284 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



cycle the deviation from perfect radial symmetry may escape 

 notice. 



That many floral types, on the other hand, are asymmetrical 

 throughout the whole sporophyll region, including the corolla, is 

 undoubted ; although, again, owing to the general tendency of the 

 plant to promote efficiency in the floral mechanism by the reduc- 

 tion of the number of its parts, the number of members produced 

 is often insufficient to give any spiral pattern to the eye. In such 

 cases the Eiehlerian convention of assuming circles as much as 

 possible has been adopted : thus, while Aconitum cannot be regarded 

 as anything else but asymmetrical throughout and slightly 

 eccentric (perianth (3 + 5), sporophylls (8 + 13), growth-centre 

 displaced anteriorly in the plane of sepal 2), many diverse views 

 have been proposed with regard to the interpretation of the flower 

 of Tropaeolum majus. 



As in previous instances, a geometrical construction diagram may 

 be readily prepared which will illustrate the phenomena to be 

 expected in the development of an eccentrically growing asym- 

 metrical construction. Thus, fig. 98 is drawn for an eccentric 

 (3 + 5) system, and fig. 99 for a (5 + 8), in both of which, as in 

 these zygomorphic flowers, the growth-centre has been displaced 

 anteriorly owing to an increased rate of growth of the posterior 

 side of the flower, and the plane of eccentricity follows that of the 

 second sepal, which is not necessarily the median plane of the 

 diagram, but sufficiently near it for practical purposes. 



Consideration of fig. 99 at once shows important features ; the 

 curve construction, as before, is wholly built on orthogonally 

 intersecting curves, and it follows from the mathematical pro- 

 perties of the numbers of the curves employed that the quasi- 

 squares may be serially numbered. But the mathematical order 

 of such enumeration is no longer that of increasing size ; that is 

 to say, it is not the order of actually visible ontogeny, although it 

 still remains the theoretical order of the initiation of the growth- 

 centres. In such a flower, therefore, some members would appear 

 to arise as protuberances out of their normal spiral sequence, owing 

 to the fact that those on the posterior side are growing at an 

 increased rate, and those on the anterior side at a diminished rate. 



