CHAPTER II 



ON MAGNITUDE 



To terms of magnitude, and of direction, must we refer all 

 our conceptions of form. For the form of an object is defined 

 when ' we know its magnitude, actual or relative, in various 

 directions ; and growth involves the same conceptions of magnitude 

 and direction, with this addition, that they are supposed to alter 

 in time. Before we proceed to the consideration of specific form, 

 it will be worth our while to consider, for a little while, certain 

 phenomena of spatial magnitude, or of the extension of a body 

 in the several dimensions of space*. 



We are taught by elementary mathematics that, in similar 

 solid figures, the surface increases as the square, and the volume 

 as the cube, of the linear dimensions. If we take the simple case 

 of a sphere, with radius r, the area of its surface is equal to 477/^, 

 and its volume to 47rf^; from which it follows that the ratio of 



o 



volume to surface, or V/S, is ^r. In other words, the greater the 

 radius (or the larger the sphere) the greater will be its volume, or 

 its mass (if it be uniformly dense throughout), in comparison with 

 its superficial area. And, taking L to represent any linear dimen- 

 sion, we may write the general equations in the form 



SocL^ F oc L\ 



or S^k.L^ and V = k'.L^; 



and (S^ X L. 



From these elementary principles a great number of conse- 

 quences follow, all more or less interesting, and some of them of 

 great importance. In the first place, though growth in length (let 



* Cf. Hans Przibrain, Anwendung elementarer Mathematik auf Biologische 

 Probleme (in Roux's Vortrdge, Heft ra), Leipzig, 1908, p. 10. 



