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.
Fig. 13. Polygon of frequency of numbers of ray flowers of the white daisy gathered at random from various german localities. From 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 sum of the two preceding. Also the ratios of these numbers, namely, 2, 3, 5, 8, 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