THE PRINCIPLE OF NUMBIR. ^9 



parts^ and one over. If these seven whorls were broken up 

 and arranged spirally, they would be represented by ^\ ; and 

 then there would be eight coils in the cycle. The presence 

 of seven and not eight whorls is due to the fact that in 

 rearranging them, so to say, in a verticillate manner, and by 

 necessarily shifting the position of the parts, a certain por- 

 tion of the spiral line is lost in forming each whorl, as the 

 angular divergence between two parts in a whorl is 120°, 

 but on the spiral it is nearly 123° ; so that by the time the 

 twenty-first organ is arrived at, only seven circles have been 

 completed. 



Similarly, in Bumex, if we supply the theoretically lost 

 corolla, the flower would consist of twenty-one parts exactly.* 



Another and somewhat frequent origin of the number 

 three in Dicotyledons is due to what I have called sym- 

 metrical reduction : when not only the different species of a 

 genus may have the number of parts of their floral whorls 

 ranging from 5 to 4 or 3 ; but such variations may occur on 

 the same plant. Thus Butacece (following the Gen. Plant.) 

 has 34 genera with 5-merous floAvers ; 18 genera with species 

 varying from 5 to 4-merous ; 16 are 4-merous ; 3 range from 

 5 to 3-merous ; 2 from 4 to 3-merotis, and 1 is 3-merous 



Tetramerous Whorls. — That a true quaternary arrange- 

 ment is due to an opposite condition of the foliage seems 

 borne out by statistics, though quinary flowers are not at all 

 uncommon as well. Thus of Butacece there are 6 genera 

 with opposite leaves and 4-merous flowers ; 2 only with 

 5-merous, and 2 with 4-5-merous flowers. On tlie other 

 hand, there are 25 genera with alternate leaves and 5-merous 

 flowers. 



* High spirals can bo otherwise treated, as in the case of Cliimojwn- 

 thiis, where whorls of fives are made out of a spiral system of Ix (^'^^ 

 below, p. 38) . 



