Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/127

Rh PHYLLOTAXIS.] BOTANY 117 be seen at once that the leaves are arranged in orthostichies marked I.-V., and that these divide the circumference into five equal portions. But the divergence between leaf 1 and leaf 2 is equal to -|ths of the circumference, and the same is the case between 2 and 3, 3 and 4, (fee. The divergence, then, is -|, and from this we learn that, start ing from any leaf on the axis, we must pass twice round the stem in a spiral through five leaves before reaching one directly over that with which we started. When the leaves or scales ars alternate, and run in a single series, they are unijugate ; when the leaves are opposite, and there are twoseries, the arrangement is bijugate ; while in the case of whorled leaves the arrangement may be trijugate or quadrijugate. The line which, winding round an axis either to the right or to the left, passes through the points of insertion of all the leaves on the axis is termed the genetic or generating spiral ; and that margin of each leaf which is towards the direction from which the spiral proceeds is the kathodic side, the other margin facing the point whither the spiral passes being the anodic side. 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 the generating spiral. Thus, in fig. 132 there are thirteen leaves which are numbered in their order, and five turns of the spiral marked by circles in the centre Fig. 132. Fig. 133. Fio. 132.- Cycle of thirteen leaves placed closely together so as to form a rosette, as in Sempervivum. A is the very short axis to which the leaves are attached, The leaves arc numbered in their order, from below upwards. The circles in the centre indicate the five turns of the spiral, and show the insertion of each of the leaves. The divergence is expressed by the fraction 5-13ths. FIG. 133. Cone of Abies alba with the scales or modified leaves numbered in the order of their arrangement on the axis of the cone. The lines indicate a rec tilinear series of scales, and two lateral secondary spirals, one turning from left to right, the other from right to left. ( T 5 3- indicating the arrangement); but this could not be detected at once. So also in Fir cones (fig. 133), which are composed of scales or modified leaves, the generating spiral cannot be determined easily. But in such cases a series of secondary spirals or parastichies are seen running parallel with each other both right and left, which to a certain extent conceal the genetic spiral. Thus, in fig. 133, it will be found that there are five secondary spirals run ning 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 interven ing between every two scales in each of these spirals, the common difference being five. Again, it will be found on examination that there are 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 G, 14, 22, 30, &c.; the third through 3, 11, 19, 27, 35, 43, and so on. Thus it is that, by counting the secondary spirals, all the scales may be numbered, and by this means the generating spiral may be discovered. From the number of secondary spirals the angle of divergence may be easily calculated, the sum of those which wind in both directions giving the denominator of the fraction, while the smaller of the two numbers representing those winding in each direction is the numerator. Thus in the in stance last mentioned the angular divergence is T 6 ^-. In the cone of the American larch (fig. 134) there is a quincuncial arrangement of scales marked by the fraction -|. There are five vertical ranks, as marked in the tabular numerical Fig. 134. view at the side of the cone, which represents the unwound surface of the COne, viz., 2, 7, 12 ; 4, 9, 14 ; Cone of a species of Larch i 0110010 KiAiKil (Larix microcarpa). Thesur- 1, O, 11 ; O, O, lo ; 0, 1U, 10, tlie face of the cone is supposed common difference in each row be- to be unwound and the scales numbered so far as seen. ing 5. On looking at the cone we find also parallel oblique ranks, two of which, ascending to the left, are marked by the numbers 1, 3, 5, which, if the diagram is coiled round a cylinder, continue in the numbers 7, 9, 11, 13, 15 ; and 2, 4, 6, 8, 10, continued into 12, 14. There are thus two left- handed spirals, with 2 as the common difference in the numbering of the scales. Again, three oblique parallel spirals ascend to the right, marked by the numbers 1, 4, 7, running into 10, 13; 3, 6, 9, 12, going on to 15; and 5, 8, 11, 14; here the common difference in the number ing of the scales is 3, corresponding with the oblique right- handed spirals. All the constant divergences found in phyllotaxis may be represented as successive convergents of the continued fraction 1 o+1+l + l + l, &c., where a may have the values 1, 2, 3, 4, &amp;lt;tc. The actual fractions thus resulting are when a = 1,...^, f, f, f-, ^, &c. n 9 1123 5 ,{,,, a ^,..-2-, -j, -g, -g, YTJ-, &amp;lt;vc. The arrangement is 2-5ths, in the five-ranked series. n 3 11 tt O,...ir, -T) MS 5, &C. The spiral is not always constant throughout the whole length of an axis. The angle of divergence may alter either abruptly or gradually, and the phyllotaxis thus becomes very complicated. This change may be brought about by arrest of development, by increased development of parts, or by a torsion of the axis. The former are exemplified in many Crassulaceae and Aloes. The latter is seen well in the Screw pine (Pandanus). In the bud of the screw pine the leaves are arranged in three orthostichies with the phyllotaxis J, but by torsion the developed leaves become arranged in three strong spiral rows running round the stem. These causes of change in phyllotaxis are also well exemplified in the alteration of an opposite or verti- cillate arrangement to an alternate, and vice versa ; thus the effect of interruption of growth, in causing alternate leaves to become opposite and verticillate, can be distinctly shown in Rhododendron ponticum. Again, parts which are usually opposite or verticillate become alternate by the vigorous development of the axis, as in Hippuris, arid also in Lysimachia vulgaris, where on different parts of the same stem there may be seen alternate, opposite, and