346. If there be a system of conics having a common self-conjugate triangle, any line passing through one of the vertices of this triangle is cut by the system in involution.

For, if in aa2 + bß* + cy2 we write a = kß, we get (ak2 + b) B2 + cy",

a pair of points evidently always harmonically conjugate with the two points where the line meets 8 and y.

Thus, then, in particular, a system of conics touching the four sides of a fixed quadrilateral cuts in involution any transversal which passes through one of the intersections of diagonals of the quadrilateral (Ex. 3, Art. 146). The points in which the transversal meets diagonals are the foci of the system, and the points where it meets opposite sides of the quadrilateral are conjugate points of the system.

Ex. 1. If two conics U, V touch their common tangents A, B, C, D in the points a, b, c, d ; a', b', c', d'; a conic S through the points a, b, c, and touching D at ď, will have for its second chord of intersection with V, the line joining the intersections of A with bc, B with ca, C with ab.

Let V meet ab in a, ß, then, by this article, since ab passes through an intersection of diagonals of ABCD (Ex. 2, Art. 263), a, b ; a, ẞ belong to a system in involution, of which the points where ab meets C and D are conjugate points. But (Art. 345) the common chords of S and V meet ab in points belonging to this same system in involution, determined by the points a, b; a, ß, in which S and V meet the line ab. If then one of the common chords be D, the other must pass through the intersection of C with ab.

Ex. 2. If in a triangle there be inscribed an ellipse touching the sides at their middle points a, b, c, and also a circle touching at the points a', b', c', and if the fourth common tangent D to the ellipse and circle touch the circle at d', then the circle described through the middle points touches the inscribed circle at d'. By Ex. 1, a conic described through a, b, c, will touch the circle at d', if it also pass through the points where the circle is met by the line joining the intersections of A, bc; B, ca; C, ab. But this line is in this case the line at infinity. The touching conic is therefore a circle. Sir W. R. Hamilton has thus deduced Feuerbach's theorem (p. 127) as a particular case of Ex. 1.

The point d' and the line D can be constructed without drawing the ellipse. For since the diagonals of an inscribed, and of the corresponding circumscribing quadrilateral meet in a point, the lines ab, cd; a'b', c'd', and the lines joining AD, BC; AC, BD all intersect in the same point. If then a, ẞ, y be the vertices of the triangle formed by the intersections of be, b'c'; ca, c'a'; ab, a'b'; the lines joining a'a, b'ß, c'y meet in d. In other words, the triangle aẞy is homologous with abc, a'b'c', the centres of homology being the points d, d'. In like manner, the triangle aßy is also homologous with ABC, the axis of homology being the line D.

SS.

CHAPTER XVII.

THE METHOD OF PROJECTION.*

347. WE have already several times had occasion to point out to the reader the advantage gained by taking notice of the number of particular theorems often included under one general enunciation, but we now propose to lay before him a short sketch of a method which renders us a still more important service, and which enables us to tell when from a particular given theorem we can safely infer the general one under which it is contained.

If all the points of any figure be joined to any fixed point in space (0), the joining lines will form a cone, of which the point is called the vertex, and the section of this cone, by any plane, will form a figure which is called the projection of the given figure. The plane by which the cone is cut is called the plane of projection.

To any point of one figure will correspond a point in the other. For, if any point A be joined to the vertex O, the point a, in which the joining line OA is cut by any plane, will be the projection on that plane of the given point A.

A right line will always be projected into a right line.

For, if all the points of the right line be joined to the vertex, the joining lines will form a plane, and this plane will be intersected by any plane of projection in a right line.

Hence, if any number of points in one figure lie in a right line, so will also the corresponding points on the projection; and if any number of lines in one figure pass through a point, so will also the corresponding lines on the projection.

*This method is the invention of M. Poncelet. See his Traité des Propriétés Projectives, published in the year 1822, a work which I believe may be regarded as the foundation of the Modern Geometry. In it were taught the principles, that theorems concerning infinitely distant points may be extended to finite points on a right line; that theorems concerning systems of circles may be extended to conics having two points common; and that theorems concerning imaginary points and lines may be extended to real points and lines.

348. Any plane curve will always be projected into another curve of the same degree.

For it is plain that, if the given curve be cut by any right line in any number of points, A, B, C, D, &c. the projection will be cut by the projection of that right line in the same number of corresponding points, a, b, c, d, &c. ; but the degree of a curve is estimated geometrically by the number of points in which it can be cut by any right line. If AB meet the curve in some real and some imaginary points, ab will meet the projection in the same number of real and the same number of imaginary points.

In like manner, if any two curves intersect, their projections will intersect in the same number of points, and any point common to one pair, whether real or imaginary, must be considered as the projection of a corresponding real or imaginary point common to the other pair.

