they form ãconjugateã spirals, one being the ãmirror-imageã of the other. Just as in the hairy coat of the animal each hair, on either side of the median ãparting,ã tends to have a certain definite direction of its own axis, inclined away from the median axial plane of the whole system, so is it both with the bony core of the horn and with the consolidated mass of hairs or hair-like substance which constitutes its sheath; the primary axis of the horn is more or less inclined to, and may even be nearly perpendicular to, the axial plane of the animal.
The growth of the horny sheath is not continuous, but more or {614} less definitely periodic: sometimes, as in the sheep, this periodicity is particularly well-marked, and causes the horny sheath to be composed of a series of all but separate rings, which are supposed to be formed year by year, and so to record the age of the animal1.
Just as we sought for the true generating curve in the orifice, or ãlip,ã of the molluscan shell, so we might be apt to assume that in the spiral horn the generating curve corresponded to the lip or margin of one of the horny rings or annuli. This annular margin, or boundary of the ring, is usually a sinuous curve, not lying in a plane, but such as would form the boundary of an anticlastic surface of great complexity: to the meaning and origin of which phenomenon we shall return presently. But, as we have already seen in the case of the molluscan shell, the complexities of the lip itself, or of the corôÙreôÙsponôÙding lines of growth upon the shell, need not concern us in our study of the development of the spiral: inasmuch as we may substitute for these actual boundary lines, their ãtrace,ã or projection on a plane perpendicular to the axisãin other words the simple outline of a transverse section of the whorl. In the horn, this transverse section is often circular or nearly so, as in the oxen and many antelopes: it now and then becomes of somewhat complicated polygonal outline, as in a highland ram; but in many antelopes, and in most of the sheep, the outline is that of an isosceles, or sometimes nearly equilateral triangle, a form which is typically displayed, for instance, in Ovis Ammon. The horn in this latter case is a trihedral prism, whose three faces are, (1) an upper, or frontal face, in continuation of the plane of the frontal bone; (2) an outer, or orbital, starting from the upper margin of the orbit; and (3) an inner, or ãnuchal,ã abutting on the parietal bone2. Along these three faces, and their corôÙreôÙsponôÙding angles or edges, we can trace in the fibrous substance of the horn a series of homologous spirals, such as we {615} have called in a preceding chapter the ãensemble of generating spiralsã which constitute the surface.
In some few cases, of which the male musk ox is one of the most notable, the horn is not developed in a continuous spiral curve. It changes its shape as growth proceeds; and this, as we have seen, is enough to show that it does not constitute a logarithmic spiral. The reason is that the bony exostoses, or horn-cores, about which the horny sheath is shaped and moulded, neither grow continuously nor even remain of constant size after attaining their full growth. But as the horns grow heavy the bony core is bent downwards by their weight, and so guides