PRELIMINARY NOTE ON THE SHAPES OP LEAVES. 81 



point, varies from 3 to 21. The phyllotactic spirals are, in fact, 

 usually I", |, f , ^, 2X5 or the successive convergents of the con- 

 tinued fraction -, (fee. 



2 + f ^ 



1. With a l^rd 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 fth 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 Feplis 

 Portula, Linn., is a very close approximation to this idea, but in 

 the leaf measured the angle was 75°. 



With the §th 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., - - - - - 42° 



Hieracium aurantiacum, Linn., - - - 42° 



Vaccinium uliginosicm, Linn., . _ . 4.4° 



Antennaria dioica, R. Br., - - - - 47° 



The geometrical idea seems to be followed very closely in these 

 forms. 



Taking T%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 Arctostavhylos Uva-ursi, Spreng., - - - 23° 

 Taraxacum officinale, Web., - - - - 26° 

 Rypochmris radicata, Linn., . . . . 26° 

 Arabis hirsuta, Scop., ----- 27° 



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 V.itis Idcea, Linn., - - -, - 52° 

 Corrigiola littoralis, Linn., - - - - - 13° 



