RULE FOR DETECTING SPIRAL ARRANGEMENT. ; 9 
a some particular spiral is drawn, and the rules are shown 
to be tru his instance, and the learner is forced to make his 
induction fro a simple instance. 
he following proof seems to me capable of comprehension by an 
intelligent learner, and it has the advantage of showing how far the 
rules depend on the spirals found in nature . belonging to the particular 
series of fraction 4, 4, 3, as distinguished from an 
he rules generally given are :—Detect two leaves, (c) and (p), 
nearly above the leaf from which you start ages ; count the number 
wi 
mental spiral you are seeking, while the less of these two numbers 
gives you its numerator. 
The student is then shown satisfactorily how to count the leaves 
from a to c by th mber of parallel secondary spirals, but the 
original rule is generally left as an article of faith. 
ose to prove that in any sn om whether 1 in B 
I prop 
the vee ict series 3, 4, %, &c., or ina 
1 m of the number of ite iabweek aand 
c, and ekibae A and D, gives the numerator of the 
fundamental fraction. 
The sum of the number of leayes between a D- 
and c, and between a and p, gives its denominator. 
B wn 
e e th y : 
by supposition the leaf which is most 5 eetamaalatr 
above a to the right without being actually above 
similarly 8 is the leaf which is most Jon dedintaty eG 
above ¢ without being exactly above it * ; moreover 
it diverges to the right - Us just as p diverges to the 
right of a. Hence B has the same position with 
respect to c that p has with respect to a, and, since 
by supposition the spiral is uniform, t the number of A 
leaves and of coils between c and B is the same as the 
number between a and pb. 
Hence in either case, leaves or coils, to count from a to o, ina then 
from a to p, is the same as to count aes a toc and then from c to B, 
that is, to count the whole way from 
Hence the sum of the eB aae of coils from a to c and from a 
to p is the number of coils from a to B, or the numerator of the 
fundamental fraction sought. 
nd the sum of the numbers of leaves from a to c and from a 
(not inclusive) is the number of leaves from a to B (not peels a 
therefore gives the denominator of the fundamental fracti 
It should, then, be noticed that in the series 3, 1 , §, each 
numerator (after the first tw 0) is the same as the fachte of the 
last fraction but one, or of the less approximating of the two con- 
* Note.—If this be oe suppose that ag other leaf Lspibeee A cag B ober 
more nearly above c ie right than x», then this new d 
nearly above a to the left chi c is, and herefo ore © cena ne te “the leaf pater 
nearly above a diverging to the left which we supposed it t 
