132 
ber as the denominator next but one preceding it. ^ ieie 
remains one more remarkable connection ^ ^tinned 
that these fractions are the successive conver gents of the continued 
fraction 
2 + 1 
1 + 1 
That is to say, if we reduce, by the ordinary rules for simplify- 
ing fractions, the portions 
2+1 2+1 
1+1 
1 
and so on, the resulting fractions will he the same as those 
SW 9 n i have said that the above series of fractions represent rte 
arrangements which exist in nature, and it is not usua^o fi d 
any species departing from the anrangement ^*“7“ 
nWfloteristic of it: in other words, the phyliotaxis oi a y 
%%%%£?* that species. The following are illnstra- 
tions : — 
Iris, or Flag. The glumes (chaff) of all grasses. Some orchids. 
Gar’ex, or Sedge. Leaves of several grasses. 
Oak Hawthorn. This is one of the commonest arrangements. 
Holly, White Lily, Greater Plantain. A common arrangemc 
amongst mosses. 
Convolvulus tricolor. Many orchids. Male fern. , * 
Scales of Spruce fir-cone. Ribwort Plantain {Plants UncUata). 
Yucca. Some mosses. 
Hoary Plantain ( Plantago media). 
10 If now, a semicircle be described, and one extremity of 
its diameter represent the position of any leaf, assumed the 
first ; and if a radius be drawn at the 180 
angular distance of 120° from this point, 
then the point where the radius meets the 
circumference will be the position ot the 
second leaf of the tristichons arrangement. 
The opposite extremity of the diameter 
will be that of the second leaf of the di- 
stichous arrangement. And these poims 
form the extreme positions for the second 
leaves of spirals of the primary series, 
corresponding to the fractions x and 4 re 
T 
T1‘ 
8 
7T 1 
. 99 ° 30 ' 6 " + 
90 ° 
Fig. 1. 
corresponding to the fractions x and 2 re- 120°, 
spectively. No second leaf ever lies nearer to the first than , 
