
15.50
7.82
15.33
0.85
0.60
1.25
0.84
0.75
1.17
100
100
Oz. av.
Total weights, . .
51-81
Uz. av.
40.43
It will be seen from this Table that “ Master Magrath,"
although a slightly built dog, possessed an amount of mus-
cular force greatly in excess of ordinary Greyhounds; and a
similar conclusion follows from the examination of the muscles
of the shoulder joint. From an examination of the dog during
398
ANIMAL MECHANICS.
:
life, I had expressed the opinion that his chief muscular
peculiarity consisted in the development of the shoulder
muscles--a conclusion which was fully borne out by the post
mortem examination. If we regard - Master Magrath” as the
unit of comparison, we find the percentage of the average
Greyhound to be-
In Hip Joint, . . 78.04 per cent.
In Shoulder Joint, .75.56
LU
These figures show the muscular superiority of “Master
Magrath," and the greater development of the muscles of
his shoulder.
In the following Table, I have collected into groups the
muscular percentages of various Canidæ, for the purpose of
comparison :
Muscular Type of Canida-Hip Joint.

Animal.
Abductors.'
Flexors,
Extensors. Adductors
Totia.]
weight.
25.85
Oz. av.
51.81
40.43
14.72
1. “Master Magrath,"
2. Average Greyhound,
3. Irish Terrier, ..
4. Australian Dingo, .
5. Pyrenean Mastiff, .
6. European Wolf,.
7. Indian Jackall, . .
8. Bengal Fox,. . .
Percentage. Percentage. Percentage. | Percentage.
19.45 14.13 40.57
18.06
16.25
38.01 27.68
21.94 15.83 36.41 25.82
19.37 15.87 42.39 22.37
15.82 15.51 41.90
26.77
21.45
34.14
17.43 17.33 39.36 25.88
17.73 15.60 41.14
17.31
27.10
mondini
25:53
This Table shows the great muscularity of the Greyhounds,
for the only dog that excels them is the Mastiff, which is a
much larger and heavier animal. It may be worth adding
that “Master Magrath's” running weight was 54 lbs.
ANIMAL MECHANICS.
399
The muscular type of the shoulder joint in the Canidæ is
shown in the following Tables :-
Muscular Type of Greyhound-Shoulder Joint.
Muscle.
“Master
Magrath."
Average
Grey-
hound.
Per-
centage.
1.48
.
Per-
centage.
1.93
1.65
1.381
Scapular.
1. Trapezius clavicularis, .
2. Cleidomastoideus, . . .
Trapezius inferior, . . .
4. Trapezius superior, . .
5. Omo-Atlanticus, ...
6. Serratus magnus, ...
7. Rhomboideus, · · · ·
I 27
.
2.40
ririnoit
1.98
9.38
3.86
Abductors.
1. Deltoideus, .......
2. Teres major,. ..
3. Supraspinatus, ..
4. Infraspinatus, ..
5. Teres minor,...
Asinino
OWN
Flexors.
1. Triceps longus, . . . . .
2. Tricipiti accessorius, ...
3. Subscapularis, . . . . .
4. Latissimus dorsi,....
18.70 1
0.38 }
22.67
4.40
9.29
11.40
Extensor's.
1. Biceps humeri, · · ·..
2. Brachiæus, . . . . . .
1.84
1.34
2.40
1.40
Adductor's.
1. Coracobrachialis,
0.21
2. Pectoralis major, .... | 21.07
10.23
100
100
0%. av.
Total weights, ..
63.45
Oz.ay.
47.94
In the following Table, I have placed the percentages of
groups of muscles, of various Canidæ, arranged in the same
manner as in the corresponding Table for the Felidæ :--
400
ANIMAL MECHANICS.
Muscular Type of Canido-Shoulder Joint.

Animal.
Scapular. Abductors. |
Flexors.
Extensors.
Adductors.
Total
weight.
Per cent.
19.23
Per cent.
34.64
Per cent. | Per cent.
3.18
21.28
Oz. av.
63.45
Per cent.
21.67
20.37
26.25
20.38
20.35
1. “Master Magrath,”
2. Average Greyhound,
3. Irish Terrier, ..
4. Australian Dingo, .
5. Pyrenean Mastiff, .
6. European Wolf, .
7. Indian Jackall,. .
8. Bengal Fox, ..
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yun ve to
Hosen
24 13
22.09
(c.) Muscular Types of Man, and of the Quadrumans.-
I shall now proceed to the comparison of the muscular types
of Man and the Quadrumans-a subject full of interest to
comparative anatomists.
The comparison includes the following animals :- .
1. Man (mean of three subjects).
2. Gorilla.*
3. Chimpanzee (mean of two).
4. Macacus Rhesus (mean of two).
5. Macacus cynomolgus.
6. Cynocephalus Hamadryas (Hepi, or Lion Baboon).
7. Papio Maimon (Mandrill).
8. Ateles ater (mean of two). (Spider Monkey.)
I shall commence with the hip joint:-
* I am indebted to Mr. T. J. Moore, Director of the Liverpool Museum, for
the opportunity of dissecting a young female Gorilla, which was brought home
in spirits by Captain J. B. Walker, F. R. G. S., from Africa.

ANIMAL MECHANICS.
401
Muscular Types of Man and the Quadrumans-Hip Joint.
Muscle.
Man.
GO-
rilla,
Chim Hama- Cyno Man- | Bhesus. | Ateles,
panzee. | dryas. | molgus. drill.
Per
Per
cent.
Per
cent.
Per
cent.
Per
cent.
Per Per Per
Abductors.
cent. cent. cent.
1. Glutæus primus, . .118.28 13.29
2. Tensor vaginæ femoris, | 2.33 | 2.10 Sp.
11.67
3. Glutæus secundus (a), 10.8971
4. Glutæus secundus(B), 1.43
:}13.98 14.58
5. Glutæus tertius, · ·| 3:52 | 1.75 | 4.17
6. Glutæus quartus, . .
7. Obturator internus, .1 3.22 11.92 | 2.08
9.33 5.38 | 3.72 6.78
13.03 13.80 13.85 11.60
1.93 1.93 2.06 1.71
2.13 1.79 3.14 2.15
4.96
9.79
0.89
1.27
3.93
4.94
5.15
Flexors.
1. Rectus femoris, ..!
2. Sartorius, · · · ·|
3. liacus, . . . . .1 4.257
4. Psoas, . . . . .)
| 3.75
| 2.50
[11.67
5.55 | 5.79 | 6.19 6.49
0.98 | 2.34 | 1.37 | 2.48 | 3.93
11.47 10.49 8.47 8.99 11.18
4.89
Extensors.
1. Biceps femoris, . .
2. Semitendinosus, . .
3. Semimembranosus, .
4. Gracilis, . . . . .
5. Quadratus femoris, .
So
1 6.24
isritari
ܠ܊ ܝܼ ܝܼ -
16.56 17.34 13.36
15.40
| 4.45 | 5.13
15.32 17.24
1,37 1.49
o trio
4.70
5.08
5:59
o -
2.29
| 9 97
19.18
1
Adductors.
1. Adductor primus, . .
2. Adductor secundus(a)||16,58
3. Adductor secundus (B)
4. Adductor tertius, . .
5. Adductor quartus, .1 3.82
6. Adductor quintus, .
7. Obturator externus, . 1.30
* 26.55
| 2.39)
$ no
1 23.00 23.69 |
17.59 24.14 |
2.91! |
| 3.33] |
- 10.45 | 1.15 ||
0.83 0.65 0.96 1.490.71 11.78
2.50 12.me
2.50 | 2.75 1.93 2.58 1.93 1 3.44
1.35
2.10
100
100 / 100
| 100
100
100
100
h
Oz, av. | 02. av.
Total weights, .. 93.17
Oz. av. Oz. av. Oz. av. 02. ay. Oz. av. Oz. ar.
2.40 24.34 7.25 | 1748 | 18-12 7.87
The peculiarities of the Hip joint muscles in Man are the
following:
1°. The enormous size of the Gluteus primus, and its com-
plete separation from the Tensor vaginæ femoris; in all the
Quadrumana, these two muscles form a continuous thin sheet.
D
2 D
402
ANIMAL MECHANICS.
2°. The complete separation of the iliacus and psoas
magnus.
3º. The separation of Glutæus secundus (a) and ().
4°. The absence of Glutæus quartus.
5º. The perfect differentiation of the adductors.
The broad differences in type will be best seen from the
following Table, in which the muscles are arranged in groups
as before:-
Muscular Types of Man and the Quadrumans-Hip
Joint.
PO

Animal.
Abductors.
Flexors.
Extensors. | Adductors.
Total
weight.
Oz. av.
93.17
1. Man, , . . .
2. Gorilla, . .
3. Chimpanzee,. .
4. Hamadryas, ..
5. Cynomulgus,. .
6. Mandrill,. . .
7. Rhesus, . . .
8. Ateles, . . . .
Per cent.
39 67
33 04
32.50
26.42
22.90
22.77
22,24
20.84
Per cene.
18.21
13. 12
17.92
18.00
Per cent.
16.68
24. 12
20.83
34.59
31.45
34.72
32.62
25.79
Per cent.
25.44
29.72
28.75
20.99
ning
27.
14.99
17.66
21.60
27.48
31.77
7.87
In this Table, it will be observed that the flexors and
adductors are much alike, and that man's superiority consists
in the increase of the abductors at the expense of the ex-
tensors. This is mainly due to the developement of the Glu-
tæus primus, which is essential to man's erect posture; and
to the diminution of the biceps femoris, which is essentially a
brute or quadrupedal muscle.
Taking together the Glutæus primus and tensor vagince
femoris, we obtain the following instructive comparison :-
ANIMAL MECHANICS.
403
Comparison of Man and Quadrumans-Glutceus primus and
Tensor vagince femoris.

Percentage, Relative Successive
of Hip
Propor- 1 Differ-
Joint
tions. ences.
Muscles.
100
.
74 6
566
45.3
33.0
25.4
43.4
11.3
1. Man, . . . .
2. Gorilla, ...
3. Chimpanzee,..
4. Hamadryas, . .
5. Rhesus, . . .
6. Cynomolgus, . .
7. Ateles, . . . .
8. Mandrill, . ..
.
.
oooww to A
26.1
gono
.
24. I
18. I
This Table shows that the difference between Man and the
Gorilla is greater than the differences between the Quadru-
mans compared with each other.
The remarkable differences between Man and the Quad-
rumans, found in the muscles of the Hip joint, are further
illustrated by an examination of the muscles of the foot.
These muscles may be divided into three distinct groups,
viz.:
1°. Walking muscles.
2°. Grasping muscles.
3º. Extending muscles.
So far as the presence or absence of individual muscles is
concerned, the peculiarities of Man's foot consist in-
1°. The presence of the peroncus tertius.
2°. The absence of peroncus quinti.
3°. The absence of extensor internodii hallucis.
The following Tables enable us to compare the relative
developement of the various muscles, or groups of muscles :-
2 D 2
404
ANIMAL MECHANICS.
Muscular Types of Man and Quadrumans-Foot.
1un
DO
WITO
112

Man.
Gorilla.
Chim-
panzee.
Hamam
dryas.
13.68
16.32
Walking Muscles.
1. Gastrocnemius, . . . . .
2. Solæus, . . . . . . .
3. Plantaris, . . . . . . .
4. Tibialis posticus, ....
14.
16.32
0.48
22.52
17.00
4. 20
8.60
8.49
9.18
4. 20
CA
2.28
4.82 | 17.94
1.48 | 4.24
0.51
8.16
| 15.30
, 0.58
Grasping Muscles.
1. Flexor digitorum longus; .
2. Flexor hallucis longus, ..
3. Flexor digitorum brevis, . .
4. Fl. dig. long. accessorius, .
Extending Muscles.
1. Tibialis anticus,. ....
2. Extensor digitorum longus, .
3. Extensor hallucis longus, .
4. Extensor secundus internodii
hallucis, . . . . . .
5. Peronæus longus, . . . .
6. Peronæus brevis,....
7. Peronæus tertius,
8. Peronæus quinti, ....
9.94
3.60
! 10.85
15. 19
2.83
10.20 14.51
6.121 4.96
3.06 1 1.71
3.63
5.53
3.38
2.04
8.16
4,08
6.10
4.19
2.60
Small.
100
100
100
100
Oz. av.
Total weight, ..
31-10
Oz. ay.
1:06
Oz. av.
1 0.98
Oz. av.
5.24
Taking the sums of the muscles in groups, we find
S
Muscular Types of Man and Quadrumans-Foot.

Walking.
Grasping. Extending.
Total
weight.
Per cent.
28.68
34.43
04. av.
31.10
1.06
33.66
0.98
1. Man, ...
2. Gorilla, ..
3. Chimpanzee, .
4. Hamadryas, .
5. Cynomolgus,.
6. Mandrill, . .
7. Rhesus, ..
8. Ateles,.
Per cent.
62:23
36.79
42.30
47.92
40.27
40.59
44.65
54.96
Per cent.
9.09
28.78
24.04
20.61
23.53
24.54
23.43
20,21
5.24
31.47
36.20
34.87
31.92
.
24.83
ANIMAL MECHANICS.
405
The following facts are evident from an inspection of this
Table:-
1°. The great developement of the walking muscles in Man.
2°. The great deficiency of the grasping (or hand-like)
action in the foot of Man.
We may also remark that in both these respects the New
World Monkeys come nearer to Man than either the Gorilla
or Chimpanzee.*
The deficiency of grasping action in the foot of Man,
stated as a physical constant characteristic of him, fully jus-
tifies the title of Quadrumanous applied by Cuvier to the
Apes, and denied by him to Man.
In the next Table, I have resolved the walking muscles
into their component parts :-
S
Walking Muscles of Man and Quadrumans.
IN

Gastro-
cnemido-
Solæus.
Tibialis
posticus.
Plantaris.
X
52.15
28.30
32.64
0
1. Man, . . . .
2. Gorilla, ...
3. Chimpanzee,. .
4. Hamadryas, . .
5. Cynomolgus, .
6. Mandrill, . . .
7. Rhesus . . .
8. Ateles, . ....
39.52
34.17
33.76
36.72
48.57
1.48
8.49
9 IR 0.48
4.20
2.71 13 39
3.14 3.69
3.14
6.39
S
If we now examine the muscles of the Shoulder joint and
Hand, we shall find the same profound differences between
Man and the Quadrumans as in the Hip joint and Foot.
In the Shoulder muscles, Man is characterised by the
presence of
1°. Levator anguli scapulæ ;
2º. Distinct Pectoralis major and minor;
* Professor Huxley, some years ago, directed my attention to the remarkable
power of standing and walking erect, possessed by the Spider Monkey; an obser-
vation fully confirmod on mechanical grounds by the foregoing Table.
406
y
ANIMAL MECHANICS.
and by the absence of
1° Occipital slip of Rhomboideus.
2º. Omo-Atlanticus.
3º. Tricipiti accessorius.
4°. Pectoralis quartus.
The following Tables show the relative developement of
the various muscles and groups of muscles :-
Muscular Types of Man and Quadrumans-Shoulder Joint.
Man. | Gorilla.
Chim-
panzee.
Hama-
dryas.
Per cent.
9.33
2.49
3.07
7.64
1.96
Per cent. Per cent. Per cent.
8.73
9.98 1.7.68
0.60 | 4.50 3.37
3.75
8.46
3.29
OO, WWAO
0.19
1.70
0.54
0.15
0.24
14.02
.
15.32
.
.
.
4. 21
I 96
8.98
3.49
2.49
3.99
1.00
7.45
2.75
4.54
6.04
0.77
4.56
1.54
3.91
1.20
Scapular Muscles,
1. Trapezius, . . . . . .
2. Cleidomastoides, .. .
3. Rhomboideus, ....
4. Serratus magnus, · · ...
5. Levator anguli scapulæ, ..
6. Omo-Atlanticus, . . . .
7. Subclavius, . . . . . .
Abductor's.
1. Deltoideus,
2. Teres major,
3. Supraspinatus, . . . . .
4. Infraspinatus, . . . . .
5. Teres minor, . .....
. Flexors.
1. Triceps longus, . . . .
2. Tricipiti accessorius, ...
3. Subscapularis, . . . . .
4. Latissimus dorsi,
Extensors.
1. Biceps humeri, ..
2. Brachiæus, , . .
Adductors.
1. Coracobrachialis, . . . .
2. Pectoralis major, . . . .
3. Pectoralis minor, ...
4. Pectoralis quartus, . ...
8.95
3.99
0.74
5.99
12.22
9.04
Į 1.63
4.52
1.51
6.92
12.50

5.04
7 25
9.35
10.15
7.68
5.12
1
5.99
3.74
1.46
10.45
2.17
1.20
9.34
๐๒ ๒ ฯ
1.51
096
100
100
100
100
Total weight, ..
0%. av.
77.11
Oz. av.
3.32
Oz. av.
4.01
v2.&V.
25.80
ANIMAL MECHANICS.
407
Taking the sums of the muscles in groups, I find the fol-
lowing Table :-
Muscular Types of Man and Quadrumans-- Shoulder Joint.

