232 Church . — The Principles of Phyllo taxis. 
Finally, take the case of 8 curves crossing 8, and number in the 
same way by differences of 8 along both series. It immediately 
becomes clear that there are 8 similar series : all other spirals have 
been eliminated; there is no ‘genetic-spiral’ at all, but only a system 
of alternating circles of members of absolutely identical value in each 
circle. We have now, that is to say, systems of true whorls , and also 
learn in what a true whorl consists — the members must be exactly and 
Fig. 37. Curve-system (7 +8) : anomalous type. 
mathematically equal in origin — while the expression a successive whorl 
is a contradiction in terms. 
From such simple and purely geometrical considerations it thus 
follows that the so-called ‘genetic-spiral’ is a property solely of inter- 
secting curve-systems which only possess 1 as a common factor, and 
is therefore only existent in one case out of three possible mathematical 
forms (Figs. 35, 36, 38). While if these four systems were subjected to 
