Fig. 242.
magnitude, = γ. Let AB and ED meet, when produced, in C; and call the angle ACE (or xCy) = β. Make the angle yCz = angle xCy, = β. Draw EG, so that the angle yEG = γ, meeting Cz in G; and draw DF parallel to EG. It is then easy to show that AEDB and EGFD are similar quadrilaterals. And, when we consider the quadrilateral AEDB as having infinitesimal sides, AE and BD, the angle γ tends to α, the constant angle of an equiangular spiral which passes through the points AEG, and of a similar spiral which passes through the points BDF; and the point C is the pole of both of these spirals. In a particular limiting case, when our quadrilaterals are all equal as well as similar,—which will be the case when the angle γ (or the angles EAC, etc.) is a {502} right angle,—the “spiral” curve will be a circular arc, C being the centre of the circle.
Another, and a very simple illustration may be drawn from the “cymose inflorescences” of the botanists, though the actual mode of development of some of these structures is open to dispute, and their nomenclature is involved in extraordinary historical confusion[496].
Fig. 243. A, a helicoid, B, a scorpioid cyme.
In Fig. [243]B (which represents the Cicinnus of Schimper, or cyme unipare scorpioide of Bravais, as seen in the Borage), we begin with a primary shoot from which is given off, at a certain definite angle, a secondary shoot: and from that in turn, on the same side and at the same angle, another shoot, and so on. The deflection, or curvature, is continuous and progressive, for it is caused by no external force but only by causes intrinsic in the system. And the whole system is symmetrical: the angles at which the successive shoots are given off being all equal, and the lengths of the shoots diminishing in constant ratio. The result is that the successive shoots, or successive increments of growth, are tangents to a curve, and this curve is a true logarithmic spiral. But while, in this simple case, the successive shoots are depicted as lying in a plane, it may also happen that, in addition to their successive angular divergence from one another within that plane, they also tend to diverge by successive equal angles from that plane of reference; and by this means, there will be superposed upon the logarithmic spiral a helicoid twist or screw. And, in the particular case where this latter angle of divergence is just equal to 180°, or two right angles, the successive shoots will once more come to lie in a plane, but they will appear to come off from one another on alternate sides, as in Fig. [243] A. This is the Schraubel or Bostryx of Schimper, the cyme unipare hélicoide of Bravais. The logarithmic spiral is still latent in it, as in the other; but is concealed from view by the deformation resulting from the helicoid. The confusion of nomenclature would seem to have arisen from the fact that many botanists did not recognise (as the brothers Bravais did) the mathematical significance of the latter case; but were led, by the snail-like spiral of the scorpioid cyme, to transfer the name “helicoid” to it.
In the study of such curves as these, then, we speak of the point of origin as the pole (O); a straight line having its extremity in the pole and revolving about it, is called the radius vector; {503} and a point (P) which is conceived as travelling along the radius vector under definite conditions of velocity, will then describe our spiral curve.
Of several mathematical curves whose form and development may be so conceived, the two most important (and the only two with which we need deal), are those which are known as (1) the equable spiral, or spiral of Archimedes, and (2) the logarithmic, or equiangular spiral.
