614 THE SHAPES OP HOKNS [ch. 



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 animal*. 

 Just as we sought for the true generating curve in the orifice, 

 or "lip," 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 corresponding 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 "trace," or projection on a plane perpendicular to the 

 axis — in 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 Amnion. 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 "nuchal," 

 abutting on the parietal bonef. Along these three faces, and 

 their corresponding angles or edges, we can trace in the fibrous 

 substance of the horn a series of homologous spirals, such as we 



* In the case of the ram's horn, the assumption that the rings are annual is 

 probably justified. In cattle they are much less conspicuous, but are sometimes 

 well-marked in the cow; and in Sweden they are then called "caK-rings," from 

 a belief that they record the number of offspring. That is to say, the growth of 

 the horn is supposed to be retarded during gestation, and to be accelerated after 

 parturition, when superfluous nourishment seeks a new outlet. (Cf. Lonnberg, 

 P.Z.S., p. 689, 1900.) 



t Cf. Sir V. Brooke, On the Large Sheep of the Thian Shan, P.Z.8., p. 511, 1875. 



