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 PãPÿ£¢1ã₤, PãPÿ£¢2ã₤, 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 ûÊlteren Gebirgsschichten, Abh. Berlin. Akad., Phys. Kl. pp. 135ã158, 1830; Ann. Sc. Nat. XXVIII, pp. 5ã43, 1833; cf. Elie de Beaumont, Sur lãenroulement des Ammonites, Soc. Philom., Pr. verb. pp. 45ã48, 1841.
511 Biblia Naturae sive Historia Insectorum, Leydae, 1737, p. 152.
512 Alcide DãOrbigny, Bull. de la soc. gûˋol. Fr. XIII, p. 200, 1842; Cours ûˋlûˋm. de Palûˋontologie, II, p. 5, 1851. A somewhat similar instrument was described by Boubûˋe. in Bull. soc. gûˋol. I, p. 232, 1831. Naumannãs Conchyliometer (Poggend. Ann. LIV, p. 544, 1845) was an application of the screw-micrometer; it was provided also with a rotating stage, for angular measurement. It was adapted for the Study of a discoid or ammonitoid shell, while DãOrbignyãs instrument was meant for the study of a turbinate shell.
513 It is obvious that the ratios of opposite whorls, or of radii 180ô¯ apart, are represented by the square roots of these values; and the ratios of whorls or radii 90ô¯ apart, by the square roots of these again.
514 For the correction to be applied in the case of the helicoid, or ãturbinateã shells, see p. 557.
515 On the Measurement of the Curves formed by Cephalopods and other Mollusks. Phil. Mag. (5), VI, pp. 241ã263, 1878.
516 For an example of this method, see Blake, l.c. p. 251.