Scapu-
lar.
Abduc-
tors.
Flexors.
Exten-
sors.
Adduc-
tors.
Total
weight.
Per
Per Per Per Per
cent. cent. cent. cent. cent.
1. Man,, 24.68 28.10 23.34 9.80 14.08
2. Gorilla, .. 24.40
| 25.45
12.80 12.05
3. Chimpanzee, 33.17 1 19.95 22.94 | 9.73
4. Hamadryas,
5. Cynomoigus,
12.15 19.54
6. Mandrill, . .
14.07 15.50
7. Rhesus, ..!
| 31.46 11.42
8. Ateles, . . i 18.
| 21.93 | 31.86 | 14.86 | 12.41
28.07
10.73
14.21
14.61
16.50
1 7.34
On examining these Tables, we find in Man and the
Gorilla a remarkable developement of the abductor muscles,
due mainly to the great size of the Deltoid muscle, which
corresponds to the Glutæus maximus in the hip. This is evi-
dent from the following Table :-
S
Comparison of the Deltoid Muscle in Man and the
Quadrumans.

Percent-
age of Relative
Shoulder | Proportion.
Muscles,
Succes-
sive Dif-
ferences.
15.32
100 00
91.51
1, Man, , . . .
2. Gorilla, . . .
3. Ateles,. . . .
4. Chimpanzee,. .
5. Hamadryas, ..
6. Mandrill,. . .
7. Rhesus, ..
8. Cynomolgus,. .
48 63
ovision
46.67
| 36.62
32.51
8.74
10. I 2
1.96
4.76
5.29
41.91
This Table shows that the Gorilla and Spider Monkey
approach nearer to Man, with regard to relative size of the
Deltoid muscle, than the other Quadrumans do.
In examining the hand of Man and the Quadrumans, the
following well-known differences are found:-
408,
ANIMAL MECHANICS.
.
1n
1°. Man posseses a distinct flexor pollicis longus,* which
does not occur in any Quadruman.
2°. Man possesses two distinct extensors of the phalanges of
the thumb; while the Quadrumans possess either one or none.
3º. In Man the Indicator belongs to the second finger
only; while in the Quadrumans it belongs to the second and
third fingers jointly.
4º. In Man the Auricularis belongs to the fifth finger only;
the Chimpanzee and Ateles resemble Man in this respect;
and in the other Quadrumans, the Auricularis belongs to the
fourth and fifth fingers jointly. .
The following Tables show the relative developement of
the various muscles and groups of muscles :-
Muscular Types of Man and Quadrumans—Hand.

Man.
Gorilla.
Chim-
panzee.
Hama-
dryas.
Per cent. Per cent. Per cent. Per cent.
5 40 7.17 17.778.24
8.06 9.73 689 / 16.49
2.58 0.75 l 1.72 3.59
17.71 l 18.92
21.53 lo
14.65
34.49
12.02
| 28.
4.24 ] | 33.25
Walking Muscles.
1. Flexor carpi radialis, .
2. Flexor carpi ulnaris, ...
3. Palmaris longus, . . . .
Grasping Muscles.
1. Flexor digitorum sublimis, .
2. Flexor digitorum profundus,
3. Flexor pollicis longus, .
Extending Muscles.
1. Extensor carpi radialis (a), .
2. Extensor carpi radialis (B),
3. Extensor carpi ulnaris, . .
4. Extensor digitorum longus, .
5. Auricularis, . . . . .
6. Extensor assis metacarpi pol-
licis, . . . . . . . .
7. Extensor primi internodii
pollicis, . . . . . .
8. Extensor secundi internodii
pollicis, . . . . . . .
9. Indicator, . . . . . . .
2.56
6.03
5.12
aco woo wiò
ci is too it
5.17
3.45
9 49
1.72
4.31
ܚ ܩܟ ܩܩܳܙ
1 62
4.48
0.67
2.41
1.53
1.02
2 59
1.72
0.90
1.08
1.66
100
100
100
100
Total weight, ..
Oz, av.
12:03
Oz. av.
0.98
1 Oz. av.
Oz. ay.
5.58
1-16
* For a remarkable exception to this rule, vide pp. 122–3.
ANIMAL MECHANICS. .
409
Taking the sums of the muscles in groups, I find the fol-
lowing results :-
Muscular Types of Man and the Quadrumans-Hand.


Walking. | Grasping. Extending.
Total
weight.
Per cent.
Per cent.
40-48
43.48
30.18
1. Man, . . . .
Gorilla, ..
3. Chimpanzee, . .
4. Hamadryas, ..
5. Cynomolgus, . .
6. Mandrill, . . .
7. Rhesus, . . .
8. Ateles,. . . .
Per cent.
16.04
17.65
16.38
28.32
15.54
19.15
22.77
16.24
52.17
49.14
40.15
53.38
34 48
31.53
31.08
30.85
30.55
28.94
Oz. av.
12.03
098
1.16
5.58
1.48
4.44
3.47
1.97
From this Table it appears that the chief muscular peculi-
arity of Man's hand consists in the great developement of the
extensor muscles; and on analysing these muscles it will be
found that it is the extensors of the wrist that are principally
developed.
Wrist Extensors of Man and the Quadrumans.

Extensor Extensor
carpi ra carpi ra-
dialis (a). dialis (B).
Extensor
carpi
ulnaris.
Total.
9.65
Per cent. Per cent.
6.48
5.12
6.03
5.01
2.56
1. Man, . . . .
2. Gorilla, . . .
3. Chimpanzee,. .
4. Hamadryas, · ·
5. Cynomolgus, . .
6. Mandrill, . . .
7. Rhesus, . . .
g. Ateles, · · · ·
ܩ ܘܼ ܗ ܝܼ
bonbonis
Per cent.
5.65
4.60
3.45
5.55
5.40
4.95
Per cent.
21.78
12.28
14,65
16.64
15.53
16.88
17.86
13.19
3.80
fra nawe
6.30
ܘ ܡܼܲ ܀
19
The remarkable developement of the extensor muscles of
the wrist in Man is probably connected with the corre-
sponding developement of the Deltoid muscle in the shoulder,
and the combination of both is essential to characteristic
human action-such, for example, as stone-throwing and
oratorical gestures.
It is interesting to observe that the Gorilla approaches
410
ANIMAL MECHANICS,
Man in the shoulder muscles more nearly than the Chimpan-
zee does, while he deviates farther from the hand of Man
than does the Chimpanzee.
If we take the walking muscles of the hand and foot as
our standard of comparison, and inquire the proportion which
the grasping muscles bear to them, we shall obtain the follow-
ing instructive Tables, which show, in a remarkable way, the
profound difference between a hand and a foot, even in Qua-
drumanous animals :-
Comparison of Walking and Grasping Muscles in the Foot of
Man and the Quadrumans.

Walking Grasping
Muscles. Muscles.
100
14.6
1. Man, . . . .
2. Gorilla, . .
3. Chimpanzee,. .
4. Hamadryas, . .
5. Cynomolgus,. .
6. Mandrill, . . .
7. Rhesus, . . .
8. Ateles, . . . .
In the foot, the grasping power is less than the walking
power.
Comparison of Walking and Grasping Muscles in the Hand of
Man and the Quadrumans.

Walking
Muscles.
Grasping
Muscles.
1. Man, . . . .
2. Gorilla, ..
3. Chimpanzee, . .
4. Hamadryas, ..
5. Cynomolgus, . .
6. Mandrill,. . .
7. Rhesus, . . .
8. Ateles,. . . .
271.0
295.6
300.0
141.7
343.5
261.1
205.0
337.5
100
100
100
In the hand, the grasping power is greater than the walk-
ing power.
ANIMAL MECHANICS.
bermond
These Tables show-
1°. The deficiency of grasping power in feet as compared
with hands.
2° The enormous developement of the walking power in
the foot of Man.
These results possess considerable interest when taken in
conjunction with the results previously obtained by me, in
discussing the differences in the flexor tendons of the hand
and foot. It may be worth while to note, in connexion with
the exceptional position of the Llama, p. 134 (which proves
that the hand of this animal is a true foot), that the propor-
tions of the walking and grasping muscles completely confirms
this view.
Walking Grasping
Power. Power.
Llama (Hand), ... 100 90.3
Taking, as before, the walking muscles of the hand as our
standard of comparison, we find the following proportions for
the extensors of the wrist :-
Comparison of Walking and Extending Muscles in the Wrist of
Mn and the Quadrumans.
Walking Extensors
Muscles. of Wrist.
100
135.7
100
1710
89.5
100
1. Man, ..
2. Gorilla, . .
3. Chimpanzee, .
4. Hamadryas, .
5. Cyn::molgus,..
6. Mandrill,. .
7. Rhesus, . .
8. Ateles, . . .
58.8
100.0
The next example of muscular type that I shall give, is
the remarkable case of the hind limbs of the Struthionida.
412
ANIMAL MECHANICS.
I have had an opportunity of examining the following speci.
mens of this interesting group of Birds :-
1°. Three Ostriches.
2°. Four Emus.
3º. Two Rheas.
4º. One Cassowary
In the Hip joint of the Struthionidæ, the following peculi-
arities are observable :---
1°. The Agitator caudæ and Tensor femoris vaginæ form an
outer sheet of powerful muscles, which overlies the glutäal
muscles.
2°. The Glutæus quartus muscle is always present.
3º. The Iliacus muscle is also present.
4º. The Adductor tertius (brevis) is invariably absent.
5° The Rectus femoris is present in the Ostrich and Rhea,
and has relation in these, as in other birds, to the Solous
muscle, and to the perforate flexors of the most important toe.
This muscle is totally wanting in the Emu and Cassowary.
6º. The Biceps femoris acquires an extraordinary develope-
ment, intimately related to the unusual running power of this
remarkable group of birds.
The following Table shows the muscular type of the
Ostrich :-
ANIMAL MECHANICS.
413
Muscular Type of Ostrich-Hip Joint.

Male. | Female.
Female.
Mean.
Per cent. Per cent. Per cent. Per cent.
15.64
12 17
10.05 10.52)
Abductors.
1. Agitator caudæ, . . . .
2. Tensor vaginæ femoris, ..
3. Glutæus primus, · · · ·
4. Glutæus secundus, . . . .
5. Glutæus tertius, . . .
6. Glutæus quartus, . . .
7. Obturator internus, ...
6.70
0.51
0.89
9.04
o rioo
o o o x
oo mico
0.98
8.96
Flexors.
1. Rectus femoris, . . . . .
2. Sartorius, . . . . . .
3. Iliacus, .......
rico
noo o
3.02
8.86
rooi
1.13
Extensors.
1. Biceps femoris, . . . . .
2. Semitendinosus, . . . .
3. Semitendinoso-accessorius, .
4. Semimembranosus, ...
20.98
6.51
2.28
7.12
22.71 | 21.62
8.19 | 6.87
0.52 0.49
2.50
2.53
Adductors.
1. Adductor primus, . . . .
2. Adductor secundus, ...)
3. Adductor quartus, . . . .
4. Adductor quintus, . . .
5. Obturator externus, ..
oo na mi
6.76
3.84
8.68
2.43
1.29
0.88
8.03
2.89
1,82
1.22
2.14
1.37
100
100
100
100
Total weight, , .
Oz. av. 1 Oz. av. Oz. av.
196.75 133.96 | 284.61
In the next Table, I show the corresponding muscular
type of the Emu :-
414
ANIMAL MECHANICS,
Muscular Type of Emu-Hip Joint.

Male.
Male.
Male.
Young
Female.
Mean.
Abductors. Per cent. Per cent. | Per cent. Per cent. Per cent.
1. Agitator caudæ,. ..
23.68
2. Tensor vaginæ femoris, }
25.97
| 23.75 34.36
3i Glutæus primus, .. 1.99 1.25
1.47 1.50
4. Glutæus secundus, . . 10.15 8.58
5. Glutæus tertius, . . 0.40 0.29
13.32 0.34
6. Glutæus quartus, ..
2.19
1.95
7. Obturator internus, il 5.27
5.25 5.23
1.28
; ܝܺܝ ܘܗ̇ ܗ̇ ܝܿܢ
9.88
Flexors.
1. Rectus femoris, . . . , None. None. | None. | None. | None.
2. Sartorius, .... 12.34 | 13.00 | 12.81 | 9.96
11.91
3. Iliacus, .....
* L * 0.15 0.15
| 17.15
17:
Extensors.
1. Biceps femoris, ... | 18.71
2. Semitendinosus, . . 8.76
3. Semitendinoso-accesso-
1.19
rius, . . . .
4. Semimembranosus,...
9.97
20.51
4.47
I. IO
0,81
11.99
2.49
2.07
2.03
7.28
7.II
Adductors.
1. Adductor primus, ..
2. Adductor secundus, il
3. Adductor quartus, ..
4. Adductor quintus,..
5. Obturator externus, .
3.77
2.37
1.41
2.0i
WE
100
100
100
100 1 100
Oz. av.
50-25
Oz. ay.
72-30
Oz. av.
95.62
Oz. ay.
33-34
Total
The following Table shows the muscular type of the
Rhea :-
ANIMAL MECHANICS.
415
Muscular Type of Rhea—Hip Joint.

Male.
Female.
Mean.
Abductors.
1. Agitator caudæ, ...
2. Tensor femoris vaginæ, .
3. Glutæus primus, . . .
4. Glutæus secundus, . . .
5. Glutæus tertius, . . . .
6. Glutæus quartus, , . . .
7. Obturator internus, ...
Per cent. Per cent. Per cent.
23.12 25.54
0.89 1.20
9.61
1902
0.59
#ico ooo
0.89
8.65
0.90
8.09
Flexors.
1. Rectus femoris, . . . . .
2. Sartorius, ·
3. Iliacus, .
0.46 / 0.47
| 10.76 / 8.35
0.31 10,23
0.46
9.56
0.27
Extensors.
1. Biceps femoris, . . . . .
2. Semitendinosus, ...
3. Semitendinoso-accessorius, .
4. Semimembranosus, ...
*4.60 | 19.48 | 17.04
8. 34. 11.00
9.92
1,92 1,06
2.73
2.40
1.49
12.07
8 14
Adductors.
1. Adductor primus, . . . .
2. Adductor secundus, ...)
3. Adductor quartus, . . . .
4. Adductor quintus, . . . .
5. Obturator externus, . .
2.26
0,28
1.32
100
100
100
Oz. av.
Oz. av.
47.73
Total weight, ..
26'02
The muscular types of the Cassowary and of the other
Struthionidæ are given in the following Table :---
416
21
1
T
YTN.
ANIMAI, MECHANICS.
11
Muscular Types of Struthionidae-Hip Joint.

Ostrich.
Emu.
Rhea.
Casso-
wary.
24.04
.
Abductors. ·
1. Agitator caudæ, ....l
2. Tensor vaginæ femoris, . .1
3. Glutæus primus, ....
4. Glutæus secundus, . . . .
5. Glutæus tertius, ...
Glutæus quartus, . . . .
7. Obturator internus, ...
Per cent. Per cent. Per cent. Per cent.
24.33
25.08
1.49
10.90
0.60
0.32
1,01
2.53
pomidoooo
8.79
5.44
.
Flexors.
1. Rectus femoris, . . . . .
2. Sartorius, . . . . : :
3. Iliacus, . . . . . . .
1
noo mi
None.
II.91
0.15
0.461 None.
9.56 15.82
0.27
0.08
18.57
Extensors.
1. Biceps femoris, . . . . .
2. Semitendinosus, . . . .
3. Semitendinoso-accessorius, .
4. Semimembranosus, . . .
21.62
6.87
0.49
17.04
9.92
1.49
2.40
2.50
Adductors,
9.07
ง
1. Adductor primus, .
2. Adductor secundus, ..
3. Adductor quartus, . . .
4. Adductor quintus, . . .
5. Obturator externus, · ·
8.03
2.89
1.82
1.22
7.08
2,69
- H )
3.05
1.64
0.37
0
100
100
100
100
Oz. av.
Total weight, . . . 53.63
11
If we combine these results in groups as before, we find
the following results :-
ANIMAL MECHANICS.
417
Muscular Type of Struthionidae-Hip Joint.

Abductors.
Flexors.
Extensors. Adductors.
Per cent.
41.17
1. Ostrich, ...
2. Emu, · · · ·
3. Rhea, . ...
4. Cassowary, ..
43.26
Per cent.
13:39
12.06
10.29
15.90
Per cent.
31.48
29.60
30 85
25.10
Per cent.
13.96
15.08
14.13
13.24
44. 73
45.76
If we compare this Table with the preceding Tables, we
shall see that the peculiarities of the Struthionidæ consist in
the development of the abductor muscles (which give the
power of standing erect) and of the extensor muscles (which
measure the power of running). This will be better seen by
the following comparison of the most important individual
muscles, on which these actions mainly depend :-
Comparative View of the Standing and Running Powers of the
Struthionide and other Animals.

