40 THE REV. GEORGE HENSLOW ON THE ORIGIN OF THE 
number of leaves constituting а cycle, which cycle has itself been determined by the 
number of pairs of leaves of the originally decussate arrangement. 
It may be again observed that when numbers of the ** secondary " series, viz. 3, 1, 2, үү; 
&c., occur, as in the Jerusalem Artichoke, they do so from an analogous process arising 
out of whorls of threes. Further, in tracing the phyllotaxis of plants of paleeozoic ages, 
it appears that, besides representatives of the usual or primary and secondary series, some 
instances occur (as in species of Lepidodendron) which have resulted from the breaking- 
up of whorls of fours, giving rise to the tertiary series 1, $, 3, &c. ; while even a quaternary 
series, 1, 1, тг, «с., finds а representative in Knorria taxina, the phyllotaxis of which is 
said to be 25 *. At the present day, however, any other than the primary series is com- 
paratively rare. 
Correlation between Internodes and Angular Divergences.—If a stem grow rapidly, or 
at least develop internodes of considerable length (as is the case with the Jerusalem Arti- 
choke), the fraction representing the subsequent arrangement of the leaves upon it will 
be a “low ” one of the series. If, however, the internodes be very short, then, conversely, 
the leaves or their equivalents will be represented by one of the higher fractions, such 
as 125, S, 21, &c., as is well seen in cones, involucres, &c. There appears at present to 
be no better explanation of this correlation than the one already alluded to, namely the 
object of avoiding interference of one another. This, however, the author is inclined to 
regard as a distinct factor in the causes which regulate leaf-arrangement. It will be 
easily seen that if leaves are on the ? plan, with very short internodes, the 6th leaf 
will lie immediately upon the surface of the first, what would be called “ opposite” in 
flowers, especially if the leaves be sessile; but if they be on the т; or, better still, -%, 
then the 14th or 22nd leaf will be freer from the first, as seen in the scales of a cone. 
The Divergence зг determines all higher ones of the primary series.—In fig. 1 it will 
be seen that the following numbers occur in the same vertieal line,—0, 5, 8, 13, 16 
[=2 х 8], and 21. Or, if written thus, 0, 5, 5 + 3, 8 + 5, 13 +3, 16 + 5, &е., the 
numbers сап easily be ascertained to any height required. If this be done, it will be | 
discovered that the numbers 34, 55, 89, 144, which form the denominators of the 
fractions higher than $$, do not fall in this same vertical line; and the question might 
be asked, on the supposition advanced in this paper, how it ever happens that higher 
fractions than 5) are represented in nature. The reply is, that while the numbers 
5, 8, 13, and 21 give rise to the divergences 2, $, 1%, and 38; respectively, yet this last, 
зт, supplies all the higher fractions. That isto say, if the spiral arrangement represented 
by эт be projected as a helix, and the numbers noted down, to (say) the 90th leaf, 
then the 34th, 55th, and 89th leaf will be found lying nearest to the vertical line, 
alternately right and' left of it, while each of these leaves approximates nearer and nearer 
to it respectively, as will be seen by referring to fig. 2. Moreover, as the initial leaves 
of successive cycles of arrangement represented by the fractions y and higher ones, 
cannot be in a truly vertical line, it becomes often difficult to fix upon the leaf most 
nearly vertical; in fact it is almost, if not quite, arbitrary whether we select the 34th, 
55th, or 89th leaf. We may go further, and say that the discovery or proof of a gene- 
ж Alluded to in Prof. Balfour's * Paleontological Botany.’ 
