34:8 NOTE TO CHAP. vm. OF THE INTRODUCTION. [APPENDIX. 



as is well known, a curve of the second order is completely de- 

 termined by five points or five tangents. But the length of the 

 calculation or construction of a curve thus determined is also 

 well known. The geometry of position, however, proves two 

 very important principles, which render it easy to construct to 

 the five given points or tangents any number of new points or 

 tangents, and thus quickly draw the curve itself. The reader 

 already acquainted with these principles will also probably re- 

 member how much auxiliary demonstration their proof in the 

 analytical geometry requires. The first of these, due to Pascal, 

 is, that the three pairs of opposite sides of a hexagon inscribed 

 within a conic section intersect upon a straight line. The 

 second, due to Brianchon^ is, that the three principal diagonals 

 of the circumscribing hexagon, which unite every pair of oppo- 

 site angles, intersect in one and the same point. Both prin- 

 ciples are easily deduced from the circle. It will be observed 

 that they are independent of the relative dimensions, centre, 

 axes, and foci of the curves. For this very reason they are of 

 the greatest generality and significance, so that an entire theory 

 of the conic sections can be based upon them. Thus Pascal's 

 principle solves the important problem of tangent construction 

 from a given point, even when the curve is given by five points 

 only, without completely constructing it. 



This problem of tangent construction to curves of the second 

 order can in many cases be solved by the aid of a principle 

 which expresses one of the most important properties of the 

 conic sections, but which, nevertheless, is seldom found in text- 

 books upon analytical geometry, because its analytical proof is 

 somewhat complicated, and little suited to set forth the property 

 in its proper light. 



For example : If through a point A [Fig. 4] in the plane of 

 but not lying upon a curve of the second order, we draw se- 

 cants, every two secants determine four points, as K, L, M, N, 

 upon the curve. Any two lines joining these four points, as 

 IiM and KN or KM and LIsT, intersect in a point of a 

 straight line a a, which is the polar of the given point A ; that 

 is, which intersects the curve in the two points of tangency 

 G G. Thus the lines through A and the intersections of a a 

 with the curve are the tangents to the curve through A. If 

 the point A were within the curve, this line a a would not in- 

 tersect it. This construction can be used in order to draw 



