xvii] THE COMPARISON OF RELATED FORMS 1043 



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\ h\ 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 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 AOh be to AOh'. 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' AOh' AOc' 



Observed values 12° 28-5° 88° — — 157° 



Calculated „ — — — 21-5° 5M° — 



Observed „ — — — 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 shghtly curved, and 

 therefore that it is curvilinear and not rectihnear 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 coordinates. 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 where they arise successively 

 from the midrib, as they do in most dicotyledons. 



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 is 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, outhne 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 



* "Node," in the botanical, not the mathematical, sense. 



