480 



MOTION. 



and as the lengths of the steps are greater in leap- (which from the principles assigned we may 

 ing than in running, the velocity is not very sen- consider to be the case,) since it is the dif- 

 sibly diminished, although the duration of each ference only between the velocities V and V' 



step" is increased ; that is, for example, when we 

 substitute three leaping, instead of four running, 

 steps in a second. In Table 12 we find that the 

 greatest length of step in running is im.5i2,and 

 in Table 13 that the greatest length of step in 

 leaping is l m .977, consequently four of the for- 

 mer exceed three in the latter mode of progres- 

 sion in a second by O m .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- 



tliat is efficient, the effective force will vary as 

 the square of the efficient velocity, that is 



F : :V-- 



hence F 



and V = 



(41) 



v/F 



or, the utmost velocity with which any animal 

 not impeded moves, is to its velocity when im- 

 peded by a given resistance, as the square root 

 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')* 



V V = ; then since F = <p ' 



pendently of which there is another important the product of the moving force into its velo- 

 vanation, which is that in leaping the swinging city, or the momentum of impulse FV, 



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 running 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. 



The 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 



wju _ 



yz 



momentum of 



(V w), and making this a 



maximum, we find u = V, and V = .^V. 

 Therefore the effective work of an animal is a 

 maximum when it is so loaded, that with its 

 whole force in action its velocity amounts to 

 one-third of the greatest velocity which it is 

 capable of exerting without any load at all. 

 In a series of experiments made on men and 

 horses, by drawing a lighter a!on^ r a canal, 

 and working several days consecutively, the 

 force was measured by the curvature and weight 

 of a track rope, as well as by a spring steel- 

 yard ; and the product of this force multiplied 

 by the velocity per hour was considered as 

 the momentum. By these experiments the 

 forces of men were found very nearly as 

 (V V') 2 , and those of horses, loaded so as 

 not to be able to trot, as (V V') 1 ' 7 to 

 (V V') 1 ' 8 , results which agree very closely 

 with the theory. In the application of these 

 formulse let us suppose a man's power to be 

 70 pounds, and his utmost speed in walking to 

 be six feet per second, hence <p will equal 70, 

 and V equal 6, therefore F = ^ = 31$, which 

 is the greatest force a man can exert in walking, 

 and he will move at the rate of iV = 2 feet in 

 a second.f The strength of a horse may be 

 easilv computed in a similar manner it is 

 generally reckoned to be six times that of man, 

 or about 420 pounds at a dead pull then its 



This is a formula of Ruler's, who has given 



account of the importance of this subject, we anot h er expression 4. A ^-1 \ : by this 



shall in conclusion briefly investigate the man- 

 ner in which animal force is estimated, and 

 under what circumstances it may be employed 

 to the greatest advantage. Thus, let <p 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, 



formula the greatest mechanical effect is when 

 V = %/ i V r , hut it does not agree with the expe- 

 riments of Schulze and others as near as the one 

 in the text See Schulze on the Strength of Men 

 and Horses, Acud. Bail. 1783. 



t Acconlinn to Ruchanen, the force expended in 

 pumping is 1742, by a winch 2856, in ringing 3883, 

 and in rowing 4095. Buchanen on human labour, 

 Report. 15, 319. 



