depends on the accidental position of the centre of projection
whether the projections be two conics 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
remained 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 intersection 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 relation 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 Clifiord. Invisible points occur in pairs of conjugate points, for a line loses always two visible points of intersection with a curve simultaneously. This is analogous to the fact that an algebraical equation with real coefficients has imaginary roots in pairs. Only onereal 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'ugates having a real point in common.
Iihe 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 conics 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 seq.). 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 they 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, clqinpident or invisible points by the involution of which they are t e oci.
Now any two pairs of conjugate points determine an involution (G- § 77 l6l)-Hence
any point-pair, whether real or invisible, is completely determined by any two pairs of conjugate points of the involution fzhich has given the point-pair as foci and may therefore be replaced them.
yTwo 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 involution 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 R on 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.,
To draw some conic which shall determine on a line s a given involution.
We have here to reconstruct the fig. 9, having given on the line s an involution. Let Q, Q'vandV R, R' (fig. 9) be two pairs of conjugate points in this (3
involution. We take any
point B and join it to R »
ancl R', and another point A
C to Q and Q'. Let BR /
CQ. at
an ' at .,
now a point P be moved " B
along s its conjugate point ' QA
P' will also move and the .f s, 1 s
two points will describe Q R' Q R
projective rows The two FIG 9
rays' AP and A'P' will
therefore describe projective pencils, and the intersection of corresponding 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 throu h the pole S (cf. fig. 9). We may now interchange A and B and find the point B'. Then BB' will also pass throu h 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 which determines on the line s the given involution. Hence-Through three points wecan always draw one conic, and only one, which 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:- It is always possible to draw a conic which 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 § 2I may now be stated:- Any two conics are 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 the following property: The lines which join any finite point to two conjugate points in the involution are at right angles to each other. Hence all circular involutions in a plane determine the same involution on the line at infinity. The latter is therefore called the circular involution on the line at injinity; and the involution which a circle determines at its centre is called the circular involution at that point. All circles determine thus on the line at infinity the same involution; in other words, they have the same two invisible points in common with the line at infinity. All circles may be considered as passing through the same two points at injinity.
These points are called the circular points at infinity, and by Professor Cayley the absolute in the plane. They are the foci of the circular involution in the line at infinity. Conversely—Every conic which passes through the circular points is a circle; because the involution at its centre is circular, hence ° angles, and this property only
conjugate diameters are at right
circles possess.-We
now see why we can draw
through any three points; these
circular points at infinity are five
only can be drawn.
Any two circles are similar and similarly situated because they have the same points at infinity (§ 21).
Any two concentric circles may be considered as having double contact at infinity, because the lines joining the common centre to the circular points at infinity are tangents to both circles at the circular points, as the line at infinity is the polar of the centre. Any two lines at right angles to one another are harmonic conjugates with regard to the rays joining their intersection to the circular points, because these rays are the focal rays of the circular involution at the intersection of the given lines. I
To bisect an angle with the vertex A means (G. § 23) to find two rays through A which are harmonic conjugates with regard to the always one and only one circle
three points together with the
points through' which one conic
