LEAVES. 45 



we shall readily discover that they are arranged spirally ; and 

 by cutting off the leaves composing one spire, we shall find that 

 they do not form a single spire, but a compound one, consisting 

 of three or four spires running parallel to each other. The 

 above are the different arrangements of leaves, but there seems 

 a tendency in opposite and verticillate leaves to assume the 

 6piral arrangement, without, however, varying their relative 

 lateral position ; and this is true of all modifications of leaves. 

 There is not only a disposition of the leaves, but of all vegeta- 

 ble productions to assume a spiral arrangement. The stems of 

 plants take, in almost all cases, more or less of the spiral growth, 

 however straight the stem may be. We may see this also in 

 the epidermis of various trees, and more particularly in the stems 

 of twining plants, as the hop and ivy. 



80# The spiral arrangement of leaves has excited much in- 

 terest recently ; and Braun, a German naturalist, has applied 

 mathematical formulae to express the elements of the spires in 

 different species, and for determining their constitution. " All 

 the spires depend upon the position of a fundamental series, 

 from which the others are deviations. The nature of the fun- 

 damental series is expressed by a fraction, of which the nume- 

 rator indicates the whole number of turns required to complete 

 one spire, and the denominator the number of scales or parts 

 that constitute it. Thus -£ T indicates that eight turns are 

 made round the axis before any scale or part is exactly vertical 

 to that which was first formed ; and the number of scales, or 

 parts that intervene before this coincidence takes place, is 21," 

 which occurs in the Corylus, Plantago lanceolata. f expresses 

 that the leaves, buds, or scales make two turns before a leaf, 

 scale, or bud is exactly above the one from which we start, and 

 that there are five of them. This is the most common variety. 

 Cherry, Althea, Potato, Peach, &c., are of this variety ; ^ in- 

 cludes the spikes of the grains ; f includes the Bay, Holly, &c. ; 

 T 5 3 expresses the elements of the spire in some of the pines. 

 Prof. Lindley remarks, that " it does not, however, appear that 

 this inquiry has led to any thing beyond the establishment of 

 the fact that, beginning from the cotyledons, the whole of the 

 appendages of the axis of plants — leaves, calyx, corolla, stamens, 

 and carpels — form an uninterrupted spire, governed by laws 

 which are nearly constant." 



81. Leaves usually present surfaces of different appearance, 



"What disposition in plants? Where may we see it? — 80. Who applied 

 mathematics to this subject? Explain its application. — 81. What is 6aid 

 of the different surfaces of leaves f 



