CHAPTER XIII 



THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: 

 WITH A NOTE ON TORSION 



We have had so much to say on the subject of shell-spirals 

 that we must deal briefly with the analogous problems which are 

 presented by the horns of sheep, goats, antelopes and other 

 horned quadrupeds; and all the more, because these horn-spirals 

 are on the whole less symmetrical, less easy of measurement than 

 those of the shell, and in other ways also are less easy of investi- 

 gation. Let us dispense altogether in this case with mathematics ; 

 and be content with a very simple account of the configuration 

 of a horn. 



There are three types of horn which deserve separate con- 

 sideration: firstly, the horn of the rhinoceros; secondly the 

 horns of the sheep, the goat, the ox or the antelope, that is to say, 

 of the so-called hollow-horned ruminants; and thirdly, the solid 

 bony horns, or "antlers," which are characteristic of the deer. 



The horn of the rhinoceros presents no difficulty. It is 

 physiologically equivalent to a mass of consolidated hairs, and, 

 like ordinary hair, it consists of non-living or "formed" material, 

 continually added to by the living tissues at its base. In section, 

 that is to say in the form of its "generating curve," the horn is 

 approximately elliptical, with the long axis fore-and-aft, or, in 

 some species, nearly circular. Its longitudinal growth proceeds 

 with a maximum velocity anteriorly, and a minimum posteriorly ; 

 and the ratio of these velocities being constant, the horn curves 

 into the form of a logarithmic spiral in the manner that we have 

 already studied. The spiral is of small angle, but in the longer- 

 horned species, such as the great white rhinoceros (Ceratorhinus), 

 the spiral form is distinctly to be recognised. As the horn 



