ON THE THEORY OF VEGETABLE SPIRALS. 363 



same relation to the greater, that the greater does to the whole ; 

 thus when a system of 8 leaves is divided into 5 and 3, as the 

 first contains the second once, so does the second the third; 

 moreover the second is opposed in direction to the first, and so 

 similarly is the third to the second. Or we might say, though 

 it is dangerous to use mathematical terms except with mathe- 

 matical precision, that every natural spiral can be divided in 

 extreme and mean ratio. 



Thus much respecting division Iby azimuths. We now come 

 to consider horizontal division, that is, division in which all 

 above and below a given horizontal circle is successively sup- 

 posed to be obliterated, so as to leave in the first instance only 

 the lower part of the spiral and in the second only the higher. 

 It may be easily shown but as before I shall not stop to give 

 the mathematical demonstration that if JVand 1) are respectively 

 the numerator and denominator of the angle of divergence of any 

 spiral, and N' and D' respectively are the numerator and deno- 



N 

 minator of the last converging fraction to _ ; then D' and D D' 



are the number of leaves in the two portions, N' and NN' 

 respectively being the corresponding number of times that the 

 spiral goes round the axis. Thus recurring to the former ex- 

 ample, the spiral whose angle of divergence is ^ is resoluble 

 into two whose fractions respectively are | and f . In figure 1 

 the line AA indicates this division, there being 3 leaves above 

 and 8 below. Of course the division might have been made so 

 as to leave 3 rows below and 8 at top. But this is scarcely 

 worth remarking ; it may be more needful to point out that in 

 examining the figure we are not, as before, to be guided by 

 the number of cells, but must look at each set of leaves as we 

 would at a spiral existing in nature. It will be observed that 

 there is a deviation from the perpendicular in both parts of the 

 figure ; a retardation, so to speak, in the one and an acceleration 

 in the other, and that these, on the whole, balance. As in the 

 former mode of division there was a certain amount of vertical 

 displacement, so here there is of horizontal. 



Let us now observe the prcerogativa which, with reference to 

 this mode of division, belongs to the natural systems. It is this, 

 that whether divided vertically or horizontally the result is the 

 same. Not so with the case of the figure; 11 was before divided 



