PHYLLOTAXIS OF OPPOSITE AND WHORLED LEAVES. 1538 
These become more complicated as the number of leaves, &c., 
in the spire is increased ; but in those cases where the leaves, 
&c., are so numerous as to be close to each other, as in the 
Screw-pine, the Pineapple (fig. 292), and in the fruit of 
Coniferous plants (fig. 293), the spiral arrangement is at once 
evident. 
By placing the fractions representing the angular divergence 
in the different varieties of Phyllotaxis side by side in a line, 
thus -—4, 4, $B) 159 Fi 349 35, WC. 5 4) Fo Tn a5. 37) We., we 
see at once that a certain relation exists between them ; for the 
numerator of each fraction is composed of the sum of the 
numerators, and the denominator of the sum of the denomina- 
tors of the two preceding fractions ; also in the first series, that 
the numerator of each fraction is the denominator of the next 
but one preceding. By applying this simple law therefore we 
may continue the series of fractions representing the angular 
divergence, &c., thus: 24, 3%, #5, &e. It should be mentioned 
with respect to the laws of Phyllotaxy, that they are frequently 
interfered with by accidental causes which produce corresponding 
interruptions of growth, so that it is then difficult, or altc gether 
impossible, to discover the regular condition. 
All the above varieties of Phyllotaxis in which the angular 
divergence is such that by it we may divide the circumference 
into an exact number of equal parts, so that the leaves com- 
pleting the cycles must be necessarily directly over those 
commencing them, are called rectiserial ; while those in which 
the divergence is such that the circumference cannot be 
divided by it into an exact number of equal parts, and thus no 
leaf can be placed precisely in a straight line over any pre- 
ceding leaf, but disposed in an infinite curve, are termed 
curviserial. The first forms of arrangement are looked upon as 
the normal ones ; the latter will show the impossibility of bring- 
ing organic forms and arrangements, in all cases, under exact 
mathematical laws. 
We have thus endeavovred to show that when leaves are al- 
ternate, the successive leaves form a spiral round the axis. The 
spire may either turn from right to left, or from left to right. 
In the majority of cases, the direction in both the stem and 
branches is the same, and it is then said to be homodromous ; 
but instances also occasionally occur in which the direction is 
different, when it is called heterodromous. 
2. Opposite and Whorled Leaves.—We have already observed 
with regard to these modifications of arrangement, that the suc- 
cessive pairs, or whorls, of leaves, as they succeed each other 
(page 148), are not commonly inserted immediately over the 
preceding, but that the second pair (fig. 287), or whorl, is placed 
over the intervals of the first, the third over those of the second, 
and so on. Here, therefore, the third pair of leaves will be 
directly over the first, the fourth over the second, the fifth over 
