354 BOTANICAL GAZETTE [NOVEMBER 
bringing the leaves or their morphological equivalents closer together, 
the parts not infrequently become arranged according to an appar- 
ently different scheme. Thus in the umbels of the wild carrot 
(Daucus Carota) peduncles are usually so arranged that they seem 
to be placed at the intersections of what appear to be lines arranged 
in the form of logarithmic spirals 
pene (see CHurcH 1), eight running in 
DISTRIBUTION OF RAYS FOR TWO : B . he other; 
LOTS OF 250 HEADS EAcH; One direction and five in t : 
MATERIAL COLLECTED aT DEN- oor, if the umbel is large, eight in one 
Nis, JULY 13 AND 14, 1905; LOT direction and thirteen in the other. 
Trou a sosepuss'raxore This arrangement might be considered 
NEAR BY as differing from the arrangement of 
eed five-ranked leaves only in the greater 
Rays Heads ai 
Sigg ——— number of intersecting spirals. 
: 2 ief i i resent 
wf ‘ point of chief interest in the p 
: . 
14 3 connection is that the ani a 
16 6 " spirals is confined rather constantly t 
ithmic 
os 3 I the lower members of the logarithm! 
I : 
£9 3 ; series I, 2, 3, 5) 8, 13, 21) 34 & 
ar rs 4 If now a daisy head be examin 
21 55 27 : : i rhaps 
22 27 12 carefully, it will generally a it 
2 : ‘ 
24 12 = always) be found that the dis a4 
28 = ei are so arranged that they appear 
: = = i ions of two sets 
a 4 4 placed at the intersections OF © ae 
29 ; 2 of spirals; or perhaps they Rok 
30 5 15 more conveniently regarded as arta® 4 
31 7 28 Ron oo of spira 
= 5 16 in either of two oppose sae 
34 7 o running from the periphery to = c tes 
= 2 of the head. I shall speak of the a 
% : though they “ 
37 I florets, therefore, as ; peer 
S ™ ~ arranged in two sets of spirals, t 
: each of these sets (which one snes 
ing on the point of view assumed) embraces all the florets. ©" 
are usually 21 spirals running one way and 13, 21, OF 34 ed 
the other way. The set of 21 spirals may be either nee ; 
or right-handed in direction; but in either case the d vie a 
the spiral is apparently correlated with the arrangement 
