THE LEAF. 29 



lowest leaf to the one next above, and continued around the 

 stem in the same direction to the successive leaves above, 

 the thread will be found to take a spiral course; thus the 

 leaves are seen to be spiral in their arrangement on the 

 stem. In the Elm the third leaf stands directly over the 

 first, and to reach it the thread has passed once around the 

 stem, or, as it is usually said, the cycle is complete when 

 the third leaf is reached, and it is expressed by the frac- 

 tion i. The numerator denotes the number of turns ; the 

 denominator, the number of leaves encountered. Experi- 

 menting in a similar manner with the Alder, the fraction i 

 is obtained, and with the Cherry, |-. In the latter case the 

 stem would be encircled twice before a leaf is found (the 

 sixth), which is inserted directly over the first, and five 

 leaves are contained in the cycle. In a similar manner the 

 fractions I with the Flax, -f^ with the Flea-bane, ^ with 

 the House-leek, ^J with cones of some Pines, and f ^ with 

 some Firs, would be obtained. 



31. If now vertical planes be passed through the points 

 of insertion of the several leaves and the axis of the shoot, 

 the angle formed by any two planes, in case of the Elm, 

 whose cycle is J, will be one-half of 360°, or 180° — that is 

 to say, the angular divergence of the leaves is 180°. 

 In like manner, the angle formed by the planes through 

 the points of insertion of the leaves and the axis of the 

 shoot of the Alder, whose cycle is i, will be found to be 

 one-third of 360°, or 120° — or the angular divergence of 

 the leaves in this case is 120°. And with the | cycle, the 

 angular divergence of the leaves is two-fifths of 360' ; with 

 the I cycle it is three-eights of 360°, and so on for every cycle. 



32. If the fractions |, \, |, |, 3%, ■^, |f , etc., be exam- 

 ined, certain definite relations will be seen to exist. For 



