xm] OF SHEEP AND GOATS 617 



curve (not unlike though generally less than that of a Dentalium 

 shell) is evident; and the spiral angle, according to the few 

 measurements I have made, is found to measure from about 

 20° to nearly 40°. In some of the large wild goats, such as the 

 Scinde wild goat, we have a beautiful logarithmic spiral, with a 

 constant angle of rather less than 70° ; and we may easily arrange 

 a series of forms, such for example as the Siberian ibex, the 

 moufflon. Ovis Ammon, etc., and ending with the long-horned 

 Highland ram : in which, as we pass from one to another, we 

 recognise precisely homologous spirals, with an increasing angular 

 constant, the spiral angle being, for instance, about 75° or rather 

 less in Ovis Ammon, and in the Highland ram a very little more. 

 We have already seen that in the neighbourhood of 70° or 80° 

 a small change of angle makes a marked difference in the appear- 

 ance of the spire ; and we know also that the actual length of the 

 horn makes a very striking difference, for the spiral becomes 

 especially conspicuous to the eye when a horn or shell is long 

 enough to shew several whorls, or at least a considerable part of 

 one entire whorl. 



Even in the simplest cases, such as the wild goats, the spiral 

 is never (strictly speaking) a plane or discoid spiral : but in 

 greater or less degree there is always superposed upon the plane 

 logarithmic spiral a helical spiral in space. Sometimes the latter 

 is scarcely apparent, for the helical curvature is comparatively 

 small, and the horn (though long, as in the said wild goats) is not 

 nearly long enough to shew a complete convolution : at other 

 times, as in the ram, and still better in many antelopes, such as 

 the koodoo, the helicoid or corkscrew curve of the horn is its 

 most characteristic feature. 



Accordingly we may study, as in the molluscan shell, the 

 helicoid component of the spire — in other words the variation in 

 what we have called (on p. 555) the angle 6. This factor it is 

 which, more than the constant angle of the logarithmic spiral, 

 imparts a characteristic appearance to the various species of 

 sheep, for instance to the various closely allied species of Asiatic 

 wild sheep, or Argali. In all of these the constant angle of the 

 logarithmic spiral is very much the same, but the shearing com- 

 ponent differs greatly. And thus the long drawn out horns of 



