TEE STATISTICAL STUDY OF EVOLUTION. 457 



on. If now the length of all these individuals be measured we shall 

 obtain a series of modes of which each corresponds to one of the 

 broods (Fig. 11). Still again, two modes may appear when the 

 material is not perfectly homogeneous although the age be constant. 

 For instance the material may contain both normal and abnormal 

 individuals. An example of this sort of polygon is given in Fig. 13. 

 A very complex curve is afforded by the number of the ray flowers of 

 composite plants. If the lappets of a thousand white daisies be 

 counted it will be found that there is not a single mode only but a 

 series of them. These modes increase in height from one extremity of 

 the range, reach a great mode at one point and then diminish again 

 (Fig. 13). It appears also that the modes do not occur at haphazard. 



25 



50 



4U 



Fig. 13. Polygon of frequency of numbers of kay flowers of the white daisy gathered 



AT RANDOM FROM VARIOUS GERMAN LOCALITIES. FEOM DATA OF LUDWIG. 



but chiefly in the series of numbers: 1, 2, 3, 5, 8, 13, 21, 44, and 65. 

 This is a mathematical series in which each term is the siun of the two 

 preceding. Also the ratios of these numbers, namely, %, %, %, %, 

 and so on, have long been known to represent the arrangement of 

 leaves on a stem; and this seems to be why the numbers of this series 

 are so prominent in the rays of the flower head. 



The comparative study of frequency polygons, such as we have been 

 making, enables us, it will be seen, to distinguish different kinds of 

 variation and to make that philosophical classification which is the 

 first step in advancing knowledge. Although the causes of variation 

 are not at once revealed, we are directed to working hypotheses that 



VOL. LIX. — 32 



