XVII] THE COMPARISON OF RELATED FORMS 733 



to maturity; but as a much more general rule, the balance 

 of forces, the ratio between radial and tangential velocities of 

 growth, remains so nicely and constantly balanced that the leaf 

 increases in size without conspicuous modification of form. It is 

 rather what we may call a long-period variation, a tendency for 

 the relative velocities to alter from one generation to another, 

 whose result is brought into view by this method of illustration. 

 There are various corollaries to this method of describing the 

 form of a leaf which may be here alluded to, for we shall not return 

 again to the subject of radial co-ordinates. For instance, the 

 so-called unsymmetrical leaf* of a begonia, in which one side of 

 the leaf may be merely ovate while the other has a cordate outline, 

 is seen to be really a case of 

 unequal, and not truly asym- 

 metrical, growth on either side 

 of the midrib. There is nothing 

 more mysterious in its conform- 

 ation than, for instance, in that 

 of a forked twig in which one 

 limb of the fork has grown 

 longer than the other. The case 

 of the begonia leaf is of sufficient 

 interest to deserve illustration, 

 and in Fig. 360 I have outlined 

 a leaf of the large Begonia dae- 

 dalea. On the smaller left-hand 

 side of the leaf I have taken at 

 random three points, a, b, c, and 

 have measured the angles, AOa, 

 etc., which th,e radii from the 



Fig. 360. Begonia daedalea. 



hilus of the leaf to these points make with the median axis. On 

 the other side of the leaf I have marked the points a', b', c', such 

 that the radii drawn to this margin of the leaf are equal to the 

 former, Oa' to Oa, etc. Now if the two sides of the leaf are 



* Cf. Sir Thomas Browne, ia The Garden of Cyrus: "But why ofttimes one 

 side of the leaf is unequal! unto the other, as in Hazell and Oaks, why on either 

 side the master vein the lesser and derivative channels stand not directly opposite, 

 nor at equall angles, respectively unto the adverse side, but those of one side do 

 often exceed the other, as the Wallnut and many more, deserves another enquiry." 



