122 BOTANICAL GAZETTE. 



will render the exact order of superposition uncertain. And cases no 

 doubt occur in which each leaf stands alone in its own vertical rank 

 and the arrangement is what is termed curviserial. Such a cycle may 

 be marked oo- co ; the angle of divergence is such that it is not pos- 

 sible to divide the circumference by it, and hence no leaf can be 

 placed exactly above another. 



12. Nevertheless, we are enabled to mark the succession of the 

 leaves and thus to determine the order of their arrangement, by 

 means of the secondary spirals. In every cycle after the first two, 

 there are one or more orders of parallel secondary spirals. These 

 pursue an oblique direction between the primary genet ating spiral 

 and the vertical ranks to which it gives rise. If we mark the alter- 

 nate leaves of any axis in the order of their succession, beginning 

 with the lowest, by the series of natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 

 9, 10, &c., we may exhibit the primary and secondary spirals and the 

 vertical ranks of the different cycles as follows : 



a. The 0-1 cycle. (Fig. 1.) 1 vertical rank, 1, 2, ' "j 

 3, 4, 5, &c. 1 



b. The 1-2 cycle. (Fig. 2.) 1 spiral of the 1st 3 

 order, 1, 2, 3, 4, 5, &c.; 2 vertical ranks, 1, 3, 5, 7, 2 

 9, &c., and 2, 4, 6, 8, 10, &c. i 



c. The 1-3 cycle. (Fig. 3.) 1 spiral of the 1st Fig, i. 



order, 1, 2, 3, 4, 5, &c.; 2 spirals of the 2d order, 1, ] \ 



3, 5, 7, 9, &c., and 2, 4, 6, 8, 10, &c.; 3 vertical i ] 



ranks, 1, 4, 7, 10, 13, &c., 2, 5, 8, 11, 14, &c., and 3, i 1 



6, 9, 12, 15, (fee. 11 



d. The 2-5 cycle. (Fig. 4.) 1 spiral of the 1st ^ ^ o ' 



order, 1,2, 3, 4, 5, &c.; 2 spirals of the 2d order, 1, > ^^ — i 



3, 5, 7, 9, &c., and 2, 4, 6, 8, 10, &c.; 3 spirals of the i i i 



3d order, 1, 4, 7, 10, 13, cfec, 2, 5, 8, 11, 14, &c., and | | | 



3, 6, 9, 12, 15, &c.; 5 vertical ranks, 1, 6, 11, 16, 21, 1 f | 



&., 2, 7, 12, 17, 22, &c., 3, 8,13, 18, 23, &c., 4, 9,14, ? j f 



19, 24, &c., and 5, 10, 15, 20, 25, &c. ' Fig. 3. 



e. The following table exhibits the successive 1 ! ^^ 1 I 



orders of spirals in each cycle as far as the 13-34 i is j i 12 



cycle. The signs f and - indicate the direction of '•,-, i V A 1 



the spirals to the right or left of the vertical line, i ^ j j 7 



as compared with the direction of the primary gen- ^ I | 4 j 



erating sjiiral. I ? 1 1 ^ 



j 1 



13. From a consideration of the origin and de- ' Fig. 4. 



