MEASUREMENT OF VARIABLE PROPERTIES 163 



(highest ordinate) is 27 cm. The arithmetical mean is 

 1,574,640:59,049 = 261 cm.i This does not represent any 

 existing prism. For the naturalist the second definition of the 

 mean is not applicable. 



A second difference exists between a curve of errors (§ 108) and 

 the variation curve of the prisms (§ 115) with reference to the 

 significance of the extreme values (limits) . In a curve of errors 

 the limits are indefinite and of secondary importance, because 

 it is always possible that a new error exceeds the greatest error 

 previously committed.^ In the variation curve of the com- 

 pound prisms, on the contrary, the extreme values are strictly 

 definite. The minimum is a^" (20 cm.) and the maximum 6^" 

 (30 cm.). If one succeeds in finding these values by observation 

 (extracting prisms from the urn), they are determined once for 

 all : it is impossible to find in the urn any prism<20 or>3o cm. 



In the example of the prisms and in all similar cases the 

 extreme values may be defined as being the direct measure 

 of the maximum and the minimum effects produced by the 

 acting combined causes. In other words, using biological 

 language, the maximum corresponds to the optimal conditions 

 of growth of the species {compound prism), whereas the minimum 

 corresponds to the most unfavourable conditions — the pessimal 

 conditions. Both extreme values are constants, between which 

 variation fluctuates. 



We shaU see below (§ 118- 119) that the extremes (especially 

 the maximum) deserve still better the name of constants when 

 variable properties of animals and plants are measured. The 

 method which I propose for the measurement of constants, the 

 description of species and the identification of specimens is 

 based upon the determination of the extreme values. 



REMARK : In § loi (Table 7, p. 127), we have seen that in 

 the series oi facial values obtained by casting two dice the extreme 

 values (2 and 12) are definite. Other examples of definite ex- 

 tremes are mentioned in § 102, examples (A) (p. 128) and (B) 

 (p. 129) ; experimental verification in § 103. In § 113, Fig. 19, 

 the maximal value is definite. 



§117.— EIGHTEENTH EXAMPLE: EXPERIMENTS 

 WITH CARDS. MEAN VALUE AND EXTREME VALUES. 

 — First Experiment : I ascribe to each card of an ordinary pack 



• The value 1,574,640 cm. is the total length of the 59,049 prisms. 



^ In the example in § io8 (measurement of a density) I have put down 

 M=io, supposing that the combined causes were lo in number. This is 

 arbitrary : I was compelled to adopt a definite value for n because I am using 

 the arithmetical method. By domg so I have arbitrarily fixed two limits 

 of error [a}" and b'"). In reality, n is an indefinite number. Instead of being 

 combined lo by lo, the causes which bring about errors may be combined 

 II by II, 12 by 12, ... , 100 by 100, etc. (See note, p. 147.) 



