CH. XIII] THE HORNS OF SHEEP AND GOATS 875 



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, by the mere reason of its 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 " sheath" 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 may be taken 

 as the "generating curve" 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 con- 

 cerned 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; they form "conjugate" spirals, one 

 being the "mirror-image" of the other. Just as in the hairy coat 

 of the animal each hair, on either side of the median "parting," 

 tends to have a certain definite direction of its owji, 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 

 less definitely periodic : sometimes, as in the sheep, this periodicity 

 is particularly well-marked, and causes the horny sheath to be com- 



* In this chapter we keep to Moseley's way of regarding the equiangular spiral 

 in space, of shell or horn, as generated by a certain figure which (a) grows, (6) revolves 

 about an axis, and (c) is translated along or parallel to the said axis, all at certain 

 appropriate and specific velocities. This method is simple, and even adequate, 

 from the naturalist's point of view; but not so, or much less so, from the mathe- 

 matician's, as we have found in the last chapter (p. 782). 



