342 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



Next, as a matter of observation also, the ratio of the con- 

 struction curves must not show any great inequality; on the 

 contrary, a very general approximation of equality in the numbers 

 of the curves appears to be the general rule. The rule appears to 

 be that one number must not be more than double the other : this 

 being again the generalisation of Schimper and Braun, which places 

 the ratio 1 : 2 as the limit. The highest range of this type of 

 ratio has been recorded as (3 : 6) for a trijugate plant oiDipsacus* 

 Hence the choice of higher plants is really restricted in the great 

 majority of cases to such combinations as — 



1 : 1 



2 : 1, 2 : 2, 2 : 3, 2 : 4 



3 : 3, 3 : 4, 3 : 5, 3 : 6, etc., 



these being the only low combinations possible. Taking these 

 nine ratios, it will be observed that three are cases of true 

 symmetry, three are Fibonacci pairs, while the (1 : 1) may also 

 be regarded as in the Fibonacci series; the (3 : 4) is the 

 commonest anomalous ratio, and the (2 : 4) the common " bijugate " 

 one. Taking only these simple expressions, then, the balance 

 of construction is in favour of the Fibonacci series, which when 

 once laid down lead on naturally to higher expansion derivatives 

 of the system, which follow with mathematical precision as con- 

 sequences of the properties of systems of intersecting spiral curves. 

 A predominance of Fibonacci ratios, so far as asymmetrical phyllo- 

 taxis is alone concerned, would thus be expected to obtain; and 

 this quite apart from any possible biological utility of the series or 

 of a spiral distribution or building mechanism, prejudices in favour 



* There is a suggestion that other ratios occurred in lower types : a wider 

 range of ratio, e.g. 1 : 3, occurs in Mosses, as also in the apical cell of the Fern ; 

 ratios of 1 : 4 also in Florideae. These and a few isolated cases (ef. 

 Gheirostrobus, Scott) require to he taken separately : the general standpoint 

 obviously being that all mathematical possibilities should he equally expected 

 to occur, and the fact that certain types obtain in present vegetation rather 

 than others may indicate the gradual eflfect of natural selection on the con- 

 struction mechanism, the general trend appearing to be, as already indicated, 

 tows^rds either symmetry or ratios of the Fibonacci aeries, 



