APPENDIX. 



NOTE TO OH AFTER TUT. OF THE INTRODUCTION UPON THB 

 MODERN GEOMETRY.* 



IT is to be regretted that, notwithstanding its beauty of form, 

 simplicity, and many happy applications in the technical and 

 natural sciences, the Modern Geometry is yet hardly known, 

 scarcely by name even, in our schools and colleges. 



The work of Gillespie upon Land Surveying, already cited in 

 the Introduction, and a treatise on Elementary Geometry by 

 William Chauvenet (Phil., 1871), are the only ones which 

 occur to us in this connection. 



It has already been stated that the modern orjpure geometry 

 of space differs essentially from the ancient, and from analytical 

 geometry, in that it makes no use of the idea of measure or of 

 metrical relations. We find in it no mention of the bisection 

 of lines, of right angles and perpendiculars, of areas, etc., any 

 more than of trigonometrical quantities, or of the analytical 

 equations of lines. We have nothing to do with right-angled, 

 equilateral, or equiangular triangles, with the rectangle, regular 

 polygon, or circle, except in a supplementary manner. So also 

 for the centre, axes, and foci of the so-called curves of the second 

 order, or the conic sections. 



On the contrary, we obtain much more general and compre- 

 hensive properties of these curves than those to which most 

 text-books upon analytical geometry are limited. 



A new path is thus opened to the conic sections, without the 

 aid of the circular cone, after the manner of the ancients, or of 

 the equations of analytical geometry. 



As a direct consequence, the principles and problems of the 

 modern geometry are of great generality and comprehensive- 



* The following remarks and illustrations are taken from the Geometrie der 

 Lage, by Beye. Hannover, 1866. 



