754 THE EQUIANGULAR SPIRAL [ch. 



recognised by Descartes, and discussed in the year 1638 in his letters 

 to Mersenne*. Starting with the conception of a growing curve 

 which should cut each radius vector at a constant angle — just as 

 a circle does — Descartes shewed how tt would necessarily follow that 

 radii at equal angles to one another at the pole would be in con- 

 tinued proportion ; that the same is therefore true of the parts cut off 

 from a common radius vector by successive whorls or convolutions 

 of the spire; and furthermore, that distances measured along the 

 curve from its origin, and intercepted by any radii, as at B, C, are 

 proportional to the lengths of these radii, OB, OC. It follows that 



Fig. 350. The equiangular spiral. 



the sectors cut off by successive radii, at equal vectorial angles, are 

 similar to one another in every respect; and it further follows that 

 the figure may be conceived as growing continuously without ever 

 changing its shape the while. 



If the whorls increase very slowly, the equiangular spiral will come to look 

 like a spiral of Archimedes. The Nummulite is a case in point. Here we have 

 a large number of whorls, very narrow, very close together, and apparently of 

 equal breadth, which give rise to an appearance similar to that of our coiled 

 rope. And, in a case of this kind, we might actually find that the whorls 

 were of equal breadth, being produced (as is apparently the case in the 

 Nummulite) not by any very slow and gradual growth in thickness of a con- 

 tinuous tube, but by a succession of similar cells or chambers laid on, round 

 and round, determined as to their size by constant surface-tension conditions 

 and therefore of unvarying dimensions. The Nummulite must always have 

 a central core, or initial cell, around which the coil is not only wrapped, but 

 out of which it springs ; and this initial chamber corresponds to our a' in the 

 expression r = a' + ad cot a. 



* (Euvres, ed. Adam et Tannery, Paris, 1898, p. 360. 



