Church —The Principles of Phyllotaxis. 231 
which is to be the prime determining factor ; that is to say, does the 
possession by the plant of a ‘genetic-spiral’ work out the subsidiary 
pattern of the parastichies, or are the parastichies the primary feature, and 
the genetic-spiral a secondary and unimportant consequence of the 
construction ? 
Now, other systems may quite as easily be drawn ; thus take next 
a system of 6 curves crossing 8. On numbering these up by differences 
of 6 and 8 respectively in either series, it will be found that this time 
all the numerals are not employed, but that there are two sets of 1, 3, 5, 
&c., and i\ 3', 5', &c., showing that pairs of members on exactly opposite 
sides of the system are of equal value. There is thus no single genetic 
spiral now present, but two equal and opposite systems — a fact which 
follows mathematically from the presence of a common factor (2) to the 
numbers 6 and 8. The existence of such factorial systems in plants has 
created much confusion, and the term bijngate applied to such a construction 
by the brothers Bravais may be legitimately retained as its designation 
(Fig. 36, system (6 + 8)). 
Again, on constructing a system of 7 curves crossing 8, and numbering 
by respective differences, this time of 7 and 8 ; as in the first case, since 
these numbers have 1 only as common factor, all the numerals are 
utilized in numbering the system ; the genetic-spiral may be traced even 
more readily than in the first example, the adjacent members along it 
being now in lateral contact, so that the resulting spiral obviously winds 
round the apex. This effect is common among Cacti, and is the result 
of a general property of these curve systems which may be summed up 
as follows : — -Given a set of intersecting curves, the same points of inter- 
section (with others) will also be plotted by another system of curves 
representing the diagonals of the first meshes, and the number of these 
curves, and also of course the difference in numerical value of the units 
along their path, will be given by the sum and dijference of the numbers 
which determine the system, for example, 5 and 8 have as complementary 
system 3 and 13 ; and also other systems may be deduced by following the 
addition and subtraction series, e. g. : — 
5 - « 
3-13 
2 — 21 
i-34. 
Whereas the (7 + 8) system gives only 1 and 15 ; the single so-called 
‘genetic-spiral,’ which includes all the points, being reached at the first 
process. Thus a Cactus built on these principles would show an obvious 
‘ genetic-spiral ’ winding on the apex and 15 ridges, which in the adult 
state become vertical as a true helical construction is secondarily produced 
as the internodes attain a uniform bulk (Fig. 37 (7 + 8)). 
R 
