618 THE SHAPES OF HORNS [ch. 



Ovis Poll, four feet or more from tip to tip, differ conspicuously 

 from those of Ovis Amnion or 0. hodgsoni, in which a very similar 

 logarithmic spiral is wound (as it were) round a much blunter cone. 



The ram's horn then, like the snail's shell, is a curve of double 

 curvature, in which one component has imposed upon the structure 

 a plane logarithmic spiral, and the other has produced a continuous 

 displacement, or "shear," proportionate in magnitude to, and 

 perpendicular or otherwise inclined in direction to, the axis of 

 the former spiral curvature. The result is precisely analogous to 

 that which we have studied in the snail and other spiral univalves ; 

 but while the form, and therefore the resultant forces, are similar, 

 the original distribution of force is not the same : for we have not 

 here, as we had in the snail-shell, a "columellar" muscle, to 

 introduce the component acting in the direction of the axis. We 

 have, it is true, the central bony core, which in part performs an 

 analogous function ; but the main phenomenon here is apparently 

 a complex distribution of rates of growth, perpendicular to the 

 plane of the generating curve. 



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

 matical treatment of a curve of double curvature 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 little 

 legs (I make them of cork), whose relative lengths are as 9 : 8 : 5, 

 and pile them up and stick them all together, we straightway 

 build up a curve of 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 growth in its own plane of the generating curve. 



