86 
NATURE NOTES 
each coil (i.e., the circumference of a circle) contains thyeeledives ; 
this same number is invariably true for all other arrangements 
(excepting the such as the leaves of the flag). 
Let another example be taken. Suppose it to be a sedge 
{Carex). Here the fourth, seventh, tenth, &c., leaves will all 
be found arranged vertically over the first ; the fifth, eighth, 
eleventh, &c., over the second ; and the sixth, ninth, twelfth, &c., 
over the third. Hence there will be only three vertical rows of 
leaves, and the name given to this arrangement is consequently 
tristichous. Moreover, it will be observed that there are but three 
leaves in each cycle, and that the cycle completes but one coil 
or circle in passing from any leaf to the next immediately over 
it : so that by adopting the method given above, of representing 
this arrangement by a fraction, the fraction will be J, and the 
angular divergence will be i of 360° or 120°. 
By extending such observations as these, we should soon 
discover other arrangements of leaves to exist in nature ; and 
we should find that their angular divergences are equally capable 
of being represented by fractions. Thus, in the garden flag 
{Iris), the leaves are on opposite sides of the stem, but are 
alternately arranged, as no two stand at the same level. This, 
therefore, will be represented by because in passing from the 
one leaf to the next, an entire semicircle is traced, and from the 
second to the third another complete semicircle ; so that the third 
leaf (which commences the next cycle) is over the first. This 
arrangement is consequently called distichous, as all the leaves 
on the stem will be in two vertical rows, and on opposite sides 
of the stem. In another kind, a cycle will coil thrice round 
the stem, and contain eight leaves; hence § will represent the 
angular divergence. Another is found to be -f-^, and several 
more exist. 
If the fractions thus constructed from actual examination 
of plants be written down in succession according as the nume- 
rators and denominators increase, they will be seen to form 
a series with remarkable connections between its component 
fractions. It will be as follows; — J, j, f, &c. It cannot fail 
to be noticed that the sum of any two successive numerators, or 
of any two successive denominators, forms that of the next 
fraction respectively, so that we might extend this series indefi- 
nitely ; thus : i, |, f, &c. It will be also 
observed that the numerator of any fraction is the same num- 
ber as the denominator next but one preceding it. There yet 
remains one more remarkable connection between them, viz., 
that these fractions are the successive convergcnts of the continued 
fraction. 
1 
2 -I 1 
1 I 1 
1 + &c. 
