159 PHYLLOTAXIS OF ALTERNATE LEAVES, 
around it. He found also, that the relation of the leaves to one 
another was constant, each being separated from the other by an 
equal distance, so that if we started with any particular leaf and 
waited until another leaf was reached which corresponded ver- 
tically with it, and then proceeded to the leaf beyond this, we 
should find that this would also correspond vertically with the 
one next above that which we started from, and so on, each 
successive leaf would be placed vertically over one of the leaves 
below, but that in all cases in the same plant, the number of 
leaves between the one started from, and that which corre- 
sponded vertically with it, would be always the same. Thus if 
we take a branch of the Apple or Cherry-tree (fig. 289), and 
commence with any particular leaf which we will mark 1, and 
then proceed upwards, connecting in our course the base of 
each succeeding leaf by a line or piece of string, we shall find 
that we shall pass the leaves marked 2, 3, 4, and 5, but that 
when we reach the one marked 6, that this will correspond 
vertically with the 1st ; and then proceeding further, that the 
7th will be directly over the 2nd, the 8th over the 3rd, the 9th 
over the 4th, the 10th over the 5th, and the 11th over the 6th 
and 1st; so that in all cases when the sixth leaf is reached, 
including the one started from, a straight line might be drawn 
from below upwards to it, and that consequently there were five 
leaves thus necessary to complete the arrangement. Bonnet 
also discovered other more complicated arrangements in which 
more leaves were necessary for the purpose. His ideas were 
little attended to at the time; but of late years by the re- 
searches of Schimper, Braun, Bravais, and others, his views 
have been confirmed and considerably extended, and it has 
been shown that the spiral arrangement is not only universal, 
but that the laws which regulate it may be reduced to mathe- 
matical precision, the formule representing the relative position 
of leaves in different plants varying, although always constant 
for the same species. The examination of these laws further 
than to show that the regular arrangement of leaves and their 
modifications is in the form of a spiral around the stem, having at 
present no very practical bearing in Botany, however interesting 
they may be in a mathematical point of view, would be out of 
place here ; we shall confine ourselves to the general discussion 
of the subject, and as alternate leaves are those which will 
enable us to do so with most facility, we shall allude to them 
first. 
1. Alternate Leaves.—If we refer again to the arrangement 
of the leaves in the Cherry or Apple, we shall find that before 
we arrive at the sixth leaf (fig. 289), which is over the first, 
the string or line used to connect the base of the leaves will have 
passed twice round the circumference of the branch. The point 
where a leaf is thus found, which is placed in a straight line, or 
perpendicularly over the first, shows the completion of a series 
