MORPHOLOGY AND MATHEMATICS. 



865 



growth is absent or at a minimum ; and about which node the rate of growth may 

 be assumed to increase symmetrically. Precisely such a case is furnished us in a leaf 

 of an ordinary dicotyledon. The leaf of a typical monocotyledon — such as a grass or 

 a hyacinth, for instance — grows continuously from its base, and exhibits no node or 

 " point of arrest." Its sides taper off gradually from its broad base to its slender tip, 

 according to some law of decrement specific to the plant ; and any alteration in the 

 relative velocities of longitudinal and transverse growth will merely make the 

 leaf a little broader or narrower, and will effect no other conspicuous alteration in its 

 contour. But if there once come into existence a node, or " locus of no growth," 

 about which we may assume the growth — which in the hyacinth leaf was longitudinal 

 and transverse — to take place radially and transversely to the radii, then we shall at 

 once see, in the first place, that the sloping and slightly curved sides of the hyacinth 

 leaf suffer a transformation into what we consider a more typical and " leaf-like " 



Fig. 8. 



shape, the sides of the figure broadening out to a zone of maximum breadth and 

 then drawing inwards to the pointed apex. If we now 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 finally the cordate leaf 

 (fig. 8). These successive changes may to some extent, and in appropriate cases, be 

 traced as the individual 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 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 asymmetrical, growth on either side of the midrib. There is 

 TRANS. ROY. SOC. EDIN., VOL. L, PART IV (NO. 27). 124 



