XIII] OF SHEEP AND GOATS 881 



and stick them all together, we straightway build up a curve ol 

 double curvature precisely analogous to the ram's horn: except 

 only that, in this first approximation, we have not allowed for the 

 gradual increment (or decrement) of the triangular surfaces, that is 

 to say, for the tapering of the horn due to the magnification of the 

 generating curve. 



In this case then, and in most other trihedral or three-sided horns, 

 one of the three components, or three unequal velocities of growth, 

 is of relatively small magnitude, but the other two are nearly equal 

 one to the other; it would involve but little change for these latter 

 to become precisely equal; and again but little to turn the balance 

 of inequality the other way. But the immediate consequence of 

 this altered ratio of growth would be that the horn would appear to 

 wind the other way, as it does in the antelopes, and also in certain 

 goats, e.g. the markhor, Capra falconeri. 



For these two' opposite directions of twist Dr Wherry has suggested a 

 convenient nomenclature. When the horn winds so that we follow it from 

 base to apex in the direction of the hands of a watch, it is customary to call 

 it a "left-handed" spiral. Such a spiral we have in the horn on the left-hand 

 side of a ram's head. Accordingly, Dr Wherry calls the condition homonymous, 

 where, as in the sheep, a right-handed spiral is on the right side of the head, 

 and a left-handed spiral on the left sid,e ; while he calls the opposite condition 

 heteronymmis, as we have it in the antelopes, where the right-handed twist 

 is on the left side of the head, and the left-handed twist on the right-hand side. 

 Among the goats, we may have either condition. Thus the domestic and 

 most of the wild goats agree with the sheep; but in the markhor the twisted 

 horns are heteronymous, as in the antelopes. The difference, as we have 

 seen, is easily explained; and (very much as in the case of our opposite spirals 

 in the apple-snail, referred to on p. 820) it has no very deep importance 



Summarised then in a very few words, the argument by which 

 we account for the spiral conformation of the horn is as follows : 

 The horn elongates by dint of continual growth within a narrow 

 zone, or annulus, at its base. If the rate of growth be identical on 

 all sides of this zone, the horn will "grow straight; if it be greater 

 on one side than on the other, the horn will become curved; and 

 it probably will be greater on one side than on the other, because 

 each single horn occupies an unsymmetrical field with reference to 

 the plane of symmetry of the animal. If the maximal and minimal 

 velocities of growth be precisely at opposite sides of the zone of 



