90 



PHYLLOTAXIS OR LEAF- ARRANGEMENT. 



hence, this arrangement is indicated by the fraction f. In other 

 words, the distance or divergence between, the first and second leaf, 

 expressed in parts of a circle, is ^ of a circle, or 360 -f- j = 144. 

 In fig. 195, a b, the spiral is ^, i.e. 

 one turn and two leaves; the third leaf 

 being placed vertically over the first, 

 and the divergence between the first 

 and second leaf being one-half the cir- 

 6 cumference of a circle, 360-f-^, = 180. 

 Again, in fig. 199, a >, the number is ^, 

 or one turn and three leaves, the an- 

 gular divergence being 120. 



172. In cases where the internodes 

 are very short, and the leaves are closely 

 applied to each other, as in the House- 

 leek, it is difficult to trace what has 

 been called the generating spiral, or 

 that which passes through every leaf 

 of the cluster. Thus, in fig. 200, there 

 are thirteen leaves which are numbered 

 in their order, and five turns of the 

 spiral marked by circles in the centre (fa indicating the arrange- 

 ment); but this could not be detected at once. So also in Fir cones 

 (fig. 201), which are composed of scales or modified leaves, the gener- 

 ating spiral cannot be determined easily. In such cases, however, 

 there are secondary spirals running parallel to each other, as is seen 

 in fig. 201, where spiral lines pass through scales numbered 1, 6, 11, 

 16, &c., and 1, 9, 17, &c., and by counting those which run parallel 

 in different directions, the number of scales intervening between 

 every two in the same parallel coil may be ascertained. Thus, in fig. 

 201, it will be found that there are five secondary spirals running 

 towards the right and parallel to each other, the first passing through 

 the scales 1, 6, 11, 16, &c.; the second through 9, 14, 19, 24, &c.; 

 the third through 17, 22, 27, 32, 37, &c.; the fourth through 30, 35, 

 40, 45, &c ; the fifth through 43, 48, 53, &c. The number of these 

 secondary spirals indicates the number of scales intervening between 

 every two scales in each of these spirals the common difference being 

 five. Again, it will be found on examination that there is a number 

 of secondary spirals running to the left, in which the common difference 

 between every two scales is eight, and that this corresponds to the 

 number of secondary spirals, the first of which passes through the 

 scales 1, 9, 17, &c.; the second through 6, 14, 22, 30, &c.; the third 



Fig. 199. Young plant of Cyperus esculentus. with leaves in three rows, or tristichous. ex- 

 pressed by the fraction J, or one turn and three leaves, a, The plant, with its leaves numbered 

 in their order. 6, Magnified representation of the stem, showing the insertion of the leaves and 

 their spiral arrangement 



