PREVAILING SYSTEMS OF PHYLLOTAXIS. 39 
ist and 2nd leaves, or between the 2nd and 3rd, is double that between the 8rd and 4th. 
These latter, it will be remembered, are separated by a long internode. The same order 
obtains with the succeeding whorls; the nodes, however, are now much more widely 
separated, while a true spiral arrangement, with the same and constant angular distance 
between all its leaves, is ultimately secured, and is henceforth continued uninterruptedly 
into the terminal bud, and represented by the fraction 2." 
Ап exactly analogous process takes place with opposite leaves in passing, for example, 
into the $ arrangement. Internodes аге, of course, developed ; and the leaves of the first 
few pairs that are separated from each other have not, if projected, an equal or correct 
angular distance of 144? between them successively ; but just as in the production of the 
$ spiral arrangement from whorls of threes, so is it here—that as the 2 plan is continued 
up into the terminal bud, the leaves “ right " themselves, as it were, and then ultimately 
come to obey this law accurately by securing a constant angular divergence. Fig. 140, 
on p. 168 of Sachs’s * Hand-book’ mentioned above, illustrates this fact, though that 
author has not detected the significance in the order of development of the leaves. 
The origin of Cycles.—It will be gathered from the foregoing remarks and from fig. 1 
that « cycle is formed from a definite number of pairs of decussating leaves. "Thus, if 
opposite leaves give rise to the 2 arrangement, then the three pairs (0, 1) (2, 3) (4, 5) are 
required to produce the first cycle. Having once, as it were, “ started " it, the subsequent 
leaves are developed in accordance with the above law, the angular divergence between 
every two consecutive leaves being 144°. Similarly, if opposite leaves pass into the $ 
arrangement, then, in addition to the first three pairs, two others (6, 7) (8, 9) are required ; 
so that if the first cycle embracing leaves 0 to 7, inclusive, be secured, the subsequent 
leaves are developed according to the same law, requiring, however, 135° as their constant 
angular divergence. 
Hence it appears that while some plants have taken three pairs of the originally 
decussating leaves to form a cycle, i. e. of the $ arrangement, other plants have 
required five pairs, others seven, while cones frequently demand eleven pairs; so that 
the 21st leaf, or the first of the second cycle, falls over that of the first cycle, or that 
indicated in the figure as 0. 
Here, again, it does not seem possible to say why one plant, or even part of a branch, 
should thus convert a definite number of pairs of opposite and deeussate leaves into a 
cycle of some alternate arrangement. We can conceive of nature distributing the leaves 
of all plants in which they are alternate simply and solely іп accordance with the law 
given above, while it might have remained indifferent as to the precise amount of the 
angular divergence. ` As a matter of fact, however, this angle is determined by the 
and the fact that the point of emergence is determined, so to say, from within the axis before the leaf is formed seems 
fatal to the condensation theory. 
:- Again, Mr. Airy takes no account of stipules, which according to his theory ought to interfere materially in leaf- 
arrangement, because, if not petiolar (as in Rosacew) they rise from the axis exactly like leaves, and may even, as in 
the Elm and Lime, grow to as much as four times the size of the leaf to which they belong. Hence with, say, two 
minute leaf-apices and four subglobular or, what would be more true to nature, conical stipules, we may ask, What 
becomes of the “ contact tery " when the ionis development proves that the leaf-arrangement has not been 
interfered with? 
G2 
