LESSON 10.] 



IN A SPIRAL OKDEK. 



73 



first, and the fourth over the second. This brings all the leaves into 

 two ranks, one on one side of the stem and one on the other ; and 

 is therefore called the two-ranked arrangement. It occurs in all 

 Grasses, — in Indian Corn, for instance ; also in the Spiderwort, the 

 Bellwort (Fig. 131) and Iris (Fig. 132), in the Basswood or Lime- 

 tree, &c. This is the simplest of all arrangements. 



186. Next to this is the three-ranked arrangement, such as ive 

 see in Sedges, and in the Veratrum or White Hellebore. The plan 

 of it is shown on a Sedge in Fig. 141, and in a diagram or cross- 

 section underneath, in Fig. 142. Here the 

 second leaf is placed one third of the way 

 rouiid the stem, the third leaf two thirds of 

 the way round, the fourth leaf accordingly 

 dii"ectly over the first, the fifth over the 

 second, and so on. That is, three leaves 

 occur in each turn round the stem, and they 

 are separated from each other by one third 

 of the circumference. 



187. The next and one of the most com- 

 mon is ihs Jive-ranked arrangement ; which 

 is seen in the Apple (Fig. 143), Cherry, 

 Poplar, and the greater part of our trees 

 and shrubs. In this case the line traced 

 from leaf to leaf will pass twice round the 

 stem before it reaches a leaf situated di- 

 rectly over any below (Fig. 144). Here 

 the sixth leaf is over the first ; the leaves 

 stand in five perpendicular ranks, equally 

 distant from each other ; and the distance 

 between any two successive leaves is just 

 two fifths of the circumference of the stem. 



188. The five-ranked arrangement Is expressed by the fraction f. 

 This fraction denotes the divergence of the successive leaves, i. e. the 

 angle they form with each other : the numerator also expresses the 

 number of turns made round the stem by the spiral line in complet- 

 ing one cycle or set of leaves, namely 5! ; and the denominator gives 

 the number of leaves in each cycle, or the number of perpendicular 



FIG. 141, Pieceof the stalk of a Sedge, with the leaves cut away, leaving their bases :' 

 tlie leaves are numbered in order, from 1 to 6. 143. Diagram or cross-section of the sam^ 

 all in one plane ; the leaves snnilarly numbered. 



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