50 THE FLOWERING PLANT. 
dicotyledons, the two first leaves (the cotyledons or seed-leaves) 
are opposite, however the following leaves may be disposed (fig. 2). 
Seedlings of mustard and cress show this very well (fig. 3). 
When the leaves are alternate, they are arranged so that their 
angular distance from one another is constant. That is to say, 
if the stem were telescoped so as to bring any two successive 
leaves to the same level, and if two radii were drawn from the 
centre of the stem at that level through the insertions of the two 
leaves, the angle enclosed by the radii would be constant, generally 
speaking, for the same plant. Diagrams of phyllotaxis represent 
the stem as being extremely conical, and looked at from above, 
the leaf insertions being marked by thick curves. ‘The simplest 
case is that of grasses, &c., where any particular leaf is succeeded 
by one placed on the opposite side of the stem. The divergence is 
evidently 180°, and this may be expressed by the fraction 4, 7.¢., 
half of the entire circumference of the stem, or 360. If a line 
is drawn round and round the stem in the same direction, cutting 
the insertions of the leaves, it will form what is called the genetic 
spiral. Of course a spiral line might be drawn according to 
pleasure in either direction, right to left, or the opposite. It is 
agreed, however, always to take the shorter course. Starting 
from any leaf and proceeding upwards, the spiral will wind once 
or more round the stem, till a leaf is reached immediately above 
the first, 7.e., in the same rank or orthostichy. Such a portion 
of the spiral is called a cycle. 
The numerator of the fraction 
4 shows us that the cycle takes 
one turn round the stem, while 
the denominator signifies that 
the cycle contains two leaves, 
and that there are two ortho- 
stichies. The series 3, 4, 2, 
3, 73, &c., Includes the com- 
monest divergences. From 2 
onwards the numerator and 
denominator of any fraction 
are respectively the sums of 
the numerators and denomi- 
nators of the two preceding 
fractions. The two-ranked (4) 
arrangement not only occurs 
in grasses, but also in many 
other monocotyledons, and in 
twigs of the lime, elm, beech, &c. The three-ranked (+) distribu- 
tion is characteristic of sedges, and the alder and aspen. Most 
Fig. 9.—? Phyllotaxis of Cherry. 
