86 RELATION OP PHYLLOTAXIS TO MECHANICAL LAWS. 



spiral paths which pass to infinity, a method may be adopted which 

 gives a mean ratio with approximate accuracy, dependent on the 

 initial error of drawing the curve-systems ; and although destitute 

 of any strict mathematical value, a simple method of comparing 

 the ratios of the lateral members to their parent axis is obtained, if 

 this secondary circle and growth-centre be regarded as representing 

 the lateral member, and affording a mean value of distances which 

 in the limit approach equality. Thus in the (3-|-5) system (fig. 28), 

 measurements may be taken in millimetres from a carefully con- 

 structed figure within the error of drawing the true log. spiral. In 

 the case figured, the radius of the circle representing member 1 

 was 22'5 mm., while the distance between the centre of this circle 

 and the centre of the whole system was exactly 45 mm., and since 

 all the orthogonal areas are similar, the ratio is a constant for every 

 member and equals 1 : 2.* 



Taking this ratio as a sufficiently correct approximation, a 

 (3-1-5) system is thus the naiural co'nseqmnce of a spiral-cortex 

 construction when the diameter of the primordium of the lateral 

 menfiber is one-half that of the axis at the poini on which it is 

 apparently inserted. 



Again, by using the metaphor of an actual spiral-vortex move- 

 mient, it is clear that the impulses which originated in the fluid 

 mass of protoplasm may be regarded as at first of the nature of 

 mathematical points from each of which a new vortex motion was 

 initiated, expanding in all directions until it came into contact with 

 adjacent vortices ; further expansion in each would then result in 

 the heaping up of a mass of protoplasm which becomes the obvious 

 external sign of the origin of a new member.f 



The diameter of the primordium is thus only the expression of 

 the distance between the new impulses, and it is the ratio of such 

 distances to the diameter of the main axis which is the fundamental 

 constant which determines subsequent phyllotaxis phenomena. As 



* Gf. Mathematical Notes. More accurately the ratio may be taken as 

 1 : 1"95, the amount of error in the geometrical construction being ^^th. 



t Hofmeister pointed out that the position of new members might be 

 indicated by changes in the tensions of the superficial cells before any elevation 

 of a primordium was observable. 



