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 concepts of magnitude and direction, 

 related to the further concept, or "dimension," of Time. Before 

 we proceed to the consideration of specific form, it will be well to 

 consider certain general phenomena of spatial magnitude, or of the 

 extension of a body in the several dimensions of space. 



We are taught by elementary mathematics — and by Archimedes 

 himself — that in similar 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 4:7rr^, and its volume to ^ttt^ ; from which it follows that the 

 ratio of its volume to surface, or V/S, is Jr. That is to say, VfS 

 varies as r; or, in other words, the larger the sphere by so much 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 dimension, we may write the general 

 equations in the form 



Soz L\ F oc L^ 



or iS = kL\ and V = k'L\ 



where k, k', are "factors of proportion," 



V V k 



and ^ cc L, or — = j-, L = KL. 



o ok 



So, in Lilliput, "His Majesty's Ministers, finding that Gulhver's 

 stature exceeded theirs in the proportion of twelve to one, concluded 

 from the similarity of their bodies that his must contain at least 

 1728 [or 12^] of theirs, and must needs be rationed accordingly*." 



* Likewise Gulliver had a whole Lilliputian hogshead for his half-pint of wine: 

 in the due proportion of 1728 half-pints, or 108 gallons, equal to one pipe or 



