ORGANS OF XUTRITIOy. 139 



11th over the 6th and 1st, so that in all cases when the sixth 

 leaf was reached including the one started from, a straight Hne 

 might be drawn from below upwards to it, and that consequently 

 there were five leaves thus necessary to complete the arrange- 

 ment. Bonnet also discovered other more complicated aiTange- 

 ments in which more leaves were necessary for the purpose. 

 His ideas were but little attended to at the time; recently 

 however by the researches of Schimper, Braun, Bravais, and 

 others, his views have been not only confirmed but consi- 

 derably extended, and it has been shown that the spiral ar- 

 rangement is not only universal, but that the laws which regu- 

 late it may be reduced to mathematical precision, the formulae 

 representing the relative position in difll^erent plants varying, but 

 being always constant for the same species. The examination 

 of these laws any further than to show that the regular an-ange- 

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

 round the stem, having at present no pi'actical bearing in Botany, 

 however interesting they may be in a mathematical point of view, 

 Avould be out of place here; we shall confine ourselves therefore 

 to the general discussion of the subject, and as alteraate 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 (^^r. 266), Avhich is over the fix'st, the 

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

 passed Uxice. round the circumference. The point where a leaf 

 is found which is placed in a straight line, or perpendicularly 

 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 dis- 

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

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

 size of the arc interposed betAveen 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 

 numerator 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 Ave have seen, ai'e 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 quincuncial, pentustichous, 

 or Jive-ranked arrangement. 



A second A-ariety of arrangement of alternate lea^-es is that 

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

 is above and directly opposite to the first (Jig. 267), and the 

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



