PRELIMINARY NOTE ON THE SHAPES OF LEAVES. 81 
point, varies from 3 to 21. The phyllotactic spirals are, in fact, 
usually 4, 2, 3, 5 o> or the successive convergents of the con- 
, &e. 

tinued fraction ! 
2+4 
1. With a ird fraction, or with three leaves in the circle, we 
should expect a leaf to have a wedge-shape at the base with an 
angle of 120 degrees. I have found no rosette with this arrange- 
ment. The leaflets of Trifolium and Oxalis, however, approxi- 
mate to this idea. 
2, With a fraction of 2th phyllotaxis we should have five leaves 
in the circle, and each should have an angle at the centre of 72°. 
Now to apply this to actual leaves there are two methods. By 
averages, or by taking any actual leaf chosen under no special 
rule. The latter method has been adopted. The leaf of Peplis 
Portula, Linn., is a very close approximation to this idea, but in 
the leaf measured the angle was 75°, 
With the 3th phyllotaxis, or eight leaves in the whorl, the angle 
at the centre should be an eighth of 360°, or 45°. I found by 
measurement— 
In Bellis perennis, Linn.,  - : = 5 - ea" 
Hieracium aurantiacum, Linn., - - a 
Vacciniwm uliginosum, Linn., - - - 44° 
Antennaria dioica, R. Br, - - = a7? 
The geometrical idea seems to be followed very closely in these 
forms. 
Taking +5,th, then for 13 leaf bases to fully occupy the circle 
each should have an angle of 27° to 28°. I found that the angle 
was— 
In Arctostaphylos Uva-ursi, Spreng.,  - - are) 
Taraxacum officinale, Web., - - Sea te 
: Hypocheris radicata, Linn., - - 3 26. 
Arabis hirsuta, Scop., - - Eero ha! ORS 
_* This general result of very fairly close correspondence of course 
struck me as remarkable. In the short time available to me I 
had simply measured only two other obovate leaves. These 
; were— 
“Vaccinium Vitis Idea, Linn, - - - - 52° 
Corrigiola littoralis, Linn., - - - - - 13° 
7 ¥. 
