102 



or 4 leaves ; and these whorls cross one another at a constant 

 angle of divergence from one extremity to the other of the 

 aggregation. 



3. In the greater number of plants with alternate inser- 

 tions the divergence of the primitive spire is an irrational 

 angle equal to 137 30' 28", which is but the small segment 

 of the circumference divided into mean and extreme propor- 

 tion : this angle corresponds to the series 1, 2, 3, 5, 8, 13, &c. 



4. Other arrangements may occur, but much more seldom, 

 in which the divergence, still irrational, may equal 99 30' 6", 



and correspond to the series 1, 3, 4, 7, 11 ; 77 57' 19", 



and correspond to the series 1, 4, 5, 9 ; 151 8' 8", 



corresponding to the series 2, 5, 7> 12, &c. 



5. The constancy of each of these angles is not affected, at 

 least in its mean value, by the inequality in height of the suc- 

 ceeding internodes and other local disturbing causes. 



6. The insertions may present false whorls when they are 

 grouped in twos or threes at the same height of the stem. 



7. The primitive spiral is prolonged down to the subter- 

 ranean stem, retaining the same invariable divergence, and 

 sometimes even to the organs of the flower. 



8. The direction of the spiral generally appears indifferent 

 on the central stem and on the branch ; it seems to be in no 

 certain relation to the direction of the convolution of the 

 voluble stems or to the torsion of the fibres on themselves ; 

 but on the branch this direction is determined by the posi- 

 tion of the first leaf to the right or to the left of the primitive 

 leaf. 



9. All the series above cited may give rise to bijugate sy- 

 stems more or less frequently, and which will most frequently 

 be present on plants with opposite stem leaves ; the ordinary 

 system may also give rise to corresponding trijugate aggrega- 

 tions. 



10. The phenomenon of the convergence of two spirals 

 into one must be explained by the partial abortion of one of 

 these spirals, or by the cohesion of the two into one ; a whole 

 series may also suffer complete abortion ; and this remark ren- 

 ders the existence of most of the recurrent series, which do not 

 appear to belong to the scries above enumerated, very doubt- 

 ful. 



