146 THE LEAVES. 



which runs into 12, 14, and so on, if the axis Tbe 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 secondary 

 spirals is the same as the common difference of the numbers on 

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

 ondary 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 superposed in the vertical ranks. We take, 

 for example, the very simple cone of the small-fruited American 

 Larch (Fig. 175), which usually completes only two cycles, for 

 we see that the lowest, one intermediate, and the highest scale, on 

 the side towards the observer, stand in a vertical row. Marking 

 this lowest 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 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, we find three of 

 these, and therefore proceed to number the lowest one by adding 

 this common difference, viz. 1, 4, 7, 10; then, passing to the one 

 next to it, on which the number 3 has already been fixed, we carry 

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

 ready fixed, we continue the numbering, 8, 11, &c. This gives us, 

 in the vertical rank to which No. 1 belongs, the sequence 1, 6, 11, 

 showing that the arrangement is of the quincunciai (f ) order. It 

 is further noticeable that the smaller number of parallel secondary 

 spirals, 2, agrees with the numerator of the fraction in this the f 

 arrangement ; and that this number added to that of the parallel 

 secondary spirals which wind in the opposite direction, viz. 3, 

 gives the denominator of the fraction. This holds good through- 

 out, so that we have only to count the number of parallel second- 

 ary spirals in the two directions, and assume the smaller number 

 as the numerator, and the sum of this and the larger number as the 

 denominator, of the fraction which expresses the angular diver- 

 gence sought. For this we must take, however, the order of sec- 



