XIV] OE PHYLLOTAXIS 649 



result is to give us arrangements corresponding to the middle 

 diagrams in Fig. 327, which are the configurations in which the 

 quadrilateral outlines approach most nearly to a rectangular 

 form, and give us accordingly the least possible ratio (under the 

 given conditions) of sectional boundary-wall to surface area. 



The manner in which one system of spirals may be caused to 

 slide, so to speak, into another, has been ingeniously demonstrated 

 by Schwendener on a mechanical model, consisting essentially 

 of a framework which can be opened or closed to correspond 

 with one after another of the above series of diagrams*. 



The determination of the precise angle of divergence of two 

 consecutive leaves of the generating spiral does not enter into the 

 above general investigation (though Tait gives, in the same paper, 

 a method by which it may be easily determined) ; and the very fact 

 that it does not so enter shews it to be essentially unimportant. 

 The determination of so-called "orthostichies," or precisely 

 vertical successions of leaves, is also unimportant. We have no 

 means, other than observation, of determining that one leaf is 

 vertically above another, and spiral series such as we have been 

 dealing with will appear, whether such orthostichies exist, whether 

 they be near or remote, or whether the angle of divergence be 

 such that no precise vertical superposition ever occurs. And 

 lastly, the fact that the successional numbers, expressed as 

 fractions, 1/2, 2/3, 3/5, represent a convergent series, whose final 

 term is equal to 0-61803..., the sectio aurea or "golden mean" of 

 unity, is seen to be a mathematical coincidence, devoid of 

 biological significance ; it is but a particular case of Lagrange's 

 theorem that the roots of every numerical equation of the second 

 degree can be expressed by a periodic continued fraction. The 

 same number has a multitude of curious arithmetical properties. 

 It is the final term of all similar series to that with which we have 

 been dealing, , such for instance as 1/3, 3/4, 4/7, etc., or 1/4, 4/5, 

 5/9, etc. It is a number beloved of the circle-squarer, and of all 

 those who seek to find, and then to penetrate, the secrets of the 

 Great ^ Pyramid. It is deep-set in Pythagorean as well as in 

 Euclidean geometry. It enters (as the chord of an angle of 36°), 



* A common form of pail-shaped waste-paper basket, with wide rhomboidal 

 meshes of cane, is well-nigh as good a model as is required. 



