series of logarithmic spirals (Figs. 258, 259, etc.); (2) it is charôÙacôÙterôÙisôÙtic of the growth of the horn, of the shell, and of all other organic forms in which a logarithmic spiral can be recognised, that each successive increment of growth is a gnomon to the entire pre-existing structure. And conversely (3) it follows obviously, that in the logarithmic spiral outline of the shell or of the horn we can always inscribe an endless variety of other gnomonic figures, having no necessary relation, save as a {514} mathôÙeôÙmatôÙiôÙcal accident, to the nature or mode of development of the actual structure10. {515}
Of these three propositions, the second is of very great use and advantage for our easy understanding and simple description of the molluscan shell, and of a great variety of other structures whose mode of growth is analogous, and whose mathôÙeôÙmatôÙiôÙcal properties are therefore identical. We see at once that the successive chambers of