392 
Proceedings of the Royal Society 
Let A, a, represent the same leaf in a plane development of a 
branch or fir-cone 
(regarded as cylin- 
drical) ; 0, a leaf 
which can be 
reached from A 
by m steps in 
a right-handed 
spiral, developed 
into the straight 
line AO, and by n steps from a in a left-handed spiral aO. These 
spirals may in general be chosen so that m and n are not large 
numbers (3, 5, 8, 13, &c., being very common values) ; but they 
must (and can always) be so taken that m spirals parallel to aO, 
and n parallel to AO, shall separately include all the leaves on the 
stem or cone. 
If m and n have a common factor A, there will be A — 1 leaves 
(besides A) which are situated exactly on the line A a, and there- 
fore the arrangement is composite, or has A distinct fundamental 
spirals. If m' and n' be the quotients of m and n by A, they are to 
be treated as m and n are treated below ; and this case thus merges 
into the simpler one, so that we need not allude to it again. 
It is obvious that, in seeking the fundamental spiral, we must 
choose the leaf nearest to A a on the side towards 0, as that suc- 
ceeding A or a. The fundamental spiral will thus be right-handed 
if P, which is nearer to A than to a, be this leaf — left-handed if 
it be p. Of course, we may have a left-handed fundamental spiral 
in the former case, and a right-handed one in the latter ; but the 
divergence in either will be greater than two right angles, and this 
the majority of botanists seem to avoid. 
Draw PQ and pq respectively parallel to a 0 and AO, then the 
requisite condition is that 
n \ s-\ ni 
— AQ - PQ, or -aq - pq , 
m n 
shall be as small as possible. 
Hence, if ^ be the last convergent to and if - > m . it is 
v n v n 
