■Church. — The Principles of Phylbtaxis. 235 



the addition of one curve only, as in the difference between the systems 

 (7 + 8), (8 + 8), &c. (Figs. 35-38)1. 4 



Thus the hypothesis of a genetic-spiral, since it entirely fails to 

 account for the arrangement of the members of all phyllotaxis systems 

 in a single spiral, may be conveniently wholly eliminated from future 

 discussions of these systems. It remains as a mere geometrical accident 

 of certain intersecting curve-systems, and the fact that such systems 

 may be very common in plant construction does, not affect the main 

 principle at all. 



On the other hand, it may be urged that in these special cases one 

 cannot get away from the fact that it does actually represent the building- 

 path as seen in the visible ontogeny of the component members, and must 

 therefore ever remain the most important feature of these systems as 

 checked by actual observation apart from theoretical considerations. But 

 even this view is not absolute ; and such a case in which, the ontogenetic 

 sequence of development is not the single spiral obtained by numbering 

 the members in theoretical series would naturally confuse the observer 

 of direct ontogeny. 



For example, in the previous cases figured the proposition of centric 

 ■ growth systems was alone considered, as being the simplest to begin 

 with ; it is obvious that even a small amount of structural eccentricity 

 will produce a very different result. Thus in Fig. 39 the (5 + 8) system 

 is redrawn in an eccentric condition, the so-called ' dorsiventrality ' of 

 the morphologist ; on numbering the members in the same manner as 

 before it is clear that the series obtained is very different from any 

 empirical ontogenetic value which would be founded on the observation 

 of the relative bulk of the members at any given moment. The occurrence 

 of such systems in plant-shoots — and it may be stated that this figure was 

 originally devised to illustrate certain phenomena of floral construction 

 in the case of Tropaeolum — gives in fact the final proof, if such were any 

 longer needed, of the simple geometrical generalization that such systems 

 of intersecting curves are always readily interpreted in terms of the 

 number of curves radiating in either direction, and not in any other 

 manner. The presence of a circular zone (whorl) or a genetic-spiral is 

 a wholly secondary geometrical consequence of the properties of the 

 numerals concerned in constructing the system. The preference of any 

 individual botanist, either in the past or at present, for any particular method 



' Cf. Relation of Phyllotaxis to Mechanical Laws. Part II, p. 109, Rising and Falling 

 Phyllotaxis. Part IV, Cactaceae. 



. Thongh the figures (35-38) have, as a matter of fact, been drawn by means of suitable ortho- 

 gonally intersecting logarithmic spirals, because these curves are easily obtained and the schemes are 

 subsequently held to be the representation of the true construction system of the plant-apex, the 

 nature of the spirals does not affect the general laws of intersection so long as this takes place 

 uniformly. 



