21^ WITHEKED LEAVES. 



oak, the poplar, or the apple, we will fasten one end of a thread to 

 the stalk of a leaf near the bottom of the stem, then carry it to 

 the next leaf above, and so from leaf to leaf, until we arrive at one 

 situated perpendicularly above the first leaf; and we shall find 

 that the thread has made two complete circuits of the stem, and 

 has embraced five leaves in its cycle, the last leaf being the sixth 

 in order from the first. 



This arrangement of leaves is represented by the fraction 

 2-5 ths, where the numerator denotes the number of spirals, and 

 the denominator the number of leaves, in the cycle. A diagram- 

 matic representation of such a stem will show five perpendicular 

 lines, upon which the leaves are placed in alternate order. Thus, 

 starting with the first line in the centre, and passing around the 

 series from right to left, we shall find the first leaf on the first 

 perpendicular, the second on the third, the third on the fifth, the 

 fourth on the second, the fifth on the fourth, and the sixth again 

 on the first line — making two entire circuits to include the five 

 leaves. 



This will be rendered more intelligible from an examination of 

 the diagram in Figure 5, which shows the leaves arranged in per- 

 pendicular lines around the stem. The same series — 2-5ths — is 

 also shown in transverse section in Figure 7 ; and the i-3rd ar- 

 rangement in Figure 6. 



The most simple order is that in which the leaves are situated 

 on opposite sides of the stem as in the lime and elm, where a 

 single spiral includes two leaves ; and this is denoted by the frac- 

 tion i. In the next form, three leaves are contained in a single 

 spiral, of which the birch is an example, and this is expressed by 

 the fraction j^. We have now obtained three fractions, which 

 may be continued in regular sequence ; the sum of the two pre- 

 ceding numerators forming the numerator of the next fraction, and 

 the sum of the two denominators being the next denominator. 

 This is termed the primary series, and the order will be as follows : 



2 3 5 8 13 21 34 55 ' 

 In some of the more complex arrangements there may be two or 

 more spirals, the directions of which may be from right to left, or 

 dextral ; or from left to right, or sinistral ; giving rise to a secondary 



