356 BOTANICAL GAZETTE [NOVEMBER 
comparison between the numbers of ray florets so far recorded as of 
taxonomic importance, inasmuch as the variations seem to be merely 
equivalent to variations in size. 
In short, the conclusions these observations seem to warrant are: 
(1) that the florets in the heads of Chrysanthemum Leucanthemum 
pinnatifidum may be regarded as arranged in either of two sets of 
spiral lines; or, what amounts to the same thing, in the intersections 
of these two sets of spirals; (2) that the number of lines in each 
set is a term of the Fibonacci series; (3) that the number is influ- 
enced by external conditions, i. e., the conditions of nutrition affecting 
size; (4) that one set is composed of 21 spirals, which are in some 
way correlated in their arrangement with the arrangement of the 
leaves; (5) that each of the 21 spirals and frequently each of the 
spirals in the other set tends to have a ray developed at its end— 
hence the modes noted by various observers; and (6) that these facts 
supply few if any data of taxonomic value. 
Turts CoLtecE, Mass, 
LITERATURE CITED 
1. CHurcn, A. H., The principles of phyllotaxis. Annals of Botany 18:227- 
243. 1904. i 
. FERNALD, M. L., Chrysanthemum Leucanthemum and the American white- 
weed. Rhodora 5:177-18r. 1903. 
———, AND Rosinson, B. L., Gray’s new manual of botany (under 
C, Leucanthemum). 7th ed. 1908. 
3- Lupwic, F., Weitere Kapitel zur mathematischen Botanik. Zeitschr. Math. 
Naturwiss. 19: 321-338. 1888. 
, Variationsstatistische Probleme und Materialien. Biometrika 1+17 
29. Igor. hemum 
Pearson, K., and Yutz, G. U., Variation in ray-flowers of Chrysant 
Leucanthemum L. Biometrika 1: 319. 1902. © : 
5. SHULL, G. H., Place constants for Aster prenanthoides. Bot. GAZE rE 38: 
333-375- I904. hemum 
Tower, W.L., Variation in the ray flowers of Chrysanthemum Leucant! es 
L. at Yellow Springs, Green Co., Ohio, with remarks upon the determin 
of nodes. Biometrika 1: 309-315. Ig02. 
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