1044 THE THEORY OF TRANSFORMATIONS [ch. 



cherry we have a "point of arrest" at the base of the berry, where 

 it joins its peduncle, and about this point the fruit (in imaginary 

 section) swells out into a cordate outhne ; while in the apple we have 

 two such well-marked points of arrest, above and below, and about 

 both of them the same conformation tends to arise. The bean and 

 the human kidney owe their "reniform" shape to precisely the 

 same phenomenon, namely, to the existence of a node or "hilus," 

 about which the forces of growth are radially and symmetrically 

 arranged. When the seed is small and the pod roomy, the seed 

 may grow round, or nearly so, like a pea ; but it is flattened and 

 bean-shaped, or elliptical like a kidney-bean, when compressed 

 within a narrow and elongated pod. If the original seed have any 

 simple pattern, of the nature for instance of meridians or parallels 

 of latitude, it is easy to see how these will suffer a conformal trans- 

 formation, corresponding to the deformation of the sphere*. 



We might go farther, and farther than we have room for here, 

 to illustrate the shapes of leaves by means of radial coordinates, 

 and even to attempt to define them by polar equations. In* a 

 former chapter we learned to look upon the curve of sines as an 

 easy, gradual and natural transition — perhaps the simplest and most 

 natural of all — from minimum to corresponding maximum, and so 

 on alternately and continuously; and we found the same curve 

 going round like the hands of a clock, when plotted on radial co- 

 ordinates and (so to speak) prevented from leaving its place. Either 

 way it represents a "simple harmonic motion." Now we have just 

 seen an ordinary dicotyledonous leaf to have a "point of arrest," 

 or zero-growth in a certain direction, while in the opposite direction 

 towards the tip it has grown with a maximum velocity. This 

 progress from zero to maximum suggests one-half of the sine-curve ; 

 in other words, if we look on the outline of the leaf as a vector- 

 diagram of its own growth, at rates varying from zero to zero in 

 a complete circuit of 360°, this suggests, as a possible and very 

 simple case, the plotting of r = sin ^/2. Doing so, we obtain a 

 curve (Fig. 500) closely resembling what the botanists call a reniform 

 (or kidney-shaped) leaf, that is to say, with a cordate outline at the 

 base formed of two "auricles," one on either side, and then rounded 



* Vide supra, p. 524. 