Ostrich.
Emu.
Rhea.
Casso-
wary.
Man.
“Mas-
ter Ma-
grath.” 1
Per
cent.
Per
cent.
Per
cent.
Per
cent.
Per
cent.
Per
Standing.
cent.
(Agitator caudæ, ..
1. Tensor vaginæ femoris, | 24.04
(Glutæus primus, ..
25.86
25.37 | 26.57 20.61 | 7.01
Running."
2. Biceps femoris, ... 21.62
18.57
17.04
15.40 5.50 21.23
In this Table, the contrast between Man and the Dog is
well brought out; and it is shown that the Struthionidæ
excel in both the respective peculiarities of Man and the Dog;
• 2 E
418
ANIMAL MECHANICS.
The last example that I shall give of the theory of mus-
cular types, is the remarkable case of the wings of birds,
which, for many reasons, is to be considered in detail.
The development of the great pectoral in birds has, of
course, been noticed by every anatomist; but the theory of
muscular types enables us to present it to the eye in a manner
that admits of immediate comparison with the corresponding
muscle in other animals; and to demonstrate the impossibility
of flight, in other animals not endowed with the peculiar mus-
cular type of birds.
The birds from which I have made my measurements are
the following :-
1. Albatross (two examples, measuring, respectively,
from tip to tip of extended wings, il ft. o in. and 9 ft. 3 in.).
2. Common Swan.
3. Dorking Cock.
4. Aylesbury Duck (two examples).
5. Gannet.
6. Curassow.
7. Jabiru (two examples).
8. Stork (two examples).
9. Common Heron.
10. Squacco Heron.
II. Flamingo.
12. White-Headed Eagle (two examples).
13. Grebe.
14. African Parrot.
15. Peregrine Falcon.
16. White Crane.
17. New Zealand Weka Rail (two examples).
ANIMAL MECHANICS.
419
Muscular Type of Birds' Wings.

II.
III.
VI.
Albatross
| Common / Dorking
(mean of
Swan. Cock.
two).
IV.
Aylesbury
Duck
(mean of
two).
Gannet. Curassow.
Scapular. Per cent. Per cent. Per cent.
1. Trapezius, ..
3.64
1.45 1 2.201
2. Rhomboideus, . 1,02 0.36 1.59
3. Serratus magnus, ' 3.04 0.83 0,61
Per cent. Per cent. Per cent.
2,28
2.05
1.84
I.10
1.39 | 2.36 | 0.74
Abductors.
1. Deltoideus, .
2. Pectoralis avium, |
3. Teres major, . .
4. Spinatus, ...)
5.07
5.11
1.43
2.03
2.73
19.94
0.53
3.75
11.28
0.49
twow
4.86
5.15
4.73
Flexors.
1. Triceps longus, . 1
2. Subscapularis, .
3. Latissimus dorsi,
w
4. 14
4.00
5.02
1.55
2.43 14.48
2.27 12.37
1 0.98
4.89
3.30
3.46
1.92
0.68
Extensors.
1. Biceps humeri, . | 1.84 | 3.57
2. Brachiæus ..1 0.80 1 1.24
2
| 2.61
0.90
0.63 3.94
| 0.63 1 0.24
L
Adductors.
1. Coraco-brachialis, 0.69 1 0.46
2. Pectoralis major, 62.67 | 62.90
3. Pectoralis minor. 1.76 1,65
0.76
35.42
1 2.58
10.90
61.96
11.88
1 0.95 2.83
1 64.42 | 55.04
| 3.14 | 2.58
100
100
100
1
100
100
1
100
100
Total weight, .
Oz. av.
14-22
02. av.
19.32
Oz. av.
13:19
| Oz. av. | Oz. av. | Oz. av.
6.13 6:35 4.07
2 E 2
420
ANIMAL MECHANICS.
Muscular Type of Birds' Wings (continued).

VII.
VIII.
IX.
Jabiru Stork
(mean of (mean of
two). I two).
Common Squacco
Heron. | Heron.
Flamingo.
ΧΙΙ.
White-
headed
Eagle
(mcan of
two).
0.85
8.45
woo on
too o
'0.2%
3.28
Scapular. Per cent. Per cent. Per cent. Per cent. Per cent. Per cent.
1. Trapezius, . 1.16 1,32 1.64 1.68 10.81 1.45
2. Rhomboideus, .1 0.82
0.62
0.97 1.01 0.81
3. Serratus magnus, 1.14
0.59
2.70 1.17 | 0.81
1.54
Abductors.
1. Deltoideus, ..
3.38 3.02 7.50
2. Pectoralis avium,
7.45 7.89 8.42 6.09
3. Teres major,..!
0.33 0.61 1.19
4. Spinatus, · · ·
3.02 4.05 4.75
Flexors.
1. Triceps longus, . 1 5.61
2.511
2. Subscapularis, l 1.72 3.12 2. I2 1.68 1.83
3. Latissimus dorsi, 1.17
0.93 1: 0.97 i 0.50 1.12
Extensors.
1. Biceps humeri, . | 2.661 2.31 1 2.99 12.02
2.02 1 2.84 I 4.55
2. Brachiæus, ..1 0.43 0.79 10.391 0.50 0.71 l 0.85
Adductors.
1. Coraco-brachialis, 2.20 1.39 1.16 10.50 1.32
2. Pectoralis major, | 62.06 65.18 67.54 72.83
| 59.
3. Pectoralis miñor, i
1.84
1.50
1.45
| 1.34 1.12 1.57
4.56
5.58
ön o
62.47
N
100
100
100
100
100
100
Oz. av.
Total weight, .
21:45
02. av.
12.89
Oz. av.
5.18
Oz. av.
2.98
Oz. av.
4-93
Oz. ay.
8.89
The first fact that strikes us, on examining the foregoing
Tables, is the extraordinary development of the great pectoral
muscle, which in many cases absorbs two-thirds of the entire
muscular force of the shoulder joint. This muscle is the
great depressor of the wing, and I have already shown what
extraordinary care is bestowed upon the placing of the socket
11
ANIMAL MECHANICS.
421
of the joint in such a position as to give the muscle the maxi-
mum amount of advantage in performing its work. The mus-
cles that lift the wing are the Deltoi l and Pectoralis avium
which latter corresponds with the Subcluvius in Mammals.
Combining together the Depressor and Levator muscles
I have found the following interesting results :-
Depressor and Levator Muscles in the Wings of Birds.
DEPRESSORS.
LEVATORS.
Ratio.
Pect. major
and
Pect. minor.
Deltoid
and
Pect. avium.
MS
IO.
vor stor
13.36
22.67
15.80 per cent.
20.70
39.09
2
·
·
8.82
19.90
14.74
12.81
10.83
1. Albatross, . . . . .
2. Swan,. . . . . . .
3. Dorking Fowl,. ,.
4. Aylesbury Duck, . .
5. Gannet, . . . .
6. Curassow, . .
7. Jabiru, .. . .
8. Stork, .. . .
9. Common Heron,
10. Squacco Heron,
11. Flamingo,.
12. White-headed Eagle, .
13. Grebe, . . . . . .
14. African Parrot,. ...
15. Falcon, . . . . . .
16. White Crane, . . . .
17. Weka Rail, . . . ,
13.0
34.54
23.07
19 21
15.70
·
·
Luig
.
.
·
66.68
68.99
74.17
63.59
·
14.71
25.04
19
15.92
I4. 54
Il 26
14.84
23.08
75.82
69.14
59.27
54.90
17.22
20,95
20.00
28
The birds in which the great pectoral becomes least are
fiers, and the Weka Rail, which uses its wings principally in
propelling itself through the long grass. The birds of
greatest soaring power are those in which the levator muscles
bear the least proportion to the depressor muscles, viz., the
Gannet, Grebe, Heron, and Albatross.
If we arrange the muscles into groups, as before, we find
the following Table :-
422
ANIMAL MECHANICS.
Muscular Type of Wings of Birds.

Scapu-| Abduc- Flexors.
Exten-
lar.
tors.
sors.
tors.
Total
weight.
1. Albatross
Per
cent.
7.70
2.64
.
.
.
.
.
.
.
.
.
Per Per
Per
Per
cent. cent. cent.
Oz. av.
10.90 2.63 65.13 14.22
19.05 8.49 4.81
119.32
28.35 | 5.38 3. II 58.76 13.19
20.25 17.83 3.51
| 6.13
1.26
26.29 | 6.504.18 60.45 4.07
19.20 | 8.50 | 3.09 66.09 (21.45
17.68 | 8.62 3.10 .68.07 12.89
14.40 6.76 3.38 70.15 / 5.18
14.26 | 4.69 74.67 | 2.98
20.58 / 8.53 3.55 | 64.91 | 4.93
20.47 | 7.58 |
62.70 8.89
13.93 6 97 1.23 | 75.82 2.44
OSS,
2. Common Swan,. . .
3. Dorking Cock, . .
4. Aylesbury Duck, .
5. Gannet,
6. Curassow,
7. Jabiru,
8. Stork, .
9. Common Heron, .
1o. Squacco Heron,
11. Flamingo, .
12. White-headed Eagle,
13. Grebe, . . .
14. African Parrot,...
15. Falcon, . . . . .
16. White Crane, ...
17. Weka Rail, . ..
.
جنب ب ب ب ب ب ب ج تب جب مجہ
2.52
22.189.27 | 3.97 | 60.28 | 1.51
23.88 7.69
55.97 | 1.89
29.56 13.04 1 6.95
0.58
If we compare the adductor muscles of any of the birds
(ranging from 40 to 76 per cent.) with the adductor muscles of
the Felidæ, Canidæ, Man, and the Quadrumans, already given,
we shall see at a glance how peculiar is the type of the mus-
cular arrangements of the wing of the Bird, and how hopeless
it would be to expect that any of the other types could attempt
to fly, even if aided by mechanical contrivances to increase
the surface exposed to the air. And yet the muscular type
of the wing of the Bird, differing profoundly as it does from
the corresponding type of the fore-limbs of the other animals,
is composed of groups of muscles bearing the same names,
and fulfilling similar functions.
The skilful artizan can produce from the same number of
wheels and pinions either a clock or a roasting jack, fulfilling
the very different functions of marking time, and of roasting
meat. An ignorant but intelligent savage, who was shown
the interior of these machines, would come to the conclusion
ANIMAL MECHANICS,
423
that they were very like each other-simply because he would
consider only their superficial resemblances; and would be
unable to appreciate the purposes which the machines were
respectively intended to fulfil. In like manner, anatomists,
from observation of apparent resemblances in the structure of
organs, such as the brain (of the specific action of whose parts
little or nothing is known), have sometimes, rashly, inferred a
greater degree of affinity between various animals than there
is any logical ground for admitting. If we confine our atten-
tion to the arrangement of muscles and bones, the objects and
uses of which are perfectly known and understood, we may
readily perceive that, under a superficial appearance of simi-
larity of parts, there really exists a profound difference of
function and intention as to the purposes to which these
organs are devoted. In the present state of anatomical
science, it seems safer to collect facts and physical constants,
characteristic of each group of organs in various animals, than
to indulge in premature speculations as to possible modes of
production of such organs. In the language of the schoolinen,
the Final Cause of the wing of a bird is to fly in air, and the
Final Cause of the fore-limb of an Otter or Seal is to swim in
water; and we can show that the intention of the organ is, in
each case, perfectly fulfilled; but it appears unscientific to
speculate as to the possible derivation of one of these struc-
tures from the other, by means of an hypothetical common
ancestor.
I shall now give the muscular type of the wings of two
* large Bats, Pteropus edulis and Epomophorus Whitii, for the
purpose of comparing it with the type of the wings of
birds :-
424
ANIMAL MECHANICS.
Muscular Type of Wings of Buts.

Pteropus
edulis.
Epomo-
phorus
Whitii.
1.80
11.80
Per cent.
5.00
2:14
II.43
2.14
1.43
2.99
4.58
2.00
2.39
.
1.43
1.44
Scapular.
Per cent.
1. Trapezius, · · · · · 4.36
2. Rhomboideus, . . . . .
3. Serratus magnus, . . . .
4. Levator anguli scapulæ,..
5. Omo-Atlanticus, . . . .
0.80
Abductors.
1. Deltoideus-
(a.) Clavicular, . . .
B.) Acromial, · · · ·
(7.) Scapular, . . . .
2. Occipito-pollicalis, . .
3. Subclavius, .
I 00
4. Teres major, , · ·
5. Supraspinatus, . . .
6. Infraspinatus, . . .
7. Teres minor,. · · ·
0.20
Flexors.
1. Triceps longus, . . . .
2. Subscapularis, ...
3. Latissimus dorsi, · · ·
Extensors.
1. Biceps humeri, . . . . . 1
2. Brachiæus, · · · · ...
0.60
Adductors.
1. Coraco-brachialis, ..... 1.00
2. Pectoralis major, .... 36.12
3. Pectoralis quartus, · · · 1.20
1.80
1.60
1.60
1.43
2.14
toni
11.18
ooo
2.86
35.01
2.85
100
100
Oz.av,
Oz. av.
0-70
Total weight, .. 2:51
This Table suggests many interesting reflections, arising
from the great number of muscles employed, as compared
with those of birds ; but we must pass on to other subjects,
noting merely the bird-like power of the adductor muscles.
In Swimming Animals we find, as might be expected, a
type of muscular structure of the fore-limb, which in many
respects approaches to that of the Flying Animals; for the
act of swimming resembles that of flying in water instead of
air. I have selected the following animals to illustrate the
swimming type of the fore limb :-
ANIMAL MECHANICS.
425 -
1. The Green Turtle.
2. The Hawk's Bill Turtle.
3. The Logger Head Turtle.
4. The Alligator (6 feet in length).
5. The Otter.
6. The Seal.
Muscular Type of Swimming Animals.

Hawk's- Logger- | Alliga- | Otter.
Green
Turtle.
bill
Turtle. |
Head
Turtle.
tors
Otter,
Seal.
111.86
7.92
XN
7.67
0.52
3.09
Scapular.
Per cent. Per cent. Per cent. Per cent. Per cent. Per cent.
1. Trapezius, . '. ...
11.60 14.88 6.61
'1.62
1, 12
2. Rhomboideus, ...
10.57 12.18 | 3.77 1 5.19
0.57
3. Serratus magnus,
2,28 | 4.86
17.44
| 20.12 |11.98
4. Omo-Atlanticus, . . .
0.77
2.31 1.2.67
Cleidomastoideus, ..
1.67 17.30
6. Levator anguli scapulve,
1.72
Abductors.
1. Deltoideus, . . . . 5.71 5.81 5.86
3.50
2. Pectoralis avium, .. 4.62 | 5.57
4.37
3. Teres major, .... 2.88 3.04
,1.35
4. Supraspinatus, ..
7.39 4.55 1.36
(4.35
5. Infraspinatus, . .
N1.80
6. Teres minor, . . . .]
Trace.
Flexors.
1. Triceps longus,. 0.27 | 0.26 | 0.30 [ 8.11 | 4.42
2. Subscapularis, . . . 110.39 | 1,72 14 29 i 4.08 | 5.31 (15.41
3. Latissimus dorsi,. . .1 6.26 | 5. ing 15.47 4.61 | 3.68 2.19
4. Tricipiti accessorius, .
| 1.91
1.64
Extensors.
1. Biceps humeri, ...
7.07 2.23 1.64 0.95
2. Brachiæus, · · · · ·11.41 3.34
12,07
2.48 1 1 30 0.73
Adductors.
1. Coraco-humeralis, ..!
16.00
8.3311
31 2.77
2. Coraco-capsularis, .. 0.33
0.501
3. Pectoralis major-
(a.) . . . . .
27.22
18.13)
2. 22
2.70|3730 20.64 25.60
4. Pectoralis minor, ..112.52 11.73
5. Pectoralis quartus, · ·
10.39 4.74
.II | 7.80
4.26
26.64
(B
)
.
.
.
.
.
[ 13.76)
100
100
100
| 100 | 100
| 100
Oz. ay. | Oz. av. | Oz. av. | Uz, av.
Total weight, . . 10513 | 24-69 162.66 20:59
Uz. &V. | Oz, av.
18-84 24.72
426
ANIMAL MECHANICS.
The relations among the types of these animals may be
seen from the following Table, in which the muscles are
arranged in groups, as before :-
Muscular Types of Flying and Swimming Animals.

Scapular. | Abductor.
Flexor.
Extensor.
Adductor.
Per cent.
18.76
22. 11
3.40
Per cent.
12.98
13.57
16.92
1. Pteropus edulis, . .
2. Epomophorus Whitii,
3. Green Turtle, ..
4. Hawk's-Bill Turtle,
5. Logger-Head Turtle,
6. Alligator, . . . .
7. Otter, . . . . .
8. Seal, . . . . .
Per cent.
18.16
14. 29
16.30
21.81
18.60
14.52
I1.00
10.75
Per cent.
11.78
9.28
3.67
13.37
6.48
Per cent.
38.32
40.72
57.71
51.19
43.42
a. 14
4.71
8.78
23.90
36.33
33.57
49.07
2.94
31.03
1.68
30.34
The muscular types of these various animals are, of
course, very diverse; but they all agree in the development
of the adductor muscles used in flying or swimming ; and
no other animals, except birds, present an approach to this
type.
I shall conclude this section on muscular types by a few
observations on the “undeveloped” fore-limbs of the Stru-
thionidæ, as I believe that a profound study of the muscular
types of these limbs leads us up to the conclusion, that they
could never have been intended for wings at all, but are pre-
pared for uses quite distinct from flying.
As the question is one of some theoretical interest, I shall
use all the materials at my disposal, and allow my readers to
draw their own conclusions from a consideration of all the
facts I am able to lay before them.
As it is a well known, although unexplained fact, that the
Ostrich and Rhea form a natural group on the one hand, con-
trasted with the Emu and Cassowary on the other,* I shall
discuss the muscular types of their wings separately.
* Among muscular peculiarities not noticed by previous anatomists, I may
mention that in the Ostrich and Rhea, the Obturator intcrnus muscle is single,
ANIMAL MECHANICS.
427
Muscular Type of Wings of Ostrich and Rhea.

