VARIATION AND CORRELATION OF THE NUMBER OF 
UMBEL RAYS OF SOME UMBELLIFERAE*. 
By C. cool and A. N. KOOPMANS, 
Laboratory of Microscopical Anatomy, Technical High School, Delft. 
(1) Since the investigations by Ludwigf into the variation of the number of 
umbel rays of the umbels of Umbelliferae, but little is to be found about this subject 
in biological literature, although the reading of the chapter " Summationskurven 
der Umbelliferen" of the article above-mentioned might suggest various problems. 
For example, in the variation curves of Ludwig the numbers of the Fibon- 
nacci series do not exclusively appear as maxima, but in some cases also the 
numbers of the first parallel series (1, 3, 4, 7, 11, etc.) appear. VoglerJ also finds 
in his investigations into Astrantia major that principal series as well as parallel 
series occur in this case. 
The few investigations by Ludwig on the Umbelliferae are hardly sufficient 
to allow us to state that modes which arise in the numbers of the first parallel 
series of these variation curves are with any degree of certainty only apparent 
modes, or to assert that they have resulted from the mixing of plants of different 
local races. On the contrary it is very probable that these numbers of the first 
parallel series are a characteristic phenomenon of the variation curves of the 
number of umbel rays of many species of Umbelliferae. 
Of the many countings which we made in the summer of 1912 to find out 
something more about this subject, only those relating to three species of plants, 
Pastinaca sativa, Anthriscus sylvestris and Sium latifolium, are used here. 
Only numbers of umbel rays of terminal umbels and those of lateral umbels 
of the first order were counted. 
• * Our thanks are due to Prof. Dr G. van Iterson who kindly revised this paper, 
t Bot. CeJitralblatt, Bd. lxiv. 
J Beihefte zum botanischen Centralblatt, Bd. xxiv. 
