THEIR ARRANGEMENT. 



145 



nodes are very short, so that the leaves touch one another, or 

 nearly so, we may readily perceive what leaves are superposed ; 

 but it is then difficult to follow the succession of the intermediate 

 leaves. When this cannot be directly done, however, the order 

 may be deduced by simple processes. 



242. Sometimes we can readily count the number of vertical 

 ranks, which gives the denominator of the fraction sought. Thus, 

 if there are eight, we refer the case to the $ arrangement in the 

 regular series ; if there are thirteen, to the T 5 arrangement, and 

 so on. 



243. Commonly, however, when the leaves are crowded, the 

 vertical ranks are by no means so manifest as two or more orders 

 of oblique series, or secondary spirals^ which are at once seen to 

 wind round the axis in opposite directions, as in the Houseleek 

 (Fig. 174 ; where the numbers 1, 6, 11 belong to a spire that winds 

 to the left, 1,9, 17 to another which winds to the right, and 3, 6, 

 9, 12 to still another that winds in the same direction) : they are 

 still more obvious in Pine-cones (Fig. 175, 176). These oblique 

 spiral 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, these 

 numbers will necessarily stand in arithmetical progression on the 

 oblique ranks, and have certain obvious relations with the primary 

 spiral which originates them ; as will be seen by projecting them 

 on a vertical plane. 



244. Take, for example, the quincuncial (|) arrangement, 

 where, as in the annexed diagram, the ascend- 

 ing spiral, as written on a plane surface, ap- 

 pears in the numbers 1, 2, 3, 4, 5, 6, and so 



on : 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. 175, where the lowest scale, 1, is 

 turned directly towards the observer), 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, ascending to the left ; 

 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, 



FIG. 175. A cone of the small- fruited American Larch (Larix microcarpa), with the scales 

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



13 



