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is also eight. The position of the several beginnings of the 8 spirals (viz. on Nos. 1. to 8.) is shown in C; and in D we have the numbers (169. 106. 43. &c.) which respectively begin the second series of each spiral.
To discover the primary spiral, we may fix on any scale as a point of departure (No. 1.), and then, by numbering the scales on two of the secondary spirals (as 1. 9. 17. &c. and 1. 6. 11. &c.) which proceed in opposite directions, we may afterwards very readily place the numbers on all the scales. The easiest method of obtaining the common differences (viz. 8 and 5), for the purpose of numbering the scales in the two cases selected, is to draw a circle round the cone, and count the number of each of the two kinds of spirals intersecting it (which will be 8 of the first and 5 of the second). When a secondary spiral perfects a complete coil (as 1. 9. &c. 161. 169.), the number of the spirals of the same kind is readily seen; but the former mode for obtaining this number will apply equally well to cases where the cone is too short for the coils to be completed.
It is obvious that the hypothesis of the spiral arrangement of the foliaceous organs of plants is a mathematical question having but little relation to Botany. Those who wish to investigate it, will find all that is known of it in Steinheil's Observations upon the Theory of Phyllotaxis, in Ann. Sc. n. s. IV. 100. 142, Bravais, sur la disposition des feuilles curvisériées, ib. VII. 42. and VIII. 161., Link, Elementa Botanica, ed. 2: II. 448., Dutrochet, Mémoires, I. 238.
In their normal state leaves are obviously distinct, both from each other and from the stem. But, in some cases, adhesions of various kinds occur, and give them a new character. Thus, in Cardui, and many other thistle-like plants, the elongated bases of the leaves adhere to the stem, and become what is called decurrent. The elevated lines upon the stem, thus formed, are called by Link and Klotzsch sterigmata : vera, when traversed by a cord of vessels; spuria, when mere elevated cellular plates. In Bupleurum perfoliatum the lobes of the base of the leaf not only cohere with the stem, but projecting beyond it, grow together, so as to resemble a leaf through which the stem has pierced : this is called being perfoliate. Frequently two opposite leaves grow together at the base, as in Caprifolium perfoliatum; to this modification the latter term is often also applied, but that of connate is what more properly belongs to it.
The anatomical structure of the leaf is this:- From the medullary sheath diverges a bundle of woody tissue, accompanied by spiral vessels : this passes through the bark, and proceeds, at an angle more or less acute, to a determinate distance from the stem, branching off at intervals, and, by numerous ramifications, forming a kind of network. At the point of the stem whence the bundle of fibro-vascular tissue issues, the cellular tissue of the bark also diverges, accompanying the fibro-vascular tissue, expanding with its ramifications, and filling up their interstices. The tissue that proceeds from the medullary sheath, after having passed from the origin of the leaf to its extremity, doubles back upon itself, forming underneath the first a new layer of fibre, which, upon its return, converges just as the first layer diverged, at length combining into a single bundle, corresponding in bulk and position to that which first emerged, and finally discharging itself into the liber. If, therefore, a section of the leaf and stem be carefully made at a nodus, it will be found that the bundle of woody tissue which forms the frame-work of the leaf communicates above with the medullary sheath, and below with the liber. This is easily seen in the spring, when the leaves are young; but is not so visible in the autumn, when their existence is drawing to a close. The double layer of fibrovascular tissue is also perceptible in a leaf which has laid during the winter in some damp ditch, where its cellular substance has decayed, so that the cohesion between the upper and lower layers is destroyed, and the latter can be easily separated. The curious Indian leaves which have the property of opening, upon slight violence, like the leg of a silk stocking, so that the hand may be thrust between their upper and lower surfaces, derive that singular separability from an imperfect union between the layer of excurrent and recurrent fibre. De Candolle remarks, that, when the fibres expand to form the limb of a leaf, they may (whether this phenomenon occurs at the extremity of a petiole, or at the point of separation from the stem) do so after two different systems: they may either constantly preserve the same plane, when common flat leaves are formed; or they may expand in any direction, when cylindrical or swollen or triangular leaves are the result. (Organogr. p. 270.)
The cellular tissue of which the rest of the leaf is composed is parenchyma, which Link then calls diachyma, or that immediately beneath the two surfaces corter, and the intermediate substance diploe. De Candolle calls these two, taken together, the mesophyllum. The whole is protected, in leaves exposed to air, by a homogeneous cuticle of indurated organic mucus (p. 1.), and a coating of epidermis, furnished with stomates; but in submersed leaves the parenchyma is naked, no epidermis overlying it.
The general nature of the parenchymatous part of leaves has been explained, both by Link and others, and figured by Mohl, firstly in 1828 (Uber die Poren des Pflanzenzellgewebes, tab. i. fig. 4, &c.), and afterwards in his elaborate enquiry into the anatomy of Palms. A very complete account is that of Adolphe Brongniart, in 1830 (Annales des Sc. vol. xxi. p. 420.), of which much of what follows is an abstract.
The epidermis is a layer of vesicles adhering firmly to each other, and sometimes but slightly to the subjacent tissue, from which they are entirely different in form and nature: in form, for their cellules are depressed, and, in consequence of the variety of outline that they present, arranged in meshes either regular or irregular; and in nature, because these bladders are perfectly transparent, colourless, and probably filled with either air or rarefied fluid, — for the manner in which light passes through them proves that they do not contain dense fluid. They scarcely ever contain any organic particles, and are probably but little permeable either to fluids or gaseous matters; while, on the other hand, the vesicles of the subjacent parenchyma are filled with the green substance which determines the colour of the leaf. The epidermis is not always formed of a single layer of vesicles, but in some cases consists of two, or even three. No trace is discoverable of vessels either terminating in or beneath the cuticle ; Brongniart states this most explicitly, and my own observations are in accordance with his: an opinion, therefore, which some botanists have entertained, that spiral vessels terminate in the stomates (D. C. Organogr. p. 272. &c.), must be abandoned. At the margin of a leaf the epidermis is generally harder than elsewhere, and sometimes becomes so indurated as to assume a finty texture, as in the Aloe, and many other plants.
Stomates are found upon various parts of the epidermis : in some plants only on that of the under side of leaves, in others on the upper also; in floating leaves upon the latter only. When leaves are so turned that their margins are directed towards the earth and the heavens, the two faces are then alike in appearance, and are both equally furnished with stomates. In succulent leaves they are said to be either altogether absent or very rare; but this is not exactly the fact. They are fewer and smaller, and perhaps more imperfect, in succulent than in other parts, but by no means absent. According to the observations of De Candolle (Organogr. p. 272.), they are, in the Orange and Mesembryanthemum, as ten in the former to one in the latter.
I have remarked (Bot. Reg. 1540.) the singular fact, that certain plants have the power of forming stomates on the upper surface of their leaves, if from any cause their leaves are inverted. Thus the stomates are usually upon the under side of leaves, where also the veins are more prominent, and hairs appear exclusively, if hairs are found upon only one of the two surfaces. In Alströmeria that side of the leaves which is organically the undermost becomes, in consequence of a twist in the petiole, the uppermost, and that side which is borne uppermost is turned undermost; and then the organic underside, being turned uppermost, has no stomates ; while the organic upper side, being turned downwards, although under other circumstances it would have neither stomates, hairs, nor elevated veins, acquires ail those characters in consequence of its inversion. A very curious observation, in connection with this subject, has been made by Mirbel, in his memoir upon the structure of Marchantia polymorpha.
The young bulbs by which this plant is multiplied are originally so homogeneous in structure, that there is no apparent