XVII] THE COMPARISON OF RELATED FORMS 1045 



off with no projecting apex*. The ground-ivy and the dog-violet 

 (Fig. 501) illustrate such a leaf; and sometimes, as in the violet, 

 the veins -of the leaf show similar curves congruent with the outer 

 edge. Moreover the violet is a good example of how the reniform 

 leaf may be drawn out more and more into an acute and ovate 

 form. 



From sin^/2 we may proceed to any other given fraction of ^, 

 and plot, for instance, r = sin 5^/3, as in Fig. 502 ; which now no 

 longer represents a single leaf but has become a diagram of the 



Fig. 501. Violet leaf. 



Fig. 500. Curve resembling the out- 

 line of a reniform leaf: r=:8in^/2. 



five petals of a pentamerous flower. Abbot Guido Grandi, a Pisan 

 mathematician of the early eighteenth century, drew such a curve 

 and pointed out its botanical analogies ; and we still call the curves 

 of this family "Grandi's curvesj*." 



The gamopetalous corolla is easily transferred to polar coordinates, 

 in which the radius vector riow consists of two parts, the one a 

 constant, the other expressing the amplitude (or half-amplitude) of 

 the sine-curve ; we may write the formula r = a + b cos nO. In 

 Fig. 503 w ^ 5 ; in this figure, if the radius of the outermost circle 

 be taken as unity, the outer of the two sinuous curves has a:b as 



* Fig. 500 illustrates the whole leaf, but only shows one-half of the sine-curve. 

 The rest is got by reflecting the moiety already drawn in the horizontal axis 

 (^=7r/2). 



t Dom. Guido Grandus, Flores geometrici ex rhodonearum et cloeliarum curvarum 

 descriptione resultantes . . . , Florentiae, 1728. Cf. Alfred Lartigue, Biodynamique 

 generale, Paris, 1930 — a curious but eccentric book. 



