difficulty by bolding the cone horizontally, and looking parallel 
with its axis, when the twenty-one rows of vertical scales will 
be observed, somewhat in appearance like the rows of grains 
in a head of Indian corn. 
22. I have said that the 22nd scale will be found 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 /y 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). 
23. 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 \, i, f, andf; 
whereas y^- , y 8 y, &c., represent spirals in which no two leaves are 
ever in the same vertical line exactly. 
24. As a general rule, all leaf-arrangements on stems with 
well-developed internodes can be represented by some one of 
the fractions \, f-, and | : whereas those with undeveloped 
internodes, as in the scales of cones, thistle-heads, &c., are re- 
presented by higher members of the series, such as yy, jt, if; &c. 
25. 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 is the case, the leaves of each group 
stand over the intervals of the group below it ; i.e., they alter- 
nate with the leaves of the groups both above and below it. 
26. 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 organs , 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 conspicuously 
the case in all primroses and cowslips, and other members of 
