XI] OF SUNDRY SPIRALS 751 



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

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

 example of the remarkable curve known as the equiangular or 

 logarithmic spiral. But before we enter upon the mathematics of 

 the equiangular 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 appear- 

 ance, 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 Httle) to consider 

 one part of the existing structure as older than another; through 

 and through, the newer particles have been merged and commingled 

 among the old ; the outhne, such as it is, is due to forces which for 

 the most part are still at work to shape it, and which in shaping it 

 have shaped it as a whole. But the horn, or the snail-shell, is 

 curiously different; for in these the presently existing structure is, 

 so to speak, partly old and partly new. It has been conformed by 

 successive and continuous increments; and each successive stage of 

 growth, starting from the origin, remains as an integral and un- 

 changing portion of the growing structure. 



We may go further, and see that horn and shell, though they 

 belong to the living, are in no sense alive*. They are by-products 

 of the animal; they consist of "formed material," as it is sometimes 

 called ; their growth is not of their own doing, hut comes of Hving 

 cells beneath them or around. The many structures which display 

 the logarithmic spiral increase, or accumulate, rather than grow. 

 The shell of nautilus or snail, the chambered shell of a foraminifer, 

 the elephant's tusk, the beaver's tooth, the cat's claws or the 

 canary-bird's— all these shew the same simple and very beautiful 

 spiral curve. And all alike consist of stuff secreted or deposited by 

 living cells ; all grow, as an edifice grows, by accretion of accumulated 



* For Oken and Goodsir the logarithmic spiral had a profound significance, for 

 they saw in it a manifestation of life itself. For a like reason Sir Theodore Cook 

 spoke of the Curves of Life; and Alfred Lartigues says (in his Biodynamique generale, 

 1930, p. 60): "Nous verrons la Conchy liologie apporter une magnifique contribution 

 a la Stereodynamique du tourbillon vital." The fact that the spiral is always 

 formed of non-living matter helps to contradict these mystical conceptions. 



