886 THE SHAPES OF HORNS [ch. 



surface, such as to correspond as nearly as possible with the sinuous 

 outhne*. Finally, after making a plaster oast of this sectional 

 surface, I drew its contour-lines (as shewn in Fig. 437 B) with the 

 help of a simple form of spherometer. It will be seen that in great 

 part this diagram is precisely similar to St Venant's diagram of the 

 •cross-section of a twisted triangular prism; and this is especially 

 the case in the neighbourhood of the sharp angle of our prismatic 

 section. That in parts the diagram is somewhat asymmetrical is 

 not to be wondered at: and (apart from inaccuracies due to the 

 somewhat rough means by which it was made) this asymmetry can 

 be sufficiently accounted for by anisotropy of the material, by 

 inequalities in thickness of different parts of the horny sheath, and 

 especially (I think) by unequal distributions of rigidity due to the 

 presence of the smaller corrugations of the horn. It is on account 

 of these minor corrugations that in such horns as the Highland 

 ram's, where they are strongly marked, the main St Venant effect 

 is not nearly so well shewn as in smoother horns, such as those of 

 Ovis Ammon and its congeners f. 



The distribution of forces which manifest themselves in the 

 growth and configuration of a horn is no simple nor merely super- 

 ficial matter. One thing is coordinated with another; the direction 

 of the axis of the horn, the form of its sectional boundary, the 

 specific rates of growth in the mean spiral and at various parts 

 of its periphery — all' these play their parts, controlled in turn by 

 the supply of nutriment which the character of the adjacent tissues 

 and the distribution of the blood-vessels combine to determine. 

 To suppose. that this or that size or shape of horn has been pro- 

 duced or altered, acquired or lost, by Natural Selection, whensoever 

 one type rather than another proved serviceable for defence or 

 attack or any other purpose, is an hypothesis harder to define and 

 to substantiate than some imagine it to be. 



There are still one or two small matters to speak of before we leave 

 these spiral horns. It is the way of sportsmen to keep record of big 

 game by measuring the length along the curve of the horn and 

 the span from tip to tip. Now if we study such measurements (as 



* This is not difficult to do, with considerable accuracy, if the clay be kept 

 well wetted or semi-fluid, and the smoothing be done with a large wet brush. 



t The curves are well shewn in most of Sir V. Brooke's figures of the various 

 species of Argali, in the paper quoted above, on p. 877. 



