in tangential coordinates, yield another theorem, the reciprocal of the former. Thus (Art. 266) we proved that if three conics (S, S+ LM, S+LN) have two points (S, L) common to all, the chords in each case joining the remaining common points (M, N, M-N), will meet in a point. Consider these as tangential equations, and the pair of tangents drawn from L is common to the three conics, while M, N, M- N denote in each case the point of intersection of the other two common tangents. We thus get the theorem, "If three conics have two tangents common to all, the intersections in each case of the remaining pair of common tangents, lie in a right line." Every theorem of position (that is to say, one not involving the magnitudes of lines or angles) is thus twofold. From each theorem another can be derived by suitably interchanging the words "point" and "line"; and the same equations differently interpreted will establish either theorem. We shall in this chapter give an account of the geometrical method by which the attention of mathematicians was first called to this "principle of duality."

300. Being given a fixed conic section (U) and any curve (S), we can generate another curve (s) as follows: draw any tangent to S, and take its pole with regard to U; the locus of this pole will be a curve s, which is called the polar curve of S

with regard to U.

The conic U, with regard to which the pole is taken, is called the auxiliary conic.

We have already met with a particular example of polar curves (Ex. 12, Art. 225), where we proved that the polar curve of one conic section with regard to another is always a curve of the second degree.

We shall for brevity say that a point corresponds to a line when we mean that the point is the pole of that line with regard to U. Thus, since it appears from our definition that every point of s is the pole with regard to U of some tangent to S, we shall

The method of reciprocal polars was introduced by M. Poncelet, whose account of it will be found in Crelle's Journal, vol. IV. M. Plücker, in his "System der Analytischen Geometrie," 1835, presented the principle of duality in the purely analytical point of view, from which the subject is treated at the beginning of this chapter. But it was Möbius who, in his "Barycentrische Calcul," 1827, had made the important step of introducing a system of coordinates in which the position of a right line was indicated by coordinates and that of a point by an equation.

briefly express this relation by saying that every point of s corresponds to some tangent of S.

301. The point of intersection of two tangents to S will correspond to the line joining the corresponding points of s.

This follows from the property of the conic U, that the point of intersection of any two lines is the pole of the line joining the poles of these two lines (Art. 146).

Let us suppose that in this theorem the two tangents to S are indefinitely near, then the two corresponding points of s will also be indefinitely near, and the line joining them will be a tangent to s; and since any tangent to S intersects the consecutive tangent at its point of contact, the last theorem becomes for this case: If any tangent to S correspond to a point on s, the point of contact of that tangent to S will correspond to the tangent through the point on s.

Hence we see that the relation between the curves is reciprocal, that is to say, that the curve S might be generated from s in precisely the same manner that s was generated from S. Hence the name "reciprocal polars."

302. We are now able, being given any theorem of position concerning any curve S, to deduce another concerning the curve s. Thus, for example, if we know that a number of points connected with the figure S lie on one right line, we learn that the corresponding lines connected with the figure s meet in a point (Art. 146), and vice versâ; if a number of points connected with the figure S lie on a conic section, the corresponding lines connected with s will touch the polar of that conic with regard to U; or, in general, if the locus of any point connected with S be any curve S', the envelope of the corresponding line connected with s is s', the reciprocal polar of S.

303. The degree of the polar reciprocal of any curve is equal to the class of the curve (see note, Art. 145), that is, to the number of tangents which can be drawn from any point to that curve.

For the degree of s is the same as the number of points in which any line cuts s; and to a number of points on s, lying on a right line, correspond the same number of tangents to S passing through the point corresponding to that line. Thus, if S be a

conic section, two, and only two, tangents, real or imaginary, can be drawn to it from any point (Art. 145); therefore, any line meets s in two, and only two points, real or imaginary; we may thus infer, independently of Ex. 12, Art. 225, that the reciprocal of any conic section is a curve of the second degree.

304. We shall exemplify, in the case where S and s are conic sections, the mode of obtaining one theorem from another by this method. We know (Art. 267) that "if a hexagon be inscribed in S, whose sides are A, B, C, D, E, F, then the points of intersection, AD, BE, CF, are in one right line." Hence we infer, that "if a hexagon be circumscribed about s, whose vertices are a, b, c, d, e, f, then the lines, ad, be, cf, will meet in a point" (Art. 265). Thus we see that Pascal's theorem and Brianchon's are reciprocal to each other, and it was thus, in fact, that the latter was first obtained.

In order to give the student an opportunity of rendering himself expert in the application of this method, we shall write in parallel columns some theorems, together with their reciprocals. The beginner ought carefully to examine the force of the argument by which the one is inferred from the other, and he ought to attempt to form for himself the reciprocal of each theorem before looking at the reciprocal we have given. He will soon find that the operation of forming the reciprocal theoren will reduce itself to a mere mechanical process of interchanging the words "point" and "line," "inscribed " and "circumscribed," "locus" and "envelope," &c.

If two vertices of a triangle move along fixed right lines, while the sides pass each through a fixed point, the locus of the third vertex is a conic section. (Art. 269).

If, however, the points through which the sides pass lie in one right line, the locus will be a right line. (Ex. 2. p. 41). In what other case will the locus be

a right line? (Ex. 3, p. 42).

