182 THE QUANTITATIVE METHOD IN BIOLOGY 



of development of the measured specimens. A comparison 

 between two mean values is, in reality, a comparison between 

 the conditions of existence of two groups of specimens. (See 



§ 120.) 



EXAMPLE : In Mnium hornum the length of the longest leaf of the fertile 

 stem varies between 5-17 and 8-o6 mm. These figures are specific constants. 

 If a certain number of specimens collected from one spot are measured, the 

 extremes may be, for instance, 5-50 and 7*00, the mean being 625 mm. A 

 second series of specimens, collected from a second spot (even in the same 

 locality) may give : extremes 6 and 750, the mean being 6-75 mm., etc. Since 

 no variation steps exist in the property under consideration, the mean values 

 obtained (and also the extremes in each series) depend on the conditions of 

 existence. These conditions are more or less variable from one spot to 

 another. 



When, on the contrary, a biological mean coincides with a 

 variation step, it is really the measure of a definite something, 

 of a specific energy which exists in the measured property, 

 independently of any external cause. Such a mean value is 

 (for the naturalist) comparable with the mean of a curve of 

 errors : it is a constant. 



EXAMPLES : Variation steps of the first degree. In Senecio jacobcsa 

 the number of marginal florets is almost always 13 (Fibonacci term) ; the 

 figures 12 and 14 are very rare. Series of specimens collected from various 

 spots ar>d various localities always give the mean 13. This mean is a con- 

 stant, it may be surmised that many properties of animals and plants, which 

 are practically constant in a species or a genus, are in reality variation steps 

 of the first degree of a certain series. 



Variation steps of the second degree. I have cultivated Chrysanthemum 

 carinatum in several series under various conditions of existence. In one of 

 the series, the variation curve was comparatively symmetrical, the most 

 frequent value (hump of the curve) being 21 (Fibonacci term). The mean 

 value was, approximately, 21. Such a coincidence between the mean and a 

 variation step is rather rare when the steps are of the second degree. 

 Ordinarily the mean coincides approximately with one of the transitory values 

 between two steps. In Chrysanthemum segetum, DE VRIES obtained the 

 following variation curve (number of marginal florets of the terminal flower- 

 head 1 ) : 



Florets . . 12 13 14 15 16 17 18 19 20 21 22 

 Specimens . i 14 13 4 6 9 7 10 12 20 i 



The arithmetical mean is 17-5. The curve is two-humped (dimorphic) : 

 the humps coincide with the variation steps 13 and 21 and represent there- 

 fore constants, whereas the mean has no definite significance. (See also 

 § 129.) 



Variation steps of the third degree. In Centaur ea cyanus (under ordinary 

 conditions of existence) the variation curve of the number of marginal florets 

 of the terminal flower-head is one-humped, the hump coinciding with the 

 values 10 or 11, the mean coinciding almost exactly with the hump. In 

 reality, the mean is intermediate between two variation steps 8 and 13, which 

 are indicated by a sudden falling of the curve between y and 8 and between 

 13 and 14. The mean does not represent a constant: it is, however, influ- 



'^ Archiv fur Entwickelungsmechanik, Bd. II., p. 52, 1896. (Quoted after 

 VERNON, loc. cit., p. 50.) I have obtained several similar curves for Chry- 

 santhemum carinatum. Unfortunately the figures are not at my disposal. 



