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 aÿ£¢ãý , bÿ£¢ãý , cÿ£¢ãý , such that the radii drawn to this margin of the leaf are equal to the former, Oaÿ£¢ãý to Oa , etc. Now if the two sides of the leaf are {734} mathôÙeôÙmatôÙiôÙcally similar to one another, it is obvious that the respective angles should be in continued proportion, i.e. as AOa is to AOaÿ£¢ãý , so should AOb be to AObÿ£¢ãý . 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 | AOaÿ£¢ãý | AObÿ£¢ãý | AOcÿ£¢ãý | |
|---|---|---|---|---|---|---|
| Observed values | 12ô¯ | 28.5ô¯ | 88ô¯ | ã | ã | 157ô¯ |
| Calculated values | ã | ã | ã | 21.5ô¯ | 51.1ô¯ | ã |
| Observed values | ã | ã | ã | 20ããã | 52ããã | ã |
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 apôÙproxôÙiôÙmate 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 ãpoint of arrest,ã 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