Lines joining corresponding Vertices of Conjugate Triangles meet in a Point, Proof by Reciprocation of Anharmonic Properties, of Theorems concerning Confocal Conics, . Three Points whose Polars with regard to two Conics are the same, their eight Points of Contact lie on the same Conic (see also p. 288), Asymptotes, together with two Conjugate Diameters, form Harmonic Pencil, 273 Lines from two fixed Points to a variable Point, how cut any Parallel to System of Conics through four Points cut any Transversal in involution, Tangent to any Conic cuts off constant Area from similar and concentric Conic, Determination of Radii of Curvature, Excess of sum of two Tangents over included Arc, constant when Vertex moves on Theorems on complete Figure formed by six Points on a Conic,. 317 ANALYTIC GEOMETRY. CHAPTER I. THE POINT. ART. 1. GEOMETRICAL theorems may be divided into two classes: theorems concerning the magnitude of lines, and concerning their position; for example, that "the square of the hypotenuse is equal to the sum of the squares of the sides," is a theorem concerning magnitude; that "the three perpendiculars of a triangle meet in a point," is a theorem concerning position. Theorems of the former class can easily be expressed algebraically. To take the example already given, if the lengths of the sides of a right-angled triangle be a, b, c, the proposition alluded to is written c2 = a2 + b2. The learner is probably already familiar with this application of algebra to geometry, as the propositions of the Second Book of Euclid all relate merely to the magnitude of lines, and the demonstration of them is much simplified by the use of algebraical symbols. But it is by no means so easy to see how to express algebraically theorems involving the position of lines. Accordingly, although algebra was, soon after its introduction into Europe, applied to the solution of the first class of questions, its use was not extended to this latter class until the year 1637, when Des Cartes, by the publication of his " Géométrie,” laid the foundation of the science on which we are about to enter. 2. The following method of determining the position of any B point on a plane is that introduced by Des Cartes, and generally used by succeeding geometers. We are supposed to be given the position of two fixed right lines, XX', YY', intersecting in the point O. Now, if through any point P we draw PM, PN, parallel to YY' and XX', it is plain that, if we knew the position of the point P, we should know the lengths of the parallels PM, PN, or, vice versâ, that if we knew the lengths of PM, PN, we should know the position of the point O. Y P N X M letter x, and the point P is said to be determined by the two equaa, y = b. tions x = 3. The parallels PM, PN, are called the co-ordinates of the point P; that parallel to YY' is often called the ordinate of the point P; and that parallel to XX' the abscissa. The fixed lines XX' and YY' are termed the axes of co-ordinates, and the point O, in which they intersect, is called the origin. The axes are said to be rectangular or oblique, according as the angle at which they intersect is a right angle or oblique. It will readily be seen that the co-ordinates of the point M on the preceding figure are x = a, y = 0; that those of the point N are x = 0, y = b; and that those of the origin itself are x = 0, y = 0.. 4. In order that the equations x = a, y = b, should only be satisfied by one point, it is necessary to pay attention, not only to the magnitudes, but also to the signs of the co-ordinates. If we paid no attention to the signs of the co-ordinates, we might measure OM = a and ON = b, on either side of the origin, |