[542] A similar hexagonal pattern is obtained by the mutual repulsion of floating magnets in Mr R. W. Wood’s experiments, Phil. Mag. XLVI, pp. 162–164, 1898.
[543] Cf. D’Orbigny, Alc., Tableau méthodique de la classe des Céphalopodes, Ann. des Sci. Nat. (1), VII, pp. 245–315, 1826; Dujardin. Félix, Observations nouvelles sur les prétendus Céphalopodes microscopiques, ibid. (2), III, pp. 108, 109, 312–315, 1835; Recherches sur les organismes inférieurs, ibid. IV, pp. 343–377, 1835, etc.
[544] It is obvious that the actual outline of a foraminiferal, just as of a molluscan shell, may depart widely from a logarithmic spiral. When we say here, for short, that the shell is a logarithmic spiral, we merely mean that it is essentially related to one: that it can be inscribed in such a spiral, or that corresponding points (such, for instance, as the centres of gravity of successive chambers, or the extremities of successive septa) wall always be found to lie upon such a spiral.
[545] von Möller, V., Die spiral-gewundenen Foraminifera des russischen Kohlenkalks, Mém. de l’Acad. Imp. Sci., St Pétersbourg (7), XXV, 1878.
[546] As von Möller is careful to explain, Naumann’s formula for the “cyclocentric conchospiral” is appropriate to this and other spiral Foraminifera, since we have in all these cases a central or initial chamber, approximately spherical, about which the logarithmic spiral is coiled (cf. Fig. [309]). In species where the central chamber is especially large, Naumann’s formula is all the more advantageous. But it is plain that it is only required when we are dealing with diameters, or with radii; so long as we are merely comparing the breadths of successive whorls, the two formulae come to the same thing.
[547] Van Iterson, G., Mathem. u. mikrosk.-anat. Studien über Blattstellungen, nebst Betrachtungen über den Schalenbau der Miliolinen, 331 pp., Jena, 1907.
[548] Hans Przibram asserts that the linear ratio of successive chambers tends in many Foraminifera to approximate to 1·26, which = ∛2; in other words, that the volumes of successive chambers tend to double. This Przibram would bring into relation with another law, viz. that insects and other arthropods tend to moult, or to metamorphose, just when they double their weights, or increase their linear dimensions in the ratio of 1 : ∛2. (Die Kammerprogression der Foraminiferen als Parallele zur Häutungsprogression der Mantiden, Arch. f. Entw. Mech. XXXIV p. 680, 1813.) Neither rule seems to me to be well grounded.
[549] Cf. Schacko, G., Ueber Globigerina-Einschluss bei Orbulina, Wiegmann’s Archiv, XLIX, p. 428, 1883; Brady, Chall. Rep., p. 607, 1884.
[550] Cf. Brady, H. B., Challenger Rep., Foraminifera, 1884, p. 203, pl. XIII.
[551] Brady, op. cit., p. 206; Batsch, one of the earliest writers on Foraminifera, had already noticed that this whole series of ear-shaped and crozier-shaped shells was filled in by gradational forms; Conchylien des Seesandes, 1791, p. 4, pl. VI, fig. 15a–f. See also, in particular, Dreyer, Peneroplis; eine Studie zur biologischen Morphologie und zur Speciesfrage, Leipzig, 1898; also Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen, Tübinger zool. Arbeiten, III, p. 35, 1899.