XIV] OK PHYLLOTAXIS 643 



cones of the 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 denomi- 

 nator of each fraction is the numerator of the next ; and further, 

 that each successive numerator, or denominator, is the sum of 

 the preceding two. Our immediate problem, then, is to determine, 

 if possible, how these numerical coincidences come about, and 

 why these particular numbers should be so commonly met with 



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



1 



j This celebrated series, which appears in the continued fraction 1 -i- 1 etc. 



1 + 

 and is closely connected with the Sectio au.rea or Golden Mean, is commonly called 

 the Fibonacci series, after a very learned twelfth century arithmetician (known also 

 as Leonardo of Pisa), who has some claims to be considered the introducer of 

 Arabic numerals into cluristian Europe. It is called Lami's series by some, after 

 Father Bernard Lami, a contemporary of Newton's, and one of the co-discoverers 

 of the parallelogram of forces. It was well-known to Kepler, who, in his paper 

 De nive se.rangula (cf. supra, p. 480), discussed it in connection with the form of 

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

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

 Pflanzenreich, Bot. Centralhl. lxviii, p. 7, 1896). Professor William AUman, 

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

 speculating on the same facts, put forward the curious suggestion that the cellular 

 tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra, 

 and that of the monocotyledons or endogens of icosahedra [On the mathematical 

 connexion between the parts of Vegetables: abstract of a Memoir read before the 

 Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, 

 Organogenic ve'ge'tale, i, p. 534). 



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