If two sides of a triangle pass through fixed points, while the vertices move on fixed right lines, the envelope of the third side is a conic section.

If the lines on which the vertices move meet in a point, the third side will pass through a fixed point.

In what other case will the third side pass through a fixed point? (p. 49).

If two conics touch, their reciprocals will also touch; for the first pair have a point common, and also the tangent at that point common, therefore the second pair will have a tangent common and its point of contact also common. So likewise if two conics have double contact their reciprocals will have double contact.

If a triangle be circumscribed to a conic section, two of whose vertices move on fixed lines, the locus of the third vertex is a conic section, having double contact with the given one. (Ex. 2, p. 250).

If a triangle be inscribed in a conic section, two of whose sides pass through fixed points, the envelope of the third side is a conic section, having double contact with the given one. (Ex. 3, p. 250).

305. We proved (Art. 301, see figure, p. 282) if to two points P, P', on S, correspond the tangents pt, p't', on s, that the tangents at P and P' will correspond to the points of contact p, p', and therefore Q, the intersection of these tangents, will correspond to the chord of contact pp'. Hence we learn that to any point Q, and its polar PP', with respect to S, correspond a line pp' and its pole q with respect to s.

Given two points on a conic, and two of its tangents, the line joining the points of contact of those tangents passes through one or other of two fixed points. (Ex., Art. 286, p. 262).

Given four points on a conic, the polar of a fixed point passes through a fixed point. (Ex. 2, p. 153).

Given four points on a conic, the locus of the pole of a fixed right line is a conic section. (Ex. 1, p. 254).

The lines joining the vertices of a triangle to the opposite vertices of its polar triangle with regard to a conic meet in a point. (Art. 99).

Inscribe in a conic a triangle whose sides pass through three given points. (Ex. 7, Art. 297, p. 273).

Given two tangents and two points on a conic, the point of intersection of the tangents at those points will move along one or other of two fixed right lines.

Given four tangents to a conic, the locus of the pole of a fixed right line is a right line. (Ex. 2, p. 254).

Given four tangents to a conic, the envelope of the polar of a fixed point is a conic section.

The points of intersection of each side of any triangle, with the opposite side of the polar triangle, lie in one right line.

Circumscribe about a conic a triangle whose vertices rest on three given lines.

306. Given two conics, S and S, and their two reciprocals, s and s'; to the four points A, B, C, D common to S and S correspond the four tangents a, b, c, d common to s and s', and to the six chords of intersection of S and S, AB, CD; AC, BD; AD, BC correspond the six intersections of common tangents to s and s'; ab, cd; ac, bd; ad, bc.*

If three conics have two common tangents, or if they have each double contact with a fourth, their six chords of intersection will pass three by three through the same points. (Art. 264).

Or, in other words, three conics, having each double contact with a fourth, may be

If three conics have two points common, or if they have each double contact with a fourth, the six points of intersection of common tangents lie three by three on the same right lines.

Or three conics, having each double contact with a fourth, may be considered

A system of four points connected by six lines is accurately called a quadrangle, as a system of four lines intersecting in six points is called a quadrilateral.

considered as having four radical centres.

If through the point of contact of two conics which touch, any chord be drawn, tangents at its extremities will meet on the common chord of the two conics.

If through an intersection of common tangents of two conics any two chords be drawn, lines joining their extremities will intersect on one or other of the common chords of the two conics. (Ex. 1, p. 250).

If A and B be two conics having each double contact with S, the chords of contact of A and B with S, and their chords of intersection with each other, meet in a point, and form a harmonic pencil. (Art. 263).

If A, B, C be three conics, having each double contact with S, and if A and B both touch C, the tangents at the points of contact will intersect on a common chord of A and B.

as having four axes of similitude. (See Art. 117, of which this theorem is an ex tension).

If from any point on the tangent at the point of contact of two conics which touch, a tangent be drawn to each, the line joining their points of contact will pass through the intersection of common tangents to the conics.

If on a common chord of two conics, any two points be taken, and from these tangents be drawn to the conics, the diagonals of the quadrilateral so formed will pass through one or other of the intersections of common tangents to the conics.

If A and B be two conics having each double contact with S, the intersections of the tangents at their points of contact with S, and the intersections of tangents common to A and B, lie in one right line, which they divide harmonically.

If A, B, C be three conics, having each double contact with S, and if A and B both touch C, the line joining the points of contact will pass through an intersection of common tangents of A and B.

307. We have hitherto supposed the auxiliary conic U to be any conic whatever. It is most common, however, to suppose this conic a circle; and hereafter, when we speak of polar curves, we intend the reader to understand polars with regard to a circle, unless we expressly state otherwise.

We know (Art. 88) that the polar of any point with regard to a circle is perpendicular to the line joining this point to the centre, and that the distances of the point and its polar are, when multiplied together, equal to the square of the radius; hence the relation between polar curves with regard to a circle is often stated as follows: Being given any point O, if from it we let fall a perpendicular OT on any tangent to a curve S, and produce it until the rectangle OT.Op is equal to a constant k3, then the

locus of the point p is a curve s, which is called the polar reciprocal of S. For this is evidently

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