860 THE SPIRAL SHELLS [ch. 



consequence of the structural uniformity or homogeneity of the 

 orgai^ism ; we have merely to suppose (as this uniformity would 

 naturally lead us to do) that the rate of increase is at each instant 

 proportional to the whole existing mass. For if Vq, Vi, etc. be 



Fig. 427. Cydammitia sp. (J>iagramniatic.) 



the volumes of the successive chambers, let Fj bear a constant 

 proportion to Fq, so that V^ = qV^, and let Fg bear the same 

 proportion to the whole pre-existing volume: then 



n = ? (Fo + Fi) =^ q (F„ + ?F„) = ?F„ (1 + q) and VJV^ = 1 + ?. 



This ratio of 1/(1 + q) is easily shewn to be the constant ratio 

 running through the whole series, from chamber to chamber; and 

 if this ratio of volumes be constant, so also are the ratios of corre- 

 sponding surfaces, and of corresponding linear dimensions, provided 

 always that the successive increments, or successive chambers, are 

 similar in form. 



We have still to discuss the similarity of form and the symmetry 

 of position which characterise the successive chambers, and which, 

 together with the law of continued proportionahty of size, are the 

 distinctive characters and the indispensable conditions of a series 

 of "gnomons." 



The minute size of the foraminiferal shell or at least of each 



