170 THE QUANTITATIVE METHOD IN BIOLOGY 



by the number of measured specimens developed at each 



Suppose, on the other hand, that the relation between the 

 length A and temperature (between 10° and 20° C.) is expressed 

 by the following table ^ : — 



Length A.g (for instance, 10 cm. 



11 , 



12 , 



13 , 



14 . 



15 . 



^ ( 



^8 ( 



K ( 

 X 



10 



^ii( 



coincides with 10° and 20° C. 



II" 

 12" 

 13° 

 14° 



19° C. 

 18° C. 

 17" c. 

 16° c. 



15° (=n 



From the above figures the following variation curve (curve 

 of frequency) of A is deduced : — 



"•10 



(V = )Ai 



Length 

 (10 cm. 



(II > 

 (12 , 



(13 : 

 (14 , 

 (15 , 



Groups 

 a+k= 1+ 1= 

 b + j= 10 + 10 = 



c + i= 45+ 45 = 



d + h — 120 + 120 = 



e +^ = 210 + 210 = 



/= 



Total 



Frequency 



2 



20 



90 



240 



420 



252 



1024 = 21° specimens 



This curve is asymmetrical : the most frequent value 



10 

 (14 cm.) is the length of the spikes which have been developed 

 either at 14° or at 16° C. The arithmetical mean of A is 1377 

 cm. It may be possible to express (in a rather complicated 

 way) the relations between the mean value and the data. The 

 latter are : (i) the variation curve of temperature (absolute 

 values and frequencies) ; (2) the values A.^, A, . . . and the 

 curve which expresses the relation between these values and 

 the temperatures between 10° and 20° C. 



In the present state of biological science, however, the 

 problem is insoluble, because we have no exact information 



^ Number of specimens (total 1024). 



' Proceeding in Fig. 25 from both limits f and T° towards the highest ordi- 

 nate, we meet on each side successively the values Xi, Xj . . . An- The 

 length Xj (practically o) coincides with the critical temperatures f and T". 

 The length X^ coincides with two temperatures (see Fig. 25) which are nearer 

 the optimum {d°), etc. The length X^ (=X;u) coincides with d°. 



