Chitrch . — The Principles of Phyllotaxis . 235 
the addition of one curve only, as in the difference between the systems 
(7 + 8), (8 + 8), &c. (Figs. 35-38) \ 
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 J 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 
1 Cf. Relation of Phyllotaxis to Mechanical Laws. Part II, p. 109, Rising and Falling 
Phyllotaxis. Part IV, Cactaceae. 
Though 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. 
