138 



THE LKAVES. 



15 



13 



12 



10 



ascending to the left 



the vertical ranks thus formed, beginning with the lowest (which 



we place in the middle- column, that it 

 may correspond with the Larch-cone, Fig. 

 208, where the lowest scale, 1, is turned 

 directly towards the observer), are neces- 

 sarily the numbers 1, 6, 11 ; 4, 9, 14 ; 2, 7, 

 12 ; 5, 10, 15 ; and 3, 8, 13. But two 

 pai'allel oblique ranks are equally apparent, 

 viz. 1, 3, 5, which, if we coil the diagram 

 round a cylinder, will be continued into 7, 9, 11, 13, 15 ; and also 

 2, 4, 6, 8, 10, which runs into 12, 14, and so on, if the axis be further 

 prolonged. Here the circumference is occupied by two secondary 

 left-hand series, and we notice that the common difference in the 

 sequence of numbers is two : that is, the number of the parallel sec- 

 ondary spirals is the same as the common (hfference of the numbers 

 on the leaves that compose them. Again, there are other parallel 

 secondary spiral ranks, three in number, which ascend to the right ; 

 viz. 1, 4, 7, continued into 10, 13 ; 3, 6, 9, 12, continued into 15 ; 

 and 5, 8, 11, 14, &c. ; where again the common difference, 3, accords 

 with the number of such ranks. This fixed relation enables us to 

 lay down the proper numbers on the leaves, when too crowded for 

 directly following their succession, and thus to ascertain the order of 

 the primary spiral series by noticing what numbers come to be super- 

 posed in the vertical ranks. We take, for example, the very simple 

 cone of the small-fruited American Larch (Fig. 208), which usually 

 completes only two cycles ; for we see that the lowest, one interme- 

 diate, and the highest scale, on the side towards the observer, stand 

 in a vertical row. Marking this lowest scale 1, and counting the 

 parallel secondary spirals that wind to the left, we find that two 

 occupy the whole circumference. From 1, we number on the scales 

 of tliat spiral 3, 5, 7, and so on, adding the common difference 2, at 

 each step. Again, counting from the base the right-hand secondary 

 spirals, we find three of them, and therefore proceed to number the 

 lowest one by adding this common difference, viz. 1, 4, 7, 10 ; then, pass- 

 ing to the next, on which the No. 3 has already been fixed, we carry 

 on that sequence, 6, 9, &c. ; and on the third, Avhere No. 5 is already 

 fixed, we continue the numbering, 8, 11, «&;c. This gives us, in the 

 vertical rank to which No. 1 belongs, the sequence 1, G, 11, showing 



FIG. 203. A cone of the small-fruited American Larch (Larix Americana), with the scales 

 numbered, exhibiting the five-ranked arrangement, as iu the annexed diagram. 



