131 
some importance may be here conveniently made; viz., that 
each coil ( i.e . the circumference of a circle) contains three leaves ; 
this same number is invariably true for all other arrange- 
ments of the “ primary ” series, as will be hereafter described. 
6. 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 f , and the 
angular divergence will be f of 360°, or 120 degrees. 
7. 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 f, because in passing 
from 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 f 
will represent the angular divergence. Another is found to be 
i 5 3 , and several more exist. 
8. 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 f, f, f, f ; such I have else- 
where* proposed to call the primary series. 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 : •§-, f, f, f, T B F , ^ T , if, ff, ff, &c. It will be also 
observed that the numerator of any fraction is the same num- 
* On the Variations of the Angular Divergences of the Leaves of Helianthus 
tuberosus. By the Rev. George Henslow. Transactions of the Linnean 
Society , vol. xxvi. p, 647. 
VOL. VI. 
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