POLE AND roL.x1i.] that in a circle all angles at the circumference standing upon the same are are equal. If two points S1, S2 on a circle be joined to any other two points A and B on the circle, then the angle illclllilcil by UN‘ 1'it)'s S,.t and S113 is equal to that between the rays .‘2_ and $213, so that as A moves along the circumfer- nice the rays S,. and S.:A describe a--pial and therefore pi-ojective pencils. The circle can thus be generated by two pro_jeeti'e pencils, and is a curve of the second order. If we join a point in space to all points on a circle, we get a P (circular) cone of the second order (§ livery plane section of this cone is a conic. This conic will be an ellipse, a parabola, or an Fig. 21. hypci-hula, according as the line at infinity in the plane has no, one, or two points in common with the conic in which the plane at infinity cuts the cone. It follows that our curves of second order may be obtained as sections of a circular cone, and that they are identical with the “Conic Sections" of the Greek inatlie- inatieians. §61. Any two tangents to a parabola are cut by all others in projective rows ; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows tliercfore similar. Hcnce— ’l'iiE0r.i£.i.—Thc larzgents to a parabola cut cach other propor- Iionally. Pom: A.‘D PoL.u:. §6'2. “'0 return once again to the remarkable figure (fig. 20) which we obtained in § 55. If a foiir-side be circumscribed about and a four-point inscribed in a conic, so that the vcrticcs of the second are the points of con- tact of the si-lcs of the first, then the triangle. formed by the dia- gonals of the first is the same as that formed by the diagonal points of the otl1ei'. Such a triangle will be called a polar triangle of the conic, so that l’Qll in fig. 20 is a polar triangle. It has the property that on the side 1) opposite P meet the tangents at A and B, and also those at C and I). From the harmonic properties of fonr—points and four-sides it follows further that the points L, M, where it cuts the lines AB and CD, are harmonic conjugates with regard to All and CD respectively. If the point P is given, and we draw a line through it, cutting the conic in A and J3, then the point Q harmonic conjugate to P with regard to AP», and the point H where the tangents at A and B mect, are determined. But they lie both on p, and therefore this line is determined. If we now draw a second line through P, cut- ting the conic in C and I), then the point M harmonic conjiwate to 1’ with regard to CD, and the point G where the tanrrcnts ztit C and D meet, must also lie on p. As the first line through°P already determines p, the second may be any line through P. Now every two lines through P determine a four-point ABCD on the conic and therefore a polar—trianglc which has one vertex at P and its oppo: ite side at 1). This result, together with its reciprocal, give the theorem :— Tiiiaoiimi. ——A ll polar-triangles which have one 'i°crtc.r in common /([1 cc also the opposite side in common. All 1'mlar-tr1'a:z_r/lcs which have one side in common hare also the opposile rent: 2: in conunon. §63. To any point P in the plane of but not on a conic cor- responds thus one line 32 as the side opposite to P in all polar tri- angles which have one vertex at P, and reciprocally to every line p corresponds one point P as the vertex opposite to p in all triangles which have p as one side. _ We call the line p the polar of P, and the point 1’ the pole of the line 12 with regard to the conic_ If a point lies on the conic, we call the tangent at that point its 1”01-11‘; flml reciprocally we call the point of contact the pole of tangent. § 64. From these definitions and former results follow- Tlieorem.—Thc polar of any point I‘ not On the (‘tonic is 3 line 1;’ which has the following properties :- Theorcm.—The pole of any line 19 not a tangent to the conic is a point P, which has the following pro- perties :— GEOMETRY 1. On every line through P which cuts the conic, it contains the har- monic conjugate of 1’ with regard to those points on the conic. 2. If tangents can be drawn from P, their points of contact lie on p. 3. Tangents drawn at the points where any line through 1’ cuts the conic, meet on it; and conversely, 4. If from any point on it tangents be drawn, their points of contact will lie in a line with P. 5. Any four-point on the conic which has one diagonal point at P has the other two lying on 1). The truth of 2 follows from 1. 399 1. Of all lines through a point on p from which two tangents may be drawn to the conic, it contains the line _wlneh is harmonic conjugate to p, with regard to the two tangents. 2. If 1) cnts the conic, the tangents at the intersections meet at P. 3. The point of contact of tangents drawn from any point on p to the conic lie in a line with P; and con- versely, 4. Tangents drawn at points where any line through 1’ cuts the conic meet on p. 5. Any four-side circuinscrihcd about a conic which has one diagonal on 1) has the other two meeting at P. If T he a point where p cuts the conic, then one of the points where PT cuts the conic, and which are harmonic conjugates with regard to PT, coincides with T; hence the other docs—tliat is, PT touches the curve at T. That 4 is true follows thus: If we draw from a point H on the polar one tangent a to the conic, join its point of contact A to the pole P, determine the second point of intersection B of this line with the conic, and draw the tangent at B, it will pass through I l, and will therefore be the second tangent which may be drawn from If to the curve. § 65. The second property of the polar or pole gives rise to the thcorem— Thcorem.—Fi'oni a point in the plane of a conic, two, one, or no tan- gents may be drawn to the conic, according as its polar has two, one, or no points in common with the curve. Tlieorem.—A line in the plane of a conic has two, one, or no points in common with the conic, according as two, one, or no tangents can be drawn from its pole to the conic. Of any point in the plane of a conic we say that it was without, on, or within the curve according as two, one, or no tangents to the curve pass through it. within the conic from those without. is known from elementary geometry. The points on the conic separate those That this is true for a circle That it also holds for other c.onics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on. The fifth property of pole and polar stated in §64 shows how to find the polar of any point and the pole of any line Ly aid of the straiglit-edge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the four-points formed by pairs of these lines and the conic (fig. 21). These constructions also solve the problem :— I’roblcm.—From a point without a conic, to draw the two tan- gents to the conic by aid of the straight-edge only. _ For we need only draw the polar of the point in order to find the points of contact. § 66. The property of a polar-triangle may now be stated thus-- T/icorc2n.—In a polar-triangle each side is the polar of the oppo- site vertex, and each vertex is the pole of the opposite side. If P is one vertex of a polar-triangle, then the other Vertices, Q and R, lie on the polar p of 1’. One of these vcrtices we may chose arbitrarily. For if from any point Q on the polar a sccaiit be drawn cutting the conic in A and D (fig. 22), and if the lines joining these points to P cut the conic again at B and C, then the line BC will pass through Q. Hence P and Q are two ‘:B of the vertices on the polar-triangle which is determined by the four-point ABCD. The third vertex R lies also on the line 72. It follows, therefore, also- Fig. 523. T ii E0iiE.i.—If Q is a point on the polar of P, then P is a point on the polar of Q ; aml reciprocally, If q is a line through the pole of p, then p is a line through the pole of q. This is a very important theorem. It may also be stated tlius— TiiEoIiE.i.—If a point moves along a line describing a row, its polar turns about the pole Of the line describing a pencil. This pencil is projcctirc to the row, so that the cross-ratio of four poles in a row equals the cr0ss—ratio of its four polars, which pass through the pole of the row. To prove the last part, let us suppose that P, A, and B in fig. 22
remain fixed, whilst Q moves along the polar p of P. This will