880 THE SHAPES OF HORNS [ch. 



the appearance 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 horn or shell is long enough 

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

 convolution. 



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

 never a plane but always a gauche spiral : 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 

 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 

 corkscrew curve of the horn becomes its most characteristic feature. 

 So 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. 816) the angle j^. 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 enveloping angle of the cone differs greatly. Thus the long 

 drawn out horns of Ovis Poli, four feet or more from tip to tip, 

 diifer conspicuously from those of Ovis Ammon or 0. hodgsoni, in 

 which a very similar logarithmic spiral is wound (as it were) round 

 a much blunter cone. 



Let us continue to dispense with mathematics, for the mathe- 

 matical treatment of a gauche spiral is never very simple, and let 

 us deal with the matter by experiment. We have seen that the 

 generating curve, or transverse section, of a typical ram's horn is 

 triangular in form. Measuring (along the curve of the horn) the 

 length of the three edges of the trihedral structure in a specimen of 

 Ovis Ammon, and calling them respectively the outer, inner, and 

 hinder edges (from their position at the base of the horn, relatively 

 to the skull), I find the outer edge to measure 80 cm., the inner 

 74 cm., and the posterior 45 cm. ; let us say that, roughly, they are 

 in the ratio of 9:8:5. Then, if we make a number of little 

 cardboard triangles, equip each with three httle legs (I make them 

 of cork), whose relative lengths are as 9:8:5, and pile them up 



