We have had so much to say on the subject of shell-spirals that we must deal briefly with the analogous problems which are presented by the horns of sheep, goats, antelopes and other horned quadrupeds; and all the more, because these horn-spirals are on the whole less symmetrical, less easy of measurement than those of the shell, and in other ways also are less easy of inôÙvesôÙtiôÙgaôÙtion. Let us dispense altogether in this case with mathematics; and be content with a very simple account of the configuration of a horn.
There are three types of horn which deserve separate consideration: firstly, the horn of the rhinoceros; secondly the horns of the sheep, the goat, the ox or the antelope, that is to say, of the so-called hollow-horned ruminants; and thirdly, the solid bony horns, or ãantlers,ã which are charôÙacôÙterôÙisôÙtic of the deer.
The horn of the rhinoceros presents no difficulty. It is physiologically equivalent to a mass of consolidated hairs, and, like ordinary hair, it consists of non-living or ãformedã material, continually added to by the living tissues at its base. In section, that is to say in the form of its ãgenerating curve,ã the horn is apôÙproxôÙiôÙmateôÙly elliptical, with the long axis fore-and-aft, or, in some species, nearly circular. Its longitudinal growth proceeds with a maximum velocity anteriorly, and a minimum posteriorly; and the ratio of these velocities being constant, the horn curves into the form of a logarithmic spiral in the manner that we have already studied. The spiral is of small angle, but in the longer-horned species, such as the great white rhinoceros (Ceratorhinus), the spiral form is distinctly to be recognised. As the horn {613} occupies a median position on the head,ãa position, that is to say, of symmetry in respect to the field of force on either side,ãthere is no tendency towards a lateral twist, and the horn accordingly develops as a plane logarithmic spiral. When two horns coexist, the hinder one is much the smaller of the two: which is as much as to say that the force, or rate, of growth diminishes as we pass backwards, just as it does within the limits of the single horn. And accordingly, while both horns have essentially the same shape, the spiral curvature is less manifest in the second one, simply by reason of its comparative shortness.
The paired horns of the ordinary hollow-horned ruminants, such as the sheep or the goat, grow under conditions which are in some respects similar, but which differ in other and important respects from the conditions under which the horn grows in the rhinoceros. As regards its structure, the entire horn now consists of a bony core with a covering of skin; the inner, or dermal, layer of the latter is richly supplied with nutrient blood-vessels, while the outer layer, or epidermis, develops the fibrous or chitinous material, chemically and morphologically akin to a mass of cemented or consolidated hairs, which constitutes the ãsheathã of the horn. A zone of active growth at the base of the horn keeps adding to this sheath, ring by ring, and the specific form of this annular zone is, accordingly, that of the ãgenerating curveã of the horn. Each horn no longer lies, as it does in the rhinoceros, in the plane of symmetry of the animal of which it forms a part; and the limited field of force concerned in the genesis and growth of the horn is bound, accordingly, to be more or less laterally asymmetrical. But the two horns are in symmetry one with another;