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and the leaves be then broken off, it will be found that a perfectly spiral line will have been formed. Upon this supposition, opposite or whorled leaves are to be considered the result of a peculiar non-developement of internodes, and the consequent confluence of as many nodes as there may be leaves in the whorl. Rhododendron ponticum will furnish the student with an illustration of this: on many of its branches some of the leaves are alternate and others opposite; and several intermediate states between these two will be perceivable. In many plants, the leaves of which are usually alternate, there is a manifest tendency to the approximation of the nodes, and consequently to an opposite arrangement of the leaves, as in Solanum nigrum, and many other Solanaceæ ; while, on the other hand, leaves which are usually opposite, separate their nodes and become alternate, as in Erica mediterranea : but this is more rare.
The best argument in support of the hypothesis that all whorls arise from the non-developement of internodes and confluence of nodes, is, however, to be derived from flowers, which are several series of whorls, as will be seen hereafter. In plants with alternate leaves, the flowers often change into young branches, and then the whorls of which they consist are broken, the nodes separate, and those parts that were before opposite become alternate; and in monstrous Tulips, the whorls of which the flower consists are plainly seen to arise from the gradual approximation of leaves, which in their unchanged state are alternate.
A most elaborate memoir has been written by a German naturalist named Braun, to prove, mathematically, not only that the spiral arrangement is that which is everywhere visible in the disposition of the appendages of the axis, but that each species is subject to certain fixed laws, under which the nature of the spires, and in many cases their number, are determined. The original appeared in the Nova Acta of the Imperial Academy Nature Curiosorum; and a very full abstract of it has been given by Martins, in the first volume of the Archives de Botanique, from which I borrow what follows:
The scales of the fruit of Coniferous plants are nothing but capellary leaves, which do not form, like the floral envelopes of other plants, a complete cavity surrounding the sexual organs on all sides, but which are slightly concave, and protect them on one side only. This point admitted, if we consider attentively the cone of a Pine, or of a Spruce Fir, we are at once led to inquire whether the scales are arranged in spires or in whorls. Breaking through its middle a cone of Pinus Picea (Silver Fir), we remark three scales, which at first sight appear to be upon the same plane; but a more attentive examination shows that they really originate at different heights, and moreover, that they are not placed at equal distances from each other; so that we cannot consider them a whorl, but only a portion of a very close spiral. But, considering the external surface of the cone viewed as a whole, we find that the scales are disposed in oblique lines, which may be studied - 1. As to their composition, or the number of scales requisite to form one complete turn of the spire; 2d. As to their inclination, or the angle, more or less open, which they form with their axis ; 3d. As to their total number, and their arrangement round the common axis, which constitutes their co-ordination. Finally, we may endeavour to ascertain whether the spires turn from right to left, or vice versa.
He then proceeds to show, that the spiral arrangement is not only universal, but subject to laws of a very precise nature. The evidence upon which this is founded is long and ingenious, but would be unintelligible without the plates which illustrate it. I must, therefore, content myself with mentioning the results. Setting out from the Pine cone above referred to, he found that several series of spires are discoverable in the arrangement of their scales, and that there invariably exists between these spires certain arithmetical relations, which are the expression of the various combinations of a certain number of elements, disposed in a regular manner. All the spires depend upon the position of a fundamental series, from which the others are deviations. The nature of the fundamental series is expressed by a fraction, of which the numerator indicates the whole number of turns required to complete one spire, and the denominator the number of scales or parts that constitute it. Thus c 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.
The following are some of the results thus obtained by Braun, in studying the composition of the spires of different plants:
in Asarum, Aristolochia, Lime tree, Vetch, Pea, the spikes of all grasses.
} is rare in Diotyledons, and generally changes into more complicated spires. It exists in Cactus speciosus, and some others.
f is the most common of all, and represents the quincunx. Mezereum, Lapsana communis, Polemonium cæruleum, Potato, are examples.
is also common, as in the Bay-tree, the Holly, common Aconite, and the tuft of radical leaves of Plantago media.
is exists where the leaves are numerous and their intervals small. Wormwood, common Arbutus, dwarf Convolvulus, and the tufts of leaves in London Pride and Dandelion, are instances.
in Woad, Plantago lanceolata, the bracts of Digitalis lanata.
in Sempervivum arboreum, the bracts of Plantago media, and of Protea argentea.
f} was found on an old trunk of Zamia horrida, and two species of Cactus (coronarius and difformis).
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. No application of the doctrine appears practicable, except to assist in the distinction of species, for which it would be well adapted, if the determination of the series with the requisite precision were less difficult; this is shown in the following instances of differences in the fundamental spire in nearly allied species,
Pinus pinaster, }} -sylvestris, }-cembra, a 1 - larix, d - microcarpa z.
Betula alba and pubescens, de and ji - fruticosa generally, is
Corylus avellana, l-americana and tubulosa, il in their male catkins.
The whole of this curious question has been simplified by Professor Henslow, in observations printed for private circulation; and I am happy to be able, by the permission of their author, to lay them in this place before the public.
“ The scales on a cone of the Spruce Fir (Abies excelsa) are placed spirally round the axis, at equal intervals; and after eight coils of the spiral, the twenty-second scale ranges vertically over the first. If this arrangement be referred to a cylinder, and then projected on a plane cutting its axis at
right angles, the angular distance (Divergence) between two contiguous scales, seen from the centre, is z of the circumference. Hence the divergence of the generating or primary spiral 1. The various peculiarities of the secondary spirals which result from the above arrangement, may be seen by inspecting fig. 54.
A. If any figure in this circle represent the divergence of a spiral, the same will also represent the number of coils which that spiral must make before the twenty-second scale upon it comes vertically over the first.
B. The figures in this circle (corresponding to the several divergencies in A.) show the number of similar and parallel spirals which must be coiled round the cylinder, in order that every scale may range upon them.
The same figures also indicate the height of each spiral — viz.: either the comparative lengths of the vertical lines between scales 1. and 22. or the distance between two horizontal circles through scales 1. and 2.; and, lastly,
These figures are the common differences in the different arithmetic series apparent on the consecutive scales of each
C is the arrangement of the first twenty-one scales on the generating spiral.
D shows the number on the scales which begin a second series of each kind of spiral, i. e. the numbers on their twentysecond scales.
N. B. The number on the scale which begins a fresh series of any spiral is found by the formula (a + 21 B) where (a) = the number on the scale beginning a former series of the spiral, and B the common difference of the numbers on two contiguous scales.
Ex. Gr. Considering the spiral (fig. 55.) through the scales 1. 9. 17. &c., 153. 161. 169. &c. A. Ist, Its divergence (from 1 to 9) is 100—20, and, 2d, It must coil once towards the left, or twenty times towards the right (of a spectator at the axis) before it passes through the twenty-second scale upon it (viz. No. 169.), which ranges vertically over the first. B. Ist, There are seven other similar spirals parallel to it. 2d, Their height (as from 1 to 169) = eight times the height from 1 to 22; and, 3d, The common difference of the numbers of the scales