It is necessary at the outset to explain a few technical terms used by botanists. The part of the stem or axis to which a leaf is attached, or from which it springs, is called a node ; the space or part of the axis between one node and the other, above or below, is called an internode, and these spaces are longer or shorter in different plants.

The relative positions of leaves on the axis, or the way in which they are distributed on the stem, is technically called phyllotaxis from two Greek words which signify leaf, and order or arrangement. Numerous treatises have been written on this subject by different observers, and it is considered to have relation to certain mathematical principles.

A very common arrangement is that which is called alternate (Fig. 1), in which the leaves stand singly on the nodes. Some common plants are examples, as the poplar, the oak, the apple, &c.

In other cases, a node appears to support two leaves --- one on one side, another on the other side of the axis : the term opposite is applied to such cases (Fig. 2).

In some cases a superficial examination might lead to an erroneous conclusion, the two leaves being nearly but not strictly in opposition ; the term sub-opposite would be the correct expression in such, there being in reality a short internode between the two. Two of our native orchids, such as the twayblade (Listera), not uncommon in shady meadows and woods, may be mentioned as examples ; and in a series of specimens a few may occur in which the leaves obviously come under the first or alternate arrangement, the internode being longer than usual.

It is, however, worthy of notice that usually the pairs of opposite leaves alternate with each other --- that is to say, if we place the stem before us, and observe a pair of leaves one on the right, the other on the left, the next pair will stand one in front and the other behind ; the successive pairs of leaves are then described as decussate (Fig. 3), but even this may not be strictly true, and several pairs of leaves may intervene between those which are properly at right angles to each other.

Take, again, the common bed-straw (Galium), or madder, and we find the leaves arranged after what is known as the verticillate, or whorled manner (Fig. 4). Here we find that more than two leaves appear to come from the same transverse zone, or node. In these cases also we find alternation of leaves in the successive whorls --- that is, each leaf usually stands opposite the spaces between the leaves of the next whorl.

It has been already stated that the length of the internodes materially affects the habit or external aspect of plants, and there may be such difference of their length on the same axis and at different periods of its growth. In some of our common native species of buttercup, the leaves on the lower part of the stem appear to be in close tufts ; those further up are more widely separated, the internodes being longer.

If in the case of alternate leaves, like those of the elm or lime tree, we suppose a line drawn round the stem, and touching the point of attachment of each leaf, it will be seen to be a spiral line ; or fasten one end of a thread to the stalk of a leaf low down on the stem, then carry it to the next leaf above, and give it a twist or turn round the base of it, and so on ; the nature of the line of connection can then be seen : there is, in fact, a helix which, in passing round the stem, is more or less regular. A horizontal projection of this is called the genetic spiral, and it is best understood in the case of alternate leaves.

It will be necessary to allude here to an expression used in connection with this subject. The term cycle has reference to the different leaves which are included in the complete circuit of the spiral --- that is to say, those leaves from the first to the one which stands right above it.

A technical name is also given to that part of the stem or axis included between one leaf and the next which stands directly over it : this is called the angular divergence. Thus, where the leaves are in two rows, the space between two opposite leaves is just one-half of the circle or circumference of the stem, and where there are three rows it is one-third ; the expression "1/2" is applied in the first case, and "1/3" in the second. The upper figure (numerator) shows the number of turns in the helix, and the lower (denominator) the number of leaves embraced in the cycle.

A circle contains 360°; the term "1/2" indicates angular divergence equal to 180°, or one-half of 360° ; "1/3" corresponds to 120°, the third of a circle.

In many plants --- such as the apple, peach, cherry, poplar, &c. --- the leaves present a five-ranked arrangement. Beginning with one leaf, two circuits round the stem are necessary before reaching the leaf directly above the one from which the line began. The fraction "2/5" is used to indicate this --- that is to say, two turns round the stem, and the sixth leaf directly above the first ; therefore 5 leaves in the cycle. Figs. 5 and 6 show a spiral and horizontal projection of a 2/5 arrangement.

When we make three turns round the stem before reaching a leaf right above the first, the expression is "3/8", 3 being the number of turns, and 8 the number of leaves in the cycle.

There are other arrangements, and all may be set down here :

1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55, &c.

Now, on examining these expressions, an interesting result comes out --- a fraction has its numerator equal to the sum of the numerators of the two preceding, and the same is true of the denominator.

The more simple arrangements are of frequent occurence ; where the internodes are very short, the leaves are crowded, and the analysis of such cases is more difficult, as in the rosettes presented by the leaves of some sedums or stonecrops, and of sempervivum or houseleek (Fig. 7, showing 5/13 arrangement), and the cones of firs.

Nevertheless, the general spiral arrangements are in such cases obvious enough ; instead of one simple spiral there are several parallel or secondary spirals, more or less numerous. This is best seen in any large, or even small, fir-cone.

Some of the spirals run from left to right, others the reverse. In such cases the primary or generating spiral has reference to a full series of the leaves on the axis, the spiral line passing through every leaf ; the secondary spirals are only partial --- that is, do not embrace every leaf or scale. In such cases, the fundamental spiral cannot be easily followed; but an examination of the secondary spirals will give assistance in this. These secondary spirals vary in number according as the fractional sign of the primary spiral is higher.

The cone of Pinus strobus --- white or Weymouth pine --- is in various works used to illustrate this subject, and the same example may be adopted here. Fig. 8 represents one side of the cone with the scales numbered, and beside it a projection of the arrangement 5/13, the normal in this cone, the number 14 being directly above the scale number 1, the cycle consisting of 13 scales, the spirals being 5. A line which passes to the left through the numbers 1, 2, 3, 4, &c., makes five turns round the cone before it ends at 14, directly above 1. There are 5 parallel spirals of the order 1, 6, 11, &c. (these give the numerator), and 8 of the order 1, 9, 17, &c. ; then 8 and 5 give 13, so that the primary spiral expressed by 5/13 may be got from the number of secondary spirals parallel to one another.

