232 



Church.— The Principles of Phylloiaxis. 



Finally, take the case of 8 curves crossing 8, and number in the 

 same way by differences of 8 along both series. It immediatelj» 

 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 



--V'"' ' 



i « ';'v-v--f- 



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 ' geneticr-spiral' is a property solely of inter- 

 secting curve-systems which only possess i 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 



