166 THE QUANTITATIVE METHOD IN BIOLOGY 



by step to very complicated cases, in which chance brings about 

 a large number of events, the diversity of which is expressed 

 by a curve of errors (Fig. i8, p. 146), or by a variation curve 

 (dice, § loi ; prisms, § 115 ; cards, § 117). 



Let us now suppose that we have measured a variable pri- 

 mordium in numerous specimens belonging to a subspecies or a 

 monotypic species ; for instance, the length of the adult spike of 

 a certain race (subspecies) of rye. 



Each observed figure is the measure of an object (individual, 

 spike) which is in a state of equilibrium. The equilibrium is 

 brought about by a series of reactions which take place in the 

 course of the development of the measured object. (See § 45, 

 growth of a petal.) In these reactions two groups of causes 

 are operative : (i) specific causes, which depend on the living 

 substance of the species under consideration ; (2) external 

 causes, which are numerous ; for instance, temperature, light, 

 water, food, etc. (See § 119.) Each cause has an influence 

 upon the observed values. The causes are combined in many 

 different ways according to the laws of chance ; their resultant 

 is different from one specimen to another. The figures of the 

 measured property, considered as a whole, are represented by a 

 variation curve which is governed by chance. 



It has often (not always) been observed that the curve 

 obtained is one humped and symmetrical (or nearly so), corre- 

 sponding to the expression (| + J)" (see note, p. 147) and similar 

 to the variation curve of the prisms in § 115 and the curve of 

 errors in § 108. (See QUETELET, § 116, p. 161.) 



What is the significance of the mean value and the extremes 

 in a variation curve of a property of a living species — ^in other 

 words, in a BIOLOGICAL variation curve ? 



We have seen in § 108 that in a curve of errors the mean 

 value is the measure of something independent of chance. If 

 the combined causes (chance) were not operative while we are 

 determuiing the specific weight of a solid (§ 108), no errors 

 would be committed and in each determination a value would 

 be obtained equal to the mean. In an inorganic variation 

 curve (for instance, the curve of the compound prisms, § 115), 

 on the contrary, each specimen is brought into existence by the 

 very same causes which result in variation by their various 

 combinations. If these causes (components of chance) were 

 not operative, 1 no mean value would be obtained, for the simple 

 reason that no specimens (compound prisms) would exist. 

 Therefore the mean value is an indirect expression of a com- 

 plicated something. (Compare § 116, p. 162.) 



A mean value calculated from a biological variation curve 



^ The causes alluded to were acting and varying continually while we were 

 extracting the simple prisms from the first urn. 



