THEIR AEKANGEMENT. 135 



202) and some other Monocotyledonous plants. Taking any leaf we 

 please to begin with, an(i numbering it 1, we pass round one third of 

 the circumference of the stem as we ascend to leaf No. 2 ; another 

 third of the circumference brings us to No. 3 ; another brings us 

 round to a point exactly over No 1, and here No. 4 is placed. No. 5 

 is in like manner over No. 2, and so on. They stand, therefore, in 

 three vertical rows, one of which contains the numbers 1, 4, 7, 10 ; 

 another, 2, 5, 8, 11 ; the third, 3, 6, 9, 12, and so on. If we draw a 

 line from the insertion of one leaf to that of the next, and so on to 

 the third, fourth, and the rest in succession, it will be perceived that 

 it winds around the stem spirally as it ascends. In the first or dis- 

 tichous mode, the second leaf is separated from the preceding by half 

 the circumference of the stem ; and, having completed one turn 

 round the stem, the third begins a second turn. In the tristichous^ 

 each 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 arc interposed 

 between the insertion of two successive leaves, in the first is J, in the 

 second ^, of the circle. These fractions severally represent, not 

 only the angle of divergence, but the whole plan of 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 tlu-ee- 

 ranked (J) is the next, and the one in which the spiral character of 

 the arrangement begins to be evident. To this succeeds 



240. The pentastichous, quincunctal, or jive-ranked arrangement 

 (Fig. 204, 205). This is much the most common case in alternate- 

 leaved Dicotyledonous plants. The Apple, Cherry, and Poplar 

 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 ascending spiral line has 

 made two revolutions round the stem, and on it the five leaves are 

 equably distributed, at intervals of f of the circumference. The 

 fraction | accordingly expresses the angular divergence of the suc- 

 cessive leaves ; the numerator indicates the number of turns made 

 in completing the cycle, and the denominator gives the number of 

 leaves in the cycle, or the number of vertical ranks of leaves on such 



