138 



THE LKAVKS. 



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, G, ] 1 ; 4, 9, 14 ; 2, 7, 

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

 parallel 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 difference 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, G, 9, 12, continued into 15 ; 

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

 Avith 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 Ave 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 Avind to the left, Ave find that two 

 occupy the Avhole circumference. From 1, Ave number on the scales 

 of that 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, Ave 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 Avhich the No. 3 has already been fixed, Ave carry 

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

 fixed, Ave continue the numbering, 8, 11, &c. This gives us, in the 

 vertical rank to Avhich No. 1 belongs, the sequence 1, 6, 11, showing 



FIG. 208 A rone of the small-fruited American Larch (T.arix Americana), with the scales 

 numbered, exhibiting tlie five-ranked arrangement, as ia the annexed diugram. 



