796 THE EQUIANGULAR SPIRAL [ch. 



only one of which much use has been made is that which Moseley 

 first employed, namely, the simple method of determining the 

 relative breadths of the whorl at distances separated by some 

 convenient vectorial angle such as 90°, 180°, or 360°. 



Very elaborate measurements of a number of Ammonites have 

 been made by Naumann*, by Grabau, by Sandberger f, and by 

 Miiller, among which we may choose a couple of cases for considera- 

 tion f. In the following table I have taken a portion of Grabau's 



Ammonites intuslabiatus 

 Ratio of breadth of 



* C. F. Naumann, Ueber die Spiralen von Conchylien, Ahh. k. sacks. Ges. 1846, 

 pp. 153-196; Ueber die cyclocentrische Conchospirale u. iiber das Windungsgesetz 

 von Planorhis corneus, ibid, i, pp. 171-195, 1849; Spirale von Nautilus u. Ammonites 

 galeatus, Ber. k. sacks. Ges. ir, p. 26, 1848; Spirale von Amm. Ramsaueri, ibid, xvi, 

 p. 21, 1864. Oken, reviewing Naumann's work (in Isis, 1847, p. 867) foretold how 

 some day the naturalist and the mathematician would each learn of the other: 

 "Um die Sache zu Vollendung zu bringen wird der Mathematiker Zoolog und 

 Physiolog, und dieae Mathematiker werden miissen." 



I G. Sandberger, Clymenia subnautilina, Jakresber. d. Ver. f. Naturk. im Herzogtk. 

 Nassau, 1855, p. 127; Spiralen des Ammonites Amaltkeus, A. Gaytani und Goniatites 

 intumescens, Ztschr. d. d. Geolog. Gesellsch. x, pp. 446-^49, 1858. Also Miiller, 

 Beitrag zur Konchyliometrie, Poggend. Ann. lxxxvi, p. 533, 1850; ibid, xc, p. 323, 

 1853. These two authors upheld the logarithmic law against Naumann and Grabau. 



J See also Chr. Petersen, Das Quotientengesetz, eine biologisch-statistische Unter- 

 suckung, 119 pp., Copenhagen, 1921; E. Sporn, Ueber die Gesetzmassigkeit im 

 Baue der Muschelgehaiiser, Arch. f. Entw. Meek, cviii, pp. 228-242, 1926. 



