— OE 
PREVAILING SYSTEMS OF PHYLLOTAXIS. 4l 
rating spiral for any higher than $ is impossible; or perhaps it would be better to say 
that, for any other arrangement than those represented by the fractions 3, 3, 2, and 8, 
the phyllotaxis is more or less uncertain, and the idea of a definite or exact generating 
spiral becomes more or less evanescent. In the latter case the reason for hesitating 
to pronounee which is most nearly vertical over the first selected, arises from the fact 
that neither the axis nor the leaves or scales are ever developed with mathematical 
accuracy, notwithstanding the fact that the position of the leaves may be mathematically 
projected. 
It will, then, be seen how all apparently higher arrangements than +8; are due to that 
divergence itself; for this is the highest that can be determined from opposite leaves *. 
Moreover, to say that the angular divergence represented by зё furnishes all the higher 
ones, is, to some extent, if not inexact, yet a somewhat incomplete statement; for in 
certain cases ту has the same effect, since this fraction is also irrational to the circum- 
ference. Thus in some cones of the Scotch Fir, while the 13th scale is very nearly over 
‚ the first, even the 34th leaf may not be far from the same vertical line, and so render it 
doubtful as to which fraction may best represent the phyllotaxis. However, it must be 
remembered that no higher denominator than 21 can be secured from opposite leaves, as 
stated above and shown by fig. 1. 
The reason why тү is the most usual kind of phyllotaxis with closely compacted 
scales seems due to the fact that i£ forms the nearest approximation to the alternation 
of whorled leaves, only one row of scales lying between those containing Nos. 0 and 21 
respectively. | 
Theoretical considerations.-—Of course it is here only advanced as a theory that nearly 
all spiral arrangements of Dicotyledons, at least, have been originally derived from 
opposite leaves ; for although they have such in the cotyledons of their embryos, yet the 
very first leaves of the plumules (as is well seen in a germinating acorn) at once assume 
their own peculiar spiral arrangement. Yet at the same time it must be remembered 
that the theory is founded upon the fact that when a branch has opposite leaves below 
and alternate above, the latter are developed according to the law expressed in fig. 1. 
The law of embryology, that a member of a higher grade passes in its embryonic state 
through the conditions of a lower grade when in its adult state, would seem to hint at 
this being at least probable, and that opposite leaves were probably the original and 
normal state of the primogenitors of dicotyledonous angiosperms. 
. Moreover, when we remember that whorled leaves were apparently very frequent in 
the carboniferous epoch, and that instances of symmetrical reduction are common 
amongst plants, е. g. in the floral whorls of Circea as compared with Epilobium (in which 
* It may be said on à priori grounds that the failure of the theory in deducing any higher fractions than >; from 
opposite leaves is unfavourable to its acceptance; but it must be remembered that no theory is to be tested by what 
may be imagined it should account for, but by what it does. Now this theory, the author maintains, does account 
for all that is necessary ; and any assertion that 12 and higher fractions ought to arise directly out of opposite leaves 
is a merely subjective impression not necessarily requiring to be correlated with any objective fact. If they can be 
satisfactorily accounted for, аз the author maintains they ean, as stated, from the #- arrangement, then all that is 
required has been done. 
