THE PRINCIPLE OF NUMBER. 15 



it, as I have explained elsewhere : * thus Fig. 3 shows how 

 the varieties of imbricate aestivations are deducible from 

 the f type (a), hy shifting the edge of the 2nd member under 

 the 4th (6, " vexillary "), the 3rd under the 5th (c, " imbricate 

 proper "), and the 1st under the 3rd (d, " contorted "). 



Similarly ternary ortrimerous wdiorls are almost universal 

 amongst flowers of Monocotyledons, and the ^ type of phyllo- 

 taxis is equally common in the foliage. It has been seen 

 that the ^ type cannot be deduced from opposite leaves, and 

 consequently never occurs, as far as I know, amongst the 

 foliage of Dicotyledons. The comparatively few genera in 

 this class with ternary flowers is therefore in accordance 

 witli the views herein expressed ; and wdiere they occur, as 



Fig. 3.— Floral ^Estivations. 



in Berheris, there are special features which lead one to 

 believe they are not due to the ^ type at all, but to the 

 breaking up of a high continuous spiral into groups of 

 threes, as will be explained hereafter. 



If, however, we take a theoretical departure from a 

 single cotyledon, as occurs in Monocotyledons, then the next 

 leaf can be at either of the limiting positions of the angular 

 distances of 180° or 120°, but not less ; for if it were less 

 than 120°, there would be four leaves in any projected 

 circle, and this would immediately introduce a member of 

 the series -^-, ^, -f-, etc., as shown above. The consequence is 



* See my paper, On the Orvjin of Floral J^stivations, Trans. Liu. Soc, 

 2nd series, Botany, vol. i. p. 177. 



