1876.] Mathematical Nature of Phyllotazis. 327 
Thus by mechanical twisting, if the twist is equal in all parts 
of the stem, we get on one side of the natural position the num- 
ber of rows 5 and 2, and on the other side 8, 3,and 1. Hence if 
we begin with the most twisted position and come toward the 
natural position, we get the numbers 
On one side . : Here 2 5 i 
On the other : : ` i g: ng: 
Now these series of numbers indicate the approach towards the 
untwisted position. What would be the number of ranks in that 
theoretically perfect untwisted state? As both these series of 
numbers are increasing, that is, the number of ranks decreases as 
you twist either way, you may infer that in the untwisted state 
the number of ranks is prodigious or innumerable. Carrying on 
the series by adding zigzag as the lines are dotted, we should 
1 2 5 13 34 89 288 
ea 8 ae ee 
_ Hence we say that the slightest conceivable twist in one direc- 
tion makes the number of ranks 377, a little more in that direc- 
hon gives 144, 55, 21, 8, 3, 1, while the slightest twist in the op- 
posite direction gives us 233, a little more 89, 34, 13, 5, 2, 1. 
There is, however, a mystery in the space between 233 and 
377, between twisting one way and twisting the other. Let us 
not seek to solve it by running the number of ranks up higher, to 
610, 987, 1597, ete., but approach it in another way. : 
In the stem twisted one way, the angle between the leaves is 
+ the whole circumference, or 2, or +55, or 43, ete. ; with the stem 
twisted the other way the angle is }, or 2, or sy, Or 35 etc., the 
‘ireumference. Let us set these in double rows : — 
Twisted one way i ee Fe a 
Twisted the other way . ‘ 4 & dr B tee mH 
Or putting them in decimals we shall see how they converge 
towards the same value: — 
Twisted one way . . 5 4 «88 882 -.38202 
Twisted the other way. 83 .875 .8809 .38818 .38194 
: Take the high fraction 1427 in the upper series and turn it 
into decimals, we get .88196603. If the leaves were at this angle 
they would form 4181 rows or ranks, and the least twist would 
Produce the lower numbers. Let us now attempt to find some 
