NATURAL SCIENCES OF PHILADELPHIA. 



529 



TABLE II. 



CALCULATED VALUES FOR THEORETICAL EGG. 



N 0TE- The length and thickness of the egg are the same for both tables, 

 viz. : length 12-76 ; thickness 8*86. The polar angle at the point of greatest 

 thickness is designated by y, but has not the same value in each table, being 

 a measured value in Table I. and a calculated value (56 34') in Table II. 

 The distance B' + /> cos y for this angle is, by calculation, 5-79. 



By these tables we perceive that the first part of the first section agrees 

 verv closely, from the apex to the widest part, with the theoretical curve. 

 The* second part is less satisfactory. The average real section would nowhere 

 differ from the theoretical curve by much more than the thickness of the egg- 

 shell (about 12-100 inch). 



ON THE SIGNIFICATION OF THESE COMPARISONS. 



Position of the Vitellus.A belief in the adaptability of polar formulse to 

 explain some processes of nature was somewhat vaguely expressed by Gran- 

 dus. James Bernoulli seems to have been strongly, though vaguely, impressed 

 with the idea of an important meaning in the logarithmic spiral. Moseley 

 and Naumann have demonstrated its existence in several shells, and similar 

 results have since been obtained.* Possibly the mechanical properties of this 

 spiral are involved among the causes which give the first direction to the wind- 

 ings of shells. Naumann suggests that all spirally wound conchylia may begin 

 with a logarithmic spiral. The law of the growth of the animal may then, 

 perhaps, be subsequently instrumental in determining the form of the shell. f 

 Lotze says, in discussing the mathematical explanation of organic forms, that 

 in pure mathematics it is not absolutely essential that the origin of co-ordi- 

 nates be in any particular place, but where an explanation of the nature of 

 phenomena is required, the origin must be taken where, in the Real, the 

 centre of emanation of action resides ; the direction and connection of the 

 co-ordinates must correspond with those of the operating forces.f 



The position of the vitellus in the pole of the theoretical egg may, possi- 

 bly, exhibit that coincidence of mathematical and organic system which is 

 intended by Lotze. It would be desirable to have correct observations of the 

 positions of different parts within the egg, in order to attempt the discovery 



* Grandu?, A Collection of Geometrical Flowers: Abridged Phil. Trans., vol. vi. p. 67, 1723. 

 Bernoulli, Leipzig Acts., 1692. Moseley, Phil. Trans., 183S. Naumann, Abh. d. Jablonowskischen 

 Gesellsch., Leipzig, 1846. Also Abh. d. Math. Pbys. Kl.d. K. S. Gesellsch., Leipzig, 1852. Sand- 

 berger. ueber die Spiralen von Ammonites, &c, Zeitschr. d. Deutsch. Gcol. Gesellsch., vol. x. 1858, 

 p. 446. 



f I am not aware to what extent the views of Moseley concerning the growth of shells have 

 been adopted by naturalists. Naumann, as far as he expresses any opinion, seems to agree with 

 Moseley on this subject. To me, Moseley's explanation of the growth of shells, and of the man- 

 ner in which their mathematical properties suit the life and growth of the animal, appear very 

 interesting aod important, and, did our limits permit, would well deBerve to be noticed here at 

 length. From the paper of Sandberger's just cited, I am led to believe that the determination of 

 the equations of the windings of shells is now recognized by scientific conchologists as a valuable 

 descriptive method. 



% Fhyeiologie, p. 330. 



1862.] 



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