134 
three And. moreover, the angular divergence of any leaf from 
the next in succession will be found in a similar manner to he 
that fractional part of 360°. Similarly, just as all angnlar 
divergences of the leaves of the primary series lie between 120 
and 180° inclusively, all those of the leaves of the secondary 
series lie between 90° and 130° ; the limiting point bemg at 
an angular distance from the first leaf of 99 30 + • 50 
it must be observed that the fractions of the secondary series 
are the successive convergents of the continued fraction . 
i 
1 + L 
1 + &C. 
14. In a manner analogous to the above, we might construct 
a tertiary series, commencing with the fractions J, , 1, and which 
would then appear as follows : — i, i, f, A; ts> tt, & c - Sucn 
a series, however, does not exist in nature, as far as I am aware. 
Having, then, before us three analogous series, it is obvious 
that we might construct any number of such series, and finally 
all would be represented by the algebraical forms, where a is 
any number : 
— — — - — 3 &c. 
. i 2&4“1 K/7.4- 
a+ 1 
3a+2 
5a+ 
These fractions being the successive convergents of the con- 
tinued fraction 
0 + 1 
1+1 
1 + &C. 
15. In all the preceding investigations, I have supposed the 
space between any two successive leaves on the stem to have 
been sufficiently developed to enable me to trace an imaginary 
spiral line through the leaves. But it sometimes happens that 
such spaces, called internodes, are so short or are practical y 
wanting, that the leaves become crowded together, so that it is 
quite impossible to say which is the second leaf after having 
fixed upon some one as the first. This is especially apparent in 
the case of fir-cones, where the scales may be considered as the 
representatives of leaves, and which, though ciow e , are 
arranged in a strictly mathematical order. , . .. 
I6. S If a cone of the Norway spruce fir be held vertically , 
the scales upon it will be observed to run in a series of parallel 
spirals, both to the left hand and to the right. This is a result ot 
their being crowded together, as well as of their definite arrange- 
