28 ON MAGNITUDE [ch. 



We learn in elementary mechanics the simple case of two similar 

 beams, supported at both ends and carrying no other weight than 

 their own. Within the limits of their elasticity they tend to be 

 deflected, or to sag downwards, in proportion to the squares of their 

 linear dimensions ; if a match-stick be two inches long and a similar 

 beam six feet (or 36 times as long), the latter will sag under its own 

 weight thirteen hundred times as much as the other. To counteract 

 this tendency, as the size of an animal increases, the limbs tend to 

 become thicker and shorter and the whole skeleton bulkier and 

 heavier; bones make up some 8 per cent, of the body of mouse or wren, 

 13 or 14 per cent, of goose or dog, and 17 or 18 per cent, of the body 

 of a man. Elephant and hippopotamus have grown clumsy as well as 

 big, and the elk is of necessity less graceful than the gazelle. It is of 

 high interest, on the other hand, to observe how little the skeletal 

 proportions differ in a httle porpoise and a great whale, even in the 

 limbs and hmb-bones ; for the whole influence of gravity has become 

 neghgible, or nearly so, in both of these. 



In ifhe problem of the tall tree we have to determine the point 

 at which the tree will begin to bend under its own weight if it be 

 ever so little displaced from the perpendicular*. In such an 

 investigation we have to make certain assumptions — for instance 

 that the trunk tapers uniformly, and that the sectional area of the 

 branches varies according to some definite law, or (as Ruskin 

 assumed) tends to be constant in any horizontal plane; and the 

 mathematical treatment is apt to be somewhat difficult. But 

 Greenhill shewed, on such assumptions as the above, that a certain 

 British Columbian pine-tree, of which the Kew flag-staff, which is 

 221 ft. high and 21 inches in diameter at the base, was made, could 

 not possibly, by theory, have grown to more than about 300 ft. It 

 is very curious that Galileo had suggested precisely the same height 

 (ducento braccie alta) as the utmost limit of the altitude of a tree. 

 In general, as Greenhill shewed, the diameter of a tall homogeneous 

 body must increase as the power 3/2 of its height, which accounts 

 for the slender proportions of young trees compared with the squat 



* In like manner the wheat-straw bends over under the weight of the loaded 

 ear, and the cat's tail bends over when held erect — not because, they "possess 

 flexibility," but because they outstrip the dimensions within which stable equi- 

 librium is possible in a vertical position. The kitten's tail, on the other hand, 

 stands up spiky and straight. 