Rhea,
Ostrich, Ostrich,
No. 1 No.2"
(female). (female).
Rhea,
No. 1
(male).
Rhea,
No. 2
(female).
No. 3
(male).
Per cent.
4.75
3.27
1.65
Per cent.
2.71
1.93
2 97
1,16
2,62
2,13
co no *
Locom
5747
Scapular.
Per cent. Per cent. ' Per cent.
1. Trapezius, . . . . 4.14
3.50
2. Rhomboideus, . . .
2 06
0.54 0.87
3. Serratus magnus, . .
3.93
Abductors.
1. Deltoideus, . . . .
2. Pectoralis avium, . .3.53
3. Teres major, ... 1 13.58
4. Spinatus, .....
Flexors.
1. Triceps longus, .. 2.97 4.47
2. Subscapularis, . . . 12.37 10.78
3. Latissimus dorsi, ... 5.15 4.03
Extensor's.
1. Biceps humeri, ...
4.581 | 6
2. Brachiæus, · · · ·
1.52
Adductors.
1. Coraco-brachialis, . 1.41
6.99
2. Pectoralis major,..
11.46 )
13.07 11.79
3. Pectoralis minor, ..
0.55 }
9.67
4.50
10.90
5.69
6.55
دی به
6.88
1.18
7.35
1.29
1.09
3:32
13.50
9.93
| 10 20
100 1 100
1 100
| 100
100
Total weight, ..
oz. av.
19.82
Oz. av.
9.18
Oz. av.
2.29
Oz. av.
4.22
Oz. av.
3.88
-
-
It will be seen from this Table that the Ostrich and Rhea
belong to distinct types, well marked, but agreeing with each
i
as in other birds, and possesses a single tendon only; whereas in the Emu and
Cassowary the Obturator internus is double, viz. :-
(a.) — with puboischial origin, and inserted an inch higher up than (1)
on the femur, so that the tendons cross each other.
(B.) — with ilioischial origin.
These muscles are quite distinct in their actions, and must have some unknown
relations to peculiarities in the gait and habit of the Emu and Cassowary, which
do not occur in the Ostrich and Rhea. This muscular peculiarity alone place
the Emu and Cassowary far remote from all other birds.
428
III
ANIMAL MECHANICS.
other in a manner that shows a wide variation from the type
of ordinary wings. This will appear better by arranging the
muscles, as before, into groups.
Muscular Type of Wings of Ostrich and Rhea.

Scapular. Abductor.
Flexor.
Extensor. Adductor.
Per cent.
Ostrich, No. 1,. . .
Ostrich, No. 2, . . .
Rhea, No. 1, ...
Rhea, No. 2, , , ,
Rhea, No. 3, ...
5.24
Per cent.
53.49
52.72
44. IO
sino con
Per cent.
20.49
19.28
23.58
21.09
6.10
6.55
8.06
Per cent.
13.42
14. 16
18.78
45.50
16.82
50.46
ထဲထဲ
20.
On comparing this Table with that already given for com-
mon birds, we find the columns of abductor and adductor
muscles precisely reversed : so that it may be asserted that
the wings of the Ostrich and Rhea are constructed on a type
the very opposite of that of the wings of common birds. The
wings recorded in the preceding Table are utterly incapable
of being used in flight, and it is extremely difficult to under-
stand how they could possibly represent common wings altered
from want of use, and so undeveloped.
This argument becomes stronger from an examination of
the wings of the Emu and Cassowary, which are formed on a
type quite different from that of the Ostrich and Rhea, and
equally irreconcileable with the type of ordinary wings. This
is shown in the following Table :-
ANIMAL MECHANICS.
429
Muscular Type of Wings of Emu and Cassowary.

Emu,
No. 1
(male).
Emu,
No. 2
(male).
Casso-
(male).
wary,
Per cent. Per cent.
27.56 | 26.41
12.24 10.38
8.49
Per cent.
25.74
9.90
14.85
7.55
3.77
Scapular.
1. Trapezius, . . . . . .
2. Rhomboideus, .....
3. Serratus magnus, · ·..
Abductors.
1. Deltoideus, . . . . . .
2. Pectoralis avium, ...
3. Teres major, . . . . . .
4. Spinatus, . . . . . ..
Flexors.
1. Triceps longus, . . . .
2. Subscapularis, . ....
3. Latissimus dorsi, ....
Extensors.
1. Biceps,
. . .
2. Brachiæus,
dvomi
6.61
0.94
7.55
0.94
7.55
10.89
1
1.02
3.77
)
1.00
Adductors.
1. Coraco-brachialis, . . . .
2. Pectoralis major, 1
3. Pectoralis minor, . . .
5.10
8.02
8.02
1.98
| 6.93
5.10
100
. 100
100
Oz. av.
Total weight, ..
0.98
Oz. av. Oz. av.
1.06 11.01
On arranging the muscles in groups, as before, we find the
following Table :
Muscular Type of Wings of Emu and Cassowary.

Scapular.
Abductor.
Flexor.
Extensor. Adductor.
Per cent.
54.08
Emu, No. 1, . . .
Emu, No. 2, . . .
Cassowary, . . .
Per cent.
21.43
18.87
25.74
Per cent.
13.27
16.04
13.86
Per cent.
1.02
3.77
1.00
Per cent.
10.20
16.04
45.28
50.49
8.91
430
ANIMAL MECHANICS.
The agreement in muscular type here shown between the
Emu and Cassowary is very interesting, and corroborates the
evidence as to their similarity drawn from other sources; but
the muscular type here given is totally different from that of
the Ostrich and Rhea, and equally irreconcileable with that of
ordinary birds. ..
From the foregoing facts, I think we are entitled to con-
clude that the fore limbs of the Struthionidæ were never
intended to be used as wings, and that they are constructed
for some other purposes, which differ essentially in the two
groups, into which these remarkable birds may be divided.
12. Principle of least-Action applied to the Arrangement of
the muscular Fibres of the Heart.—At page 137 and following
pages, I have shown the amount of work done by the heart;
and I propose here to inquire how that work is done, and to
show that it is done in conformity with the principle of
“ least action.”
The heart is composed of innumerable muscular fibres
arranged in what Borelli calls a glomifile manner, similar to a
ball of twine. More correctly, we might compare the heart
to two balls of twine, each having a cavity in its centre, and
both surrounded by a third ball of twine enveloping the whole
structure.
The muscular fibres of the heart are, however, not
continuous, as this illustration supposes, but are indepen-
dent of each other, the appearance of the whole being
glomifile.
The law of muscular contraction which has to be fulfilled
is the following: if I denote the length of a single fibre before
contraction, and l' its length after contraction, whenever the
muscular fibre is perfectly free, it is found that l' bears a con-
stant ratio to l; or
1' = nl.
(109)
ANIMAL MECHANICS.
431
10
WU11
If, from any cause, the fibre be prevented from contract-
ing to its full extent represented by (109), there is a loss of
work which might otherwise be done, and the principle of
least action is violated. We may, therefore, state the prin-
ciple of “least action,” as applied to the heart, in the follow-
ing manner :-
The arrangement of the fibres of the heart must be such as to
allow each fibre to contract to the fullest extent required by the
law of muscular contraction.
Before proceeding to prove that the actual arrangement
of the fibres of the heart is such as will satisfy the foregoing
Law, it will be worth while to show by an illustration how
easy it would be to devise a heart that would not satisfy the
Law, and would, therefore, be an imperfect contrivance.
The special work done by the heart is to contract itself
upon an internal cavity containing fluid at a given pressure,
and so to expel that fluid from the cavity. Let us imagine
that it is proposed to do this work by a number of concentric
spheres of muscular fibres placed over each other, the fibres
of each sphere being quite independent of those of the other
spheres. Let V denote the volume of the ventricles of the
heart distended with blood at the commencement of the
systole, which volume is reduced to zero at the close of the
contraction ; let r denote the radius of any sphere of muscular
fibres at the commencement, and pol the radius of the same
sphere at the end of the systole. It is obvious that r, go are
proportional to l, l', the lengths of the fibres of the sphere in
question before and after the systole. Now, as V denotes
the difference in volume of the sphere before and after its
contraction, we have
TY
PM
LO
V = 47 (73 – p3),
3
god = nr.
432
.
ANIMAL MECHANICSMT
From these equations, it is easy to see, after a few reduc-
tions, that
3
n =
1
(110)
V
1 x 30
4703
4TVU
This equation establishes a relation between n and r ; that
is, between the coefficient of muscular contraction and the
radius of the muscular sphere, and shows that if n be taken
for the fullest amount of contraction possible to the fibre, the
value of becomes fixed, and there is only one sphere of
muscular fibres that does the whole of the work possible
for it.
Solving equation (110) for r, we find
r =
30
(II)
1471 (I – no)
As an example of the foregoing we may take the human
heart, in which the united volumes of the two ventricles is
10 cubic inches; and assuming that each fibre contracts
through one-ninth of its length, we have
S
V = 1c,
8
n
=
Introducing these values into (111), we find
p = 2.002 in.
Hence, in the case supposed, the muscular sphere whose
radius is 2 inches, does its full work; the spheres of greater
radius do less than their full work, and the spheres of lesser
radius are called upon to do more than their full work. In
fact, in order to make a heart of the structure supposed a
perfect mechanism, it would be necessary to have a different
law of muscular contraction for each sphere of fibres; the
ANIMAL MECHANICS.
433
amount of contraction n proper to each sphere being deter-
mined as a function of r by means of equation (110).
We are familiar, in the construction of large guns, with
an instance that may be regarded as the converse of the sup-
posed case of the heart just given. A large piece of ordnance
cannot be made strong enough to resist the explosive force
of a heavy charge of powder, merely by increasing the thick-
ness of the gun; for according to principles similar to those
employed above (using square roots instead of cube roots), it
can be shown that the inner rings of the gun are called upon
to do more than their share of the work of resistance, while
the outer rings are called upon to do less than their share.
Hence it follows that the inside portion of a very large gun
might be torn up by the force of the explosion before the
outside portion had suffered any strain at all. In order to
prevent this, and to compel every portion of the gun to
take its proper share of the work of resistance, large guns
have been made in separate rings, the outermost rings being
shrunk on while red hot, so that when cold the inner rings
are in a state of compression, and the outer rings are in a
state of tension. When the explosion now takes place, the
work done by the outer rings is represented by the compres-
sion of the inner rings, and these latter take no share in the
work of resistance until the force of the explosion has
destroyed their state of compression caused by the resistance
of the outer rings; and by this ingenious contrivance, an
attempt is made to compel every particle of metal of which
the gun is composed to bear its own individual fair share of
the work to be done.
The celebrated Armstrong 600 pounder gun was formed
in this manner of eight distinct rings, and when the charges
of powder were gradually increased until the gun was burst,
it was found that the ist, 6th, and 8th rings were burst, the
remaining five being apparently uninjured. If the principle
S
2 F
434
ANIMAL MECHANICS.
(7
aimed at in the construction of the gun could have been
completely fulfilled in practice, all the eight rings would have
been ruptured simultaneously when the bursting charge of
the gun was reached.
Having shown by the preceding example the conditions
required by the principle of “ Least Action” when applied to
the heart, I shall now describe the actual arrangement of the
fibres of the heart, and demonstrate that the law of least
action is completely fulfilled by them. The heart has been
compared by Borelli and Winslow to two muscles enclosed
in a third, and the fibres composing these three muscles may
be classified as follows:-
1°. Common fibres passing round both ventricles.
2°. Proper fibres passing round the left ventricle.
3º. Proper fibres passing round the right ventricle.
I shall commence by describing the course of the common
fibres. The muscular fibres of the auricles and ventricles,
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forming quite distinct systems, both take origin from the
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ANIMAL MECHANICS.
435
common tendinous zone which surrounds all the orifices of
the heart, with the exception of that of the pulmonary
artery. This tendinous zone is shown in Fig. 100.* *
In this figure (AA) is the great central fibro-cartilage,
situated upon the septum with which it is incorporated, be-
tween the mitral and tricuspid orifices, and behind the right
posterior sinus of the aorta.
The left fibro-cartilage (B) which is much smaller than
the central fibro-cartilage, is seated to the left of the mitral
orifice, and behind the left posterior sinus, with which it is
closely attached.
At (CC) is shown the strong fibrous loop situated in front
and to the left of the root of the aorta, stretching from the
left to the central fibro-cartilage, with both of which it is in-
timately incorporated.
At D is shown the tendinous ring of the mitral orifice,
stretching from the left to the central fibro-cartilage.
The tendinous ring of the tricuspid orifice is shown at E,
attached at both its extremities to the central fibro-cartilage.
Lastly, at Fis shown the insular fibrous ring that surrounds
the orilice of the pulmonary artery; and is but slightly con-
nected with the central mass of fibro-cartilaginous structure.
All the muscular fibres of the ventricles take origin from
the tendinous zone, and having wound round the heart in the
manner presently to be described, either return directly to
the tendinous zone, or indirectly through the intervention of
the columnæ carnece and papillary muscles.
The common fibres of the heart arise from the outer side
of the tendinous ring, and from the outer surface of the
heart, as shown in Fig. 101,+ which represents the posterior
and anterior aspects of the dissected heart. The general
* Copied by permission from Dr. Sibson's Medical Anatomy.
+ This figure is taken, by permission, from Dr. Sibson's Medical Anatomy.
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appearance of the bundles of common fibres is well compared
by Dr. Sibson to
the overlapping
capes of a coach-
man's great cout
when twisted. As
each layer proceeds
from right to left,
it twists upon itself
and at length dips
underneath the
layers to which it
was superficial at
the beginning of
its course.
Posterior
The following
observations and
measurements were
made on the heart
of the ox, several
preparations of
which were placed
at my disposal by
Dr. Sibson, and se-
veral others inade
by myself were ex-
amined while fresh
after long continued
boiling. On tracing
the course of the
Anterior
superficial common
fibres, it is found that
Fig. 101.
they describe a complete circumference in a spiral line previous
to entering deeper into the heart in the neighbourhood of the


R
DAHA
Intern
.
.
9.
.
Med
.
probleme
Will
DIY
ow
h
S
.
ANIMAL MECHANICS.
437
vortex, from which point they return almost in a straight line
towards the tendinous zone from which they originally
started, but on the return path they now form the innermost
wall of the ventricle; so that these fibres during their spiral
course are the outermost, and during their direct course are
the innermost fibres of the heart.
The fibres which lie inside the superficial fibres, after de-
scribing a complete spiral, plunge into the interior of the
heart before reaching the vortex, and return back towards the
tendinous zone almost in a straight line, lying outside the
first group of fibres which line the cavity of the ventricle.
There is thus a sort of bag-shaped cavity formed round each
ventricle, between the outside of the heart and the walls of
the ventricle.
These bag-shaped cavities are now to be supposed occu-
pied by the proper fibres of each ventricle. I was able to
trace the complete course of the proper fibres of the left ven.
tricle, filling up the space between the spiral and direct por-
tions of the common fibres, and I found that the proper fibres
wind in a spiral line round the ventricle, and return back
towards the tendinous zone, after describing a circuinference
and one-fifth.
Thus, it will be seen that the common and proper fibres
are similarly arranged, and may be divided each into two
portions; the first portion spiral and outer, and the second or
return portion direct and inner. The common fibres, how-
ever, return to the same azimuth on the circumference of the
tendinous ring, from which they started, after having described
a complete circumference in their course; while the proper
fibres return to an azimuth on the tendinous ring 72° in ad-
vance of the azimuth from which they started, after having
described a complete circumference and one-fifth in their
course.
It is easy to see that as the common fibres describe a
T
438
ANIMAL MECHANICS.
0
complete circumference, the transverse components of the
force of contraction will equilibrate each other, and thus no
motion of rotation is given to the heart by these fibres; on
the other hand, the transverse components of the force of
contraction of the proper fibres are unsymmetrical, and they
will give a resultant twisting motion to the cavity of the ven-
tricle, the effect of which will be, to wring out completely
every drop of blood from the ventricle at the close of the
systole.
Having ascertained the arrangement of the several va-
rieties of fibres, I dissected them out carefully, and measured
their lengths. On doing so, it became apparent that the
common fibres were equal to each other in length, within the
errors of observation; and also that the proper fibres were
equal to each other in length, this length being distinct from
the length of the common fibres. This result is caused by
the peculiar shape of the heart,* which compels the fibres
that enter the interior above the vortex to describe their
spiral circumference on a more equatorial region of the heart
than the apex; so that although the return path of these fibres
is shorter than that of the superficial fibres that enter the vor-
tex, their total lengths remain the same.
From the preceding statement, it appears that each fibre is
independent of the others, and is at full liberty to contract to
the utmost extent required by the law of muscular contraction,
and thus the law of least action is fulfilled by the common
fibres and proper fibres respectively, each fibre belonging to
either group having the same length. But, in order that the
two groups of fibres, working simultaneously, may produce
the maximum of work required by the law of least action, it
is necessary that a certain relation should exist between the
S
S
* Or, to speak more correctly, the shape of the heart is the necessary con-
sequence of the law of least action, which requires the fibres of the same group
to have the same length.
ANIMAL MECHANICS
439
1
respective lengths of the two groups of fibres. This relation
may be investigated as follows:-
Let L denote the length of the common fibres passing
round both ventricles, and let the volumes of the left and
right ventricle be denoted by d and p.
Let 1 denote the length of the proper fibres that pass round
- the left ventricle only:
Let the fibres L and 1, become L' and l' when contracted ;
then it is plain that we have
S
L3 - L's ltp
78 - 113
:
(112)
If n denote the coefficient of muscular contraction, and i
it be supposed to be the same for both groups of fibres, then
equation (112) becomes
13 (1–103)
to
13 (1 - 1) = a
;
or
13
To
+
T
(113)
S
As the left-hand side of this equation depends on measure-
ments of the lengths of the two systems of fibres, while the
right-hand side depends on measurements of the volumes of
the two ventricles, it is plain that it affords us a test, within
limits of errors of observation, of whether the principle of
least action applies in all its force to the theory of the action
of the heart.
As the lengths of the fibres and sizes of the hearts
examined differed considerably, I selected a large and small
heart for measurement, with the following results :-
No. 1. No. 2.
Length of common fibres, . 13.25 in. . 8.5 in.
Length of proper fibres, · · 10.75, . 6.5 »
440
ANIMAL MECHANICS.
Hence we find
a (No. 1.) = (13.7) – 1.873
(No. 2.) = (8.) = 2.236
16.5
Mean, . 2.055
Brown, from observations made on the heart of the Ox,
states that the right ventricle exceeds the left in the propor-
tion of 110 to 100, which would make
+p_.10.
.
The following observations have been made by anatomists
on the relative volumes of the two ventricles in the human
heart:-
Human Heart.
2
1. Lower, .. .: 2.00
2. Sabatier, ..... 2.00
3. Andral, . . . . . 2.00
4. Gordon, ..... 2.25
5. Cruveilhier, · · · · 2.00 (less than.)
6. Portal, . . . . . 2.40
7. Sibson, · ·... • 2.00
8. Quain,...... 2.00 (greater than.)
9. Lieberkuhn, · · · · 2.50
Mean, ... 2.128
There are good physiological grounds for believing that,
during life and action, the volumes of the two cavities are
ANIMAL MECHANICS.
441
equal in a healthy heart ; and that the slight preponderance
in favour of the right ventricle found by anatomists depends
on the lesser thickness of its walls, which causes it to distend
more than the left ventricle, under the hydrostatical pressure
to which it is subjected by the anatomical observer. If this
be so, then the principle of least action applied to the heart
leads to the remarkable prediction, that the lengths of the
common fibres ought to bear to the lengths of the proper
fibres, the ratio of the cube root of 2 to unity,
V=V2.
>w
(114)
Assuming this Law to be correct, we can calculate from
the observed lengths of the common fibres, measured in our
two hearts, what the length of the proper fibres ought to be,
and so compare our observations with theory. I find the
following results :-
Comparison of Observed and Calculated lengths of Muscular
Fibres in the Heart of the Ox.

