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 i, 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 (a) to the 

 numbers 6 and 8. The existence of such factorial systems in plants has 

 created much confusion, and the term bijugate 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 i 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 difference 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- 8 



2 — 21 



1-34- 

 Whereas the (7 + 8) system gives only i 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 



