Ovis Ammon, etc., and ending with the long-horned Highland ram: in which, as we pass from one to another, we recognise precisely homologous spirals, with an increasing angular constant, the spiral angle being, for instance, about 75ô¯ or rather less in Ovis Ammon, and in the Highland ram a very little more. We have already seen that in the neighbourhood of 70ô¯ or 80ô¯ a small change of angle makes a marked difference in the appearance of the spire; and we know also that the actual length of the horn makes a very striking difference, for the spiral becomes especially conspicuous to the eye when a horn or shell is long enough to shew several whorls, or at least a considerable part of one entire whorl.
Even in the simplest cases, such as the wild goats, the spiral is never (strictly speaking) a plane or discoid spiral: but in greater or less degree there is always superposed upon the plane logarithmic spiral a helical spiral in space. Sometimes the latter is scarcely apparent, for the helical curvature is comparatively small, and the horn (though long, as in the said wild goats) is not nearly long enough to shew a complete convolution: at other times, as in the ram, and still better in many antelopes, such as the koodoo, the helicoid or corkscrew curve of the horn is its most charôÙacôÙterôÙisôÙtic feature.
Accordingly we may study, as in the molluscan shell, the helicoid component of the spireãin other words the variation in what we have called (on p. 555) the angle ö¡. This factor it is which, more than the constant angle of the logarithmic spiral, imparts a charôÙacôÙterôÙisôÙtic appearance to the various species of sheep, for instance to the various closely allied species of Asiatic wild sheep, or Argali. In all of these the constant angle of the logarithmic spiral is very much the same, but the shearing component differs greatly. And thus the long drawn out horns of {618} Ovis Poli, four feet or more from tip to tip, differ conspicuously from those of Ovis Ammon or O. hodgsoni, in which a very similar logarithmic spiral is wound (as it were) round a much blunter cone.
The ramãs horn then, like the snailãs shell, is a curve of double curvature, in which one component has imposed upon the structure a plane logarithmic spiral, and the other has produced a continuous displacement, or ãshear,ã proportionate in magnitude to, and perpendicular or otherwise inclined in direction to, the axis of the former spiral curvature. The result is precisely analogous to that which we have studied in the snail and other spiral univalves; but while the form, and therefore the resultant forces, are similar, the original distribution of force is not the same: for we have not here, as we had in the snail-shell, a ãcolumellarã muscle, to introduce the component acting in the direction of the axis. We have, it is true, the central bony core, which in part performs an analogous function; but the main phenomenon here is apparently a complex distribution of rates of growth, perpendicular to the plane of the generating curve.
Let us continue to dispense with mathematics, for the mathôÙeôÙmatôÙiôÙcal treatment of a curve of double curvature is never very simple, and let us deal with the matter by experiment. We have seen that the generating curve, or transverse section, of a typical ramãs horn is triangular in form. Measuring (along the curve of the horn) the length of the three edges of the trihedral structure in a specimen of Ovis Ammon, and calling them respectively the outer, inner, and hinder edges (from their position at the base of the horn, relatively to the skull), I find the outer edge to measure 80 cm., the inner 74 cm., and the posterior 45 cm.; let us say that, roughly, they are in the ratio of 9ã₤:ã₤8ã₤:ã₤5. Then, if we make a number of little cardboard triangles, equip each with three little legs (I make them of cork), whose relative lengths are as 9ã₤:ã₤8ã₤:ã₤5, and pile them up and stick them all together, we straightway build up a curve of double curvature precisely analogous to the ramãs horn: except only that, in this first approximation, we have not allowed for the gradual increment (or decrement) of the triangular surfaces, that is to say, for the tapering of the horn due to the growth in its own plane of the generating curve. {619}