XIII] OF SHEEP AND GOATS 877 



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 nuchal, abutting on the parietal 

 bone*. 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 w^e have called in a preceding chapter 

 the ''ensemble of generating spirals" which define or constitute the 

 surface. 



The case of the horn differs in ways of its own from that of 

 the molluscan shell. For one thing, the horn is always tubular — 

 its generating curve is actually, as well as theoretically, a closed 

 curve; there is no such thing as "involution," or the wrapping of 

 one whorl within another, or successive intersection of the generating 

 curve. Again, while the calcareous substance of the shell is laid 

 down once for all, fixed and immovable, there is reason to believe 

 that the young horn has, to begin with, a certain measure of flexi- 

 bility, a certain freedom, even though it be slight, to bend or fold 

 or wrinkle. And this being so, while it is no harder in the horn 

 than in the shell to recognise the general field of force or general 

 direction of growth, the actual conditions are somewhat more 

 complex. 



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 the growth of the horn in a new direction. 

 Moreover as age advances, the core is further weakened and to 

 a great extent absorbed: and the horny sheath or horn proper, 



* Cf. Sir V. Brooke, On the large sheep of the Thian Shan, P.Z.S. 1875, p. 511. 



