328 Mathematical Nature of Phyllotazis. [ June, 
simpler mode of representing these fractions .88196603 or 458, 
which are the same. 
Dividing both numerator and denominator of 4437 by 1597 
will give }+-987,; dividing each term of the last fraction by 
987 gives us 2 
LES W 
and continuing the process gives us 
bey 
bag 
equal to 
i 
pe 
} 
“Tage 
equal to 
$+ 
Hh, 
ty 
itys 
equal to 
b+ 
T+ Bha 
t+ bea, 
Now calling the fraction 
Hz, 
continued indefinitely by the name of z, it is plain that the 
phyllotactic fractions beginning with }, 4, 2, 3, %, continually ap- 
proach nearer and nearer to the value },.,, or 
- 
LAS ae 
+2 
and these values are alike. Putting the first two equal gives 
Fan oe a 
whence 
x? e—=1, r=—}+1 4/5, and }-5=} (8—V 5). 4 
This expression, } (83 — 4/5), is equal to .88196+, and ale 
presses the exact ratio of the leaves in a theoretically untwist 
stem when the number of rows is infinite. Other arrangements 
k; 
a 
3 
i 
: 
: 
i 
i i: 
a 
