XIV] OR PHYLLOTAXIS 923 



Norway spruce, Beal* found 92 per cent, in which the spirals were 

 in five and eight rows ; in 6 per cent, the rows were four and seven, 

 and in 4 per cent, they were four and six. In each case they were 

 nearly equally divided as regards direction ; for instance, of the 467 

 cones shewing the five-eight arrangement, the five-series ran in 

 right-handed spirals in 224 cases, and in left-handed spirals in 243. 

 Omitting the "abnormal" cases, such as we have seen to occur 

 in a small percentage of our cones of the spruce, the arrangements 

 which we have just mentioned may be set forth as follows (the 

 fractional number used being simply an abbreviated symbol for the 

 number of associated helices or parastichies which we can count 

 running in the opposite directions): 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 

 34/55, 55/89, 89/144. Now these numbers form a very interesting 

 series, which happens to have a number of curious mathematical 

 properties "f. We see, for instance, that the denominator of each 



* Amer. Naturalist, vn, p. 449, 1873. 



t This celebrated series corresponds to the continued fraction 1+1 etc., 



1 + 

 and converges to 1-618..., the numerical equivalent of the sectio divina, or 

 ''Golden Mean." The series of numbers, 1, 1, 2, 3, 5, 8, ..., of which each is the 

 sum of the preceding two, was used by Leonardo of Pisa (c. 1170-1250), nicknamed 

 Fi Bonacci, or films bonassi, in his Liber Abbaci, a work dedicated to magister 

 mens, summus philosophus, Michael Scot {Scritti, i, pp. 283-284, 1857). This learned 

 man was educated in Morocco, where his father was clerk or dragoman to Pisan 

 merchants; and he is said to have been the first to bring the Arabic numerals, or 

 " novem figurae Indorum,'' into Europe. The Fibonacci numbers were first so-caUed 

 by Eduard Lucas, Bollettino di Bibliogr. e Storia dei Sci. Matem. e Fis. x, p. 129, 1877. 

 The general expression for the series 



V^Y (\-V5\ 



^^{(^T-C~)"l' 



was known to Euler and to Daniel Bernoulli {Coram. Acad. Sci. Imp. Petropol. 

 1732, p. 90), and was rediscovered by Binet, C.E. xviii, p. 563, 1843; xix, p. 939, 

 1844) and by Lame, ibid, xix, p. 867, after whom it is sometimes called Lame's 

 series. But the Greeks were familiar with the series 2, 3: 5, 7 : 12, 17, etc.; which 

 converges to V2, as the other does to the Golden Mean ; and so closely related are 

 the two series, that it seems impossible that the Greeks could have known the one 

 and remained ignorant of the other. (See a paper of mine, on "Excess and Defect, 

 etc.," in Mind, xxxviii. No. 149, 1928.) 



The Fibonacci (or Lame) series was well known to Kepler, who, in his paper 

 De nive sexangula (1611, cf. supra, p. 695), discussed it in connection with the form 

 of the dodecahedron and icosahedron, and with the ternary or quinary symmetry of 

 the flower. (Cf. F. Ludwig, Kepler iiber das Vorkommen der Fibonaccireihe im 

 Pflanzenreich, Bot. Centralbl. lxviii, p. 7, 1896.) Professor William Allman, 

 Professor of Botany in Dublin (father of the historian of Greek geometry). 



