IDEAL ANGLES. 67 



of leaf-distribution, and the tendency to a biological interpretation 

 of phenomena, is responsible for the hypothesis that a nearer and 

 nearer approach to the "ideal angle" of each series implied a 

 better distribution of leaves in relation to their external environ- 

 ment, by preventing overlapping. The suggestion that biological 

 aim on the part of the plant may to a great extent control 

 the protoplasmic mechanism of phyllotaxis cannot be wholly 

 neglected; and the formation of a "concentration-system" has 

 already been placed in such a light, although it was not necessarily 

 accepted as proved. But it cannot be too strongly insisted, that 

 in any. spiral, that is to say, any asymmetrical series, whatever 

 unequal ratios the parastichies may have, every system is equally 

 an ideal one so far as leaf-distribution is concerned, in that no two 

 leaves are ever vertically superposed within the limit of practical 

 observation and, construction, a fact which follows from mathe- 

 matical deduction and geometrical construction by log. spirals.* 

 Every asymmetrical system equally obeys Hofmeister's law, the 

 logical consequence of which is, again, that no superposition ever 

 takes place. The whole theory of an ascending series reaching to a 

 perfect type of leaf-distribution thus falls to the ground ; and not 

 only so, but the symmetrical condition, which has been put forward 

 as possibly the true aim of the plant, implies an actual formation of 

 vertically superposed series of members, and therefore, according to 

 the original hypothesis of Bonnet, an immediate departure from 

 the maximum exposure. Nor is there any reason to doubt that 

 biological causes may induce such a result, when the maximum ex- 

 posure ceases to be the optimum ; the remarkable production of a 

 decussate phyllotaxis in the assimilating shoots alone of types 

 which show other xerophytic adaptations being the most obvious 

 example.-|- 



Wiesner,! who approached the subject from this very standpoint 

 of leaf-distribution, was led to very remarkable results. 



He pointed out that the series in which x had a minimum value 



* Gf. note on Mathematdcal Orthostiehies. 



t Gf. Clematis, Labiatae, Euphorbia Lathyris, Jasminvm nudiflorum, 

 Crassula perfoliata. 

 1;. Flora, 1875, tNos. 8£and 9. 