Common
Fibre.
Proper Proper
Fibre
Fibre
(observed). (calculated).
Difference.
No. 1,
10.75 in.
6.50,
10.52 in.) +0.23 in.
6.74» - 0.24 ,
No. 2,
1 8.50 ,
These differences are less than a quarter of an inch, and
will be readily admitted by practical anatomists to be well
inside the unavoidable errors of observation.
If there be any Naturalist who thinks it possible that a
mechanism such as I have here described could grow up of
itself, from chance combinations of fibres developed in a
20
y
442
ANIMAL MECHANICS.
.
long lapse of time, and without the aid of a Contriving Mind,
I would ask him to study the remarkable words of Cicero :-
“ Nihil est enim, quod ratione et numero moveri possit,
sine consilio; in quo nihil est temerarium, nihil varium, nihil
fortuitum.”
. 13. Applications of General Laws of Muscular Action.-
I shall conclude my observations on Aniinal Mechanics by
laying down in this section a few General Laws, and showing
the inferences that may be deduced from them. In drawing
these inferences, I have the advantage of being able to make
use of the experiments of Mr. W. Stanley Jevons,* of Man-
chester, and of Mr. Frank E. Nipher, † of Iowa University,
U. S. A., in addition to my own experiments and observa-
tions ; and I believe it will be admitted, from the results of
the investigations I have been able to make, that a great
future in the way of useful practical applications lies open to
the Science of Animal Mechanics.
I have been led to the establishment of the three following
Laws, the proofs of which I shall give in detail:-
t
S
LAW 1.-In comparing together different muscles, the work
done in contracting is proportional to the weight of each.
LAW II.-In comparing the same muscle (or group of
muscles) with itself, when contracting under different external
conditions, the work done is always constant in a single con-
traction.
LAW III.—When the same muscle (or group of muscles). is
kept in constant action until fatigue sets in, the total work done,
multiplied by the rate of work, is constant.
I shall now take these Laws in order, and show the evi-
dence on which each is based.
* Nuture, 30th June, 1870, page 158.
† School Laboratory, September and December, 1871, Iowa City, page 108.
ANIMAL MECHANICS.
443
Law 1. Work proportional to weight. The germ of this
Law is to be found in Borelli's great book, Vol. I.,
Props. cxxi., cxxii., cxxiii. (pp. 209–211).
I here give the statements of Borelli's Propositions.
PROPOSITION CXXI.
Si duo musculi ejusdem animalis fuerint æque crassi, scilicet
compositi ex pari multitudine fibrarum, sed earum longitudines
inæquales fuerint; suspendent quidem æqualia pondera : at po-
tentiæ motivæ, et altitudines suspensionum eandem proportionem
habebunt, quam longitudines musculorum.
TI
WN
2
.
VIAC
V
PROPOSITION CXXII.
Si vero, altitudines musculorum fuerint æquales, et crassities
eorum inæquales : potentiæ motivæ, et pondera suspensa, pro-
portionalia erunt crassitiebus musculorum; at pondera ad æquales
altitudines ascendent.
PROPOSITION CXXIII.
At, si tam altitudines, quam crassities musculorum incequales
fuerint, pondera suspensa erunt, ut crassities musculorum; alti-
tudines elevationum eorum ut longitudines musculorum; at po-
tentice compositam proportionem habebunt ex ratione crassitierum
et longitudinum.
. If in these Propositions we replace the phrase potentia
motira by its modern equivalent, Work done, we obtain the
following statement:
The Work done by the contraction of a muscle is proportional
to its length and area of cross section jointly.
But it is evident, cæteris paribus, that the weight of a
muscle is proportional to the product of its length and cross
section; so that Law I. may be fairly called Borelli's Law;
although he failed to see the important consequences that
e
444
ANIMAL MECHANICS.
flow from it. In my section on Muscular Types I have en.
deavoured to show what valuable applications may be made
of this Law by the Comparative Anatomist, when combined
with the Law of Least Action applied to muscular structure
and arrangements.
LAW II. Constancy of Work done.--The second Law may
be described as the law of constancy of work done by a group
of muscles, in a single contraction, performed under different
external conditions.
The admirable experiments published by Mr. W. Stanley
Jevons in Nature (30 June, 1870) enable me to demonstrate
the truth of this Law in a satisfactory manner.
Mr. Jevons ascertained by numerous experiments the
comparative distances to which various weights could be
thrown by hand, on level ground. The weights employed
ranged from į lb. up to 56 lbs., and were thrown as nearly as
possible in a uniform manner, and at the most advantageous
angle of projection. About 57 experiments at different times
it was quite obvious that good average results were obtained,
the correspondence of different sets being very satisfactory.
The following results were obtained :-
Mr. Jevons' Experiments (Throwing Weights).

Weight in lbs.
Average Distance
thrown in feet.
Calculated. | Difference.
mino assento
to lit + tot!
οοοοοοοο
23.05
27.15
ANIMAL MECHANICS.
445
The calculated distances contained in the foregoing Table
were obtained by Mr. Jevons, by assuming the following
empirical Law :
(115)
W
+ 9
where
X = distance thrown,
w = weight thrown,
P, 9, unknown constants to be determined.
The experiments gave Mr. Jevons eight distinct equa-
tions by which to determine the two unknown quantities p
and q; and by means of the method of least squares, Mr. Je-
vons found their most probable values to be
p = 115.7
y = 3.9
Mr. Jevons correctly stated that the agreement between
observation and calculation is so close as to prove the accu-
racy of the empirical law (115), but he professes his inability
to explain it on mechanical principles. I subsequently suc-
ceeded in doing so, and published my results in Nature. My.
investigation is as follows :
In throwing weights by the hand in the manner described
by Mr. Jevons, the arm, after a little practice, instinctively
pitches the weight at the angle (459) corresponding to the
maximum range (x), and as the maximum range is propor-
tional to the square of the velocity of projection, it may be
used to replace that quantity, in estimating the work done by
the arm. The total work done by the arm is the same as if
the weight used and the weight of the arm were concentrated
at the centre of Gyration of the loaded arm, regarded as a
compound pendulum.
446
ANIMAL MECHANICS.
Let us assume
w = weight held in hand,
a = weight of arm,
v = velocity of centre of Gyration.
By the law of Constancy of Work, we have
(W + a) v2 = constant.
V = velocity of the hand,
k = radius of gyration,
a = length of the arm.
(116)
Let
We now have the additional equations,
(117)
and since the arm may be assumed to be approximately
cylindrical, we have
(w +) a = (10 + a)kº,
w
or,
2 wt a
olms
(118)
a
W
Substituting from (117) and (118) in equation (116), we
find
Y
vº(w + ) - Const. :
1
W
+
31
and finally, writing a for V, as before explained, we have
X =
?
(119)
20 +-
3
. This equation is identical with the empirical formula (115)
employed by Mr. Jevons, and shows that his constant, q,
L
ANIMAL MECHANICS.
447
represents one-third of the weight of the moving arm ex-
pressed in pounds. The weight of Mr. Jevons' arm, includ-
ing the scapula and its muscles actually in motion, is therefore
IT.7 lbs.
The remarkable agreement between the observations and
equations (115) and (119) which may be derived directly, as
I have shown, from the Law of constancy of work done in a
single contraction of a group of muscles, proves completely
the accuracy of that Law.
Although the work done by the contraction of the muscles
is constant, the useful work done is not constant, and it may
be worth while to inquire how it varies, and whether it
admits of a maximum, a circumstance which would have an
important practical significance.
Using the same notation as before
The useful work done = wx = n.
The weight thrown = W = &.
If & and n be used to designate the abscissa and ordi-
nate of a curve, by constructing this curve of useful work
done, we may ascertain whether its ordinates admit of a
maximum or not. Multiplying Mr. Jevons' equation (115)
by w, we find
pal?
1X =
W + q
or
(126)
. & + a
This represents an equilateral hyperbola referred to axes
parallel to its asymptotes, and intersecting on the curve
itself.
In Fig. (102) I have constructed from calculation the
curve of useful effect, which is an equilateral hyperbola,
having OX and OY for asymptotes.
(D
448
TY
ANIMAL MECHANIC
FT

00.
1.
| 12 4.
-
14.
.
.
.
=
.
:: . . . •
.
..
:
1
.
**
.
| Fig. 102.
ANIMAL MECHANICS.
449
p p = 115.77
Fig. 102. Curve of useful work, n =
& +9 9 = 3.9)
and have shown the actual useful effects of each weight, by
means of the centres of the small circles corresponding to each
weight. This Figure conveys to the eye. a complete idea of the
accuracy of the Law of Constancy of Work, and of its agree-
ment with observation.
The upper line in the Figure shows the horizontal asymp-
tote, towards which the ordinate continually approaches, and
both the curve itself and the observations show that the
useful effect increases with the weight employed.
Equation (116) may be thus written,
Wv2 + av2 = constant;
in which wo? represents the useful work, and av? the work
done by the arm; in this equation, a is a constant, being
the weight of the arm, and v grows less as the weight thrown
increases; hence the work expended on the arm is less as
the weight is greater, and therefore the useful work must
increase continually with the weight, and reaches a maximum
when the weight is infinite.
In Figure 102, the useful work is represented by the
ordinate of the hyperbola, and the work done on the arm is
represented by the intercept of the ordinate between the
curve and its asymptote, which diminishes continually as the
weight (abscissa) increases.
The coordinates of the centre of the hyperbola (120) are
1
a =
B =
-9,
P,
-
and the equation of the curve, referred to its asymptotes, is
(121)
En + pq = 0.
29
450
ANIMAL MECHANICS.
L
From the preceding data, we may calculate the maximum
velocity which the muscles employed as in the experiments
are capable of giving to the hand. For, since the work done
on the arm is a maximum when w = 0, we find from equa-
tion (115) the maximum range of an infinitely small weight
x = P = '15-7 = 293 feet.
9 3.9
Now, it is well known that the velocity of projection
corresponding to the maximum range on a horizontal plane is.
V = Vgr,
(122)
from which we find
V = 30.9 feet per second,
showing that the muscles in question can produce on the hand
a velocity nearly equal to that of a body which has fallen for
one second.
LAW III. The Law of Fatigue.—The third Law. of Mus-
cular Action states that the total work done multiplied by the
rate of work is constant;—and as soon as this product attains its
constant value, fatigue sets in, and the muscles become in-
capable of giving out more work, until they have been allowed
to rest. Hence, the Third Law may be appropriately termed
the Law of Fatigue.
First Illustration of Law of Fatigue.--I shall take my first
proof of this Law from Mr. Jevons' experiments.* Mr. Je-
vons raised and lowered various weights by a pulley and cord
through the convenient range of the arm, continuing the
motion with unrelaxed rapidity until the power of the muscles
was entirely exhausted. From these experiments, he found
the following results :-
* Nature, 30th June, 1870, p. 159.
ANIMAL MECHANICS. •
451
Dir. Jevons' Experiments (lifting Weights with Pulley).

Weight
lifted.
No. of
times
lifted.
Useful
Work.
5.7
11.9
23.0
37.6
III.O
319.2
499.8
644.0
789.6
1554.0
From the manner in which these experiments were made,
it is easy to see that the weight of the arm does not enter
into the result. Let
w = weight raised; '
n = number of times it is raised, carried to point of
fatigue of muscles ;
h = height through which weight is raised ;
t = time occupied by a single lift.
Hence we have
Total work done = whn,
Rate of work · =
The rate of work is found by dividing the work done in a
given time by the time itself. Hence we find by the Law of
Fatigue
wahan
= constant;
(123)
or, since h and t are constant,
wan = A.
(124)
If we write
& = w,
n = n,
2 G 2
452
ANIMAL MECHANICS.
we have
Ein = A,
(125)
which represents a cubical hyperbola.
If we write
E = W,
n = wn (useful effect),
we find
En = A,
(126)
which represents an equilateral hyperbola.
If we write A = 19600, in equation (124), we can calculate
the following values of n, and compare them with the ob-
served values :-
Mr. Jevons’ Experiments (lifting Weights with Pulley).

W.
n (ob-
served).
n (cal-
culated).
Difference.
Lt 0.8
II.I
6.2
25.0
100.0
I+II+
ö onion
ovocour
44.3
III.O
Cubical hyperbola, w’n = 19600.
In Fig. 103, I have constructed the cubical hyperbola,
wzn = 19600,
showing both its asymptotes, and marking with small circles
the observations of Mr. Jevons; the agreement between cal-
culation and observation is sufficiently close for this class of
experiments.
The relation between the useful effect and weight lifted is
represented by an equilateral hyperbola (126), and the useful
ANIMAL MECHANICS.
453
effect diminishes as the weight increases, and becomes infinite
when the weight is zero. There is, therefore, no true maxi-
mum of effect in these experiments.
Second Illustration of Law of Fatigue.--I shall take my
second illustration of the Law of Fatigue from experiments
similar to those described (pp. 24-44), from which I obtained

.
-
200+
,
:,'.
:
.
1
- 2128
.
12
Fig. 103.
my first view of the Law. The experiments consist in hold-
ing out the arm, horizontally, either unloaded, or carrying
weights in the palm of the hand, held upwards, and noting
the time during which the arm could be held out horizontally.
The following results are the means of many observations
made on myself, and include those before described:
454
ANIMAL MECHANICS.
Dr. Haughton's Experiments (holding Arms horizontal).

