CHAPTEPv 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 investigation. 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 consideration : 

 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 physio- 

 logically 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 the horn is 

 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 loga- 

 rithmic 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 curvature is 

 distinctly recognised. As the horn occupies a median position on 

 the head — a position, that is to say, of symmetry in respect to the 

 field of force on either side — there is no tendency towards a lateral 

 twist, and the horn accordingly develops as a plane logarithmic 

 spiral. When two median horns coexist, the hinder one is much 

 the smaller of the two: which is as much as to say that the force, 



