543 Cf. DãOrbigny, Alc., Tableau mûˋthodique de la classe des Cûˋphalopodes, Ann. des Sci. Nat. (1), VII, pp. 245ã315, 1826; Dujardin. Fûˋlix, Observations nouvelles sur les prûˋtendus Cûˋphalopodes microscopiques, ibid. (2), III, pp. 108, 109, 312ã315, 1835; Recherches sur les organismes infûˋrieurs, ibid. IV, pp. 343ã377, 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 corôÙreôÙsponôÙding 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 MûÑller, V., Die spiral-gewundenen Foraminifera des russischen Kohlenkalks, Mûˋm. de lãAcad. Imp. Sci., St Pûˋtersbourg (7), XXV, 1878.
546 As von MûÑller is careful to explain, Naumannãs formula for the ãcyclocentric conchospiralã is appropriate to this and other spiral Foraminifera, since we have in all these cases a central or initial chamber, apôÙproxôÙiôÙmateôÙly spherical, about which the logarithmic spiral is coiled (cf. Fig. 309). In species where the central chamber is especially large, Naumannãs 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 û¥ber Blattstellungen, nebst Betrachtungen û¥ber den Schalenbau der Miliolinen, 331 pp., Jena, 1907.