XIV] OR PHYLLOTAXIS 931 



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 suc- 

 cessions 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 frac- 

 tion. 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 the regular pentagon and 

 dodecahedron, the triumphs of Pythagorean or Euclidean geometry. 

 It enters (as the chord of an angle of 36°) into the thrice-isosceles 

 triangle of which we have spoken on p. 762 ; it is a number which 

 becomes (by the addition of unity) its own reciprocal — its properties 

 never end. To Kepler (as Naber tells us) it was a symbol of Creation, 

 or Generation. Its recent application to biology and art-criticism 

 by Sir Theodore Cook and others is not new. Naber' s book, already 

 quoted, is full of it. Zeising*, in 1854, found in it the key to all 



* A. Zeising, Neue Lehre von der Proportion des menschlichen Korpers aus einem 

 bisher unerkannt gebliebenen die game Natur und Kunst durchdringenden morpho- 

 logischen Grundgesetze entwickelt, Leipzig, 1854, 457 pp.; ibid. Deutsche Viertel- 

 jahrsschrift, 1868, p. 261; also, posthumously, Der Goldene Schnitt, Leipzig, 1884, 

 24 pp. Cf. S. Gunther, Adolph Zeising als Mathematiker, Ztschr. f. Math. u. 

 Physik. {Hist. Lit. Abth.), xxi, pp. 157-165, 1876; also F. X. PfeiflFer, Die Propor- 

 tionen des goldenen Schnittes an den Blattern u. Stengelen der Pflanzen, Ztschr. 

 f. math. u. naturw. Unterricht, xv, pp. 325-338, 1885. For other references, see 

 R. C. Archibald, op. cit. Among modern books on similar lines, the following 

 are curious, interesting and beautiful (whether we agree with them or not): Jay 

 Hambidge, Dynamic Symmetry, Yale, 1920; C. Arthur Coan, Nature's Harmonic 

 Unity, New York, 1912. 



