XI] 



CONCERNING GNOMONS 



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 mathematical 

 properties are therefore identical. We see at once that the 

 successive chambers of a spiral Nautilus (Fig. 237) or of a straight 

 Orthoceras (Fig. 300), each whorl or part of a whorl of a peri- 

 winkle or other gastropod (Fig. 258), each new increment of the 

 operculum of a gastropod (Fig. 263), each additional increment of 

 an elephant's tusk, or each new 

 chamber of a spiral foraminifer 

 (Figs. 259 and 260), has its leading 

 characteristic at once described and 

 its form so far explained by the 

 simple statement that it constitutes 

 a gnomon to the whole previously 

 existing structure. And herein lies 

 the explanation of that "time- 

 element" in the development of 

 organic spirals of which we have 

 spoken already, in a prehminary 

 and empirical way. For it follows 

 as a simple corollary to this 

 theorem of gnomons that we must not expect to find the 

 logarithmic spiral manifested in a structure whose parts are 

 simultaneously produced, as for instance in the margin of a 

 leaf, or among the many curves that make the contour of a 

 fish. But we must rather look for it wherever the organism 

 retains for us, and still presents to us at a single view, the successive 

 phases of preceding growth, the successive magnitudes attained, 

 the successive outhnes occupied, as the organism or a part thereof 

 pursued the even tenour of its growth, year by year and day by 

 day. And it easily follows from this, that it is in the hard parts 

 of organisms, and not the soft, fleshy, actively growing parts, 

 that this spiral is commonly and characteristically found; not 

 in the fresh mobile tissues whose form is constrained merely by 

 the active forces of the moment ; but in things hke shell and tusk, 

 and horn and claw, where the object is visibly composed of parts 



33—2 



Fig. 260. Another spiral fora- 

 minifer, Cristellaria. 



