CH. XIII] THE HORNS OF SHEEP AND GOATS 613 



occupies a median position on the head, — a position, that is to say, 

 of symmetry in respect to the field of force on either side, — there 

 is no tendency towards a lateral twist, and the horn accordingly 

 develops as a flane 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 

 "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 is, accordingly, that of 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 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 sym- 

 metry 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 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 hornv sheath is not continuous, but more or 



