236 Church. — The Priiieiples of Phylloiaxis. 



of interpreting any of these systems has little bearing on the case: the 

 subject is purely a mathemaltical one; and the only view which can be 

 acceptable is that which applies equally well to all cases, in that the 

 question is solely one of the geometrical properties of lines and numbers, 

 and must therefore be settled without reference to the occurrence of 

 such constructions in the plant. 



If all phyllotaxis systems are thus to be regarded solely as Cases of 

 intersecting curves, which are selected in varying numbers in the shoots 

 of different plants, and often in different shoots of the same plant, with 

 a tendency to a specific constancy which is one of the marvellous features 

 of the plant-kingdom^ it remains now to discuss the possibility of attaching 

 a more direct significance to these curves, which in phyllotaxis construction 

 follow the lines of what have been termed the contact-parasticlms ; that 

 is to say, to consider 



I. What is the mathematical nature of the spirals thus traced ? 

 II. What is the nature of the intersection ? and 



III. Is it possible to find any analogous construction in the domain 

 of purely physical science ? 



The suggestion of the logarithmic spiral theory is so obvious that 

 it would occur naturally to any physicist: the spirals are primarily of- 

 the nature of logarithmic spirals \ the intersections are orthagdnal; and 

 the construction is directly analogous to the representation of lines 

 of equipotential in a simple plane case of electrical conduction. In 

 opposition to this most fruitful suggestion) it must be pointed out however 

 that the curves traced on a section are obviously never logarithmic spirals, 

 and the intersections cannot be measured as orthogonal. But then it 

 is again possible that in the very elaborate growth-phenomena of a plant- 

 shoot secondary factors come into play which tend to obliterate the 

 primary construction ; in fact, in dealing with the great variety of 

 secondary factors, which it only becomes possible t6 isolate when 'the 

 primary construction is known, the marvel is rather that certain plants 

 should yield such wonderfully approximately accurate systems. To begin , 

 with, logarithmic spiral constructions are infinite, the curves pass out to 

 infinity, and would wind an infinite number of times before reaching the 

 pole. Plant constructions on the other hand are finite, the shoot attains 

 a certain size only, arid the pole is relatively large. The fact that similar 

 difficulties lie in the application of strict mathematical construction to 

 a vortex in water, for example, which must always possess an axial tube 

 of flow for a by no means perfect fluid, or to the distribution of potential 

 around a wire of appreciable sizCj does not affect the essential value of 

 the mathematical conception to physicists. And, though the growth of 

 the plant is finite, and therefore necessarily subject to retarding influences 

 of some kind, there is no reason why a region may not be postulated, 



