192 THE QUANTITATIVE METHOD IN BIOLOGY 



Prunus spinosa, for instance, the number 20 prevails. In the 

 neighbourhood of Leipzig, DIETEL found in this species a 

 variation curve in which the figure 21 was predominant (LUD- 

 WIG).^ In Crati?,gus coccinea LUDWIG found a variation 

 curve with a distinct hump coinciding with the value 8. 

 According to LUDWIG it may be surmised that in the men- 

 tioned examples the ordinary cyclic disposition of the stamens 

 is transformed into a spiral one. 



By the above examples an interesting line of research is 

 indicated. 



§ 132.— METHODS FOR THE DISCOVERY OF VARIA- 

 TION STEPS. — Several methods for the discovery of variation 

 steps may be followed. I give here six methods and I suggest 

 a seventh one. 



FIRST METHOD : When a two-humped (dimorphic) or 

 multi-humped (polymorphic) variation curve is obtained, it 

 may happen that the humps coincide with variation steps. 

 This may be considered as very probable if a certain arith- 

 metical relation exists between the values which coincide with 

 the humps. In such a case it is advisable to verify the obtained 

 conclusion by carrying out a second series of measurements or 

 by applying another method. (See below.) 



SECOND METHOD : In a one-humped (nionomorphic) 

 curve the existence of variation steps may be surmised, as often 

 as a sudden decrease of the ordinates (frequencies) coincides 

 with two values between which certain arithmetical relations 

 exist, and also when arithmetical relations exist between both 

 extreme values, or between the most frequent value (hump of 

 the curve) and one of the extreme values or both extreme 

 values. In such cases verification is, of course, necessary. 

 (Example : Centaur ea cyanus, § 124, p. 180.) 



THIRD METHOD : Variation steps may be discovered by 

 comparing the value of a property in several subspecies, species, 

 genera, etc. ; in other words, by examining the variation of a 

 given property through a long series of specific forms. From 

 the predominance of certain figures the existence of variation 

 steps may be deduced. (Examples: the predominance of the 

 figures 3, 6, 9 in the number of stamens of the Monocotyledons — 

 the predominance of the Fibonacci terms in the number of rays 

 of the umbella of the Umbelliferae, etc.) 



FOURTH METHOD : Variation steps may be rendered 

 observable by cultivating plants (or breeding animals) under 

 conditions of existence as various as possible, or by observing 

 them in the state of nature under different conditions. EX- 



1 Prof. Dr F. LUDWIG, Ueber Variationskurven und Variationsfldchen der 

 Pflanzen. Botan. Centralhlatt, 1895, vol. Ixiv., pp. 103-105. 



