133 
nor further than 180°.* The positions of all the second leaves 
are upon the arc included between those extreme points (viz., 
120 and 180 degrees from the extremity of the diameter cor- 
responding to the position of the assumed first leaf). Thus: 
for the pentastichous, as we have seen, it is at an angular dis- 
tance of 144° ; for the f divergence the second leaf is at an 
angular distance of 135°, while the positions of the second 
leaves of the spirals, represented by the consecutive fractions 
to to it, &c., gradually approximate to some intermediate 
point on the arc, but which no known example ever reaches. 
That point will be understood, by mathematicians, to represent 
the “limiting” value of the continued fraction 4+i^i + &c., 
or 3 -=p of 360°, or 137° 30' 28"+ 
11. Occasionally, other fractions must be constructed to 
indicate peculiar arrangements, and which cannot be repre- 
sented by any one of the fractions of the primary series given 
above. I discovered the Jerusalem Artichoke to be a plant 
which, unlike most species having their own peculiar arrange- 
ments constantly the same, offered the most singular variety. 
Not only were some leaves opposite , i.e. in pairs at right angles, 
but also in threes , all on the same level ; and when this was 
the case, they followed the same law regulating their positions, 
as already mentioned in the case of opposite or decussate leaves; 
viz., that the leaves of each group of three alternate in position 
with those of the groups above and below them ; I have calledf 
this arrangement tricussate. But besides these two kinds, the 
leaves on many stems were arranged alternately, and could be 
represented by the fractions \, -§-, f, &c. But more than this; 
for I found that the fractions f, tt, to and others were likewise 
to be frequently obtained. Now these latter are obviously part 
of an analogous or secondary series; and if continued would 
stand thus : y, y, y, yy, yy, yy, &C. 
12. This secondary series will be seen, on comparing it with 
the primary, to differ in commencing with the fractions \, £, in 
place of i; but afterwards, each successive fraction may 
be written down as in the primary series by simply adding the 
two successive numerators and denominators respectively. 
13. If, now, we project on a plane a cycle of any one of 
the spiral arrangements represented by a fraction of this 
secondary series, as in the case of £, we shall find that a com - 
plete circumference will invariably contain four leaves instead of 
* If the second leaf be at a greater distance than 180, and not less than 
240 degrees from the first, it will be seen that the conditions are simply 
reversed, and the spiral will then run round in the opposite direction, 
t Op. cit. 
M 2 
