68 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



(2) gave the most eqvMl distribution of leaves on an axis, and made 

 use of the minimum number of members. Although Wiesner's 

 views were thus brought into line with Schimper's series, ^, \, f , 

 etc., and pointed to the ideal angle 137° 30' 28", it is clear that his 

 generalization impUes much more than the simple statement that 

 this angle is better than any of the limiting angles of other series. 

 With a given number of leaves, this angle gives the optimum 

 approach to a symmetrical construction; that is to say, reverting 

 to the metaphor of buUding one brick at a time, this angle gives the 

 optimum method of oscillation across the system while laying down 

 the stated number of units, so that radial symmetry is most nearly 

 attained. Radial symmetry is, in fact, the grand aim, and not the 

 biological requisitions of leaf -distribution, which would be equally 

 well served by any other series, and, when unsatisfactory, may be 

 readily compensated by secondary zones of elongation either in the 

 main axis or in the lateral members themselves. 



To suit the theory of Schimper, Wiesner made x = 2; but the 

 same results obviously follow when x=l, since the ratios J, f, f, f, 

 etc., are all complementary of those of the previous set, and the 



limiting angle ( ^^2~^ ) °^ 360° = 222° 29' 32" is the inverse angle 



of 137' 30' 28".* 



The objection previously taken to the Sehimper-Braun theory 

 series of fractions was that they were used either to express angular 

 measurements which could not be measured, or orthostichies which 

 could not be proved to be vertical. It has now been seen that the 

 so-called orthostichies are, in all cases of asymmetrical phyllotaxis, 

 themselves log. spiral curves, and the divergence angles between 

 them are therefore contained by curved lines. In theory, the 

 angular measurements still hold,f but they not only become im- 



* The curious fact that the ratio * ^ ~ is also that hy means of which 



Euclid constructed the pentagon {iin 18° = ^5^^-^^) formed the subject of 



botanical speculation on the part of Kepler in 1611. Ludwig, " Weiteres iiber 

 Pibonaccicurven," Bot. Gentralb., 68, p. 7. 

 t Gf. note on Oscillation Angles. 



