528 THE LOGARITHMIC SPIRAL [ch. 



obtuse one of Harpa or Dolium. In short it is obvious that all 

 the difEerences of form which we observe between one shell and 

 another are referable to matters of degree, depending, one and all, 

 upon the relative magnitudes of the various factors in the complex 

 equation to the curve. 



The paper in which, nearly eighty years ago. Canon Moseley 

 thus gave a simple mathematical expression to the spiral forms of 

 univalve shells, is one of the classics of Natural History. But 

 other students before him had come very near to recognising 

 this mathematical simplicity of form and structure. About the 

 year 1818, Reinecke had suggested that the relative breadths of 

 the adjacent whorls in an Ammonite formed a constant and 

 diagnostic character ; and Leopold von Buch accepted and 

 developed the idea*. But long before, Swammerdam, with a 

 deeper insight, had grasped the root of the whole matter : for, 

 taking a few diverse examples, such as Helix and Spirula, he 

 shewed that they and all other spiral shells whatsoever were 

 referable to one common type, namely to that of a simple tube, 

 variously curved according to definite mathematical laws; that 

 all manner of ornamentation, in the way of spines, tuberosities, 

 colour-bands and so forth, might be superposed upon them, but 

 the type was one throughout, and specific difEerences were of a 

 geometrical kind. "Omnis enim quae inter eas animadvertitur 

 differentia ex sola nascitur diversitate gyrationum: quibus si 

 insuper externa quaedam adjunguntur ornamenta pinnarum, 

 sinuum, anfractuum, planitierum, eminentiarum, profunditatum, 

 extensionum, impressionum, circumvolutionum, colorumque : . . . 

 tunc deinceps facile est, quarumcumque Cochlearum figuras 

 geometricas, curvosque, obliquos atque rectos angulos, ad unicam 

 omnes speciem redigere: ad oblongum videUcet tubulum, qui 

 vario modo curvatus, crispatus, extrorsum et introrsum flexus, 

 ita concrevitf." 



* J. C. M. Reinecke, Maris protogaei Nautilos, etc., Coburg, 1818. Leopold 

 von Buch, Ueber die Ammoniten in den alteren Gebirgsschichten, Abh. Berlin. 

 Ahad., Phtjs. Kl. pp. 135-158, 1830; Ann. Sc. Nat. xxvin, pp. 5-43, 1833; of. 

 Elie de Beaumont, Sur I'enroulement des Ammonites, Soc. Philom., Pr. verb. 

 pp. 45-48, 1841. 



•j- Biblia Naturae sive Historia Insectorum, Leydae, 1737, p. 152. 



