APPENDIX.] 



THE MODERN GEOMETRY. 



347 



other two opposite sides through C, then will the other diagonal 

 intersect the line in a point D, which for the same three points, 



FIG. 2. 



A B C D 



A, B and C, is always the same for every possible construction. 

 Moreover, these four points, A, B, C and D, are always harmonic 

 points, so that D is harmonically separated from B by the 

 points A and C. Thus, AB:BC::AD:CD. This construc- 

 tion may also be applied in surveying, as in passing around an 

 obstruction, as a wood, etc., into the same line again. 



Again : We may notice the following principle concerning 

 the triangle [Fig. 3] : 



FIG. 8. 



If two triangles, ABC and A x B! Ci, are so situated that the 

 lines joining corresponding angles, as A A 1? B BX, C C l5 meet in 

 a common point S, then will the intersections of corresponding 

 sides, as A C and A t C lf A B and A x B 1? B C and B x C t , meet in a 

 common line, as u u. The inverse also, of course, holds good : 

 that if the sides intersect on a line, the lines through the angles 

 intersect in a point. 



Another series of principles are connected with the curves of 

 the second order, or conic sections. From analytical geometry, 



