799 of a triangle a more general one. But before this could be done we had to add the line at infinity to the Hues in the given figure. In a similar manner a great many theorems relating to metrical properties can be generalized by taking the line at infinity or points at infinity as forming part of the original figure. Conversely special cases relating to measurement are obtained by projecting ! some line in a figure of known properties to infinity. This is true , for all properties relating to parallel lines or to bisection of , segments, but not immediately for angles. It is, however, possible to establish for every metrical relation the corresponding protective property. To do this it is necessary to consider imaginary elements. These have originally been introduced into geometry by aid of coordinate geometry, where imaginary quantities constantly occur as roots of equations. Their introduction into pure geometry is due principally to Poncelet, who by the publication of his great work Traite des Proprietes Project ives des Figures became the founder of projective geometry in its widest sense. Monge had considered parallel projection and had already distinguished between permanent and accidental properties of figures, the latter being those which depended merely on the accidental position of one part to another. Thus in projecting two circles which lie in different planes it depends on the accidental position of the centre of projection whether the projections be two conies which do or do not meet. Poncelet introduced the principle of continuity in order to make theorems general and independent of those accidental positions which depend analytically on the fact that the equations used have real or imaginary roots. But the correctness of this principle re mained without a proof. Von Staudt has, however, shown how it is possible to introduce imaginary elements by purely geometrical reasoning, and we shall now try to give the reader some idea of his theory. 25. IMAGINARY ELEMENTS. If a line cuts a curve and if the line be moved, turned for instance about a point in it, it may happen that two of the points of iutersi ction approach each other till they coincide. The line then becomes a tangent. If the line is still further moved in the same manner it separates from the curve and two points of intersection are lost. Thus in considering the rela tion of a line to a conic we have to distinguish three cases the line cuts the conic in two points, touches it, or has no point in common with it. This is quite analogous to the fact that a quadratic equation with one unknown quantity has either two, one, or no roots. But in algebra it has long been found convenient to express this differently by saying a quadratic equation has always two roots, but these may be either both real and different, or equal, or they may be imaginary. In geometry a similar mode of expressing the fact above stated is not less convenient. We say therefore a line has always two points in common with a conic, but these are either distinct, or coincident, or invisible. The word imaginary is generally used instead of invisible ; but, as the points have nothing to do with imagination, we prefer the word " invisible " recommended originally by Clifford. Invisible points occur in pairs of conjugate points, for a line loses always two visible points of intersection with a curve simultane ously. This is analogous to the fact that an algebraical equation with real coefficients has imaginary roots in pairs. Only one real line can be drawn through an invisible point, for two real lines meet in a real or visible point. The real line through an invisible point contains also its conjugate. Similarly there are invisible lines tangents, for instance, from a point within a conic which occur in pairs of conjugates, two con jugates having a real point in common. The introduction of invisible points would be nothing but a play upon words unless there is a real geometrical property indicated which can be used in geometrical constructions that it has a definite meaning, for instance, to say that two conies cut a line in the same two invisible points, or that we can draw one conic through three real points and the two invisible ones which another conic has in common with a line that does not actually cut it. We have in fact to give a geometrical definition of invisible points. This is done by aid of the theory of involution (G. 76 sq.). An involution of points on a line has (according to G. 77 (2)) either two or one or no foci. Instead of this we now say it has always two foci which may be distinct, coincident, or invisible. These foci are determined by the involution, but the} also determine the involution. If the foci are real this follows from the fact that conjugate points are harmonic conjugates with regard to the foci. That it is also the case for invisible foci will presently appear. If we take this at present for granted we may replace a pair of real, coincident, or invisible points by the involution of which they are the foci. Now any two pairs of conjugate points determine an involution (G. 77(6)). Hence any point-pair, whether real or invisible, is completely determined by any two pairs of conjugate points <>f the involution which has the given point-pair as foci and may therefore be replaced by them. Two pairs of invisible points are thus said to be identical if, and only if, they are the foci of the same involution. We know (G. 82) that a conic determines on every line an in volution in which conjugate points are conjugate poles with regard to the conic that is, that either lies on the polar of the other. This holds whether the line cuts the conic or not. Furthermore, in the former case the points common to the line and the conic are the foci of the involution. Hence we now say that this is always the case, and that the invisible points common to a line and a conic are the invisible foci of the involution in question. If then we state the problem of drawing a conic which passes through two points given as the intersection of a conic and a line as that of drawing a conic which determines a given involution on the line, we have it in a form in which it is independent of the accidental circumstance of the intersections being real or invisible. So is the solution of the problem, as we shall now show. 26. We have seen ( 21) that a conic may always be projected into itself by taking any point S as centre and its polar s as axis of projection, corresponding points being those in which a line through S cuts the conic. If then (fig. 9) A, A and B, B are pairs of corresponding points so that the lines AA and BB pass through S, then the lines AB and A B , as corresponding lines, will meet at a point Ron the axis, and the lines AB and A B will meet at another point R on the axis. These points R, R are conjugate points in the involution which the conic determines on the line s, because the triangle RSR is a polar triangle (G. 62), so that R lies on the polar of R. This gives a simple means of determining for any point Q on the line s its conjugate point Q . We take any two points A, A on the conic which lie on a line through S, join Q to A by a line cutting the conic again in C, and join C to A . This line will cut s in the point Q required. PROBLEM. To draw some conic which shall determine on a line s a given involution. Solution. We have here to reconstruct the fig. 9, having given on the line s an involution. Let Q, Q and R, R (fig. 9) be two pairs of conjugate points in this involution. We take any point B and join it to R and R , and another point C to Q and Q . Let BR and CQ meet at A, and BR and CQ at A . If now a point P be moved along s its conjugate point P will also move and the two points will describe projective rows. The two rays AP and A P will therefore describe projective pencils, and the intersection of corre sponding rays will lie on a conic which passes through A, A , B, and C. This conic determines on s the given involution. Of these four points not only B and C but also the point A may be taken arbitrarily, for if A, B, C are given, the line AB will cut s in some point R, As the involution is supposed known, we can find the point R conjugate to R, which we join to B. In the same way the line CA will cut s in some point Q. Its conjugate point Q we join to C. The line CQ will cut BR in a point A , and then AA will pass through the pole S (comp. fig. 9). We may now inter change A and B and find the point B . Then BB will also pass through S, which is thus found. At the same time five points A, B, C, A , B on the conic have been found, so that the conic is completely known Avhich determines on the line s the given involu tion. Hence THEOREM. Through three points we can ahuays draw one conic, and only one, ivhich determines on a given line a given involution, all the same whether the involution has real, coincident, or invisible foci. In the last case the theorem may now also be stated thus : THEOREM. It is always possible to draw a conic u-hich passes through three given real points and through two invisible points which any other conic has in common with a line. 27. The above theory of invisible points gives rise to a great number of interesting consequences of which we state a few. The theorem at the end of 21 may now be stated : THEOREM. Any two conies arc similar and similarly situated if they cut the line at infinity in the same two points real, coincident, or invisible. It follows that Any two parabolas are similar; and they are similarly situated as soon as their axes are parallel. The involution which a circle determines at its centre is circular (G. 79) ; that is, every line is perpendicular to its conjugate line. This will be cut by the line at infinity in an involution which has 1 the following property: The lines which join any finite point to two conjugate points in the involution are at right angles to each
I other. Hence all circular involutions in a plane determine the-