236 Church . — The Principles of Phyllotaxis. 
of interpreting any of these systems has little bearing on the case: the 
subject is purely a mathematical 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-par as tichies ; 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 orthogonal ; 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 to 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, and 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 size, 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, 
