[37 | 
ПІ. On the Origin of the prevailing Systems of Phyllotaxis. By the Rev. GEORGE 
| Hrnstow; M.A., F.L.S., F.G.S. 
(Plate IX.) 
Read February 4th, 1875. 
INTRODUCTION.—In my paper “ On the Variations of the angular divergences of the 
Leaves of Helianthus tuberosus” (Trans. of the Linn. Soc. vol. xxvi. p. 647), the con- 
cluding paragraph commenced as follows :—* If we limit our inquiries to the conditions 
of the primary [ordinary fractional] series alone, we find, as a fact in nature, that if we 
assume any leaf as No. 1, then No. 2 lies at an angular distance from it, between 120° 
and 180° inclusively. If we ask why it is so, no answer can be given. Further, 
starting with this condition, we find the second leaf has an inherent tendency to take 
up a definite point as its position along that are. ..... If we ask why it is so, or why 
the second leaves are not anywhere upon that arc, there is as yet no answer to this 
question any more than to the first.” 
Іп the present paper I propose endeavouring to reply to both these questions, and to 
expound what appears to me to be the fundamental origin of the various systems of 
phyllotaxis exemplified in nature. 
The Spiral arrangement is deduced from Opposite and Decussate Leaves.—1f a stem (as 
of the Jerusalem Artichoke) be found having opposite and decussate leaves from the base 
upwards, but becoming converted into an alternate and spiral arrangement above (a not 
uncommon feature), the first process of such a change is the development of short 
internodes, so as to separate the leaves of each opposite pair, as in fig. 1. Тһе numbers 
represent the leaves, which are supposed to occupy their original decussating positions. | 
The order of the uwmbers shows the true spiral arrangement now produced by this 
development of new internodes, and which are indicated by the thin lines, while the 
original and long internodes are represented by the thicker portions of the spiral. 
In other words, if a line be drawn from leaf to leaf successively, it will be spiral, 
and will pass through the leaves in the order indicated by the numbers on the projected 
helix (fig. 1). (See note, p. 45.) Е 
In following up this spiral line, it will be seen that just those numbers which are the 
denominators of the fractions of the usual (primary) series, +, 3, 3, $, 1% зг, fall over 
one another, viz. 0, 5, 8, 13, 16 [228], and 21. Similarly the number of coils of the 
helix between 0 and each of these numbers successively supplies the numerators. Thus 
there are /wo coils between 0 and 5; three coils between 0 and 8; five coils between 
0 and 13; and lastly, eight coils between 0 and 21. 
Тһе denominators of the first two fractions (2 and 3) are not represented amongst 
these numbers ; the reason for this will be explained further "s 
In the diagram the leaves are represented as retaining their originally decussating 
positions; but practically, while the internodes are being developed, the leaves shift 
their positions, as it were, and obey the following law :— 
SECOND SERIES.—BOTANY, VOL. I. G 
