CH. V] MECHANICAL RECREATIONS 97 



Models. I may add here the observation, which is well 

 known to mathematicians, but is a perpetual source of disap- 

 pointment to ignorant inventors, that it frequently happens 

 that an accurate model of a machine will work satisfactorily 

 while the machine itself will not do so. 



One reason for this is as follows. If all the parts of a model 

 are magnified in the same proportion, say m, and if thereby a 

 line in it is increased in the ratio m : 1, then the areas and 

 volumes in it will be increased respectively in the ratios m a : 1 

 and m 8 : 1. For example, if the side of a cube is doubled then 

 a face of it will be increased in the ratio 4 : 1 and its volume 

 will be increased in the ratio 8 : 1. 



Now if all the linear dimensions are increased m times 

 then some of the forces that act on a machine (such, for 

 example, as the weight of part of it) will be increased m* times, 

 while others which depend on area (such as the sustaining 

 power of a beam) will be increased only m a times. Hence the 

 forces that act on the machine and are brought into play by 

 the various parts may be altered in different proportions, and 

 thus the machine may be incapable of producing results similar 

 to those which can be produced by the model. 



The same argument has been adduced in the case of animal 

 life to explain why very large specimens of any particular breed 

 or species are usually weak. For example, if the linear dimen- 

 sions of a bird were increased n times, the work necessary to 

 give the power of flight would have to be increased no less 

 than n 7 times* Again, if the linear dimensions of a man 

 of height 5 ft. 10 in. were increased by one-seventh his height 

 would become 6 ft. 8 iu., but his weight would be increased 

 in the ratio 512 : 343 (i.e. about half as much again), while 

 the cross sections of his legs, which would have to bear this 

 weight, would be increased only in the ratio 64 : 49 ; thus 

 in some respects he would be less efficient than before. Of 

 course the increased dimensions, length of limb, or size of 

 muscle might be of greater advantage than the relative loss 

 of strength ; hence the problem of what are the most efficient 



* Helmholtz, Gesammelte Abhandlungen, Leipzig, 1881, vol. i, p. 165. 



7 



B. B. ' 



