PART VII 

 VARIATION STEPS 



§ 122.— EXAMPLES OF VARIATION STEPS : PEDI- 

 ASTRUM, PERISTOME OF MOSSES.— In Pediastrum (§ 64) 

 the segments (cells) of an adult specimen {ccenobium, Fig. 3, 

 p. 79) are 2, 4, 8 ... 2" in number. The same series of 

 values is observed in the number of teeth of the peristome of the 

 mosses. These values are the expression of certain states of 

 equilibrium (see, for instance, § 66), which are more probable 

 (more stable ?) than the intermediate values. I call them 

 variation steps. 



Although attention has been repeatedly called to this subject, 

 especially by LUDWIG and also, among others, by DE 

 BRUYKER, WASTEELS, DE VRIES, VERSCHAFFELT 

 and myself, its importance has not yet been realized by the 

 great majority of the biologists. 



In general, when a given property passes through a series of 

 values, if certain values a', a", a'" are observed more frequently 

 than the intermediate values, the terms a', a", a'" . . . are 

 variation steps. Between the values a', a", a'" . . . certain 

 arithmetical relations exist. 



§ 123.— DIVERSE SERIES OF VARIATION STEPS.— 

 Several series of variation steps have already been discovered. 

 I limit myself to five examples ^ : 



(i) The Fibonacci series {discovered and thoroughly investi- 

 gated by LUDWIG), which consists of the terms i, 2, 3, 5, 8, 13, 

 21, 34. . . . Each term is the sum of the two preceding terms, 

 the first terms being i and 2. Examples are very numerous ; 

 for instance, the number of marginal florets in many species of 

 Corymbiferous Compositse. This series seems to exist also in 

 the animal kingdom (LUDWIG) ; for instance, in certain 

 properties of the shells of many Mollusca [Pecten, VAN DER 

 GUCHT; Scalaria, MACLEOD and WASTEELS, etc.). 



(2) The series 3, 5, 7 . . . (uneven numbers). Example : 

 the number of leaflets of digitate leaves in many plants ; for 

 instance, in Trifolium pratense quinquefolium (discovered by 

 DE VRIES). 



(3) The series 5, 10, 15 . . . (multiples of 5). Example : 



1 My notes are not within my reach. 



178 



