VARIATION STEPS 179 



the number of styles in a race of Geranium with numerous styles 

 (discovered by DE VRIES). (See § 131.) 



(4) The series 3, 6, 9 (multiples of 3) . Example : the number 

 of stamens in many Monocotyledons ; for instance : Iris, 3 ; 

 Lilium, 6 ; Butomus, 9. The values i, 2 ... 4, 5 ... 7, 8 

 ... 10, II . . . are uncommon, rare or very rare in the 

 androecium of the Monocotyledons. 



(5) The series 2, 4, 8, 16, 32 . . . (powers of 2) . Examples : 

 nimiber of cells in Pediastrum ; number of teeth of the peri- 

 stome of the Mosses. 



REMARK : Certain values belong to more than one series ; for instance ■ 

 the value 3 belongs to the series i, 2 and 4 ; the value 5 belongs to the series 

 I, 2 and 3 ; the value 8 belongs to the series i and 5. (See § 126.) 



§ 124.— VARIATION STEPS OF THE FIRST, THE 

 SECOND AND THE THIRD DEGREE.— Several degrees are 

 observed in the frequency of the characteristic terms of a given 

 series of variation steps compared with the frequency of the 

 intermediate values. I discern three degrees of frequency, 

 between which aU possible transitions exist : 



Variation steps of the first degree. — The characteristic terms 

 of the series under consideration exist alone, the intermediate 

 values being very rare (or practically never observed). Ex- 

 ample : genus Senecio. In Senecio nemorensis, subspecies (?) 

 pentaglossa (ordinary form, province Luxemburg, Belgium), 

 the marginal florets are 5 in number. In a subspecies (?) which 

 I have found in Belgium, and which might be called Sen. nem., 

 subsp. triglossa (if not yet described), the number is ^.f^ In 

 Sen. nem., subsp. (?) octoglossa (mentioned in Switzerland) the 

 number is 8. In Senecio jacobcea (Flanders) the number is 13. 

 In these examples the observed figures (3, 5, 8, 13) belong to the 

 Fibonacci series. According to my (very numerous) observa- 

 tions, the. above-mentioned figures 3, 5 and 13 are almost in- 

 variable in each of the corresponding specific forms, and, 

 according to floristic data this is also the case with figure 8 in 

 subsp. octoglossa.^ The figures 2, 4, 8, 16 . . . in the peri- 

 stome of the Mosses, the step values 3, 6, 9 iw the androecium 

 of many Monocotyledons are probably almost always of the 

 first degree. (That is to say, invariable or almost invariable 

 in a given specific form.) 



Variation steps of the second degree. — The transitory values 



^ This subspecies is very abundant in a wood about i kilom. north from the 

 station Poix St Hubert (province Luxemburg), east from the railway. 



2 In the British Islands, the number of marginal florets varies in S. nemorensis 

 between 5 and 8 (seldom more than 6 or 7). (BENTHAM, British Flora, 1866, 

 pp. 254-256.) According to LUDWIG (Botan. Centralblatt, vol. Ixiv., 1895, 

 p. 100) the typical values in the mentioned species are 3 and 5, the figure 5 

 being predominant. 



