The helical arrangement (as in the fir-cone) was carefully studied in the middle of the eighteenth century by the celebrated Bonnet, with the help of Calandrini, the mathematician. Memoirs published about 1835, by Schimper and Braun, greatly amplified Bonnet’s investigations, and introduced a nomenclature which still holds its own in botanical textbooks. Naumann and the brothers Bravais are among those who continued the investigation in the years immediately following, and Hofmeister, in 1868, gave an admirable account and summary of the work of these and many other writers[573].
Starting from some given level and proceeding upwards, let us mark the position of some one leaf (A) upon a cylindrical stem. Another, and a younger leaf (B) will be found standing at a certain distance around the stem, and a certain distance along the stem, {637} from the first. The former distance may be expressed as a fractional “divergence” (such as two-fifths of the circumference of the stem) as the botanists describe it, or by an “angle of azimuth” (such as ϕ = 144°) as the mathematician would be more likely to state it. The position of B relatively to A must be determined, not only by this angle ϕ, in the horizontal plane, but also by an angle (θ) in the vertical plane; for the height of B above the level of A, in comparison with the diameter of the cylinder, will obviously make a great difference in the appearance of the whole system, in short the position of each leaf must be expressed by F(ϕ · sin θ). But this matter botanical students have not concerned themselves with; in other words, their studies have been limited (or mainly limited) to the relation of the leaves to one another in azimuth.
Whatever relation we have found between A and B, let precisely the same relation subsist between B and C: and so on. Let the growth of the system, that is to say, be continuous and uniform; it is then evident that we have the elementary conditions for the development of a simple cylindrical helix; and this “primary helix” or “genetic spiral” we can now trace, winding round and round the stem, through A, B, C, etc. But if we can trace such a helix through A, B, C, it follows from the symmetry of the system, that we have only to join A to some other leaf to trace another spiral helix, such, for instance, as A, C, E, etc.; parallel to which will run another and similar one, namely in this case B, D, F, etc. And these spirals will run in the opposite direction to the spiral ABC.
In short, the existence of one helical arrangement of points implies and involves the existence of another and then another helical pattern, just as, in the pattern of a wall-paper, our eye travels from one linear series to another.
A modification of the helical system will be introduced when, instead of the leaves appearing, or standing, in singular succession, we get two or more appearing simultaneously upon the same level. If there be two such, then we shall have two generating spirals precisely equivalent to one another; and we may call them A, B, C, etc., and A′, B′, C′, and so on. These are the cases which we call “whorled” leaves, or in the simplest case, where {638} the whorl consists of two opposite leaves only, we call them decussate.
Among the phenomena of phyllotaxis, two points in particular have been found difficult of explanation, and have aroused discussion. These are (1), the presence of the logarithmic spirals such as we have already spoken of in the sunflower; and (2) the fact that, as regards the number of the helical or spiral rows, certain numerical coincidences are apt to recur again and again, to the exclusion of others, and so to become characteristic features of the phenomenon.
The first of these appears to me to present no difficulty. It is a mere matter of strictly mathematical “deformation.” The stem which we have begun to speak of as a cylinder is not strictly so, inasmuch as it tapers off towards its summit. The curve which winds evenly around this stem is, accordingly, not a true helix, for that term is confined to the curve which winds evenly around the cylinder: it is a curve in space which (like the spiral curve we have studied in our turbinate shells) partakes of the characters of a helix and of a logarithmic spiral, and which is in fact a logarithmic spiral with its pole drawn out of its original plane by a force acting in the direction of the axis. If we imagine a tapering cylinder, or cone, projected, by vertical projection, on a plane, it becomes a circular disc; and a helix described about the cone necessarily becomes in the disc a logarithmic spiral described about a focus which corresponds to the apex of our cone. In like manner we may project an identical spiral in space upon such surfaces as (for instance) a portion of a sphere or of an ellipsoid; and in all these cases we preserve the spiral configuration, which is the more clearly brought into view the more we reduce the vertical component by which it was accompanied. The converse is, of course, equally true, and equally obvious, namely that any logarithmic spiral traced upon a circular disc or spheroidal surface will be transformed into a corresponding spiral helix when the plane or spheroidal disc is extended into an elongated cone approximating to a cylinder. This mathematical conception is translated, in botany, into actual fact. The fir-cone may be looked upon as a cylindrical axis contracted at both ends, until {639} it becomes approximately an ellipsoidal solid of revolution, generated about the long axis of the ellipse; and the semi-ellipsoidal capitulum of the teasel, the more or less hemispherical one of the thistle, and the flattened but still convex one of the sunflower, are all beautiful and successive deformations of what is typically a long, conical, and all but cylindrical stem. On the other hand, every stem as it grows out into its long cylindrical shape is but a deformation of the little spheroidal or ellipsoidal surface, or cone, which was its forerunner in the bud.
This identity of the helical spirals around the stem with spirals projected on a plane was clearly recognised by Hofmeister, who was accustomed to represent his diagrams of leaf-arrangement either in one way or the other, though not in a strictly geometrical projection[574].