xm] OF SHEEP AND GOATS 887 



they may be found in Mr Rowland Ward's book*), we shall soon 

 see that the two measurements do not tally one with the other: 

 but that a pair of horns, the longer when measured along the curve, 

 may be the shorter from tip to tip, and vice versa. We might set 

 this down to mere variability of form, but the true reason is simpler 

 still. If the axes of the two horns stood straight out, at right angles 

 to the median plane, then growth in length and in width of span 

 would go on together. But if the two horns diverge at any lesser 

 angle, then as the horns grow their spiral curvature will tend to 

 bring their tips nearer and farther apart alternately. 



There is one last, but not least curious property to be seen in 

 a ram's horns. However large and heavy the horns may be — and 

 in Ovis Poll 50 or 60 lb. is no unusual weight for the pair to grow to — 

 the ram carries them with grace and ease, and they neither endanger 

 his poise nor encumber his movements. The reason is that head 

 and horns are very perfectly balanced, in such a way that no bending 

 moment tends to turn the head up or down about its fulcrum in 

 the atlas vertebra ; if one puts two fingers into the foramen magnum 

 one may lift up the heavy skull, and find it hang in perfect equili- 

 brium. Moreover, the horns go on growing, but this equipoise is 

 never lost nor changed ; for the centre of gravity of the logarithmic 

 spiral remains constant. There are other cases where heavy horns, 

 well balanced as they doubtless are, yet visibly affect the set and 

 balance of the head. The stag carries his head higher than a horse, 

 and an Indian buffalo tilts his muzzle higher than a cow. 



A further note upon torsion 



The phenomenon of torsion, to which we have been thus intro- 

 duced, opens up many wide questions in connection with form. Some 

 of the associated phenomena are admirably illustrated in the case 

 of climbing plants ; but we can only deal with these still more briefly 

 and parenthetically. The subject has been elaborately dealt with 

 not only in Darwin's books f, but also by a great number of 

 earlier and later writers. In "twining" plants, which constitute 

 the greater number of "climbers," the essential phenomenon is a 

 tendency of the growing shoot to revolve about a vertical axis — 



* Records of Big Game, 9th edition, 1928. 



t Climbing Plants, 1865 (2nd ed. 1875); Power of Movement in Plants, 1880. 



