PREVAILING SYSTEMS ОЕ PHYLLOTAXIS. 45 
No. 34 is placed vertically over 0. Strictly, however, this should be slightly to the left 
of the vertical line (but it would be inconvenient to allow for deviations due to minutes 
and seconds); for since 34 is not rational to the circumference, or a measure of 360, that 
leaf could never fall absolutely vertically over the first (0). 
The scheme may therefore be considered as approximately representing the 13 plan. 
The figures лоно by single zigzag 1 lines are those which give rise to the series 
4 $, 6i to [= 81, 14 1%» tz == 8), 39, 33 $ [= 8 3$, «е. 
and which, as‘has been shown, are deducible from opposite leaves by omitting the lowest 
or first. 
There are several other fraetions which represent the angular divergences of leaves occa- 
sionally met with in nature, but which have hitherto not been referred to any phyllotaxis. 
Thus Hofmeister, in his * Handbuch der physiologischen Botanik, mentions the 
following :— 
forti p т. (15), Ай 209 $b 81 3517 s 
He endeavours to show how these may be derived font кодов of the ordinary series 
by a slight decrease or increase of the angular divergence, thus :— 
іі 35 or—i-— d 
т-і- Wy т=$+ ds 
rr =15+ 934 ОГ = $ — oy 
зб = $— зо ОГ = + d 
ЗЇ = 2 — тів ог = 34+ уз 
З= 3+ өз ог={— oy 
i? == 02—туув ог = тұ + біт. 
Ы. * 
To these may be added 1% and + 
Now, of мн ip P d 1i, oy ii сап һе more easily obtained from the new series, 
commencing 4 3 тт, &е., as given above. 
For 1% лї апа 15 p immediately from opposite leaves when the lowest leaf is neglected. 
Now 44 = 4 — zio з 7, and therefore differs from үр by a much smaller fraction than from 
$0ri. Again 11 differs from тг by only i+. 
So that while of the above ми fractions 2, 74 (which is simply the reverse of 
1%), and 18 are deducible from some number or other respectively of the usual series, 
3%, 11, and also 45 and H, are to be obtained by starting with fr, which, in turn, is 
directly obtainable from opposite leaves 1. 
* Excepting 14, these are indicated on the scheme by means of arrows. | 
t This fraction 1+ I cannot account for, unless it approximates the reverse of some other ; otherwise it cannot belong 
to the usual series, as there must be always 3 leaves in every projected circle of a cycle ; consequently 14 appears to 
be too high a number (of cycles) for 31 leaves. 
+ Since this paper was in type I have met with a curious exception in Lagerstrümia, in which the order of the 
leaves, instead of forming a spiral, “ oscillate” through three quarters of a circle. The first four leaves succeeding 
the opposite are as in fig. 1; but the subsequent arrangement is as follows—O, 1, 2, 3, 5, ^ 455 8, 9, 19, IL, 13, 
12, 15, 14, 16, 17, 18, 19, 21, 20. This, perhaps, may prove to be the clue to thè origin of distichous leaves 
amongst Dicotyledons. 
SECOND SERIES.—BOTANY, VOL. I. 
