[512] Alcide D’Orbigny, Bull. de la soc. géol. Fr. XIII, p. 200, 1842; Cours élém. de Paléontologie, II, p. 5, 1851. A somewhat similar instrument was described by Boubée. in Bull. soc. géol. I, p. 232, 1831. Naumann’s 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 D’Orbigny’s instrument was meant for the study of a turbinate shell.
[513] It is obvious that the ratios of opposite whorls, or of radii 180° apart, are represented by the square roots of these values; and the ratios of whorls or radii 90° apart, by the square roots of these again.
[514] For the correction to be applied in the case of the helicoid, or “turbinate” shells, see p. 557.
[515] On the Measurement of the Curves formed by Cephalopods and other Mollusks. Phil. Mag. (5), VI, pp. 241–263, 1878.
[516] For an example of this method, see Blake, l.c. p. 251.
[517] Naumann, C. F., Ueber die Spiralen von Conchylien, Abh. k. sächs. Ges. pp. 153–196, 1846; Ueber die cyclocentrische Conchospirale u. über das Windungsgesetz von Planorbis corneus, ibid. I, pp. 171–195, 1849; Spirale von Nautilus u. Ammonites galeatus, Ber. k. sächs. Ges. II, p. 26, 1848; Spirale von Amm. Ramsaueri, ibid. XVI, p. 21, 1864; see also Poggendorff’s Annalen, L, p. 223, 1840; LI, p. 245, 1841; LIV, p. 541, 1845, etc.
[518] Sandberger, G., Spiralen des Ammonites Amaltheus, A. Gaytani, und Goniatites intumescens, Zeitschr. d. d. Geol. Gesellsch. X, pp. 446–449, 1858.
[519] Grabau, A. H., Ueber die Naumannsche Conchospirale, etc. Inauguraldiss. Leipzig, 1872; Die Spiralen von Conchylien, etc. Programm, Nr. 502, Leipzig, 1882.
[520] It has been pointed out to me that it does not follow at once and obviously that, because the interspace AB is a mean proportional between the breadths of the adjacent whorls, therefore the whole distance OB is a mean proportional between OA and OC. This is a corollary which requires to be proved; but the proof is easy.
[521] A beautiful construction: stupendum Naturae artificium, Linnaeus.