DI-TRIMEROUS WHORLS AMONG THE FLOWERS OF DICOTYLEDONS. 161 
usually staminodal (e.g. Magnolia) or developed out of bracts (e. g. Calycanthus and 
Epiphyllum). 
It is therefore quite impossible for the ternary arrangements of flowers of Dicotyledons, 
having two circles of three members each (i. e. for a calyx, corolla, stamens, or carpels), 
to be derived from M tyledons ; because the } and à divergences in the foliage are 
the immediute consequence of the single cotyledon, whereas the $ divergence follows 
at once from an opposite pair of cotyledons or leaves. 
Hence it is seen why the flower of a Tulip really consists of an outer cycle (calyx) 
of three parts, an inner one of three parts, the two forming the perianth, 3, 3 stamens, 
i. e. two cycles of three parts each and three carpels. The calyx and corolla are well 
differentiated in Hydrocharis, Alisma, Tradescantia, and others. 
To place two examples, for a contrast, together—Rwmex has 3+3 .(34-3]. 34-3 . 8, 
or 21 parts in all, composed of three single cycles of the ? divergence, with three extra 
parts, i. e. 21 in all, thus forming one cycle of the ç divergence. Hydrocharis has 
3.3.(¢) 3,3,3,3.(2) 38,3; i.e. a continuous succession of single cycles of the 
$ divergence. 
It may be noted that, although the passage from opposite leaves to 2 is carried out by 
degrees, as I have shown in the case of the Jerusalem Artichoke and small species of 
Epilobium, it more often occurs suddenly without any intermediate stages, as in germi- 
nating acorns, the first leaves of the plumule being pentastichous. Nor is there any 
transition between 2 and 4 in the Laurel and Ivy described above. 
It may be desirable to add a few words in explanation of the reason why the phyllo- 
taxis of the floral whorls can change from one divergence to another. It is because they 
are all “ represented " on one and the same axis when the parts are numerous and com- 
pacted together. Thus, in the cone of the Spruce-fir, the most nearly vertical rows 
(21 in number) are in accordance with 3. The next two in height running to the right 
and left, respectively, represent ;$ and $. The next lower pair will stand for 2 and 1; 
while even } can be shown to be present when ali the scales are properly numbered. 
Now, spirals can surrender, as it were, 3, 5, or 8 elements to make any whorl; but not 
13, as a too awkward number for a whorl. It might be asked, if all the leaves or their 
representatives, as bracts &c., form a continuous spiral, say of 1i divergence (e. g. the 
stamens of Magnolia), why should such definite “ cycular " numbers as 3, 5, 8 be detached, 
as it were, and not any, different, numbers be given up to make a floral whorl? The 
answer is not forthcoming; but Nature seems to have some predilection for them. 
This is well seen in the ray-florets of Composites, when large numbers are counted. 
Thus with the Ox-eye Daisy and Sunfiower, the ray-florets gave 21 as by far the largest 
maximum ; also 13, 26 (==2 15), and 34 as much lower maxima. ls 
The petals of the Cowslip gave as maxima 3, 5, 8, 10 (=2x5); 4, 7, 9 being minima. 
Many other cases are known. A correspondent in the South of Europe tells me that 
he has observed that the maxima 21 and 34 occur in Marigolds and are correlated to 
their habitats. Thus 21 is the usual or typical number; but near the sea the maxima 
are 26 (=2 13) and 34. 
23 
SECOND SERIES.—BOTANY, VOL. VIL 
