DI-TRIMEROUS WHORLS AMONG THE FLOWERS OF DICOTYLEDONS. 159 
the climbing stem are 3. In these cases light appears to be the determining cause; but 
when whorls are formed, as in flowers, say, in 5's, the number may increase to 6's or 
diminish to 4’s, as may be seen in any corymb of Elder. Such differences are apparently 
due to varying amounts of nutriment *. 
Turning now to Monocotyledons, the ternary arrangement of the floral whorls is the 
result of the 5 arrangement. Commencing with a spiral of three leaves, at 120° apart, 
the fourth leaf would be over the first; but on projecting the spiral on to a plane, 
No. 4 takes a position 60° further on, in accordance with the * Law of Alternation” for 
whorls. No. 4, now starting the second whorl, would bring No. 7 over it, but this 
appears 60° further on and consequently stands over No. 2, and so on, as shown in 
Fig. 2. No. 21 will be seen to fall over the first, and as it is arrived at after eight 
revolutions, it represents the 34; divergence; see Fig. 2, p. 160. 
In Paris alone, so far as I know, may a quinary arrangement be found in the flower, 
a general floral formula being 4, 5 or 6. 4, 5 or 6 . 8, 10 or 12. 4 or 5. When quinary 
the members of the perianth are (according to Le Maout and Decaisne) distinctly quin- 
cuncial, or $. The explanation is as follows :—We have seen that among Dicotyledons 
a spiral arrangement can readily pass from one divergence to another, as 2 to 2 or vice 
versd, and more rarely into 3, or even 4, as in Laurel and Ivy. Similarly, } has passed 
upwards into the next divergence or ? in the floral whorls of Paris, and even in the 
foliage of some Lilies, as Lilium speciosum, and into higher ones, as 45; or =; in Pineapples. 
But, whereas 2 is the origin of all others in Dicotyledons, 4 or 4 is the origin and 
precedes all others in Monocotyledons. Hence the presence of 2 in Paris is purely 
accidental and offers no indication of a Monocotyledon giving rise to a Dicotyledon, any 
more than ternary whorls in the latter. 
Other indications of floral affinities with Dicotyledons might be thought to be trace- 
able in a few quaternary flowers occurring in Aspidistreze, Cyclanthese, and Potamogeton ; 
but if the flowers of pond-weeds be examined it will be found that they have two pairs 
of imbricated opposite leaves to the perianth, with £wo pairs of opposite stamens and a 
pistil of four carpels. These are easily deduced from whorls of threes by symmetrical 
reduction, the floral formula being 2.2.2,2.2,2. Degradations of various kinds 
among flowers of aquatic plants are a general phenomenon. 
It may be noted that while the leaves of the perianth of Potamogeton are imbricated, 
they are valvate in Cyclanthacez ; thus the latter exhibit a still further degree of degra- 
dation. I would repeat, in order to emphasize the fact, that the fundamental difference 
between the ternary whorls of flowers of Dicotyledons and those of Monocotyledons is 
thus seen to be that in the latter each whorl of three members corresponds to a single 
cycle of the 3 divergence, whereas in the former there are—at least of some whorls of 
the fowers—two circles of three members each, corresponding to a single cycle of the 2 
divergence ; or, if there be three circles of three members each, then such will be refer- 
able to a single cycle of the 2 divergence. If there be more than three circles of three 
parts each, then they are parts of the single cycle of 35, 37, or 44, &c.; the petals being 
. EUM. z 
* See paper on ** Numerical Increase, &c. cited on p. 156. 
