480 
ee ee ere in leap- 
ing than in running, the velocity is not very sen- 
sibly diminished, although the duration of each 
step is increased; that is, for example, when we 
substitute three leaping, instead of four running, 
steps ina second. In Table 12 we find that the 
t length of step in running is 1™.542, and 
in Table 13 that the greatest length of step in 
leaping is 1™.977, consequently four of the for- 
mer exceed three in the latter mode of p 
sion in a second by 0.237, or 0.76736 feet 
only ; but this loss of space is compensated by 
the greater time the pedestrian employs in tak- 
ing each step in leaping, being = 0".136. 
MM. Weber consider that leaping stands in 
the same relation to running that the grave step 
in slow walking does to quick walking. The 
difference of the grave and quick step in walk- 
ing consists in this, that in the former the oscil- 
lation of the leg has nearly completed its curve 
of vibration before it is placed on the ground, 
and therefore forms a greater angle with the 
vertical than the hind leg. The same difference 
also exists between running and leaping, inde- 
pendently of which there is another important 
variation, which is that in leaping the swinging 
leg completes its entire arc of vibration before it 
is placed on the ground, and makes the greatest 
angle with the vertical it can possibly effect. 
In running, on the contrary, the leg is set per- 
pendicularly on the ground and consequently 
the angle with the vertical =o. In the grave 
step as well as in leaping the foot is brought 
into contact with the ground preparatory to 
resting upon it; and, lastly, the duration of the 
leaping step exceeds that of the ranning step 
in the same degree that the duration of the 
grave step exceeds that of the quick step. These 
are the principal differences which distinguish 
the four modes of locomotion most usually 
adopted by man. 
he study of the mechanism of which the lo- 
comotive organs of animals is composed, of the 
laws by which their progression is accomplished, 
and of the vital force which they expend in pro- 
pelling the body from one point in space to 
another with different velocities, serves to in- 
struct alike the anatomist and the physiologist, 
the artist and the mechanician. Ignorance of 
these laws has been productive of grotesque 
delineations of the human figure as well as of 
the lower animals when represented in motion. 
We have abundant evidence of this in the pro- 
ductions of painters and sculptors, both of 
the ancient and modern schools. Locomotion 
is not only a function indispensably necessary 
for the prolongation of the lives of a vast as- 
semblage of animals, but it is also applicable as 
a force to innumerable purposes in life. On 
account of the importance of this subject, we 
shall in conclusion briefly investigate the man- 
ner in which animal force is estimated, and 
under what circumstances it may be employed 
to the test advantage. Thus, let @ denote 
the whole force of an animal when at rest, and 
let us suppose it to be incapable of any effort 
when it is moving with the velocity V; let F 
be the effective force when its velocity is V’; 
then, if the action of the force be uniform, 
MOTION. 
which from the principles assigned we may 
der to be ae since it is the dif. 
ference only between the velocities V and V’ 
that is efficient, the effective force will vary a 
the square of the efficient velocity,that is = 
9:F::(V—of:(V—Vp 
hence F = @ (yao ~~ ° : 
Vi! ue 
WT Pe. VE " i 
or, the utmost velocity with which any 
not impeded moves, is to its velocity when im 
peded by a given resistance, as the square roo! 
of its absolute force to the difference of the 
square roots of its absolute and effective forces 
Let us now investigate the velocity with which 
an animal must move and what must be its 
load, that the work performed by it may be a 
maximum. Retaining the same notation, let 
V —V’ = u; then since F = @ “=F ie 
the product of the moving force into its velo 
city, or the momentum of impulse FV’ 
will = ox (V — u), and making this ; 
maximum, we find u = 3V, and V’ = 4V 
Therefore the effective work of an animal is 
maximum when itis so loaded, that with 
whole force in action its velocity amounts t 
one-third of the greatest velocity which it | 
capable of exerting without any load at al 
In a series of experiments made on men an 
horses, by drawing a lighter along a can: 
and working several days consecutively, th 
force was measured by the curvature and weig 
of a track rope, as well as by a spring stee 
yard 5 and the product of this force multiplies 
y the velocity per hour was considered a 
the momentum. By these experiments the 
forces of men were found very nearly as 
(V — V’, and those of horses, loaded so ¢ 
wv to oe able to trot, as (V — V’jr7 
— V’)'8, results which agree close 
with the theory. In the applieatieial of the 
formule let us suppose a man’s power to 
70 pounds, and his utmost s in walking 
be six feet per second, hence @ will equal 7 
and V equal 6, therefore F = $9 = 31j, 
is the greatest force a man can exert in walki 
and he will move at the rate of }V = 2 feet 
a second.t The strength of a horse ma 
easily computed in a similar manner—it 
generally reckoned to be six times that of n 
or about 420 pounds at a dead pull—the 
* ‘This is a formula of Euler’s, who has g 
another expression > (--F): by 
formula the greatest mechanical effect is ’ 
V’ = / $V, but it does not agree with the 
riments of Schulze and others as near as the 
in the text. See Schulze on the Strength of 
and Horses, Acad. Berl. 1783. ae 
+ According to Buchanen, the force ( 
pumping is 1742, by a winch 2856, in ringing 3 
and in rowin .—Buchanen on human lab 
Repert. 15, 319. et 
and V = 
re 
he : oe 
