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[650] of a begonia, in which one side of the leaf may be merely ovate while the other has a cordate outline,
Fig. 360. Begonia daedalea.
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 illustration, and in Fig. [360] I have outlined a leaf of the large Begonia daedalea. On the smaller left-hand side of the leaf I have taken at random three points, a, b, c, and have measured the angles, AOa, etc., which the radii from the hilus of the leaf to these points make with the median axis. On the other side of the leaf I have marked the points a′, b′, c′, such that the radii drawn to this margin of the leaf are equal to the former, Oa′ to Oa, etc. Now if the two sides of the leaf are {734} mathematically similar to one another, it is obvious that the respective angles should be in continued proportion, i.e. as AOa is to AOa′, so should AOb be to AOb′. This proves to be very nearly the case. For I have measured the three angles on one side, and one on the other, and have then compared, as follows, the calculated with the observed values of the other two:
| AOa | AOb | AOc | AOa′ | AOb′ | AOc′ | |
|---|---|---|---|---|---|---|
| Observed values | 12° | 28.5° | 88° | — | — | 157° |
| Calculated values | — | — | — | 21.5° | 51.1° | — |
| Observed values | — | — | — | 20 | 52 | — |
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 approximate 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 Dionaea, 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 {735} 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 θ − A log r = c (Fig. [361]). This beautiful and