437 
Although not strictly belonging in a discussion of phyllotaxy, 
‘the arrangement of leaflets is an interesting and valuable guide to 
the study of our native trees. The palmately compound leaf is 
Tepresented only by the genus AZsculus, whose leaves are also 
‘Opposite. Of the trees with pinnately compound leaves, only two 
— Senera, /raxinus and Negundo, belong to the opposite-leaved 
Stroup, and of all the trees with compound leaves there are but 
‘two with abruptly pinnate leaves, namely, Gleditschia and Gym- 
nocladus, in which the compounding is also carried to the second 
-or third degree. 
The spiral phyllotaxy is subdivided according to the number 
‘Of leaves in a series, which may be considered as a whorl length- 
‘ened out by the growth of the stem between the times at which 
leaves were put forth. The beginnings of the several series—or, 
‘Of course, any corresponding leaves of different series—are in the 
Same perpendicular column. A series of any particular number 
Of leaves is also included in a certain number of spiral turns about 
the stem. Thus, the simplest series has its leaves arranged 1-2, 
1-2, each leaf being half a circumference from the next and the 
Series of two leaves being completed in one turn. The next 
Series runs 1-2-3, 2-3, each leaf being a third of a circumfer- 
nce from its neighbor, so that this series is also completed in one 
Spiral turn. The next series contains five leaves, but each is 2 in- 
Stead of 4 of a circumference from the next leaf above or below, 
So that the series requires two turns about the stem before we 
réach a leaf directly above no. 1. It will be noticed that nature has 
“pparently formed this third series by adding together the number 
Of turns and the number of leaves of the two preceding series, 
just as little children would add the fractions ¥% and &%, numerator 
to numerator and denominator to denominator, and get the result 
% The next series, characterized by eight leaves and three spiral . 
turns, seems, likewise, to have been formed by adding the one- 
turn and three-leaf series to the two-turn and five-leaf series. A 
series higher still has 2 +3 turnsand 5 + 8 leaves. Beyond this, 
we may still imagine nature continuing to add numerator to nu- 
Merator and denominator to denominator, but the leaves become 
So densely crowded, that we cannot bring them into order and, 
hence, we term the arrangement a fascicle, or dwuch to translate 
