496 THE LOGARITHMIC SPIRAL [ch. 



possibly be a physical or dynamical, though there may well be 

 a mathematical Law of Growth, which is common to, and which 

 defines, the spiral form in the Nautilus, in the Globigerina, in the 

 rani's horn, and in the disc of the sunflower. 



Of the spiral forms which we have now mentioned, every one 

 (with the single exception of the outhne of the cordate leaf) is an 

 example of the remarkable curve known as the Logarithmic Spiral. 

 But before we enter upon the mathematics of the logarithmic 

 spiral, let us carefully observe that the whole of the organic forms 

 in which it is clearly and permanently exhibited, however different 

 they may be from one another in outward appearance, in nature 

 and in origin, nevertheless all belong, in a certain sense, to one 

 particular class of conformations. In the great majority of cases, 

 when we consider an organism in part or whole, when we look (for 

 instance) at our own hand or foot, or contemplate an insect or 

 a worm, we have no reason (or very little) to consider one part 

 of the existing structure as older than another; through and 

 through, the newer particles have been merged and commingled, 

 by intussusception, among the old; the whole outhne, such as it 

 is, is due to forces which for the most part are still at work to 

 shape it, ami which in shaping it have shaped it as a whole. But 

 the horn, or the snail-shell, is curiously different; for in each of 

 these, the presently existing structure is, so to speak, partly old 

 and partly new; it has been conformed by successive and con- 

 tinuous increments ; and each successive stage of growth, starting 

 from the origin, remains as an integral and unchanging portion 

 of the still growing structure, and so continues to represent what 

 at some earlier epoch constituted for the time being the structure 

 in its entirety. 



In a slightly different, but closely cognate way, the same is 

 true of the spirally arranged florets of the sunflower. For here 

 again we are regarding serially arranged portions of a composite 

 structure, which portions, similar to one another in form, differ 

 in age ; and they differ also in magnitude in a strict ratio according 

 to their age. Somehow or other, in the logarithmic spiral the 

 time-element always enters in ; and to this important fact, full of 

 curious biological as well as mathematical significance, \v^e shall 

 afterwards return. 



