XII] OF THE FORAMINIFERA 595 



measured in a number of cases. Van Iterson* has done so in 

 various Miliolinidae, with such results as the following: 



Triloculina rotunda, d'Orb. 



No. of chamber 12 3 4 5 6 7 8 9 10 



Breadth of chamber in /^ — 34 45 61 84 114 142 182 246 319 

 Breadth of chamber m /x, 



calculated — 34 45 60 79 105 140 187 243 319 



Here the mean ratio of breadth of consecutive chambers may 

 be taken as 1-323 (that is to say, the eighth root of 319/34) ; and 

 the calculated values, as given above, are based on this deter- 

 mination. 



Again, Rhumbler has measured the linear dimensions of a 

 number of rotahne forms, for instance Pulvinulina menardi 

 (Fig. 259) : in which common species he finds the mean hnear 

 ratio of consecutive chambers to be about 1-187. In both cases, 

 and especially in the latter, the ratio is not strictly constant from 

 chamber to chamber, but is subject to a small secondary fluctua- 

 tion f. 



When the linear dimensions of successive chambers are in 

 continued proportion, then, in order that the whole shell may 

 constitute a logarithmic spiral, it is necessary that the several 

 chambers should subtend equal angles of revolulion at the pole. 

 In the case of the Miliolidae this is obviously the case (Fig. 311); 

 for in this family the chambers lie in two rows (Bilocuhna), or 

 three rows (Triloculina), or in some other small number of series : 

 so that the angles subtended by them are large, simple fractions 

 of the circular arc, such as 180° or 120°. In many of the nautiloid 

 forms, such as Cyclammina (Fig. 312), the angles subtended, 

 though of less magnitude, are still remarkably constant, as we 



* Van Iterson, G., Mathem. u, mikrosk.-anat. Studien iiber Blattstellungen, nebst 

 Befrachtungen iiber den Sckalenbau der Miliolinen, 331 pp., Jena, 1907. 



•f Hans Przibram asserts that the linear ratio of successive chambers tends in 

 many Foraminifera to approximate to 1-26, which =^2-, in oth^r words, that 

 the volumes of successive chambers tend to double. This Przibram would bring 

 into relation with another law, viz. that insects and other arthropods tend to 

 moult, or to metamorphose, just when they double their weights, or increase their 

 linear dimensions in the ratio of 1 : ^2. (Die Kammerprogression der Foraminiferen 

 als ParaUele zm Hautungsprogression der Mantiden, Arch. f. Entw. Mech. xxsiv 

 p. 680, 1813.) Neither rule seems to me to be well grounded. 



38—2 



