OfiGANS OF NTTEinON. 137 



ments in which more leaves were necessary for the purpose. 

 His ideas were but little attended to at the time ; hut of late 

 years by the researches of Schimper, Braun, Bravais, and others, 

 his views have been not only confirmed but considerably ex- 

 tended, and it has been shown that the spiral arrangement is 

 not only universal, but that the laws which regulate it may be 

 reduced to mathematical precision, the formulse representing the 

 relative position of leaves in different plants var}*ing, but being 

 always constant for the same species. The examination of 

 these laws any further than to show that the regular arrange- 

 ment of leaves and their modifications is in the form of a spiral 

 around the stem, having at present no practical bearing in 

 Botany, however interesting they may be in a mathematical 

 point of view, would be out of place here ; we shall confine our- 

 selves therefore to the general discussion of the subject, and as 

 alternate leaves are those which will enable us to do so with 

 most facility, we shall allude to them first. 



Alternate Leaves. — If we refer again to the arrangement of 

 the leaves in the Cherry or Apple, we shall find that before we 

 arrive at the sixth leaf {fig. 263), which is over the first, the 

 string or line used to connect the base of the leaves will have 

 passed twice round the circumference. The point where a leaf 

 is thus found, which is placed in a straight line, or perpendi- 

 cularly over the first, shows the completion of a series or cycle, 

 and thus in the Cherry and Apple the cycle consists of five 

 leaves. As the five leaves are equidistant from each other, and 

 as the line which connects them passes twice round the stem, 

 the distance of one leaf from the other will be | of its circum- 

 ference. The fraction |, therefore, is the angular divergence, or 

 size of the arc interposed between the insertion of two succes- 

 sive leaves, or their distance from each other expressed in parts 

 of the circumference of the circle, or 360°-^|= 144° ; the nu- 

 merator indicates the number of turns made in completing the 

 cycle, and the denominator the number of leaves contained in 

 it. The successive leaves as they are produced on the stem, 

 as we have seen, are also arranged in similar cycles. This ar- 

 rangement of cycles of five is by far the most common in Dico- 

 tyledonous Plants. It is termed the qidncuncial , pentasticJious, or 

 five-ranked arrangcmcn t. 



A second variety of arrangement of alternate leaves is that 

 which is called distichous or two-ranked. Here the second 

 leaf is above and directly opposite to the first {fig. 264), and 

 the third being in like manner opposite to the second, it is 

 placed vertically over the first, and thus completes the cycle, 

 which here consists of but two leaves ; the fourth leaf again is 

 over the second, and the fifth over the third and first, thus com- 

 pleting a second cycle; and so on with the successive leaves. 

 Here one turn completes the spiral, so that the angular diver- 



