180 THE QUANTITATIVE METHOD IN BIOLOGY 



between the variation steps are rather common, but distinctly 

 less frequent than the characteristic step values. Examples : 

 number of leaflets in Tri/olium pratense quinquefolium : the 

 figures 3, 5, 7 . . . are predominant ; leaves with 4, 6, 8, . . . 

 leaflets being less numerous, although not uncommon (DE 

 VRIES). In Chrysanthemum segetum (number of marginal 

 florets) the Fibonacci terms 13 and 21 are much more frequent 

 than the transitory values {DE VRIES). This is also the case 

 with the number of marginal florets in many other Composites 

 (LUDWIG). For instance, in Chrysanthemum carinatum the 

 terms 5, 8, 13 and 21 are distinctly predominant (DE 

 BRUYKER and myself). 



Variation steps of the third degree. — Here the limits of varia- 

 tion coincide more or less exactly with two characteristic terms 

 of a series, the most frequent value being one of the transitory 

 values. Example : the number of marginal florets in the 

 terminal flower-head of Centaurea cyanus ^ : the most frequent 

 value is about 10 or 11 (under ordinary conditions of existence). 

 Starting from this maximum the frequency of the flower-heads 

 decreases in the positive and in the negative sense till the values 

 8 and 13 are reached. Below and above these limits there is a 

 sudden decrease in the frequency: flower-heads with >I3 

 marg. florets are uncommon and such with <8 marg. florets 

 are rare. The variation of the property under consideration 

 oscillates, as it were, between two limits which coincide with two 

 variation steps. I think that numerous examples of the third 

 degree will be discovered as soon as more attention is paid to 

 this subject. 



§ 125.— SYMMETRY AND VARIATION STEPS.— Many 

 biologists look upon the Fibonacci terms and the other series 

 mentioned in § 123 as being merely the expression of certain 

 states of symmetry. (This view is not a sufficient reason to 

 overlook the whole subject.) 



In the case of Pediastrum and Euastropsis (number of cells : 

 series 2, 4, 8 . . . 2". See § 65) the observed figures are not a 

 consequence of symmetry, since they are determined before the 

 cells meet one another, thus before any symmetry exists with 

 regard to their relative position. In a bicellular specimen of 

 Euastropsis the bilateral symmetry is a state of equilibrium 

 which depends on the fact that two independent cells meet each 

 other. Similarly in Pediastrum tetras, the radial symmetry 

 depends on the fact that four cells meet each other and are 

 united into a system of equilibrium which has the form of a 



^ Specimens with blue flowers collected in ryefields in Flanders. Specimens 

 with blue and specimens with purplish flowers (subspecies atropurpurea) 

 raised from seed which was obtained from HAAGE and SCHMIDT in Erfurt. 



