1042 THE THEORY OF TRANSFORMATIONS [ch. 



shall soon see the sloping sides of the hyacinth leaf give place to 

 a more typical and "leaf -like" shape. If we alter the ratio between 

 the radial and tangential velocities of growth — in other words, if we 

 increase the angles between corresponding^radii — we pass successively 

 through the various configurations which the botanist describes as 

 the lanceolate, the ovate, and the cordiform leaf. These successive 

 changes may to some extent, and in appropriate cases, be traced as 

 the individua,l leaf grows 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 instance, the so-called 

 A unsymmetrical leaf* of a begonia, 



in which one side of the leaf may be 

 merely ovate while the other has a 

 cordate outhne, is seen to be really 

 a case of unequal, and not truly 

 asymmetrical, growth on either side 

 of the midrib. There is nothing 

 more mysterious in its conformation 

 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 illus- 

 tration, and in Fig. 499 I have 

 outlined a leaf of the large Begonia 

 daedalea. On the smaller left-hand 

 Fig. 499 Begonia daedalea. ^^^^ ^f ^he leaf I have taken at 

 random three points a, 6, c, and have measured the angles, AOa, etc., 



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

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

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

 not 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." 



