166 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



VII Multijugate Types. 



When the type of normal asymmetrical phyllotaxis is thus com- 

 pletely isolated as consisting of systems mapped out by log. spiral 

 curves in the ratio series of Braun and Fibonacci, 2, 3, 5, 8, etc.; 

 and the type of normal symmetrical phyllotaxis is equally clearly 

 delimited as a secondary construction, physiologically independent 

 of the ratio-series, though connected with it phylogenetically, the 

 greatest interest attaches to all other phyllotaxis phenomena, which 

 though less common, may throw light on the causes which tend to 

 induce symmetry, before postulating, as a last resource, some 

 hypothetical inherent tendency in the protoplasm itself. 



These types may be included under two series : firstly, the multi- 

 jugate systems of Bravais; and secondly, systems in which the 

 parastichy ratios belong to series other than that of Braun and 

 Fibonacci, e.g., the 3, 4, 7, 11 . . . .,-4, 5, 9, 14 . . . ., or still 

 higher series. 



The term multijugate was applied by the brothers Bravais to types 

 of phyllotaxis in which the numbers expressing the parastichy 

 ratios are divisible by a common factor; so that 2 (13 + 21) = (26 

 + 42), a bijugate system; while 3 (13 + 21) = (39 + 63) would be a 

 trijugate one. 



Expressed in angular measure, there is clearly no difference 

 between such divergences and the expression ^f, and in the spiral 

 theory of Schimper there was in fact>no room for such types, 

 except as anomalous expressions of transitional whorled stages or 

 " twisted whorls " of 2, 3, etc., in which successive whorls wfere 

 neither superposed nor exactly alternating.* The simple method 

 *Gf. Wydler, Mma, 1851, p. 125. 



