RELATIVE POSITIONS OF LATERAL MEMBERS, 
cases of decidedly bilateral construction the genetic spiral might be imagined just as 
well, and with the same divergence, ascending right or left, by which of course it 
loses its importance for any morphological conclusion, as much as if one supposed 
it to change its direction from leaf to leaf 
It is principally in upright axes with solitary leaves arranged in three, four, 
five, or more directions, that the spiral construction appears conformable to 
nature, and agrees with the symmetrical relationships of plants, of which more 
will be said hereafter. The spiral construction proves to be opposed to nature 
in bilateral structures, especially in creeping or climbing stems, and in lateral 
branches. 
In those cases in which the spiral construction may be employed naturally 
to elucidate the relative positions of the members, two cases may be distinguished, 
according as the divergences, on the one hand, are very unequal and change 
abruptly, or, on the other hand, are nearly or quite equal to one another or 
only change gradually. In the first case the members appear to be arranged 
irregularly and without order, as the foliage-leaves on the stem of Fritillaj^ia 
imperialis (Fig. 151), the flowers on the rachis of the raceme of Triglochin 
palustre or of many Dicotyledons. When the change of divergence on the same 
axis is abrupt, it may also ap- 
pear more natural to represent 
the phyllotaxis by two homodromal 
spirals instead of one, as in many 
species of aloe, where the shoots 
commence with leaves arranged in 
two rows, and then pass over into 
complicated divergences which lead 
finally to rosettes of leaves radiat- fig. 152.— Transverse section of a shoot of ^/ö^Äv-r^r. 
ing on all sides. This occurs, 
e.g. in Aloe ciliaris^ latifolia, hrachyphylla, Lingua, nigricans, and Serra. Fig. 
152 shows the transverse section of a shoot of the last-named species;' the 
first six leaves are arranged alternately in two rows with a constant diver- 
gence \ ; at the 7th leaf this arrangement is suddenly changed; instead of 
being placed over 5, its position is between 5 and 6; but the 8th leaf exhibits 
the divergence \ from the 7th; the 9th again changes the divergence, instead of 
being placed over 7, it is between 7 and 6; the loth leaf again diverges about \ 
from the 9th; and so on. The leaves 7-15 are evidently arranged in pairs, 
the pairs being 7, 8 ; 9, 10; 11, 12; 13, 14; each pair consists of two alternate 
'{i.e. not opposite) leaves, the divergence of which is J; but the pairs them- 
selves diverge from one another by smaller fractions. If it is desired to unite all 
the leaves from i to 15 by a genetic spiral, an abrupt alteration of the divergence 
would occur in it. The relative positions are shown, however, more simply 
and clearly if, keeping in view the bilateral origin of the shoot, two spirals are 
constructed, each of which commences from one of the original orthostichies, and, 
so to speak, continues it in a spiral curve; the one contains all the leaves with an 
even number, the other those with an uneven number; the two are homodromal, 
running in the same direction round the stem. The bilateral origin of the shoot 
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