THE PRINCIPLE OF NUMBER. S7 



less, it occurs as a " definite " namber in several instances. 

 The 12 stamens of Lythrum are, of course, two series of six 

 each. Both 12 and 24 are found in the Grassulaceoe, as in 

 Sempervivum, in which genus the petals vary from 6 to 20, 

 and the stamens from 12 to 40. This seems to show that in 

 the one case they are combinations of cycles of threes, in the 

 other, of fives ; just as Berberis illustrates the former, Chimo- 

 nanihus the latter instance. 



Indefinite Whorls. — As soon as we pass from twelve to 

 some higher number, then flowers cease to be whorled, and 

 the parts are arranged spirally, and follow more or less 

 exactly the laws of alternate phyllotaxis ; interferences occur 

 in consequence of the want of space, some secondary spirals 

 being often incomplete. Moreover, since the fibrovascular 

 cords become fused, in other words branch by chorisis, and 

 are not independent as of ordinary foliage, parts take up 

 slightly different positions to what they would if they could 

 strictly follow phyllotactical laws. 



I have alluded to what I call " symmetrical increase and 

 decrease " as causes of variation in the number of parts of 

 whorls ; and what brings about these variations in number, 

 is an excess or deficiency of nutriment and vital activity 

 respectively. There are innumerable examples of all the 

 above kinds of changes in number. In fact, if any one or 

 series of whorls of a flower be ^z-merous, it may become 

 n ± o^-merous, and will give rise to symmetrical increase or 

 decrease accordingly ; or again, three whorls of the same 

 flower may become n ± x, n ± y, n ± ;:r-merous ; when all 

 numerical symmetry between them will be destroyed. 



Similarly, if the parts be spirally arranged, the number 

 may vary from the prevailing one by increasing or decreasing 

 the length of the spiral, both in flowers of the same plant or 

 in different species of the same genus ; as, for example, may 



