THEIR ARRANGEMENT. 



143 



leaf is separated from the preceding and succeeding by one third 

 of the circumference, there are three leaves in one turn, or cycle, 

 and the fourth commences a second cycle, which goes on in the 

 same way. That is, the angular divergence, or size of the arc in- 

 terposed between the insertion of two successive leaves, in the first 

 is ^-, in the second ^-, of the circle. These fractions severally rep- 

 resent, not only the angle of divergence, but the whole plan in 

 these two modes ; the numerator denoting the number of times the 

 spiral line winds round the stem before it brings a leaf directly 

 over the one it began with ; while the denominator expresses the 

 number of leaves that are laid down in this course, or which form 

 each cycle. The two-ranked mode () is evidently the simplest 

 possible case. The three-ranked () is the next, and the one in 

 which the spiral character of the arrangement m 



begins to be evident. It is further illustrated in 

 the next, namely, 



239. The pentastichous, quincuncial, or jive- 

 ranked arrangement (Fig. 172). This is much 

 the most common case in alternate-leaved Dico- 

 tyledonous plants. The Apple, Cherry, and Pop- 

 lar afford ready examples of it. Here there 

 are five leaves in each cycle, since we must 

 pass on to the sixth before we find one placed 

 vertically over the first. To reach this, the as- 

 cending spiral line has made two revolutions 

 round the stem, and on it the five leaves are 

 equably distributed, at intervals of f of the cir- 

 cumference. The fraction f accordingly ex- 

 presses the angular divergence of the successive 

 leaves ; the numerator indicates the number of 

 turns made in completing the cycle, and the de- 

 nominator gives the number of leaves in the 

 cycle, or the number of vertical ranks of leaves 

 on such a stem. If we shorten the axis, as it was in the bud, or 



FIG. 172. Diagram of the five-ranked arrangement of the leaves, as in the Apple-tree ; a spi- 

 ral line is drawn ascending the stem and passing through the successive scars which mark 

 the position of the leaves from 1 to 6. It is made a dotted line where it passes on the opposite 

 side of the stem, and the scars 2 and 5, which come on that side, are made fainter. 173. A 

 plane, horizontal projection of the same; the dotted line passing from the edge of the first leaf 

 to the second, and so on to the fifth leaf, which completes the cycle ; as the sixth would come 

 over, or within the first. 



