734 THE THEORY OF TRANSFORMATIONS [ch. 



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 : 



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 under- 

 stand 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 



