346 NOTE TO CHAP. Vm. OF THE INTRODUCTION. [APPENDIX. 



ness. Thus the most important of those properties of the conic 

 sections which are proved in text-books of analytical geometry 

 are but special cases of its principles. A few particular ex- 

 amples taken from the Oeometrie der Lage, by Reye, which 

 could not well have been inserted in the Introduction to this 

 work, will best explain and illustrate our general remarks the 

 more so as these examples are of special interest and value to 

 the engineer. 



It is a problem of frequent occurrence in surveying to pass a 

 line through the inaccessible and invisible point of intersection 

 of two given lines. The Geometry of Measure, or ancient 

 geometry, gives us any required number of points upon this line 

 by the aid of the principle, that the distances cut off from par- 

 allel lines by any three lines meeting in a common point are 

 proportional. The Geometry of Position furnishes us with a 

 simpler solution. 



Fio. 1. 



T 



Thus the two lines <z, J being given [Fig. 1.], we have sim- 

 ply to choose any point we please, as P. From this point draw 

 any number of lines desired, in any direction intersecting the 

 given lines. Now, in any quadrilateral which any two of these 

 lines form with the two given lines a and b, we have simply to 

 draw the diagonals. The intersections of all these diagonals 

 lie in the same straight line passing through the intersection A 

 of the two given lines, and therefore determine the line re- 

 quired. Observe that the construction is entirely independent 

 of all metrical relations, and depends solely upon the relative 

 position of the two given lines. 



Again : If we take upon any straight line three points, A, B 

 and C [Fig. 2.], and construct any quadrilateral, two opposite 

 Bides of which pass through A, one diagonal through B, and the 



