[502] This is the so-called Dreifachgleichschenkelige Dreieck; cf. Naber, op. infra cit. The ratio 1 : 0·618 is again not hard to find in this construction.
[503] See, on the mathematical history of the Gnomon, Heath’s Euclid, I, passim, 1908; Zeuthen, Theorème de Pythagore, Genève, 1904; also a curious and interesting book, Das Theorem des Pythagoras, by Dr. H. A. Naber, Haarlem, 1908.
[504] For many beautiful geometrical constructions based on the molluscan shell, see Colman, S. and Coan, C. A., Nature’s Harmonic Unity (ch. ix, Conchology), New York, 1912.
[505] The Rev. H. Moseley, On the Geometrical Forms of Turbinated and Discoid Shells, Phil. Trans. pp. 351–370. 1838.
[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 “turbinate” shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a “gauche” 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 (θ). 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. R3 − R2 ⁄ R2 − R1) is now equal to ε2π sin θ cot α , θ being the semi-angle of the enveloping cone. (Cf. Moseley, Phil. Mag. XXI, p. 300, 1842.)
[508] As the successive increments evidently constitute similar figures, similarly related to the pole (P), it follows that their linear dimensions are to one another as the radii vectores drawn to similar points in them: for instance as P P1 , P P2 , which (in Fig. [264], 1) are radii vectores drawn to the points where they meet the common boundary.
[509] The equation to the surface of a turbinate shell is discussed by Moseley (Phil. Trans. tom. cit. p. 370), both in terms of polar coordinates and of the rectangular coordinates x, y, z. A more elegant representation can be given in vector notation, by the method of quaternions.
[510] J. C. M. Reinecke, Maris protogaei Nautilos, etc., Coburg, 1818. Leopold von Buch, Ueber die Ammoniten in den älteren Gebirgsschichten, Abh. Berlin. Akad., Phys. Kl. pp. 135–158, 1830; Ann. Sc. Nat. XXVIII, pp. 5–43, 1833; cf. Elie de Beaumont, Sur l’enroulement des Ammonites, Soc. Philom., Pr. verb. pp. 45–48, 1841.
[511] Biblia Naturae sive Historia Insectorum, Leydae, 1737, p. 152.