858 THE SPIRAL SHELLS [ch. 



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 determination. 

 Again, Rhumbler has measured the linear dimensions of a number 

 of rotaline forms, for instance Pulvinulina menardi (Fig. 363) : in 

 which common species he finds the mean linear 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 fluctuation*. 



When the linear dimensions of successive chambers are in con- 

 tinued 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 revolution at the pole. In the case of the 

 Miliolidae this is obviously the case (Fig. 425); for in this family 

 the chambers lie in two rows (Biloculina), 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. 426), the angles subtended, though of less magnitude, are still 

 remarkably constant, as we may see by Fig. 427; where the angle 

 subtended by each chamber is made equal to 20°, and this diagram- 

 matic figure is not perceptibly different from the other. In some 

 cases the subtended angle is less constant; and in these it would 

 be necessary to equate the several linear dimensions with the 

 corresponding vector angles, according to our equation r = e^cota 

 It is probable that, by so taking account of variations of 6, such 

 variations of r as (according to Rhumbler's measurements) Pul- 

 vinulina and other genera appear to shew, would be found to 

 diminish or even to disappear. 



The law of increase by which each chamber bears a constant 

 ratio of magnitude to the next may be looked upon as a simple 



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

 many Foraminifera to approximate to 1-26, which = V2; in other 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 : V 2. (Die Kammerprogression der Foraminiferen 

 als Parallele zur Hautungsprogression der Mantiden, Arch. f. Entw. Meek, xxxiv, 

 p. 680, 1813.) Neither rule seems to me to be well grounded (see above, p. 165). 



