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OBSERVATIONS FOR YOUNG BOTANISTS. 
V. — Phyllotaxis or Leak-Arrangements. 
F several leafy shoots from different plants be taken, it 
will be observed that many, probably the majority, 
have their leaves placed one at a time on the stem ; 
or, as botanists say, alternately ; e.g., the garden Hag, 
a sedge, the oak, and the holly. The rest will almost always 
have two leaves at the same position (or node), but situated on 
opposite sides of the stem : e.g., lilac, privet, or horse-chestnut. 
Of the latter, it will be also noticed that each pair of leaves stands 
at right angles to those above and below it. Such series of pairs of 
opposite leaves constitute what has been called the decussate 
arrangement. Extended observations will only strengthen the 
conclusion that leaves are for the most part alternate or opposite.'^' 
Alternate Leaves. — If one take a branch of the may or 
oak, and hold it vertically, with any selected leaf before one, 
and then pass the finger upwards along the stem from that leaf 
to the next, and thence to the third, fourth, fifth, and sixth leaf 
in succession, it will be found that the one last reached (sixth) 
is exactly over, or in the same vertical line with, the first ; and 
if we proceed further, we shall find the seventh is vertically 
over the second, the eighth over the third, and so on, the 
eleventh being, therefore, over both the sixth and first. 
The following observations will result from this examina- 
tion : — Obs. I. All the leaves on the branch are arranged in 
five vertical rows ; from this fact such an arrangement has been 
called pentastichous. Obs. 2. The imaginary line traced by the 
finger in passing from leaf to leaf successively is a spiral line. 
Obs. 3. This spiral line coils twice round the stem before 
arriving at the sixth leaf ; the portion of the spiral intercepted 
between the first and sixth leaf is called a cycle. Obs. 4. A 
cycle contains five leaves, the sixth being the first leaf of the 
succeeding cycle. 
The method adopted to represent this arrangement is by 
means of the fraction |. The numerator (2) indicates the 
number of coils in a cycle. The denominator (5) shows the ntimhev 
of leaves in a cycle. 
Let a complete cycle be projected on a plane surface, and 
represented by a “ helix ” (a spiral line like a watch-spring) 
having two complete coils, and let the corresponding positions 
of the leaves be marked upon it. Then if radii be drawn from 
the centre to the positions of the leaves, the angle between 
those drawn to any two successive leaves will be two-fifths of 
a whole circumference, or of 360° ; i.e., it will contain 144 degrees. 
From this fact, the fraction | is called the angular divergence of 
the pentastichous arrangement of leaves. An observation of 
some importance may be here conveniently made; viz., that 
* Leaves will occasionally be found grouped in threes or some higher num- 
ber : they are then said to be whorled or verticillate. 
