body slide over another; and adhesion the force, which unites two

b

Fig. 173.

Composition of Forces.

polished bodies when applied to each other, a force, which is measured by the perpendicular efa fort necessary for separating the two bodies. The more polished the surfaces in contact, the greater is the adhesion, and the less the friction; so that where the object is merely to facilitate the sliding of one surface over another, it will be always advantageous to make the

surfaces polished, or to put a liquid between them.

A beam or rod of any kind, resting at one part on a prop or support, which thus becomes its centre of motion, is a lever. The ten inch

W

F

Fig. 174.

Lever of the first kind.

beam, P W, Fig. 174, is a lever, of which F may be considered the prop or fulcrum; P, the part at which the power is applied, and W, the point of application of the weight or resistance.

In every lever we distinguish three points;-the fulcrum, power, and resistance; and, according to the relative position of these points, the lever is said to be of the first, second, or third kind. In a lever of the first kind, the fulcrum is between the resistance and power, as in Fig. 174; F being the fulcrum on which the beam rests and turns; P, the power; and W, the weight or resistance. We have numerous familiar examples of this lever;-the crowbar in elevating a weight; the handle of a pump; a pair of scales; a steelyard, &c. A lever of the second

kind has the resistance W, Fig. 175, between the power P and the fulcrum F; the fulcrum and power occupying each one extremity. The rudder of a ship, a wheelbarrow, and

nut-crackers, are varieties of this kind of lever. In a lever of the third kind, the power P is between the resistance W, and the fulcrum F, Fig. 176; the resistance and the fulcrum occupying each one extremity of the lever. In the last two levers, the weight and the power change places. Tongs and shears are levers of this kind; also, a long ladder raised against a wall by the efforts of a man: here the fulcrum is at the part of the ladder which rests on the ground; the power is exerted by the man; and the resistance is the ladder above him.

In all levers are distinguished,-the arm of the power and the arm

of the resistance. The former is the distance comprised between the power and the fulcrum, P F, Figs. 174, 175, and 176; and the latter is the distance W F, or that between the weight and the fulcrum. When, in the lever of the first kind, the ful

crum occupies the middle, the lever is said to have equal arms; but if it be nearer the power or the resistance, it is said to be a lever with unequal arms.

The length of the arm of the lever gives more or less advantage to the power, or the resistance, as the case may be. In a lever of the first kind, with equal arms, complete equilibrium would exist, provided the beam were alike in every other respect. But if the arm of the power be longer than that of the resistance, the resistance is to the power as the length of the arm of the power is to that of the arm of the resistance; so that if the former be double or triple the latter, the power need only be one-half or one-third of the resistance, in order that the two forces may be in equilibrium. A reference to the figures will exhibit this in a clear light. The three levers are all presumed to be of equal substance throughout, and to be ten inches, or ten feet, in length. The arm of the power, in Fig. 174, is the distance P F, equal to eight of those divisions; whilst that of the resistance is W F, equal to two of them. The advantage of the former over the latter is, consequently, in the proportion of eight to two, or as four to one; in other words, the power need only be one-fourth of the resistance, in order that the two forces may be equilibrious. In the lever of the second kind, the proportion of the arm P F of the power is to that of the resistance, W F, as ten-the whole length of the lever-to two; or five to one; whilst, in the lever of the third kind, it is as two to ten, or as one to five; in other words, to be equilibrious, the power must be five times greater than the resistance. We see, therefore, that in the lever of the second kind, the arm of the power must necessarily be longer than that of the resistance, since the power and the fulcrum are separated from each other by the whole length of the lever; hence this kind of lever must always be advantageous to the power; whilst the lever of the third kind, for like reasons, must always be unfavourable to it, seeing that the arm of the resistance is the whole length of the lever, and, therefore, necessarily greater than that of the power.

It can now be understood why a lever of the first kind should be most favourable for equilibrium; one of the second for overcoming resistance; and one of the third for rapidity and extent of motion: for whilst, in Fig. 176, the power is moving through the minute arc at P,

in order that the lever may assume the position indicated by the dotted lines F w, the weight or resistance is moving through the much more considerable space W w.

The direction in which the power is inserted into the lever likewise demands notice. When perpendicular to the lever, it acts with the greatest advantage, the whole of the force developed being employed in surmounting the resistance; whilst if inserted obliquely a part of the force is employed in tending to move the lever in its own direction; and this part is destroyed by the resistance of the fulcrum.

Lastly: the general principles of equilibrium in levers consist in this; that whatever may be the direction in which the power and resistance are acting, they must always be to one another inversely as the perpendiculars drawn from the fulcrum to their lines of direction. In Fig. 176, for example, the line of direction of the upper weight is W w; that of the power P p; and, to keep the lever in equilibrium in this position, the forces must be to one another inversely as F w to F p.

