38 ON MAGNITUDE [ch. 



power of contraction." This hypothesis, which is so easily seen on 

 physical grounds to be unnecessary, has been amply disproved in 

 a series of excellent papers by Felix Plateau*. 



From the impulse of the preceding case we may pass to the momentum 

 created (or destroyed) under similar circumstances by a given force acting 

 for a given time : mv = Ft. 



We know that . m oc P, and t = Ifv, 



so that lH = Fllv, or v^ = F/l\ 



But whatsoever force be available, the animal may only exert so much of 

 it as is in proportion to the strength of his own limbs, that is to say to the 

 cross-section of bone, sinew and muscle ; and all of these cross-sections are 

 proportional to P, the square of the linear dimensions. The maximal force, 

 -f'max, which the animal dare exert is proportional, then, to l^; therefore 



Fraa^JP = coustant. 



And the maximal speed which the animal can safely reach, namely 

 F'max = ^max/^, IS also constaut, or independent {ceteris paribus) of the dimensions 

 of the animal. 



A spurt or effort may be well within the capacity of the animal but far 

 beyond the margin of safety, as trainer and athlete well know. This margin 

 is a narrow one, whether for athlete or racehorse; both run a constant risk 

 of overstrain, under which they may "pull" a muscle, lacerate a tendon, or 

 even "break down" a bonet. 



It is fortunate for their safety that animals do not jump to heights pro- 

 portional to their own. For conceive an animal (of mass m) to jump to 

 a certain altitude, such that it reaches the ground with a velocity v; then 

 if c be the crushing strain at any point of the sectional area (A) of the limbs, 

 the limiting condition is that mv = cA. 



If the animal vary in magnitude without change in the height to which 

 it jumps (or in the velocity with which it descends), then 



m P , 



coc 2 oc ^"2, orZ. 



The crushing strain varies directly with the linear dimensions of the animal ; 

 and this, a dynamical case, is identical with the usual statical limitalfion of 

 magnitude. 



* Recherches sur la force absolue des muscles des Inverlebres, Bull. Acad. R, 

 de Belgique (3), vi, vii, 1^83-84: see also ibid. (2), xx, 1865; xxii, 1866; Ann. 

 Mag. N.H. xvii, p. 139, 1866; xix, p. 95, 1867. Cf. M. Radau, Sur la force 

 musculaire des insectes, Revue des deux Mondes, lxiv, p. 770, 1866. The subject 

 had been well treated by Straus-Diirckheim, in his Considerations generates sur 

 Vanatomie comparee des animauz articuUs, 1828. 



t Cf. The dynamics of sprint -running, by A. V. Hill and others, Proc. R.S. (B), 

 cii, pp. 29-42, 1927; or Muscular Movement in Man, by A. V. Hill, New York, 

 1927, ch. VI, p. 41. 



