932 ON LEAF-ARRANGEMENT [ch. 



morphology, and the same writer, later on, declared it to dominate 

 both architectm^e 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 

 accommodable unto this number, were inexcusable Pythagorisme." 

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

 biological question of phyllotaxis seems to depend on the assertion, 

 .first made by Chauncey Wright*, 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 f; 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 (K), in which the first two or any two successive ones 

 divide the circumference. Now 5/8 and all successive fractions 

 differ inappreciably from Z." To this view there are many simple 

 objections. In the first place, even 5/8, or 0-625, is but a moderately 

 close approximation to the "golden mean"; and furthermore, 

 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 comparatively 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 which cannot commend itself to a plain student 

 of physical science : nor is the hypothesis rendered more acceptable 

 when Sir T. Cook qualifies it by telHng us that "all that a plant 

 can do is to vary, to make blind shots at constructions, or to 

 ' mutate ' as it is now termed : and the most suitable of these con- 

 structions will in the long run be isolated by the action of Natural 

 Selection." Thirdly, we must not suppose the Fibonacci numbers 



♦ On the uses and origin of the arrangement of leaves in plants, Mem. Amer. 

 Acadfix, p. 380, 1871, Cambridge, Mass. Cf. J. Wiesner, Ueberdie Beziehungen der 

 Stellungsverhdltnisse der Laubbldtter zur Beleuchtung, Wien, 1902. 



t This is what Ruskin spoke of as " the vacant space " ; Mod. Painters, v, chap, vi, 

 p. 44, 1860. Leonardo had in like manner explained the leaf-arrangement as 

 serving to let air pass between the leaves, keep one from overshadowing another, 

 and let rain-drops fall from the one leaf to the one below. 



