650 ON LEAF-ARRANGEMENT [ch. 



into the thrice-isosceles triangle of which we have spoken on 

 p. 511 ; 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 morphology, 

 and the same writer, later on*, declared it to dominate both archi- 

 tecture and music. But indeed, to use Sir Thomas Browne's 

 words (though it was of another number that he spoke) : " To 

 enlarge this contemplation into all the mysteries and secrets ac- 

 commodable unto this number, were inexcusable Pythagorisme." 

 If this number has any serious claim at all to enter into the 

 biological question of phyllotaxis, this must depend on the fact, 

 first emphasized by Chauncey Wright f, that, if the successive 

 leaves of the fundamental spiral be placed at . the particular 

 azimuth which divides the circle in this "sectio aurea," then no 

 two leaves will ever be superposed ; and thus we are said to have 

 " the most thorough and rapid distribution of the leaves round the 

 stem, each new or higher leaf falling over the angular space 

 between the two older ones which are nearest in direction, so as 

 to divide it in the same ratio (/i), in which the first two or any 

 two successive ones divide the circumference. Now o/S and all 

 successive fractions differ inappreciably from /i." To this view 

 there are many simple objections. In the first place, even 5/8, 

 or -625, is but a moderately close approximation to the "golden 

 mean" ; in the second place the arrangements by which a better 

 approximation is got, such as 8/13, 13/21, and the very close 

 approximations such as 34/55, 55/89, 89/144, etc., are compara- 

 tively rare, while the much less close approximations of 3/5 or 

 2/3, or even 1/2, are extremely common. Again, the general 

 type of argument such as that which asserts that the plant is 

 "aiming at" something which we may call an "ideal angle" is 

 one that cannot commend itself to a plain student of physical 

 science: nor is the hypothesis rendered more acceptable^ when 

 Sir T. Cook qualifies it by telling us that " all that a plant can do 



* Deutsche Vierteljahrsschrift, p. 261, 1868. 

 t Memoirs of Amer. Acad, ix, p. 389. 



