MATHEMATICS 37 



invalidate the absolute truth of the determinations of 

 geometry themselves, which are more accurately con- 

 formed to, the more the nature of any material ren- 

 ders it able to approach more nearly to the perfection 

 desired. 



Geometry arose through desires and efforts to measure 

 land accurately, and the properties of angles and triangles 

 actually serve this process now. One of the most useful pro- 

 perties of triangles consists in the fact that two of them, 

 however different in size, are in other respects exactly 

 similar to each other if the angles of one are severally 

 equal to the angles of the other. It is by the aid of 

 such considerations that many of the most important and 

 prodigious scientific measurements have been effected.* 



Euclid's work treats not only of lines, angles, triangles, 

 circles, &c., but of the geometrical properties of solid 

 figures of several different shapes. 



Greek geometers occupied themselves, in a purely 

 speculative manner, with the different methods in which 

 a circular cone may be cut. The investigation of the 

 various kinds of curves which may be produced at the 

 edge of such a cone by cutting across t it in different 

 directions, constituted the study known as " Conic 

 Sections." The importance of these investigations will 

 become clear when we have to consider falling and other 

 movements of various bodies. 



Very many geometrical propositions which were long 

 thought incapable of investigation and solution save by 

 the method proper to geometry, were subsequently 

 found capable of more convenient treatment by the aid 

 of algebra, a change which has produced most impor- 

 tant results in the study of astronomy. 



* See post, p. 177. f See post, p. 65. 



