24 ON MAGNITUDE [ch. 



should be a steeper one, and the staying power should be less, in 

 the smaller than in the larger individual. This is the case of long- 

 distance racing, where the big winner puts on his big spurt at the 

 end. And for an analogous reason, wise men know that in the 

 'Varsity boat-race it is judicious and prudent to bet on the heavier 

 crew. 



Leaving aside the question of the supply of energy, and keeping 

 to that of the mechanical efficiency of the machine, we may find 

 endless biological illustrations of the principle of simihtude. 



In the case of the flying bird (apart from the initial difficulty of 

 raising itself into the air, which involves another problem) it may 

 be shewn that the bigger it gets (all its proportions remaining the 

 same) the more difficult it is for it to maintain itself aloft in flight. 

 The argument is as follows : 



In order to keep aloft, the bird must communicate to the air 

 a downward momentum equivalent- to its own weight, and there- 

 fore proportional to tlie cube of its own linear dimensions. But 

 the momentum so communicated is proportional to the mass of 

 air driven downwards, and to the rate at which it is driven : the 

 mass being proportional to the bird's wing-area, and also (with 

 any given slope of wing) to the speed of the bird, and the rate 

 being again proportional to the bird's speed; accordingly the 

 whole momentum varies as the wing-area, i.e. as the square of the 

 linear dimensions, and also as the square of the speed. Therefore, 

 in order that the bird may maintain level flight, its speed must 

 be proportional to the square root of its linear dimensions. 



Now the rate at which the bird, in steady ffight, has to work 

 in order to drive itself forward, is the rate at which it commmiicates 

 energy to the air; and this is proportional to mV^, i.e. to the 

 mass and to the square of the velocity of the air displaced. But 

 the mass of air displaced per second is proportional to the wing- 

 area and to the speed of the bird's motion, and therefore to the 

 power 2| of the hnear dimensions ; and the speed at which it 

 is displaced is proportional to the bird's speed, and therefore to 

 the square root of the hnear dimensions. Therefore the energy 

 communicated per second (being proportional to the mass and to 

 the square of the speed) is jointly proportional to the power 2| of 

 the linear dimensions, as above, and to the first power thereof : 



