MATHEMATICAL NOTES ON LOG. SPIEAL SYSTEMS. 331 



of construction from the origin, and 6 is measured from the line 

 joining these points. The curve touches the sides of the mesh 

 where they are met by the two intermediate spirals which deter- 

 mine the " centre of construction." 



From the character of the above equation it follows that the 

 curve is symmetrical with respect to the line joining the origin to the 

 centre of construction. 



This mathematical deduction is of the greatest botanical interest, 

 in that it brings out the remarkable fact that every lateral 

 primordium is primarily hilaterally symmetrical with regard to a 

 radius drawn through the centre of construction and the centre 

 of the main axis ; and thus, whether in a whorled or spiral phyllo- 

 taxis system, its primary structural peculiarities will be identical. 

 Thus, no change whatever is involved in the properties or shape 

 of the lateral members themselves, when the phyllotaxis system 

 passes from an asymmetrical construction to a symmetrical one ; 

 that is to say, change of symmetry in the radial axis system does 

 not directly affect the symmetry of the appendage, and whatever 

 the curve-ratio of the construction, the leaf-members would be 

 equally isophyllous, although eccentric growth of the whole shoot, 

 by affecting the shape of the ovoid curve itself, involves anisophylly. 

 On the other hand, the contact-pressures of adjacent growing 

 primordia, which cause them to approach the shape of the quasi- 

 square meshes, result in making the primordia secondarily 

 asymmetrical to a certain extent when the curve-system is 

 asymmetrical. 



The general result of this mathematical investigation is to 

 establish the fact that certain essential properties are common to 

 all leaf-primordia expressed as quasi-circles ; and these may now 

 be expressed in botanical phraseology. 



I. All such appendages are hilaterally symmetrical about a 

 median line, the radius drawn through their own centre of 

 construction and the growth-centre of the axis itself. The 

 appearance of radial flattening to which they are subject in the 

 main growing system thus exaggerates this symmetry in two 

 orthogonal directions — one a radius of the system, the other a 

 circular path of the same central system. 



