MORPHOLOGY AND MATHEMATICS. 867 



The agreement is very close, and what discrepancy there is may be amply 

 accounted for, firstly, by the slight irregularity of the sinuous margin of the leaf ; 

 and secondly, by the fact that the true axis or midrib of the leaf is not straight but 

 slightly curved, and therefore that it is curvilinear and not rectilinear triangles 

 which we ought to have measured. When we understand these few points regarding 

 the peripheral curvature of the leaf, it is easy to see that its principal veins approxi- 

 mate closely to a beautiful system of isogonal co-ordinates. It is also obvious that 

 we can easily pass, by a process of shearing, from those cases where the principal 

 -veins start from the base of the leaf to those, as in most dicotyledons, where they 

 arise successively from the midrib. 



It may sometimes happen that the node, or " point of arrest," is at the upper 

 instead of the lower end of the leaf-blade ; and occasionally there may be a node at 

 both ends. In the former case, as we have it in the daisy, the form of the leaf will 

 be, as it were, inverted, the broad, more or less heart-shaped, outline appearing at 

 the upper end, while below the leaf tapers gradually downwards to an ill-defined 

 base. In the latter case, as in Dionma, we obtain a leaf equally expanded, and 

 similarly ovate or cordate, at both ends. We may notice, lastly, that the shape of 

 a solid fruit, such as an apple or a cherry, is a solid of revolution, developed from 

 similar curves and to be explained on the same principle. In the 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 outline ; 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. 



Most of the transformations which we have hitherto considered (other than that 

 of the simple shear) are particular cases of a general transformation, obtainable by 

 the method of conjugate functions and equivalent to the projection of the original 

 figure on a new plane. 



Appropriate' transformations, on these general lines, provide for the cases of a 

 coaxial system where the Cartesian co-ordinates are replaced by coaxial circles, or 

 a confocal system in which they are replaced by confocal ellipses and hyperbolas. 



Yet another curious and important transformation, belonging to the same class, 

 is that by which a system of straight lines becomes transformed into a conformal 

 system of logarithmic spirals : the straight line Y— AX=c corresponding to the 

 logarithmic spiral 0— A log r—c (fig. 10). This beautiful and simple transformation 

 lets us at once convert, for instance, the straight conical shell of the Pteropod or 

 the Orthoceras into the logarithmic spiral of the Nautiloid. 



These various systems of co-ordinates, which we have now briefly considered, are 

 sometimes called " isothermal co-ordinates," from the fact that, when employed in 