In applying these mechanical principles to the illustration of muscular motion, we must, in the first place, regard each movable bone as a lever, whose fulcrum or centre of motion is in its joint; the power at the insertion of the muscle; and the resistance in its own weight and that of the parts which it supports. In different parts of the skeleton we find the three kinds of levers. Each of the vertebræ of the back forms, with the one immediately beneath it, a lever of the first kind,the fulcrum being seated in the middle of the under surface of the body of the vertebra. The foot, when we stand upon the toe, is a lever of the second kind,-the fulcrum being in the part of the toes resting upon the soil; the power in the muscles inserted into the heel, and the resistance in the ankle-joint, on which the whole weight of the body rests. Of levers of the third kind we have numerous instances; of which the deltoid, to be described presently, is one. In this, as in other cases, the applicability of the principle, laid down regarding the arms of the lever, &c., is seen, and we find, that, in the generality of cases, the power is inserted into the lever so near to the fulcrum, that considerable force must be exerted to raise an inconsiderable weight;— that so far, consequently, mechanical disadvantage results; but such disadvantage enters into the economy of nature, and is attended with so many valuable concomitants as to compensate richly for the expense of power. Some of these causes, that tend to diminish the effect of the forces, we shall first consider, and afterwards attempt to show the advantages resulting from these and similar arrangements in effecting the wonderful, complicate operations of the muscular system. In elucidation of this subject, we may take, with Haller,1 the case of the deltoid--the large muscle, which constitutes the fleshy mass on the top of the arm, and whose office it is to raise the upper extremity. Let W F, Fig. 177, represent the os humeri, with a weight W at the elbow, to be raised by the deltoid D. The fulcrum F is necessarily, in this case, in the shoulder joint; and the muscle D is inserted much

Elementa Physiologie, lib. xi. 2.

nearer to the fulcrum than to the end of the bone on which the weight
rests; the arm of the power P F,-(supposing, for a moment, that it
is acting at this part with every advantage, which we shall see pre-
sently it is not,)-is, consequently,
much shorter than that of the resist-
ance W F, which, as in all levers of
the third kind, occupies the whole.
length of the lever. In estimating

the effect from this cause alone upon
the power to be exerted by the del- w
toid, we may suppose, that the arm

of the power is to that of the resist

ance as 1 to 3;-the deltoid being inserted into the humerus about onethird down. Now, if we raise a

W

Fig. 177.

P

P

DS
AF

Action of the Deltoid.

weight of fifty-five pounds in this way, and add five pounds for the weight of the limb-(which may be conceived to act entirely at the end of the bone-the power, which the deltoid must exert to produce the effect, is equal not to sixty pounds, but to three times sixty or one hundred and eighty pounds.

Figures 177 and 178 exhibit the disadvantages of the deltoid, so far as regards the place of its insertion into the lever; but many muscles have insertions much less favourable than it. The biceps, D, for example, in Fig. 179,-the muscle which bends the forearm on the arm,-is attached to the forearm ten times nearer the elbow-joint or fulcrum than to the extremity of the lever; and if we apply the argument to it, supposing the weight of the globe, in the palm of the hand, to be fifty-five pounds and the weight of the limb five pounds, it would have to act with a force equal to sixty times ten, or six hundred pounds, to raise the weight.

Muscles, again, are attached to the bones at unfavourable angles. If they were inserted at right angles in the direction P P, Fig. 177, the whole power would be effectually applied in moving the limb. On the other hand, if the muscle were parallel to the bone, the resistance would be infinite, and no effect could result. In the animal it rarely

happens, that the muscle is inserted at the most favourable angle: it is generally much smaller than a right angle. Reverting to the deltoid, this muscle is inserted into the humerus at an angle of about ten deFig. 179.

grees. Now, a power acting obliquely upon a lever, is to one acting perpendicularly, as the sine of inclination, represented by the dotted line Fs, Fig. 177, to the whole sine P P. In the case of the deltoid, the proportion is as 1,736,482 to 10,000,000. Wherefore, if the muscle had to contract with a force of one hundred and eighty pounds, owing to the disadvantage of its insertion near the fulcrum, it would have, from the two causes combined, to exert a force equal to 1,058 pounds.

Again, the direction in which the fibres are inserted into the tendon. has great influence on the power developed by the muscle. There are few straight muscles, in which all the fibres have the same direction as

Fig. 180.

90°

70'

45

30°

26°

the tendon. Fig. 180 will exhibit the loss of power, which the fibres must sustain in proportion to the angle of insertion. The fibre t F would, of course, exert its whole force upon the tendon, whilst the fibre t 90°, by its contraction, would merely displace the tendon. Now, the force exerted is, in such case, to the effective force,that is, to that which acts in moving the limb,-as the whole sine t F is to the sines of the angles at which the fibres join the tendon represented by the dotted lines. Borelli and Sturm have calculated these proportions as follows:-At an angle of 30°, they are as 100 to 87; at 45° as 100 to 70; at 26° as 100 to 89; at 14° as 100 to 97; and at 8° as 100 to 99.

T t

Insertion of Fibres into Tendon.

14.0

8°

F

The largest angle, formed by the outer fibres of the deltoid, is estimated by Haller at 30°: the smallest about 8°. If this disadvantage