502 



THE LOGARITHMIC SPIRAL 



[CH. 



right angle, — the "spiral" curve will be a circular are, C being the 

 centre of the circle. 



Another, and a very simple illustration may be drawn from the "cymose 

 inflorescences" of the botanists, though the actual mode of development of 

 some of these structures is open to dispute, and their nomenclature is involved 

 in extraordinary historical confusion*. 



In Fig. 243 B (which represents the Gicinnus of Schimper, or cyme unipare 

 scorpioide of Bravais, as seen in the Borage), we begin with a primary shoot 



from which is given off, at a certain definite 

 angle, a secondary shoot : and from that in turn, 

 on the same side and at the same angle, another 

 shoot, and so on. The deflection, or curvature, 

 is continuous and progressive, for it is caused by 

 no external force but only by causes intrinsic in 

 the system. And the whole system is sym- 

 metrical: the angles at which the successive 

 shoots are given off being aU equal, and the 

 lengths of the shoots diminishing in constant 

 ratio. The result is that the successive shoots, 

 or successive increments of growth, are tangents 

 to a curve, and this curve is a true logarith- 

 mic spiral. But while, in this simple case, 

 the successive shoots are depicted as lying in 

 a, plane, it may also happen that, in addition to their successive angular 

 divergence from one another within that plane, they also tend to diverge 

 by successive equal angles from that plane of reference; and by this 

 means, there will be superposed upon the logarithmic spiral a helicoid twist 

 or screw. And, in the particular case where this latter angle of divergence 

 is just equal to 180°, or two right angles, the successive shoots will once more 

 come to he in a plane, but they will appear to come off from one another on 

 alternate sides, as in Fig. 243 A.' This is the Schravbel or Bosiryx of Schimper, 

 the cyme unipare he'licoide of Bravais. The logarithmic spu-al is still latent 

 in it, as in the other ; but is concealed from view by the deformation resulting 

 from the hehcoid. The confusion of nomenclature would seem to have arisen 

 from the fact that many botanists did not recognise (as the brothers Bravais did) 

 the mathematical significance of the latter case; but were led, by the snail- 

 like spiral of the scorpioid cyme, to transfer the name "hehcoid" to it. 



Fig. 243. A, a helicoid, B, 

 a scorpioid cyme. 



In the study of such curves as these, then, we speak of the 

 point of origin as the pole (0) ; a straight hne having its extremity 

 in the pole and revolving about it, is called the radius vector; 



* Cf. Vines, The History of the Scorpioid Cyme, Journ. of Botany (n.s.), x, 

 pp. 3-9, 1881. 



