Church . — The Principles of Phyllotaxis . 239 
retical generalizations of the mathematical conception of uniform growth, 
and would be at the same time in closest agreement with the facts of 
observation ; while no other mathematical scheme could be drawn which 
would include primordia arranged in such contact relations and at the 
same time give an orthogonal construction. If, that is to say, the quasi- 
circle can be established as the mathematical representative of the 
primordium of a lateral appendage, the orthogonal construction, which 
is the one point most desired to be proved, will necessarily follow. 
Fig. 41. Quasi-circles of the systems (2 + 2), (1 + 1) and (1 4- 2) arranged for illustration in the 
plane of median symmetry. C' , C " , C r,f , the centres of construction of the respective curves. 
(After E. H. Hayes.) 
It remains therefore now to discuss the nature of the curves denoted 
by the term quasi-circles ; their equations may be deduced mathematically, 
and the curves plotted on paper from the equations. These determinations 
have been made by Mr. E. H. Hayes. Thus a general equation for the 
quasi-circular curve inscribed in a mesh made by the orthogonal inter- 
