242 Church . — The Principles of Phyllo taxis. 
result in their becoming obvious to the eye. The fact that the quasi-circle 
hypothesis satisfies all the demands of centric growth systems, whether 
symmetrical or asymmetrical, as exhibited in the fundamental character 
of foliar appendages, and that these characters may be deduced as the 
mathematical consequences of the simple and straightforward hypothesis 
of placing centres of lateral growth in a centric system which is also grow- 
ing, may be taken as a satisfactory proof of the correctness of the original 
standpoint. And it is difficult to see what further proof of the relation 
between a leaf-primordium as it is first initiated, and the geometrical 
properties of a quasi-circle growth system is required ; but it still remains 
to connect this conception with that of orthogonal construction. 
This however naturally follows when it is borne in mind, firstly that no 
other asymmetrical mathematical growth-construction is possible, except 
the special quasi-square system which will include such quasi-circles ; and 
secondly, that the contact-relations of the quasi-circles in these figures are 
identical with those presented by the primordia in the plant, and could only 
be so in orthogonal constructions. It thus follows that with the proving of 
the quasi-circle hypothesis, the proof is further obtained that the intersection 
of the spiral paths must be mutually orthogonal ; and it becomes finally 
established that in the construction of a centric phyllotaxis system, along 
logarithmic spiral lines, the segmentation of the growth system at the 
hypothetical growth-centre does follow the course of paths intersecting 
at right angles ; and the principle of construction by orthogonal trajectories, 
originally suggested by Sachs for the lines of cell-structure and details 
of thickened walls, but never more fully proved, is now definitely estab- 
lished for another special case of plant-segmentation, which involves the 
production of lateral appendages without any reference to the segmentation 
of the body into ‘ cell ’ units. 
But even this is not all ; the point still remains, — What does such 
construction imply in physical terms? Nor can it be maintained that the 
present position of physical science affords any special clue to the still 
deeper meaning of the phenomena. The fact that the symmetrical con- 
struction in terms of logarithmic spirals agrees with the diagram for dis- 
tribution of lines of equipotential and paths of current flow in a special case 
of electric conduction, while the asymmetrical systems are similarly homo- 
logous with lines of equal pressure and paths of flow in a vortex in a perfect 
fluid, the former a static proposition, the latter a kinetic one, may be only 
an ‘ accident.’ On the other hand it must always strike an unprejudiced 
observer that there may be underlying all these cases the working of some 
still more fundamental law which finds expression in a similar mathematical 
form. 
In conclusion, it may be noted that if the proof here given of the 
principle of plant construction by orthogonal trajectories is considered satis- 