Weight.
Time.
Useful
Effect.
o lbs.
420 secs.
275
200
275
400
441
454.5
434
363
14
1
323
The useful effect is estimated by taking the product of the
weight and the time during which it is held in the horizontal
position.
Using the notation employed in p. 25; let
w= weight held in hand.
a = weight of arm.
t = time of holding arm horizontal, until fatigue sets in.
x = distance of centre of gravity of loaded or unloaded
arm from centre of glenoid cavity.
w = the unknown angular velocity.
The moment of the weights equilibrated by the action of
the muscles is
moment = (w + a) x.
The total work done will be proportional to this moment
multiplied by the time.
Hence
Total work done = w (w + a) xt,
Rate of work = w (w + a) x.
Hence, by the Law of Fatigue,
w? (W + a)2 x t = const.
(127)
ANIMAL MECHANICS.
455
455
Now, we have
(w + a) x = wat a,
where a is the length of the arm, supposed to be an uniform
cylinder ; therefore
(60 + a) 2 = a(60 +)
= al w
+
Sint
substituting in equation (127), we obtain (assuming w? to be
an unknown constant)
o ?
w talt= A.
(128)
2
.
This equation represents a cubical hyperbola, whose
asymptotes are
. t = 0.
(129)
,
1
Solving equation (128) for t, we find
(130)
w
+
The useful effect is found by multiplying (130) by w, and is
1. Aw
Useful effect = wt = -
(131)
(1,2
20
+
S
This equation represents a cuspidal cubic, whose shape I
shall presently describe.
I find the following values of A and a to give me the best
agreement between theory and observation, viz. :-
A = 7730.
a = 8.5 lbs.
456
ANIMAL MECHANICS.
Using these values I find the following comparison of
theory and observation :-
No. 1.-Dr. Haughton's Experiments.-(Time of holding Arm ·
horizontal).

t (observed).
t (calculated).
Difference.
en Om Nm
420 secs.
427 secs.
280
198
147
11 +
O OWONTv
we a
são
+ +
ää с
31.6,
+ 1
23.31
No. 2.-Dr. Haughton's Experiments.(Useful Effect). -
(Holding Arm horizontal).

Useful Effect
observed.
Useful Effect
calculated.
Difference.
275
400
441
280
395.5
441
454
430
366
6.5
454.5
434
363
II
14
322
326
The cubical hyperbola (130) is shown, drawn to scale in
Fig. (104); its asymptotes (129) being OX and OY; and the
small circles as before denoting the actual observations.
ANIMAL MECHANICS457
U
.

:
i.
:
retum
2007
•
..
!
-
Ti 2 3
7.5
11.
1.7
Fig. 104.
• Fig 104.--Dr. Haughton's Experiments-(Arms horizontal).
{ Cubical Hyperbola
Cubical Hyperbola t =
para
A = 7730.
a = 8.5 lbs.
The general form of the cuspidal cubic (131) whose ordi-
nates represent the useful effect, is shown in Fig. (105), in
458
ANIMAL MECHANICS.
which X'OX is the single asymptote, corresponding to t = 0,
and ApA is the double asymptote corresponding to
W
+
- =
0,
and having a cusp at negative infinity.

.
..:
III
..
..
m
.
12
.
:
.
..AY,
Fig. 105.
S
There is an hyperbolic branch lying between pX' and pA'.
The curve passes through the origin 0, and the positive
weights are measured on OX, while negative weights are
measured on OX'. The asymptote AA' lies at a distance from
YY' equal to “; which signifies that if the hand were lifted
by a weight equal to that of half the arm, it could be held
out for an infinite time, without being tired, as is evident.
The portion of the curve with which we are concerned
lies between OX and OY.
The ordinate (or useful effect) attains a maximum mm',
when the weight Om' is equal to half the weight of the arm;
and the point i corresponding to On' equal to the weight of
ANIMAL VI.
NIM
459
MECHANICS.
the arm, is a point of inflexion on the curve where the cur-
vature changes sign, and the curve becomes asymptotic to
the line 0x
The following are the values of the abscissæ and ordinates
at these important points :-
ola
Om - On' = a
mm' = 4 nn' - 44
(132)
1271
- za
90
The tangent drawn at the point of inflexion cuts off an
abscissa on the axis OX equal to 4d. The readiest mode of
approximating to the best numerical values of A and a, is by
comparing the co-ordinates of the maximum effect (m) with
the equations (132).
In Fig. (106) I have drawn, on a larger scale, the por-
tion of the cuspidal cubic (131) corresponding to positive
values of w and t, and have also shown the actual observations
by means of the small circles. The agreement between theory
and observation is very satisfactory.

4807Y:
4007.
:
300
200+)
.
100 #
.
.
Lo 1 2. 3. 4.
5
7
. 11 -
14
.
.
Fig. 106.
460
ANIMAL MECHANICS.
Fig. 106.-Dr. Haughton's Experiments --(Holding Arm hori-
zontal)-Useful Effect.
-
Aw
wt =
A = 7730.
a =
8.5 lbs.
A valuable series of observations of the same kind have
been made by Mr. Jevons,* of which the following are the re-
sults :-
Mr. Jevons' Experiments-(Holding. Arms horizontal).

Weight.
Time.
Useful Effect.
321
438
321.2 secs.
218.99
147.99
87.4 »
60 3 1
32.5 »
$92
612
603
455
I found equations (130) and (131) to represent the time
useful effect best, when
A = 13340.
a =. 10.9 lbs.
By using these values I find the following comparison of
theory and observation.
No. 1.—Mr. Jevons' Experiments--- Time of holding Arm
horizontal).

w. 1 t (observed).
+ (calculated).
Difference.
KOVAN-
321 2 secs.
218.91
147.9 %
87.4 »
60.3 ?
32.5 1
320.7 secs. (
240.3 .
149.4
86.1 99
55.9 29
35.2 19
+ 0.5
- 21.4
- 1.5
+ 1.3
+ 4.4
- 2.7
* Nature, June 30, 1870, page 158.
ANIMAL VI
461
MECHANICS.
No. 2.-Mr. Jevons' Experiments—(Useful Effect)-(Holding
Arms horizontal).

Useful Effect | Useful Effect
(observed). (calculated).
Difference.
321
438
320.7
480.7
597.5
592
612
603
602.5
+ 0.3
- 42.7
- 5.5
+ 9.5
- 38.7
* 44.1
558.9
493 7
455
The maximum useful effect corresponds to a weight
5.45 lbs., equal to half the weight of the arm, and its value
is (132)
Maximum useful effect = = 612.
in
III
m
20

.
300
-
-
i
.
.
.
.
.
.
1 2
4
10
.14
27.
Fig. 107
462
ANIMAL MECHANICS.
In order to show the agreement of theory and observation,
I have drawn in Fig. (107) the cubical hyperbola (130), -
and have represented Mr. Jevons' observations with small
circles.
Fig. 107.---Mr. Jerons' Experiments-(Holding Arms hori-
zontal).
A A = 13340
a = [0.9 lbs
w xala
IT
Third Illustration of Law of Fatigue.--I shall take the
third illustration of the Law of Fatigue from the experiments
performed by Mr. Nipher.*
Mr. Nipher lifted a weight (w) through a height (h) in the
time (t), a number of times (n), until the muscles of the arm
became fatigued. The arm was raised from the vertical to
the horizontal position. The lifts were made isochronous
with the alternate beats of a pendulum, and during the other
alternate beats, the arm dropped to the side, the weight being
caught upon a cushion attached to the leg; and when the
time (t) exceeded one second, the weight was supported upon
a framework during the interval of rest. The experiments
were made at intervals not less than three hours, and were so
arranged as to eliminate from each series, as far as possible,
the increase in the power of the muscles produced by the
66 training."
The following Table contains the results of one set of ex.
periments :
W = kilograms, h = 0.74 metre, t = 1.164 secs.
# Iowa School Laboratory, vol. 1, p. 108 (1871),
ANIMAL MECHANICS.
463
Mr. Nipher's Experiments—(Raising Weights at given rate)
WIuu

w.
n (right arm).
| n (left arm).
255
213
oovac AWNE
31.7
do courwv
24.7
17.8
12.8
In this case, the work done involves the raising of the
arm as well as the raising of the weight; let a represent the
weight of the arm, then we have, supposing the arm to be an
uniform cylinder,
Total work done = (x +") hn,
al
2
=
10
+
a
kate of work
- (0 + )
1 W
ola
Hence, by the Law of Fatigue,
1 219 t
-
= constant,
or, since h and t are constant,
a12
w+
) n = A.
.
(133).
2
. . This, as before, represents a cubical hyperbola similar to
equation (130).
I find, by trial, the following values of the constants : com
Right arm.
Left arm.
A = 1000 : A = 850
a = . 2.0
a = 2.2
464
ANIMAL MECHANICS
P
Calculating by these constants from equation (133) I ob-
tain the following comparison of Theory and Observation.
Mr. Nipher's Experiments (Raising Weights at same rate).
h = 0.74 metre.
t = 1.164 SCC.

RIGHT ARM.
LEFT ARM.
w.
n (obscrved). n (calculated).
Difference.; n (observed). n (calculated). Difference.
-
-
+
-
+
255
97
250
TIL
62.5
200.4
88
61
5.0
14.0
1.5
2.2
a
37.7
+ tot + 1 1 1 +
50.
32.7
29.3
1.5
It+II! +
OOO-HOWw
20.
21.
31.7
24.7
17.8
12,8
9.5
1. I
0.2
-
12.8
I 2.4

.
.
:
.
.
I
200
:::
.
.**
".
:
.
.
.
.
.
.
.
.
.
:
.
.
::
:::
-
.
1
:
5
.
-
.
.
.
.
.
I.
:...
5.:.:. :6
2
3
.
3:
.
.
"..
Fig. 108.
ANIMAL MECHANICS
465

:.
.
:
:
300M
...,
.
....
.
.
$
.
.
..'.
'
.
::.
:.
1
...2
6
.
X
811
Fig. 109.
In Figs. 108 and 109 I have constructed the Cubical
Hyperbolas for each arm, and shown the actual observations
by small circles. The agreement between theory and obser-
vation is as close as can be expected.
Mr Nipher himself represents the observations empirically,
by means of an Equilateral Hyperbola, which, for the portion
of the curves under observation, does not differ materially
in shape from the Cubical Hyperbola, indicated by the Law
of Fatigue as the true form of the curve.
2 H
466
ANIMAL MECHANICS.
On comparing these experiments with those made by
Mr. Jevons and myself, the small value of the weight of the
arm appears striking. Converting kilograms into pounds,
we find-
Mr. Jevons' arm = 10.9 lbs.
Dr. Haughton's arm = 8.5 . . .
. . Me Ninher's right arm = 4.41 »
y left arm = 4.85 ,
.
• In Mr. Nipher's experiments, however, there may have
been some peculiarity in his mode of raising the weight,
which prevented the full weight of the arm from coming into
play. We have already seen that, in Mr. Jevons' experi-
ments made in raising weights by means of a pulley, the
weight of the arm disappeared altogether from the result,
which is expressed by the equation (124).
The useful effect obtained by Mr. Nipher's experiments is
found from (133), by calculating the product of w and n.
This gives the formula, ..
Aw
Useful effect =.wn = =
w
(134)
to
IN
This represents a cuspidal cubic, as before, having a
maximum ordinate corresponding to the weight,
а
w =-
2
I find, by using the numerical constants already given, the
following comparison between theory and observation :-
ANIMAL MECHANICS,
467
No, I.---Mr. Nipher's Experiments (Useful Effect)–Right Arm
(raising Weights at given rate).

204
Useful
Effect
(observed).
Useful
Effect
(calcu-
lated),
Difference,
kil.
i
.
N
Oxoo
cov anawn mo
ܗ ܟܗ. ܛ ܟ̣
150.8
146.5
129
116.6
102.4
++$1111+
No, 2.--Mr. Nipher's Experiments (Useful Effect)--Left Arm
(raising Weights at given rate).

20:
Useful
Effect
(obseryed).
Useful
Effect
(calcu
lated).
Difference,
+
213
IS
150
126,8
123.5
106.8
89.6
į i tot el
76
'.
The maximum effects, calculated from (132), are
Maximum effect 4 (250 (Right arm)
2a193.2 (Left arm) ſ
20
20
Fourth Illustration of Law of Fatigue. Another, and very
interesting, method of testing the Law of Fatigue consists
in raising weights held in the hand or hands, from the vertical
+
468
ANIMAL MECHANICS.
to the horizontal position, at a given rate, until fatigue sets in,
and then repeating the experiments with the same weight, at
a different rate. In every case, the time of raising the weight
is made equal to the interval of rest that elapses until the next
lift.
The following observations were made by me on Dr. Alex-
ander Macalister. The weight used was 14 lbs., and the
height through which it was lifted was 30 inches. One arm
only (the right) was used in the experiments.
In the following Table, I have compared the results,
observed with the results calculated from the Law of Fatigue,
and the agreement is highly satisfactory.
n represents the number of lifts before fatigue ;
t represents the time occupied by the lift, which is made
equal to the time of repose.
Dr. Macalister (Raising Weights at varying rates).
2 = 14 lbs. h = 30 inches.

92. (obser-
ved).
n (calcu-
lated).
Differ-
ence.
36.0
39.0
ܩܟ݂ ܝܲܢ ܗܲ ܡܶܢ ܒ݁ܶܝ
0.65 seis
0,75
1.18
1.40
2.00
36. 1
39.7
49.5
51.4
WWPOO
51.0
51.7
53.0
44.5
öömoniooni
50.6
ww O Awaran
44.5
38.1
37.8
27.5
ܗ
29.8
The column of calculated values of n was found as fol-
lows :- In raising the arm and weight from the vertical to the
horizontal position in the time t, the muscles perform two
distinct kinds of work, dynamical and statical; the dynamical
work being measured by weight multiplied by height lifted,
and the statical work being measured in a manner similar to
ANIMAL MECHANICS.
469
the work of holding out the loaded arms horizontally. In
other words, the arm and weight must be not only lifted but
sustained during the time t.
Using the same notation as before, we have
Dynamical work = (10 + a) xn ; (135)
(w + a) x
Rate of Dynamical work !
If the arm were held in the horizontal position during the
time t, the statical work would be represented by
w (w + a) xt,
and the total statical work would be
w (w + a) xnt.
But the arm is not held horizontally, but sustained for equal
times in every position of the quadrant from vertical to hori.
zontal, while the arm and weight are passing from the vertical
to the horizontal position.
Hence, the statical work done bears a certain fixed pro-
portion to the statical work of holding the arm and weight
horizontally, and
Statical work = w (w + a) xnt « K;
’ (136)
Rate of statical work = w (w + a) x * R."
The coefficient K is thus found: let a denote the angle
between the vertical and the axis of the arm in any position ;
and let the whole time t be divided into m parts; and let the
loaded arm rest at the angle o during the mth part of the
time t. The work done in the position 0 is (resolving the
weight along the tangent to the circle),
| 0 (0 + a) 2m x sin 6 x 8.
Let us now divide the quadrant into m equal parts, and
470
ANIMAL MECHANICS.
suppose that the arın in rising from the vertical to the hori-
zontal position rests at each division of the quadrant, for the
mth part of the time to The angle 0 in our last equation will
have, successively, the following value:-
2T
37T
11177
2m
27
29.12
201
Let us write
20n'
then the total work done will be
w (w + a) an (sin + sin 20 + &c. + sin mp).
т
Comparing this with equation (136), we find
K =
(sin ¢ + sin 24 + &c. + sin mp),
+ S1
(137)
where m is to be made indefinitely large ; but
منتششد
2
2
sin o + sin 20 + &c. + sin mo .= sind
sin
JOIN
• or,
.
sin mg
.
?
R=
sin (m + 1)
2_
m sin ?
Substituting, in this equation,
for °, we obtain
VCU
sin
(
+ 1) +
41
2* sin
1
K =
R
:
an sin
41
and, finally, making m very large, we find
ANIMAL MECHANICS.
471
4 sina ?
4
2
K =-
(138)
TT
'
Substituting this value of K in eqriations (136), we obtain
Statical work
= 20 (v + a) ænt ;
(139)
Rate of statical work = ~
(W + a) x.
The two kinds of Work, Dynamical and Statical, just
described, are performed by the muscles of the shoulder,
until fatigue occurs. The Law of Fatigue applied to this case
gives us the following result :-
Let
W = Total work of both kinds :
R = Rate of work of both kinds;
W, = Total dynamical work ;
Ri = Rate of dynamical work ;
W, = Total statical work ;
R, = Rate of statical work.
i
The Law of Fatigue gives the following equation :
WR = W R + W,R2 = constant. (140) .
Substituting, in this equation, the dynamical and statical
work and rate of work given in equations (135) and (139),
we obtain
: '(0 + a)* x(**)* t} n = constant. (141)
In the present experiments (w + a) x is a constant, and :
therefore (141) may be thus written,
ANIMAL MECHANICS.
At
r2 =
0
(142)
20
It i
The values of n in the experiments made on Dr. Macal-
ister were calculated from (142), using the constants,
A = 64.5,
24% = 0.6221.
If n, t be made the ordinate and abscissa of a curve,
equation (142) will represent one of the central cubics,
having its centre at the origin, and the axis of x asymptotic
at positive and negative infinity. The ordinate will have a
maximuin value for co-ordinates.
t = t
“
= + 1.6075 secs.
2W
n
= +
Ап
—
= + 51.84.
4w
The curve (142), on the side of positive axes is repre-
sented in Fig. 110, and the actual observations are marked
by small circles.

