820 THE EQUIANGULAR SPIRAL [ch. 



drive in a "right-handed" screw, the terms right-handed and left- 

 handed are purely conventional; and the mathematicians and the 

 naturalists, unfortunately, use them in opposite ways. Thus the 

 mathematicians call the snail-shell or the joiner's screw kiotropic; 

 and Listing for one has much to say about lack of precision or even 

 confusion on the part of the conchologists and the botanists, from 

 Linnaeus downwards, in their attempts to deal with right-handed 

 and left-handed spirals or screws*. The convolvulus twines to the 

 right, the hop to the left; vine- tendrils are said to be mostly right- 

 handed. At any rate, Clerk Maxwell spoke of hop-spirals and vine- 

 ^irals, trying to avoid the confusion or ambiguity of left and right. 

 Some climbing plants are one and some the other; and, the architect 

 shews httle preference, but builds his spiral staircases or twisted 

 columns either way. But in all these, shells and all, the spiral runs 

 one way; it is isotropic, while the fir-cone shews spirals running both 

 ways at once, and we call them heterotropic, or diadroinic. 



When we find a "reversed shell," a whelk or a snail winding the 

 wrong way, we describe it mathematically by the simple statement 

 that the apical angle (^) has changed sign. Such left-handed shells 

 occur as a well-known but rare abnormality; and the men who 

 handle snails in the Paris market or whelks in Billingsgate keep 

 a sharp look-out for them. In rare instances they become common. 

 While left-handed whelks (Buccinum or Neptunea) are very rare 

 nowadays, it was otherwise in the epoch of the Red Crag; for 

 Neptunea was then extremely common, but right-handed specimens 

 were as rare as left-handed are today. In the beautiful genus 

 Ampullaria, or apple-snails, which inhabit tropical and sub-tropical 

 rivers, there is unusual diversity; for the spire turns to the right 

 in some species, and to the left in others, and again some are flat 

 or "discoid," with no spire at all; and there are plenty of half-way 

 stages, with right and left-handed spires of varying steepness or 

 acutenessf; in short, within the limits of this singular genus the 

 apical angle (y^) may vary from about ± 35° to ± 125°. But we 

 need not imagine that the direction of gK)wth actually changes 

 over from right-handed to left-handed; it is enough to suppose 



* See Listing's Topologie, p. 36; and cf. Clerk Maxwell's Electricity and 

 Magnetism, i, p. 24. 



t See figures in Arnold Lang's Comparative Anatomy (English translation), 

 II, p. ICl, 1902. 



