128 PHYLLOTAXY, OB LEAF-ARRANGEMENT. 



ranks are a necessary consequence of the regular ascending 

 arrangement of parts with equal intervals over the circumference 

 of the axis ; and, if the leaves are numbered consecutively, their 

 numbers will necessarily stand in arithmetical progression on the 

 oblique ranks, and have certain obvious relations with the pri- 

 mary spiral which originates them, as will be seen by projecting 

 them on a vertical plane. 



245. Take, for example, the f arrangement, where, as in the 

 diagram annexed to Fig. 244, the primitive spiral, written on a 

 plane surface, appears in the numbers, 1, 2, 3, 4, 5, 6, and so 

 on : the vertical ranks thus formed are necessarily the numbers 

 1-6-11 ; 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, will be continued into 7-9-11-13-15 ; and 

 also the 2-4-6-8-10 continues into 12-14, and so on, if the axis 

 be prolonged. Here the circumference is occupied by two secon- 

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

 in the sequence of numbers is two ; that is, the number of the 

 parallel secondary 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 ; 3-6-9-12, 

 continued into 15 ; and 581114, &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 they are too crowded for directly following their succes- 

 sion, and thus to ascertain the order of the primary spiral series 

 by noticing what numbers come to be superposed in the verti- 

 cal ranks. Thus, in the small cone of the American Larch 

 (Fig. 244) , which usually completes only three heights of leaves, 

 the lowest, highest, and a middle one make a vertical row 

 which faces the observer. Marking this first scale 1 , and count- 

 ing 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 that spiral 3-5-7, and so on, adding the com- 

 mon 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 com- 

 mon difference, viz. 1-4-7-10 ; then, passing to the next, on 

 which the No. 3 has already been fixed, we carry on that se- 

 quence, 6-9, &c. ; and on the third, where 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 1-6-1 1 , 

 showing that the phyllotaxy is of the five-ranked, or f order. 



