718 CURVE line y ^ ) the two intersections are the point (x=-r, y = /3), and the point at infinity oil the line y = ft. It is moreover to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly ; to support the theorem in its universality, it is necessary to take account of these Various circumstances. The foregoing notion of a point at infinity is a very im portant one in modern geometry ; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line. This admits of an easy illustration in solid geometry. If with a given centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line, i.e., this projection is the intersection of the given plane with the plane through the centre and the given line. Say the projection is always a line, then if the figure is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane. 1 . Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it in m points, real or imaginary, distinct or coincident. Descartes in the Geomttrie defined and considered the remarkable curves called after him ovals of Descartes, or simply Cartesians, which will be again referred to. The next important work, founded on the Geometric, was Sir Isaac Newton s Enumeratio linearum tertii ordinis (1706), establishing a classification of cubic curves founded chiefly on the nature of their infinite branches, which was in some details completed by Stirling, Murdoch, and Cramer; the work contains also the remarkable theorem (to be again referred to), that there are five kinds of cubic curves giving by their projections every cubic curve whatever. Various properties of curves in general, and of cubic curves, are established in Maclaurin s memoir, "De linearum geometricarum proprietatibus generalibus Tractatus " (posthumous, say 1746, published in the 6th edition of his Algebra). We have in it a particular kind of correspond ence of two points on a cubic curve, viz,, two points corre spond to each other when the tangents at the two points again meet the cubic in the same point. The Geometric Descriptive by Monge was written in the year 1794 or 1795 (7th edition, Paris, 1847), and in it we have stated, in piano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, .viz., " Given a surface of the second order and a circumscribed conic surface which touches it .... then if the conic surface moves so that its summit is always in the same plane, the plane of the curve of contact passes always through the same point." The theorem is here referred to partly on account of its bearing on the theory of imaginaries in geometry. It is in Brianchon s memoir " Sur les surfaces du second degre" " (Jour. Polyt., t. vi. 1806) shown how for any given position of the summit the plane of contact is determined, or reciprocally ; say the plane XY is deter mined when the point P is given, or reciprocally ; and it is noticed that when P is situate in the interior of the surface the plane XY does not cut the surface ; that is, we 1 More generally, in solid geometry infinity is a plane, its intersec tion with any given plane being the right line which is the infinity of this given plane. have a real plane XY Intersecting the surface in tho imaginary curve of contact of the imaginary circumscribed cone having for its summit a given real point P inside the surface. Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic ; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other. The term " pole " was first used by Servois, and " polar " by Gergonne (Gerg., t. i. and iii., 1810-13); and from the theorem we have the method of reciprocal polars for the transformation of geometrical theorems, used already by .Brianchon (in the memoir above referred to) for the demonstration of the theorem called by his name, and in a similar manner by various writers in the earlier volumes of Gergonne. We are here concerned with the method less in itself than as leading to the general notion of duality. And, bearing in a somewhat similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the memoir by Gaultier, on the graphical construction of circles and .spheres (Jour. Polyt., t. ix., 1813). The well-known theorem as to radical axes may be stated as follows. Consider two circles partially drawn so that it does not appear whether the circles, if completed, would or would not intersect in real points, say two arcs of circles ; then we can, by means of a third circle drawn so as to intersect in two real points each of the two arcs, determine a right line, which, if the com plete circles intersect in two real points, passes through the points, and which is on this account regarded as a line pass ing through two (real or imaginary) points of intersection of the two circles. The construction in fact is, join the two points in which the third circle meets the first arc, and join also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the " radical axis of the two circles ; " it is a line determined by a real construc tion and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may be) intersections of the given circles. The intersections which lie on the radical axis are two out of the four intersections of the two circles. The question as to the remaining two intersections did not present itself to Gaultier, but it is answered in Poncelet s Traite des proprietes projectiles (1822), where we find (p. 49) the statement, "deux circles places arbitrairemenfc sur un plan . . . ont id^alement deux points imaginaires communs a 1 infini;" that is, a circle qua curve of the second order is met by the line infinity in two points ; but, more than this, they are the same two points for any circle what ever. The points in question have since been called (it is believed first by Dr Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently spoken of as the points I, J ; and we have thus the circle characterized as a conic which passes through the two circular points at infinity ; the number of conditions thus imposed upon the conic is = 2, and there remain thrae arbitrary constants, which is the right number for the circle. Poncelet throughout his work makes continual use of the foregoing theories of imaginaries and infinity, and also of the before-mentioned theory of reciprocal polars. Poncelet s two memoirs " Sur les centres des moyennes harmoniques," and " Sur la theorie generale des polaires reciproques," although presented to the Paris Academy in
1824 were only published (Crelle, t. iii. and iv., 1828,