.
.
.
•
.
.
..:
-
0.65 1
8.0.75
:1.13
: 1.40,
2.00
2.84
3.68
5:09
:X
.
Fig. 110.
ANIMAL MECHANICS.
473
Fig. 110.-Dr. Macalister (Raising Arm from vertical to hori-
zontal position, at varying rate).
-
At
1.
l
il
ress
*
4 = 64.5
2w = 0.622
2W
-or}
I tj
The foregoing observations and theory establish the im-
portant fact, that there exists a certain rate of lifting the
loaded arm to the horizontal position, which will give a
maximum amount of work. This rate is
t = * = 1.6075.
2W
The observations also give us the value of the important
constant w, used in pp. 24-44; substituting for 77, we find
w = 0.9772.
Having obtained the preceding results, I next made a series
of observations on my son, Mr. J. Gilbert Haughton (æt. 21),
who is accustomed to the use of 10 lb. dumb-bells. Taking a
10 lb. dumb-bell in each hand, he raised both arms to the
horizontal position, with palms upward; the time of rais-
ing (t) being constant for each experiment, but varied from
one experiment to another; the interval between each effort
of raising was also made equal to the time t. The following
results were obtained :-
.12
Mr. J. Gilbert Haughton (Raising Weights at varying rates).
W = 10 lbs.
h = length of arm.

in (observed).
n (calculated). Difference
13.8
en fins
0.53 sec.
1.1955
2 143 ,
3.00 99
6.00 ,
13.0
21.5
21,0
17.0
10.2
21.0
20.9
monte caroo
11 tot
18.6
o
10.8
- 0.6
a
The calculated values of n were found from equation (142),
using the constants
A = 27.5.
26
me = 0.630.
This curve is shown in Fig. Ill, and the observations
are marked by small circles.

."
-
es
1.
1
fi
::::
:
:
. 0.53
..
.1.192.143
SALA
M
SELV
Fig. ur.
ANIMAL MECHANICS.
475
Fig. 111.-Mr. J. Gilbert Haughton (Raising both Arms loaded,
from vertical to horizontal position, at varying rates).
At A = 27.50 )
20 = 0.630
1 26
The maximum work possible, and the time of lift corre. -
sponding to it, are found as before, viz. :
n = * = 21.83.
40
t = mcm = 1.5873 secs.
20
The value of w found from the last of these equations is
w = 0.9896.
Mr. Nipher's experiments on raising the loaded arın from
the vertical to the horizontal position, at varying rates, gave
him the following results :-
No. 1.-Mr. Nipher (Raising Weight with Right Arm, at
varying rates).
w = 7 kilos. h = 0.74 metre.
val2012
set

nz (observed).
n (calculated)
Difference
22.2
1.164 secs.
1.50 34
15.8
22.8
18,5
17.3
22.
21.7
18. 2
Nritvoro
au WA WILO
ооооооо
15.3
to a cour N J OCH
OOW a
15.5
13.8
TI! +++
14.3
12.8
9.4
3.4
W
476
ANIMAL MECHANICS.
I have calculated the column for n, from equation (142),
using the constants
A = 30.4.
20
26 = 0.665.
TO
im
The maximum value of n, and the time of lift correspond-
ing to it, are
n = AF = 22.82
4W
n
=
te
- 1.50 secs.
20
and the value of w deduced from the above is
w = 1.0472.
Mr. Nipher's results, from experiments made with his
left arm, are given in the following table :-
No. 2.—Mr. Nipher (Raising weights with Left Arm, at
varying rates).
w = 7 kilos. h = 0.74 metre.

