503 See, on the mathôÙeôÙmatôÙiôÙcal history of the Gnomon, Heathãs Euclid, I, passim, 1908; Zeuthen, Theorû´me de Pythagore, Genû´ve, 1904; also a curious and interesting book, Das Theorem des Pythagoras, by Dr. H. A. Naber, Haarlem, 1908.
506 It will be observed that here Moseley, speaking as a mathematician and considering the linear spiral, speaks of whorls when he means the linear boundaries, or lines traced by the revolving radius vector; while the conchologist usually applies the term whorl to the whole space between the two boundaries. As conchologists, therefore, we call the breadth of a whorl what Moseley looked upon as the distance between two consecutive whorls. But this latter nomenclature Moseley himself often uses.
507 In the case of Turbo, and all other ãturbinateã shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a ãgaucheã spiral, such that the radius vector no longer revolves in a plane perpendicular to the axis of the system, but is inclined to that axis at some constant angle (ö¡). The figure still preserves its continued similarity, and may with strict accuracy be called a logarithmic spiral in space. It is evident that its envelope will be a right circular cone; and indeed it is commonly spoken of as a logarithmic spiral wrapped upon a cone, its pole coinciding with the apex of the cone. It follows that the distances of successive whorls of the spiral measured on the same straight line passing through the apex of the cone, are in geometrical progression, and conversely just as in the former case. But the ratio between any two consecutive interspaces (i.e. Rÿ£¢3ã₤ãã₤Rÿ£¢2ã₤ãã₤Rÿ£¢2ã₤ãã₤Rÿ£¢1) is now equal to öçÿ£¢2üã₤sinã₤ö¡ã₤cotã₤öÝã₤, ö¡ being the semi-angle of the enveloping cone. (Cf. Moseley, Phil. Mag. XXI, p. 300, 1842.)