XIV] OR PHYLLOTAXIS 933 



to liave any exclusive relation to the Golden Mean; for arithmetic 

 teaches us that, beginning with any two numbers whatsoever, we are 

 led by successive summations toward one out. of innumerable series 

 of numbers whose ratios one to another converge to the Golden 

 Mean*. Fourthly, the supposed isolation of the leaves, or their most 

 complete "distribution to the action of the surrounding atmosphere" 

 is manifestly very little affected by any conditions which are confined 

 to the angle of azimuth. For if it be (so to speak) Nature's object 

 to set them farther apart than they actually are, to give them freer 

 exposure to the air or to the sunlight than they actually have, then 

 it is surely manifest that the simple way to do so is to elongate 

 the axis, and to set the leaves farther apart, lengthways on the 

 stem. This has at once a far more potent effect than any nice 

 manipulation of the "angle of divergence." 



Lastly, and this seems the simplest, the most cogent and most 

 unanswerable objection of them all, if it be indeed desirable that 

 no leaf should be superimposed above another, the one condition 

 necessary is that the common angle of azimuth should not be a 

 rational multiple of a right angle — should not be equivalent to 



— I - j . One irrational angle is as good as another : there is no 



special merit in any one of them, not even in the ratio divina. We 

 come then without more ado to the conclusion that while the 

 Fibonacci series stares us in the face in the fir-cone, it does so for 

 mathematical reasons; and its supposed usefulness, and the hypo- 

 thesis of its introduction into plant-structure through natural 

 selection, are matters which deserve no place in the plain study 

 of botanical phenomena. As Sachs shrewdly recognised years ago, 

 all such speculations as these hark back to a school of mystical 

 idealism. 



* Thus, instead of beginning with 1, 1, let us begin 1, 7. The summation-series 

 is then 1, 7, 8, 15, 23, 38, 61, 99, 160, 259, ..., etc.; and 99/160=0-618... and 

 259/160= 1-619...; and so on. But after all, the old Fibonacci numbers are not 

 far away. For we may write the new series in the form : 

 7 (0, 1, 1, 2, 3, 5, 8, ...) 

 + 1 (1,0, 1, 1,2,3,5,'...). 