It has been already stated that in the primary series any numerator is the same number as the denominator of the fraction next but one preceding. This relation does not hold in the others ; "but if it be remembered that the denominators can be formed by adding the numerator and denominator of the corresponding fraction of the preceding series, the true and general relation at once appears."

The following are examples: --- The denominator of the fraction 3/8 supplies the numerator to the fraction 8/21 ; but iu the secondary series the denominator is 11 (i.e., 8 + 3); so also in the tertiary series the denominator of the corresponding fraction is 14 --- that is, 11 + 3 is equal to 8 + 3 + 3. The fourth fractions may, therefore, stand thus :

3/8, 3/(8+3), 3/(8+3+3), 3/(8+3+3+3), &c.

Mr. Henslow shows by a diagram (Fig. 9) that "the angular distances included by the limiting positions of the second leaves of all generating spirals, commencing at 0, decrease according as the spirals belong to the secondary, tertiary, or quaternary series ; so also does the number of leaves in a single coil increase correspondingly ; and, therefore, the higher the series, the more nearly does any spiral belonging to it approach the verticillate condition, provided the internodes be but slightly developed."

There appears to be a relation between the folding or mutual relation --- technically called œstivation or prefloration --- of the parts of the flower when in bud and the laws of phyllotaxis or leaf-arrangement. In many such flower-buds the arrangements 2/5, 3/8, 5/13 may be recognised; but to enter into details would necessitate the use of technicalities foreign to the subject of this article.

It may be stated here that in some of the lower forms of plants, such as mosses, ferns, &c., the angle of divergence of the leaf-organ is related to the principle of growth. In the 1/2 arrangement the cell at the end of the axis is divided into two. When the segmentation or division of the apical cell is in three rows, each new division-wall of the cell at the apex being parallel to the last division-wall but two, two rows of leaves are formed, arranged spirally with the divergence 1/3. The segmentation, then, of the apical cell has a relation to the leaf-arrangement in Cryptogams --- mosses, &c. ; in Phanerogams --- flowering plants --- the same relation does not hold.

Attention may now be directed to the number 4, or a multiple of it in some of the lower forms of plants. The instances are so numerous that a few examples may suffice. It is also worthy of notice that division of protoplasm into two is not uncommon. This is well illustrated in the development of the reproductive spores, as they are called. The gills of mushrooms are covered with numerous club-shaped cells --- sporophores or spore-bearers --- on the summit of which the spores are in groups of 4. In others, such as the well-known mushroom called the morell, there are eight spores in an oblong case or cell. In one of our most common sea-weeds, the Fucus vesiculosus, or bladder fucus, so-called from numerous air-vesicles on it, the contents of oogonium divide into eight portions (Fig. 11).

The spores or seed-like organs of mosses, are produced in fours. In the common male fern, Aspidium filix mas., they follow the same law.

This is also illustrated in the development of the grains of the dust-like pollen, which is shaken out of the flowers of the higher plants, or those which have obvious flowers. This is well seen in the earlier stages of the pollen in mallow. After the four very young grains escape from the mother-cell and are free, they increase in size, and the surface becomes rough with projecting points. In some cases the grains, after escaping from the parent cell, even when mature, remain bound together, giving, thus, composite pollen. This occurs in species of typha (cat's-tail) or bullrush. An early stage of the pollen thus remains persistent (Fig. 12).

We may finally bring under notice the very notable law which prevails in the number of the teeth which surround the mouth of the ripe capsules or cases which contain the spores of mosses. It may be stated, in passing, that these teeth, forming what is called the peristome, are highly sensitive to moisture, folding over the mouth of the capsule, or unfolding outwards, according to the state of the atmosphere as regards moisture or dryness.

The numbers of these teeth, when present, are 4, or some power of 4 up to 64, being 4 in tetraphis (Fig. 1 3) ; 8 apparently in some species of orthotrichum (Fig. 14); 16 in grimmia and others ; in zygodon, the outer teeth are considered to be of 32 primary divisions, united 2 or 4 together, so as to represent 16 or 8 plain teeth. In polytrichum there are 64 (rarely 32) teeth. Where there are two rows of teeth, which is a frequent character, the law also prevails, and those of the one row alternate with those of the other.

If we examine the parts of the flower in the higher orders of plants, we observe also that certain numbers prevail, but they are less constant. In those which are called monocotyledonous, in which there is apparently one lobe in the seed, the three-ranked arrangement prevails, as in crocus, iris, tulip, &c. The four- and five-ranked, on the other hand, are most frequent among dicotyledonous plants. Fuchsia, epilobium, &c., have the whorls of the flower in groups of 4. In primroses, and many others, the number 5 prevails --- that is, the quinary; among them, however, there are some exceptions, the number 3 being seen in magnolia, barberry, &c.

As an example of the three-ranked arrangement in the monocotyledonous division, we may take the flower of a hyacinth ; we observe on the outside 3 parts, or "sepals" as they are called, forming the "calyx" or cup. More internally, we see other 3 --- the petals --- the corolla, or coloured part of the flower. Next we find 6 "stamens" in two rows, an outer and inner, and in the centre of the flower a seed-vessel of 3 pieces conjoined; all these alternate with each other. The same succession of parts may be observed in complete flowers of dicotyledons, the numbers in each whorl being, however, mostly 4 or 5, although there are exceptions to this.