n (observed).
n (calculated).
Difference.
1,164 secs.
1.50
ipimtino no
12.8
19.7
16.2
13.5
11.7
18.8
15.7
12.9
anttino
OO OOO
Tilti
OON Nama
11.8
10.8
I have calculated the column for n, from equation (142),
using the constants
ANIMAL MECHANICS.
477
A = 26.2
:
20
<= 0.666.
The maximum value of n, and the time of lift corresponde
ing to it, are
n = A = 19.65.
n
40
t = =
4W
=
1.50 secs.
The value of w is the same as in the experiments made on
the right arm, viz. :-
w = 1.0472.
Combining together all the values of w, we obtain the
mean value of this important constant.
Value of w.
1. Dr. Macalister, .... 0.9772
2. Mr. J. Gilbert Haughton, . 0.9896
3. Mr. Nipher, . . . . . i 1.0472
Mean, . . . . . 1.0046
The angular velocity w, turns out, therefore, to be equal
to unity, so that the arc described in a second would be equal
to the radius, and the time of describing the quadrant would
be
- = 1.5708 sec.,
and the maximum effect would be produced when the arms
are raised to the horizontal position at the rate corresponding
to an angular velocity equal to unity.
478
ANIMAL MECHANICS,
-
Law of Refreshment. In considering the Law of Fa.
tigue of muscles, an interesting question arises as to the
law by means of which the fatigued muscles are restored to a
condition in which they are able to resume work as before,
This law may be called the Law of Refreshment, and its in-
vestigation requires the solution of the following problem :
At what rate does fresh arterial blood supplied to muscles
restore to them the power of again giving out work, of which
they have been deprived by fatigue ?
In the case of the heart, we have a muscle that never
tires during life, and we may therefore assume that during
each cycle of work and rest, a supply of fresh arterial blood
is given to the heart, sufficient to enable it to commence
again, quite refreshed, a second cycle, and so on for ever, or
at least for eighty or ninety years of a long life. If we define
the Coefficient of Refreshment to be the work restored to the
muscle in foot pounds per ounce per second, we obtain the
coefficient of the heart's refreshment from the following data,
already given (pp. 144-5).
Work done by heart in one cycle, equal to 3 ounces lifted
through 9.92 feet = 3 * 9.92 ft. lbs,
Assuming the heart to beat 92 times in one minute, the
AY
A
.
TY
ei
work done by heart in one second - 2*** 9.92 ft, Ibę.
The weight of the heart is 9:39 03,
Hence,
The work done by the heart per punca per second is
13
equal to
172 *_3 * 9.0%
60 X 16 * 9.39
0.9977 ft, lb.
At.
This amount of work per punge per second must be given
1
UM
S
ANIMAL MECHANICS,
479
back to the heart by the arterial blood, because the heart
never tires; hence
Coefficient of Refreshment of human heart = 0.2377 ft. Ib.
Next to the action of the heart, the best example we have
of muscles continuing to act for a long time without fatigue,
is the case of the muscles used in the act of walking. Let us
examine the Coefficient of Refreshment for these muscles. A
man walking at the most convenient pace for a long day's
work walks at the rate of one mile per 17 minutes. Hence
the distance travelled in one second will be
5280
17.5 * 60'
The work done (p. 54) will be, at this pace, good part
of the weight, which may be taken at 150 lbs., lifted through
the above space. Hence, the work done in one second by
the walking muscles, is
5280 X 150
= 32.752 ft. lbs,
17.5 * 60 * 23.03
The muscles more or less employed in the act of walking
are those of the hip, knee, and ankle-joints. Taking the
weights of these muscles from the tables (pp. 401, 404), we
obtain for each side 134.27 oz. This weight doubled, or
248.54 02., is the total weight of muscles employed in doing
the work just calculated. Hence we obtain
Coefficient of Refreshment
of walking muscles
32.952
**0.1318 ft. Ib.
248.54
The following experiments were undertaken in order to
obtain additional data to determine the Coefficient of Refresh.
ment. Dr. Macalister, holding 10 lbs. in each hand, at a
480
ANIMAL MECHANICS
given signal, raised both arms to the horizontal position, and
kept them in that position until compelled by fatigue to lower
the weights. After a fixed interval of rest, the operation was
repeated, and so on; the successive times of holding up the
arms were noted, and formed, as may be well supposed, a
series which diminished rapidly at the commencement, and
afterwards more slowly. On plotting the experiments, taking
the number of the observation as the x, and the time of hold-
ing up the arms as the y of the curve, it became apparent that
the form of the curve was hyperbolic, and convex towards
the axis of X. We can find the horizontal asymptote of this
during which the fatigue incurred is exactly counterbalanced
by the refreshment supplied in each cycle of rest and work.
When this condition is attained, the shoulder muscles can
continue to work for hours without sensible fatigue, as in the
case of the muscles employed in walking.
Experiments No. I.-Interval of Rest = 60 seconds.
ist Effort, .. 60 seconds. (mean).
6th , .. 25 · ·
31st, . 16 .. .. 1
56th .. .. 14 ..
S
These results are the means of many observations made
on different days; and I selected the 6th, 31st, and 55th ob-
servations in order to simplify the calculations. It is evident
that the hyperbola which ultimately coincides with the curve
of observations must have for its equation,
(ix + a) (y - 3) = k> ;
(143)
and we are to determine a, b, and k', from assuming any
ANIMAL MECHANICS.
481
three points on the .curve to be known. Expanding (143),
we find
2y + ya - xß = kx + aß.
If we substitute in this equation the corresponding co-
ordinates
* = 5 x = 30 = 55
Y = 25 y = 10 y = 14
we obtain three linear equations to determine a, B, and ke.
The values of these constants are
B = 10.85
a = 9.29
ki= 202.6.7.
We have now to determine the work done, when the
condition of dynamical equilibrium is attained, and the re-
freshment restores the work during each cycle. By the equa-
tions which lead to (127), the work done is
Work = w (w + a) xt
where
U-I.
w = weight held in hand.
a = weight of arm.
X = distance of centre of gravity of loaded arm from
centre of glenoid cavity.
In addition to the work done during the time t, which
denotes the time of holding the arm horizontally, we must
take account of the work done in lifting the loaded arms from
the vertical into the horizontal position, which is evidently
Hence
(w + a)x.
Total work = (w + a) x (t + r)
(144)
% I
482
ANIMAL MECHANICS.
If we assume the weight of the arm to be 10 lbs., and its
length 2 feet, we have
w = 10 lbs.
a = 10 lbs.
x = 1.5 ft.;
and equation (144) gives us
Total work = 30 (t + I).
Substituting for t, its value 10.85 = ß, we have
Total work = 30 x 11.85 ft. lbs.
To find the Coefficient of Refreshment, we must divide this
work by the number of ounces in the muscles in action, and
by the whole time of the cycle of rest and work.
The muscles more or less engaged in lifting and holding
the loaded arm horizontally are
1. Upper half of Trapezius, )
2. Cleido-mastoid, , , Holding up scapular arch.
3. Levator anguli scapulæ,. )
4. Deltoid, · · · Holding up arm.
5. Supraspinatus, .
0. biceps humerl, · · Guarding the forearm
7. Brachiæus, . . .}
T from forcible extension.
8. Supinator radii longus, :)
9. The flexors of the wrist ) Guarding the hand from
and fingers, . .l forcible extension.
From the Tables (pp. 406, 408), we may calculate the
total weight of these muscles at 35.41 oz. Hence as the time
of the cycle of rest and work is
60 + 10.85 = 70.85;
we have
Coefficient of Refreshment = -
35.41 * 70.85
х
)
-
O.1417.
ANIMAL MECHANICS,
483
Experiments No. II.--Interval of rest = 30 seconds,
Ist Effort, · · 60 seconds... (mean).
41st 1 .. 10.47 , . . .
79th , .. 6.58 , . . .
99th , . 6.00 , ..
From these data we have the following pairs of co-ordi.
nates, to determine from equation (143) the distance ß of the
horizontal asymptote from the axis of x.
x = 40 x = 78 X = 98.
y = 10.47 %. = 6.58 y = 6.00,
The constants of the hyperbola turn out to be
B = 4.23.
a = - 1.83
k2 = 171.74.
From equation (144) we have
Work done =30 (t + I) = 30 * 5.23 ft. lbs.,
and, since the period of the cycle is 30 + 4.23 = 34.23 ; we
find as before
Coefficient of Refreshment s -
30 * 5-23
~= 0.1 296.
35.41 X 34.23
Experiments No. III.--Interval of rest = 15 seconds.
ist Effort, . 60 seconds, · (mean).
8th
10.64
43rd , . 5.93 99 0 99
92nd 5 o 4.02 1 .. »
From these data we find the following pairs of co-ordi-
nates:
Q = 7
X = 42 X = 91.
.y = 10.64 Y = 5.93 y = 4.02.
F
484
ANIMAL MECHÁNICS.
Substituting in equation (143), we obtain, finally, for the
constants of the hyperbola,
B = 1.323
a = 39.2
k= 430.58.
Hence, since the period of the cycle is
15 + 1.323 = 16.323,
we find the Coefficient of Refreshment
30
30 X 2.323
=
=0.1206
35.41 X 16.323
Collecting together the results obtained from the walking
muscles, and from the muscles of the arms we find
Coefficient of Refreshment of Voluntary Muscles. --
ft. Ibs. per oz. per sec.
1. Walking Muscles, . . . 0.1318
2. Muscles of Arm (60 secs. rest), 0.1417
3. „ „ (30 ), 0.1296
3. ? » (15.. » ), - 0.1206
Mean,
..
.
0.1309
On comparing the mean Coefficient of Refreshment of the
voluntary. muscles with the Coefficient of Refreshment of the
heart, viz., 0.2377, we see that the heart receives double the
refreshment in the same time. This interesting result is
quite in accordance with the views of those anatomists who be-
lieve that the heart receives double the supply of arterial
blood that any other muscle receives, in consequence of the
ANIMAL MECHANICS.
485
semi-lunar valves, during life, not closing the openings of the
twice the supply of arterial blood, it ought to possess a Coeffi-
cient of Refreshment double that of any other muscle.
Conclusion.— I bring my work to a close with some regret,
as it has afforded me many pleasant hours of thought and re-
search. None of my readers can see its defects more clearly
than I myself perceive them, for I have not been able to use
more than a tenth part of the materials at my disposal, and I
feel that I have failed to convey by my words much of the
interest that attaches to the problems which I have endea-
voured to solve.
This much, however, I can guarantee; all the dissections,
weighings, and observations have been made by my own
hands, with every precaution of which I could think, to ensure
accuracy. My observations have also been made without
preconceived hypotheses to guide them; and many of my
most interesting results have been forced upon my notice by
the facts placed before my eyes in the Dissecting-room and
Laboratory.
I have done my best to travel non quo itur, sed quo eundum
est; and if I have erred in attaching too much importance to
Geometry, as the Queen and Mistress of all the Sciences, I
have at least the sanction of Plato for my error, who, when
asked how the Divine Being spent His time, replied, yewuerpɛīv
Tov Deóv.
" Errare mehercule malo cum Platone, quàm--"
INDEX.
ABDOMEN, curvature of surface of, | Anomaly of flexor pollicis in man, 122.
in parturient women, 223.
Angular velocity of forearm produced
Abdominal muscles, force of, 159. I by biceps, 175.
thickness of, 160.
Anästhesia, in labour, remarks on,
vertical tension of, 223.
167.
transverse tension of, 224. Anus, sphincter of the, 208.
surface, curvatures of, 161. Application of general laws of mus-
Absolute force of muscles, 62.
cular action, 442.
Absorption, rapidity of, 150.
Arm, weight of the, 27.
Accelerator urinæ, action of, 178. Armstrong gun, bursting of, 433.
Action, principle of least, 238. Arterial blood necessary for muscular
Adductor muscles in man, 327, 346, contraction, 7.
354.
Armadillo, foot tendons of, 107.
primus, 327.
Ass, biceps of, 169.
- secundus, a, 328.
Ateles, hand tendons of, 125.
» B, 327.
- muscles of, 400.
-- tertius, 327.
Axis of maximum instability, 274.
quartus, 329.
- of neutral equilibrium, 274.
in leopard, 349.
- minimum stability, 274.
- in lion, 376.
- llama, 337, 352.
Bat's wing, muscular type of, 423.
Agitator caudæ, 363.
Baxter, work done by frog muscles, 72.
Ai, foot tendons of, 108.
Bear, tendons of foot of, 110.
Albatross wing, 282.
- hand of, 131.
- latissimus dorsi of, 256. Bengal tiger, 103.
Alligator, flexors of toes in, 117.
fox, 29.
American method of hanging, 9. Beowulf quoted, 7.
Anatomy, comparative, of the tendons Biceps, cross section of, 64.
of hand and foot, 76.
- comparative anatomy of, 173.
Andral, on right ventricle, 145.
cruris, cross section of, 67.
Angel shark, pectoral muscles of, 180. 1 of Felidæ, 179.
+
488
INDEX.
Biceps, and brachiæus compared, 176. Chimpanzee, thumb flexor of, 121.
in ungulates, 174. - muscles of, 400..
- Work done by, 34.
Ciliaris, 202.
Bicipiti accessorius, length of fibres Classification of muscles, 164.
of, 1.
Cleavage planes, 4.
Biometer, 22.
in rocks and muscles compared, 4.
Bird, muscular fibres of, 3.
Coefficient of contraction in muscle, 71.
we muscular type of wing of, 419. - refreshment, 479.
---- principle of least action shown | Coignet's lift, 51.
in the wing of, 302.
Collongues on susurrus, 15, 21.
tendons of foot in, 113.
Comparison of the strength of tena
Blood, arterial, in muscular action, 7. don, force of muscle, and the force
quantity of, 150.
consumed in friction, in man, 87,
Boat racing, work done in, 48. in mastiff, 90.
Bombay goat, 104, 132:
Constancy of work done, law of, 444.
Borelli, 5, 73.
Constancy of volume in contracting
Boxlift, -59. .
muscle, 5.
Bouillaud on right ventricle, 145. Conclusion, 485.
Bowman, 1, 3.
Condition of maximum in rotator
Brachiæus anticus, cross section of, 2. muscles, 323.
-- comparative anatomy of, 173. Contraction, muscular, nature of, 5.
Brain, weight of, 32.
Coulomb, work done by porters, 60.
Brahmin cow, biceps of, 174. Cross section of muscles, 84.
Buccinator, 206.
- of tendons, 85.
Buys Ballot, on cross section of mus w of biceps, 2.
cles, 2.
Cruveilhier on thickness of ventricles
Buzzard, pectoral muscle of, 1846 of heart, 145.
Curvatures of abdominal surfaces, 161.
Cabwheels, sound of, on London
Cygnus atrata, tendons of, 118.
pavement, 19.
Czermak, 20.
Caninus musculus, 207.
Cynocephalus hamadryas, 400.
Capillary coefficient of blood resist-
ance, 143.
D'Alembert's principle, 25.
Capuchin monkey, tendons of hand Darwin, 387.
in, 125.
Death by hanging, causes of, 8.
Carnivores, hand tendons of, 125. Deltoidal muscles, 192.
foot tendons of, 95.
Deltoid muscle, comparative anatomy
- friction of foot tendons in, 102.1 of, 193.
Cassowary, muscular type of, 429. Diameter of elementary fibres, I.
Causes of death in hanging, 8. Diaphragm, tension of, 231.
Cercocebus, hand tendons of, 124. Dilator Iridis, 200.
Chemical effects of muscular con- | Dingo, foot tendons of, 99.
traction, 5.
- muscles of, 398,
N
INDEX.
· 489
Donders on cross sections of fibres, 2. Fibrillæ, 5.
on cross sections of muscles, 64. | Final causes, 75, 156.
o estimate of blood pressure, 144. | Flexor hallucis longus, 76.
a work done by heart, 147.
digitorum, 76.
- capacity of the left ventricle, 138. - accessorius, 76.
Drop, length of, required for instan- muscles of forearm, 63.
taneous death in hanging, 11.
tendons of hand, 120.
Duncan (Matthews), and Tait, expe Foot, tendons of, 76.
riments on uterine membranes, 155. Foot, Dr., experiments by, 26.
Dynamical work done by muscle, 44. | Force of leg flexors, 68.
the muscles - absolute, of muscles, 63.
of entire body, 62.
- transfer of, caused by skew
muscles, 361.
Edentata, foot tendons of, 107.
Formula for finding length of drop in
Effects of muscular contraction, 5.
hanging, 9.
Electrical theory of muscular contrac-
for wave motion, 15.
tion, 6.
---- for finding statical work done
Ellipsoidal muscles, 210.
by muscles, 25.
Emu, muscular type of, 424.
Fox, Bengal, foot tendons in, 99.
Equivalent fibre in skew muscles, 360.
- finger flexors of, 130. ..
Experiments, Foot's, 26.
-- flexor pollicis of, 131. .
Jevons', 39, 442, 450, 460.
Friction in tendons of foot, 76, 102, 87,
- Haughton's, Dr., 454.
- in tendons, table of, 118.
1 J. G., 473.
- in tendons of fingers, 134. . -
y Captain, 26.
in long flexors of birds, 117.
» S., 26.
Frog, work done by muscles of, 72.
Macalister's, 463, 479. .
Nipher's, 442, 462, 475.
General laws of muscular action, 442.
- Ormsby's, 26.
General principles, 1.
Gig action of horse explained, 177.
Fatigue, law of, 29, 450.
Glutæus primus, 364.
Felidæ, biceps of, 179.
quartus, 371.
-- depressors of arm in, 250. --- quintus, 373.
- extensors of forearm in, 251. – secundus, 367.
- hamstrings of, 261.
tertius, 369.
- gastrocnemii of, 261.
Goat, Bombay, hand tendons of, 132.
Fibres, elementary, shape of cross - 5 foot tendons of, 103.
sections of, 2.
Gorilla, dissection of, 400.
- elementary, dimensions of, 1. | Gratiolet, on thumb of Chimpanzee,
of heart, arrangement of, 431,434. ! 121.
of heart, proper and common, Greyhound, adductor muscles of, 343,
437.
- of heart of ox, lengths of, 441. I muscular type of, 397.
354.
490
INDEX.
:
26
Guns, large, construction of, 433. Hyperbola, cubic, 43.
Hales, capacity of right and left ven- | Hyperboloid of one sheet, 233.
tricles, 144.
Indicatrix curve, 210.
on force of blood currents in
Ingram on hanging, 7.
arteries, 138.
Inherent work in a triangular muscle,
Hamadryad baboon, 400.
182.
Hamstrings of lion and tiger, 257.
Insectivora, tendons of toes in, 107,
Hand tendons, anatomy of, 120.
Involuntary muscle, 3.
Hanging, history of, 7. .
Iris, muscles of, 200.
o causes of death in, 8.
- rules for, 9.
Jabiru, foot tendons of, 114
Haughton, Dr., experiments of, 454. Jacchus, foot tendons of, 95.
- Captain, , 26. Jackal (Canes pallipes), 98, 398.
J. G.,
473. / Jaguar, tendons of foot of, 97.
- S.,
Jevons' experiments, 39, 442, 450,
Hare, tendons of toes in, 109.
460.
Hapale Jacchus, 95.
Joulin on force in parturition, 157.
Heart, coefficient of refreshment of, | Kangaroo, foot tendons of, u.
484. .
- ratio of section of muscle to
shape of, 438.
tendon in, 91.
-- muscular tissue of, 4.
| King, Croker, on hanging, 13.
weight of human, 145.
| Knorz on coefficient of muscular
mechanical work done by, 137. | force. 60.
- principle of least action showed | Koster on coefficient of muscular
in arrangement of fibres of, 430.
force, 70.
Hedgehog, tendons of toes in, 107.
Helmholz on work done by human i Labouring, force of man in a day, 60.
heart; 146.
Laennec, on the ventricle, 150.
rate of action of motor nerves of Lagothrix Humboldtii, tendons of
frog, 14.
foot in, 95.
work done byfrog's muscles, 72, | Lagrange's theorem, 152, 218.
Henke on coefficient of muscular - principle of virtual velocities
force, 69.
used, 187.
Heron, pectoral muscles of, 185. | Latissimus dorsi, human, work done
- wing of, 295–313.
by, 191.
Hip, theory of, 361.
- dorsi in tiger, 250.
- peculiarities of, in man, 401. - dorsi, comparative anatomy of,
- abductor muscles of, in lion, 374. i 250.
Hirsch on rate of action of sensitive Law of work proportional to weight,
nerves in frog, 14.
443.
Hodman, useful work done by, 58. - of constancy of work, 444.
Horse, action of, 253,
-- of fatigue, 29, 450.
INDEX.
491
WI
Law of refreshment, 478.
Man, muscular type of, 400.
-- of muscular action general, ap- | - tendons of foot of, 76.
plication of, 442.
, hand of, 121.
Least action in nature, principle of, - pectoral of, 187.
238.
muscles of, compared with those
Lehmann, on quantity of blood in of monkeys, 402.
body, 150.
- anomalous hand tendon of,
Leopard, tendons of foot in, 98.
122.
- muscles of, 391.
adductors of, 327,
- adductors of, 349, 354. :
Mansfeldt on cross section of muscle,
Levret, cubic contents of uterus, 153. 1 64.
Length of fibres, 1.
Marsupials, muscles of foot of, 11r.
Levator ani, 208.
- muscles of hand of, 133.
Lioness, friction in tendons of, 96. Mangabey, anatomy of tendons of,
- biceps of, 175.
92, 127.
- anatomy of, 390.
Martinus Scriblerus, 52.
Lion, anatomy of, 390.
Mastiff, ratio of sections of muscles
- biceps, 179.
to tendons, 89.
- pectineus of, 329.
om tendons of foot in, 101.
-- tensor vaginæ femoris, 366. -- muscles of, 398.
Lithopædus Stenonensis, on the forces Matteucci, work done by frog
of parturition, 162.
muscles, 72.
Llama, toe tendons, 119.
Mechanical conditions of equilibrium
- hand tendons, 132.
of skew muscles, 236.
~ pectineus, 329.
Moment of muscular force, 63.
- adductors, 352.
Monkeys, New World, tendons of
Loss of work in triangular muscle, foot in, 94.
· 187.
- New World, tendons of hand
Lower, on volume of ventricles, 440. in, 124.
- Old World, tendons of hand in,
Macaques, tendons of, 92, 123.
123.
Macacus cynomolgus, 400. . - Old World, tendons of foot in,
nemestrinus, 92.
92.
- hand tendons of, 123.
Motor nerves, rate of action of, 14.
rhesus, anatomy of, 400.
Mouth, muscles around, 204.
Macalister, experiments of, 468, 479. Muscles, absolute force of, 63.
Macaw, wing of, 297, 314.
- cross sections of, 84.
Maclaren's experiment, 50.
classification of, 164.
Macnamara, experiments on the ra- |
1 coefficient of the contraction of,
pidity of absorption, 150.
71.
Magrath, Master, history of, 343. ---- dynamical work done by, 62.
- muscles of, 344, 397.
- prismatic, 104.
492
INDEX.
Muscles, quadrilateral, action of, in a, Obliquus internus, tensile strain of,
plane, 232.
222.
- skew, 236.
Old World monkeys, tendons in, 94.
wer statical work done by, 29, 62. Orbicularis palpebrarum, 201.
- striped, 3.
--- malaris, 205.
- sexual differences in, 3.
oris, 204.
- uterine, weight of, 154. , Ostrich, foot tendons, 113.
unstriped, 3.
-- muscular type of, 411, 427,
Muscular fibre, nature of, 1.
Otter, tendons of foot in, 100.
- contraction, nature of, 5. Ox, foot tendons of, 103.
contraction, effect of, 5.
_ length of fibres in heart of, 441.
- susurrus, 18.
type of bat's wing, 423. Papio maimon, 400.
bird's wing, 419.
Parturition, muscular forces in, 131.
, casšowary, 429.
- total forces available in, 162. -
emu, 429.
Parkes quoted, 47.
man, 400.
Paviours, work done by, 57.
ostrich, 427.
Pectoral muscles, human work done
rhea, 427.
by, 190.
struthionidæ, 417. Pectineus muscle in lion, 328.
canidæ, 391.
muscle in llama, 329.
12 felidæ, 399.
Pedlars, work done by, 60.
„ swimming animals, 425. | Penniform muscles, 167.
- action, application of general | Pennated muscles in relation to pris-
laws of, 442.
matic, 170.
Musical note of muscular susurrus, 18. Personal equation, 14.
Mylohyoid muscle in man, 177. Phalanger, foot tendons of, 12.
Pheasant, silver, foot tendons of, 115.
Nature of muscular fibre, 1.
- silver, wings of, 299, 315.
- of muscular contraction, 5, 14. Piers Plowman, quoted, 8.
Natural selection, 158.
Poisseuille on discharge of liquids
Nervous supply of ocular muscles, through capillary tubes, 142.
203.
- on blood pressure, 142.
New World monkeys, tendons in, 94. | Poppel, cubic contents of uterus, 154,
Nipher's experiments, 442, 462, 475. Porcupine, foot tendons of, 110.
Nylghau, biceps of, 169.
--- hand tendons of, 133.
Postulate of teleology, 238.
Obturator, externus, 385.
of work done by skew muscles,
- internus, 373.
358.
- „ in Struthionidæ, 427. Pope quoted, 53.
Obliquus externus, tensile strain of, Pound, Roman, 81.
219.
Porters, work done by, 60.
INDEX.
493
10
· Position of socket in wing of bird, Rhinoceros, foot tendons of, 105.
317. .
Rhesus, action of foot in, 93.
Prairie hunters, 81. . .
Rigor mortis, 6.
Principles, general, 1.
Risorius muscle, 205.
Prismatic muscles, 165, 166. Rodents, foot tendons of, 107.
Principle of least action, 238, 302. - pectoral muscles, 193.
shown in - hand, tendons of, 132.
heart, 431.
Rosenthal, work done by frog
Propositions regarding muscular ar | muscles, 73
· rangement, 387. .
Rule for ensuring instant death in
Pyrenean mastiff, tendons of foot of, hanging, 9.
85, 88.
Running and standing power, com.
parison of, in animals, 417.
Quadrumana, hand flexors in, 123. Russell, Scott, on resistance of eight-
Quadrilateral muscles, 194.
oared boat, 51.
- muscles, examples of, 196.
Quadratus labii inferioris, 206. Sambur deer, biceps of, 173.
Sarcolemma, 3.
Rankine's principle, 47.
Sartorius, fibres of, 1.
- observation on work in carrying | Schulze on tendons of foot, 82.
loads, 57.
Section, cross, of muscle, 84.
Ratio of observed work done by
- cross, of tendon, 88.
skew muscles to maximum possible | Semitendinosus, section of, 6%.
work, 359.
Semimembranosus, section of, 67.
of section of muscle to tendon Shoulder muscles, work done by, 27.
in mastiff, 89.
---- joint, theory of, 361, 388.
of section of muscle to tendon Sibson on fibres of heart, 145.
in man, 86.
Skew muscles, postulate for, 358..
section of muscle to tendon in
y transfer of force caused
kangaroo, 91.
by, 361.
Rate of nerve action, 14.
~ muscles, action of, 232, 242.
Rectus abdominis, tensile strain of, Sloth, three-toed, foot tendons of,
219.
105.
Relation of useful work to weight, 44. Socket, position of, in wing of bird,
Reptile flexors, 117.
Refreshment, law of, 478.
Sömmerring on right ventricle, 145.
- coefficient of, 479.
Sorby on cleaved rocks, 4.
Rhomboideus muscle, anatomy of, Spengler, experiments on blood pres.
66.
sure, 149.
Rhomboidal muscles, 166.
| Sphincter muscles, 4, 196.
Rhea, foot tendons of, 75, 114.
muscles of the eye, 201.
- muscular type of, 401. .
ng mouth, 204.
317.
494
INDEX.
209.
Spider monkey, toe tendons of, 119. | Tiger, claw, cutting of, 127.
Statical work done by muscles, 24, 62. -- foot, tendons of, 102.
Standing and running powers in - latissimus dorsi of, 249.
animals compared, 417.
mwe muscles of, 391.
Squirrel, pectoral of, 193.
hamstrings of, 253.
Struthionidæ, muscular type of, 411. - hand tendons of, 127.
flexor tendons of foot in, 113. Tinnitus aurium, relation of, to susur.
- hip-joint of, 417.
rus, 17.
and other animals compared, Triangular muscles, 178.
417.
is work done by, 180.
--- tendons of, 75.
inherent work in,
Striped muscle, 3.
182.
Striping in muscle, mechanical cause
y loss of work in,
of, 4.
187.
Stokes on susurrus, 20.
Triangularis oris, 206.
Surfaces curved, muscles forming, Transfer of force in skew muscles,
361.
Susurrus, muscular, 16.
Trapezius, human, 193.
Supraspinatus, human, weight of, 31. Turner on tendons of foot, 82.
work done by, 29.
Tyndall on compressed wax cleavage,
Swan, black, 116.
Swimming animals, muscular type of, Types, muscular, 390.
425.
» of birds, 419, 420.
, of man and quadru.
Temporal muscle, human, work done mana, 401, 406.
by, 191.
o muscular, of felidæ, 391.
Tendo-palpebrarum, 202.
, of canidæ, 399.
Tension, law of, in ellipsoidal sur-
os of struthionidæ, 417.
faces, 216.
Ungulates, tendons of, 103, 131.
Tendons of hand and foot, anatomy
Unstriped muscle, 3.
of, 74.
Uterus, position of gravid, 153.
- cross sections of, 85.
- volume of $ 153.
Tension, vertical, of abdominal
surface of 1 154.
muscles, 223.
o amount of muscle in, 154.
- transverse, of abdominal muscles,
- muscular wall of, mean thick-
224.
ness, 155.
Tensor vaginæ femoris, 366.
muscular wall of, tensile strain
Terrier, Irish, 398.
on, 155.
Thickness of elementary fibre, 1.
Uterine contraction, hydrostatical
am of abdominal muscles, 160.
pressure produced by, 155.
Tiger, effects of ingrowing claw on
moscas contraction, extreme force of,
weight of muscles, 128.
157.
-.
.
INDEX.
495
Velocity of blood spouting from | Wing of albatross, 282.
human artery, 128.
- of wood pigeon, 293.
angular, produced by biceps of bird, position of socket in,
and brachiæus, *175.
317.
Ventricles of heart, thickness of, 145. — muscular type of, 419.
- of heart, volume of, 440. Wollaston on susurrus, 16.
- left, daily work of, 145.. Woman, muscles of, 3.
- right, 1 145. Work, excessive muscular, 46.
Voluntary muscle, 3.
— dynamical, done by muscle,
Vulpes Bengalensis, 99.
44.
- done by triangular muscle, 180.
Walking, 54.
penniform muscle, 168.
Weber's observations on, 54.
human heart, 137.
Wallaby, foot tendons of, 111-115.
, exponents of, 6.
hand tendons of, 133.
biceps, brachialis, 34.
Wave-motion, formula for, 15.
- proportional to weight, 443.
Weka-weka rail, 115.
- done by bird's wing and pec-
Weber on coefficient of muscular
toral, 319.
force, 69.
- statical, done by muscles, 44.
on walking, 54.
Wolf, foot tendons of, 102.
on quantity of blood in body, - hand - 129.
150.
muscles of, 398.
Weisbach on flow of water in iron Wood pigeon, wing of, 293, 312.
pipes, 45.
Weights of muscles, 30, 45.
Zebu bull, tendons of toes in, 104.
- of heart, 145.
Zygomatic muscles, 205.
THE END.

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