PHYLLOTAXIS 



43 



We may also remember that there must be 21 vertical rows of leaves. 

 These may generally be seen without much difficulty by holding the cone 

 horizontally, and looking parallel with its axis, when the twenty-one rows 

 of vertical scales will be observed on revolving it, somewhat in appearance 

 like the rows of grains in a head of Indian corn. 



I have said that the 22nd scale will be found almost immediately 

 above, but not accurately in the same vertical line, with the one selected 

 as No. 1. That it cannot be precisely so is obvious from the fact that ¥ 8 r 

 of 360°, or 137° 31' + , is not an aliquot part of a circumference; the 

 consequence is that the 22nd leaf must stand a little out of the vertical 

 line, and of course the 43rd will be double that distance, and the 64th 

 treble the amount, and so on. Hence it results that this supposed vertical 

 line is in reality a highly-elevated spiral line, and instead of there being 

 21 actually vertical rows of scales there will be 21 very elevated spirals 

 (see fig. 2). 



That the rows of leaves on any stem may be strictly vertical, the 

 arrangement must be represented by some fraction the denominator of 

 which measures 360°, such as \, ^, f, and f ; whereas ^ 8 T , &c. represent 

 spirals in which no two leaves are ever in the same vertical line exactly. 



As a general rule, all leaf-arrangements on stems with well- developed 

 internodes can be represented by some one of the fractions -J, ^, f , and f ; 

 whereas those with undeveloped internodes, as in the scales of cones, 

 thistle-heads, &c, are represented by higher members of the series, such 

 as Y %, 2D if* &c. 



I must now turn to the other condition under which leaves are 

 arranged, namely opposite. When this is the case, each pair of leaves, as 

 has been stated above, stands at right angles to the pairs above and below 

 it. Some plants have, either normally or occasionally, three or more 

 leaves on the same level. When this occurs, the leaves of each group 

 stand over the intervals of the group below it ; i.e. they alternate with 

 the leaves of the groups both above and below it. 



This kind of arrangement is best seen in the parts of flowers, all of 

 which are homologous with, or partake of, the same essential nature as 

 leaves, and which, when complete in number, are separable into four sets 

 of orga?is, called the four floral whorls ; viz. calyx of sepals, corolla of 

 petals, stamens, and pistil of carpels. It appears to be an invariable law 

 that the parts of each whorl should alternate with those of the whorls above 

 and below them. Indeed, so impressed are botanists with the persistency 

 of this law, that when the parts of any one of the floral whorls stand 

 immediately in front of the parts of a preceding external whorl, they at 

 once infer that an intermediate whorl has disappeared. This is con- 

 spicuously the case in all primroses and cowslips, and other members of 

 the family to which they belong ; wherein it will be noticed that each 

 stamen is affixed or adherent to the tube of the corolla, but immediately 

 in front of a petal, and not between two petals. That this idea of the 

 suppression of another whorl of stamens is not without foundation, it 

 may be observed that the flowers of a little denizen of damp meadows, 

 Samolus Valerandi, and akin to a primrose, have rudimentary stump-like 

 organs which stand affixed to the corolla, and alternate with the petals ; 

 while the true stamens alternate with the former ; and therefore, as in the 



