THEIR ARRANGEMENT. 137 



tions of the arrangement of alternate leaves that actually occur. 

 These higher forms are the most common where the leaves are 

 crowded on the stem, as in the rosettes of the Houseleek (Fig 207), 

 and the scales of the Pine-cones (for the ar- 

 rangement extends to all parts that are modifi- 

 cations of leaves), or where they are numerous 

 and small in proportion to the circumference of 

 the stem, as the leaves of Firs, &c. In fact, 

 when the internodes are long and the base of 

 the leaves large in proportion to the size of the 

 stem, it is difficult, and often impossible, to tell 

 whether the 9th, 14th, or 22d leaf stands ex- =»^ 



actly over the first. And when the internodes are very short, so 

 that the leaves touch one another, or nearly so, we may readily per- 

 ceive what leaves are superposed ; but it is then difiicult to follow 

 the succession of the intermediate leaves. The order, however, may 

 be deduced by simple processes. 



243. 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 f arrangement in the regu- 

 lar series ; if there are thirteen, to the j^j arrangement, and so on. 

 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. 207 ; 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 ob- 

 vious in Pine-cones (Fig. 208, 209). 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 wiU neces- 

 sarily stand in arithmetical progression on the obhque ranks, and 

 have certain obvious relations with the primary spiral which origi- 

 nates them ; as will be seen by projecting them on a vertical plane. 



244. Take, for example, the quincunical (f ) arrangement, where, 

 as in the annexed diagram, the ascending spiral, as written on a 

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



FIO. 207. An offset of the Houseleek, with the rosette of leaves unexpanded, exhibiting the 

 5-13 arrangement ; the fourteenth leaf being directly over the first. 



12* 



