VARIATION STEPS 191 



Let us see what information may be obtained from the 

 variation steps. 



LUDWIG has counted the petals of 1170 flowers of Primula 

 officinalis which were all obtained from a single meadow (near 

 Wieda).! The number varies from i to 22. The variation 

 curve has five humps, corresponding to 3, 5 (predominant), 8, 

 10 (5 X 2 ?) and 13 ; the superior limit coincides almost exactly 

 with 21. From these observations it may be concluded that 

 the number 5, which is practically constant in the species, is 

 here a term of the Fibonacci series, and it may therefore be 

 surmised that the corolla of Primula is, in reality, spiral, the 

 divergence being ordinarily 2/5. 



REMARK : For the observed facts three explanations are a priori possible : 

 (I) In the meadow where the flowers were collected certain peculiar conditions 

 of existence prevail by which a number of unusual reactions (expressed by the 

 humps of the curve) have been produced instead of the ordinary reaction 

 which results in the number 5 — thus a modification similar to the changes 

 which I have observed in Chrysanthemum carinatum (§ 128, p. 184) — and 

 in general, similar to modifications produced by cultivation. It may be 

 surmised, moreover, that not only the number of petals, but also the angle of 

 divergence has been modified. (2) In the locality mentioned a peculiar sub- 

 species exists, characterized by the fact that the Fibonacci series of the first 

 degree observed in the ordinary form (the term 5 being invariable just as in 

 Senecio Jacob csa) has been transformed into a series of the second degree 

 (comparable to Chrysanthemum carinatum and many other Compositae). 

 (3) In the mentioned meadow a mixture of several subspecies (pure lines ?, 

 races) occurs. I look upon the latter explanation as being very improbable, 

 because it implies about half-a-dozen mutations in a single property. 



In a certain species of Myosotis ^ a form exists which has 

 more than 5 petals ; the number is variable, the figure 8 being 

 predominant. Since 8 is a Fibonacci term we may conclude 

 that the ordinary figure 5 is also a Fibonacci term, and it may 

 be surmised that the corolla of Myosotis is spiral (divergence 2/5) . 



REMARK : The form of Myosotis alluded to may be looked upon as being 

 probably a subspecies because its characteristic feature is hereditary, even 

 when the plants are cultivated in a country very distant from their original 

 locality. 



In a certain form of Geranium examined by DE VRIES, the 

 number of pistils was very variable : the predominant numbers 

 were 5, 10, 15 . . . (multiples of 5). It may therefore be sur- 

 mised that the gynaecium of Geranium is really cyclic, the 

 deviating specimens having two, three or more pentamer 

 cycles of pistils. 



Number of stamens in the Rosacece : in this property, a series 

 of variation steps coinciding with the multiples of 5 exists. In 



1 Ber. deut. Bot. Gesellsch., xiv., 1896, p. 204. (Quoted according to VERNON, 

 loc. cit., p. 48.) 



21 think M. alpestris. The seeds (from which plants were raised in the 

 Botanic Garden at Ghent) were obtained from Germany. I don't remember 

 the name of the tradesman. (My notes are not within my reach.) 