Any tangent to one curve will be projected into a tangent to the other.

For, any line AB on one curve must be projected into the line ab joining the corresponding points of the projection. Now, if the points A, B, coincide, the points a, b, will also coincide, and the line ab will be a tangent.

More generally, if any two curves touch each other in any number of points, their projections will touch each other in the same number of points.

349. If a plane through the vertex parallel to the plane of projection meet the original plane in a line AB, then any pencil of lines diverging from a point on AB will be projected into a system of parallel lines on the plane of projection. For, since the line from the vertex to any point of AB meets the plane of projection at an infinite distance, the intersection of any two lines which meet on AB is projected to an infinite distance on the plane of projection. Conversely, any system of parallel lines on the original plane is projected into a system of lines meeting in a point on the line DF, where a plane through the vertex parallel to the original plane is cut by the plane of projection. The method of projection then leads us naturally to the conclusion, that any system of parallel lines may be considered as passing through a point at an infinite distance, for their projections on any plane

pass through a point in general at a finite distance; and again, that all the points at infinity on any plane may be considered as lying on a right line, since we have showed that the projection of any point in which parallel lines intersect must lie somewhere on the right line DF in the plane of projection.

350. We see now, that if any property of a given curve does not involve the magnitude of lines or angles, but merely relates to the position of lines as drawn to certain points, or touching certain curves, or to the position of points, &c., then this property will be true for any curve into which the given curve can be projected. Thus, for instance, "if through any point in the plane of a circle a chord be drawn, the tangents at its extremities will meet on a fixed line." Now since we shall presently prove that every curve of the second degree can be projected into a circle, the method of projection shows at once that the properties of poles and polars are true not only for the circle, but also for all curves of the second degree. Again, Pascal's and Brianchon's theorems are properties of the same class, which it is sufficient to prove in the case of the circle, in order to know that they are true for all conic sections.

351. Properties which, if true for any figure, are true for its projection, are called projective properties. Besides the classes of theorems mentioned in the last Article, there are many projective theorems which do involve the magnitude of lines. For instance, the anharmonic ratio of four points in a right line {ABCD}, being measured by the ratio of the pencil {0.ABCD} drawn to the vertex, must be the same as that of the four points {abcd}, where this pencil is cut by any transversal. Again, if there be an equation between the mutual distances of any number of points in a right line, such as

AB. CD.EF+k.AC.BE.DF+1.AD.CE.BF+ &c. = 0, where in each term of the equation the same points are mentioned, although in different orders, this property will be projective. For (see Art. 311) if for AB we substitute

each term of the equation will contain OA. OB. OC. OD. OE.OF

in the numerator, and OP3 in the denominator. Dividing, then, by these, there will remain merely a relation between the sines of angles subtended at O. It is evident that the points A, B, C, D, E, F, need not be on the same right line; or, in other words, that the perpendicular OP need not be the same for all, provided the points be so taken that, after the substitution, each term of the equation may contain in the denominator the same product, OP. OP. OP", &c. Thus, for example, "If lines meeting in a point and drawn through the vertices of a triangle ABC meet the opposite sides in the points a, b, c, then Ab. Bc. Ca = Ac.Ba. Cb." This is a relation of the class just mentioned, and which it is sufficient to prove for any projection of the triangle ABC. Let us suppose the point C projected to an infinite distance, then AC, BC, Cc are parallel, and the relation becomes

the truth of which is at once perceived on making the figure.

352. It appears, from what has been said, that if we wish to demonstrate any projective property of any figure, it is sufficient to demonstrate it for the simplest figure into which the given figure can be projected; e.g. for one in which any line of the given figure is at an infinite distance.

Thus, if it were required to investigate the harmonic properties of a complete quadrilateral ABCD, whose opposite sides intersect in E, F, and the intersection of whose diagonals is G, we may join all the points of this figure to any point in space 0, and cut the joining lines by any plane parallel to OEF, then EF is projected to infinity, and we have a new quadrilateral, whose sides ab, cd intersect in e at infinity, that is, are parallel; while ad, be intersect in a point ƒ at infinity, or are also parallel. We thus see that any quadrilateral may be projected into a parallelogram. Now since the diagonals of a parallelogram bisect each other, the diagonal ac is cut harmonically in the points a, g, c, and the point where it meets the line at infinity ef. Hence AB is cut harmonically in the points A, G, C, and where it meets EF.

Ex. If two triangles ABC, A'B'C', be such that the points of intersection of AB, A'B'; BC, B'C'; CA, C'A'; lie in a right line, then the lines AA', BB', CC' meet in a point.