242 Church. — The Principles of Phyllotaxis. 



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 relatioo 

 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- 



