APPENDIX.] 



THE MODERN GEOMETRY. 



through a given point tangents to a conic section by the sim- 

 ple application of straight lines. Upon every secant through 



A, moreover, there are four remarkable points, viz. : the point 

 A itself, the first intersection B with the curve, the intersection 

 with the polar, and, finally, the second intersection D with 

 the curve. These four points are harmonic points, and the 

 polar a a contains, then, every point which is harmonically 

 separated from A by the two curve points. The important 

 principles relating to the centre and conjugate diameter of 

 conic sections are merely special cases of the above important 

 principles. These last can be easily extended to surfaces of 

 the second order, as the intersection of these by a plane is, in 

 general, a curve of the second order. 



From these few examples, which might be indefinitely multi- 

 plied, it may easily be seen how very different, but not less im- 

 portant than those of analytical geometry, are the theorems of the 

 geometry of position. Thus the latter are generally proved by 

 aid of the angle which the tangents make with the line through 

 the focus, or by the distances cut off from the axes that is, by 

 metrical relations. We refer, of course, to the elements of 

 analytical geometry as contained in most text-books, and not to 

 those most fruitful and later methods whose existence are 

 chiefly due to the sagacity of Plucker (Introduction, VIIL). 



