104 STRUCTURAL BOTANY. [88,89. 



the alternate, including all cases with one leaf at 

 each node ; the opposite, including cases with two or 

 more leaves at each node. 



263. The character of the alternate type of leaf- 

 arrangement is sometimes represented by a spiral, 

 which was at one time supposed to be invariable. 

 This generating spiral, as it was called, is illustrated 

 by Figures 295-300. Take a straight leafy shoot or 

 stem of the Elm or Flax, or any other plant with seem- 

 ingly scattered leaves, and beginning with the lowest 

 leaf, pass a thread to the next above, thence to the 

 next in the same direction, and so on by all the 

 leaves to the top ; the thread will form a regular 

 spiral. 



264. The Elm cycle. In the strictly alternate 

 arrangement (Elm, Linden, Grasses) the spiral thread 

 makes one complete circuit and commences a new one 

 at the third leaf. The third leaf stands over the first, 

 the fourth over the second, and so on, forming two 

 vertical rows of leaves. Here (calling each complete 

 circuit a cycle) we observe, first, that this cycle is 

 composed of two leaves ; second, that the angular 

 distance between its leaves is ^ a circle (180) ; 

 third, if we express this cycle mathematically by J, 

 the numerator (1) will denote the turns or revolu- 

 tions, the denominator (2) its leaves, and the frac- 

 tion itself the angular distance between the leaves 

 (i of 360). 



265. The Alder cycle. In the Alder, Birch, Sedges, 

 etc., the cycle is not complete until the fourth leaf is 

 reached. The fourth leaf stands over the first, the 

 fifth over the second, etc., forming three vertical rows. 

 Here call the cycle % ; 1 denotes the turns, 3 the 



