The Greek geometers invented other curves; in particular, the “conchoid,” which is the locus of a point such that its distance from a given line, measured along the line drawn through it to a fixed point, is constant; and the “cissoid” which is the locus of a point such that its distance from a fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the circle at the point opposite to the fixed point. Obviously the number of such geometrical or kinematical definitions is infinite. In a machine of any kind, each point describes a curve; a simple but important instance is the “three-bar curve,” or locus of a point in or rigidly connected with a bar pivotted on to two other bars which rotate about fixed centres respectively. Every curve thus arbitrarily defined has its own properties; and there was not any principle of classification.

The principle of classification first presented itself in the Géométrie of Descartes (1637). The idea was to represent any curve whatever by means of a relation between the coordinates (x, y) of a point of the curve, or say to represent the curve by means of its equation.

Descartes takes two lines xx', ys, called axes of coordinates, intersecting at a point O called the origin (the axes are usually at right angles to each other, and for the present they are considered as being so); and he determines the position of a point P by means of its distances OM (or NP) = x, and MP (or ON) = y, from these two axes respectively; where x is regarded as positive or negative according as it is in the sense Oz or Ox' from O; and similarly y as positive or negative according as it is in the sense Oy or Oy' from O; or what is the same thing— [table]

Any relation whatever between (x, y) determines a curve, and conversely every curve whatever is determined by a relation between (z, y).

Observe that the distinctive feature is in the exclusive use of such determination of a curve by means of its equation. The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation zç2 ∕y2 is —+ ∣ i = 1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.

We obtain from the equation the notion of an algebraical or geometrical as opposed to a transcendental curve, viz., an algebraical or geometrical curve is a curve having an equation F(z, y) = 0, where F(z, y) is a rational and integral function of the coordinates (x, y) ; and in what follows we attend throughout (unless the contrary is stated) only to such curves. The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced to the foregoing rational anl integral form, and regard this as the equation of the curve. And we have hence the notion of a curve of a given, order, viz., the order of the curve is equal to that of the term or terms of highest order in the coordinates (x, y) conjointly in the equation of the curve; for instance, xy - 1 = 0 is a curve of the second order.

It is to be noticed here that the axes of coordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of coordinates; but the order is independent of the axes, and has a determinate value for any given curve.

We hence divide curves according to their order, viz., a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax + by + c = 0, or say (*&», y i 1) = 0; or by an equation of the second order, ax h + 2h,xy + δy 2 + 2fy + 2gx + c = 0, say (*$z, y, l) 2 = 0; or by an equation of the third order, &c; or what is the same thing, according as the equation is linear, quadric, cubic, &c.

A curve of the first order is a right line; and conversely every right line is a curve of the first order.

A curve of the second order is a conic, or as it is also called a quadric; and conversely every conic, or quadric, is a curve of the second order.

A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.

A curve of the order m has for its equation (*$r, y, 1) = 0; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order m. The number of coefficients is ⅜(w⅛+l)(m +2); but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as ∣(w + l)(w + 2) - 1, that is, jm(m + 3); and a curve of the order m may be made to satisfy this number of conditions; for example, to pass through ⅛m(m + 3) points.

It is to be remarked that an equation may break up; thus a quadric equation may be (ax + by + c)(a'x + b'y + c') = 0, breaking up into the two equations αz + δy + c = 0, a'x + b'y + c = 0, viz., the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz., the pair of lines is considered as a quadric curve. But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which does not break up.

The intersections of two curves are obtained by combining their equations; viz., the elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).

Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn ; in particular, if the curve of the order n is an arbitrary line (n= 1), then the order of the resultant equation is = m,; and the curve of the order m meets therefore the line in m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.

The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry—and it in fact plays an all-important part in modern geometry. As in algebra we say that an equation of the m- th order has wι roots, viz., we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of the wι-th order is met by an arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the rø-th order, as a curve which is met by an arbitrary line in m points (that is, of course, in m, and not more than m, points).

The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m = 2, if the hyperbola xy - 1 = 0 be cut by the line y = β, the resultant equation in x is βx - 1 = 0, and there is apparently only the intersection (x = 1, y = β)∙, but the theorem is, in fact, true for every line whatever: a curve of the order m meets every line whatever in precisely m points. We have, in the case just referred to, to take account of a point at infinity on the line y = β', the two intersections are the point (x=~, P y = β), and the point at infinity on the line y = β.

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 important 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, iae., 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 Géométrie 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 Géométrie, 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 correspondence of two points on a cubic curve, viz,, two points correspond to each other when the tangents at the two points again meet the cubic in the same point.

The Géométrie 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 degré” (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 determined 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 have a real plane XY Intersecting the surface in the 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 complete circles intersect in two real points, passes through the points, and which is on this account regarded as a line passing 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 construction 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 Traité des propriétés projectives (1822), where we find (p. 49) the statement, “deux circles placés arbitrairement sur un plan . . . ont idéalement deux points imaginaires communs à l’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 whatever. 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 three 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 théorie générale des polaires réciproques,” although presented to the Paris Academy in 1824 were only published (Grelle, t. iii. and iv., 1828, 1 More generally, in solid geometry infinity is a plane,—its intersection with any given plane being the right line which is the infinity oí this given plane.

1829), subsequent to the memoir by Gergonne, “Considerations philosophiques sur les élémens de la science de l’étendue” (G er g., t. xvi., 1825-26). In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find “dans la géométrie plane, à chaque théorème il en répond nécessairement un autre qui s’en déduit en échangeant simplement entre eux les deux mots points et droites ; tandis que dans la géométrie de l’espace ce sont les mots points et plans qu’il faut échanger entre eux pour passer d’un théorème à son corrélatif;” and the plan is introduced of printing correlative theorems, opposite to each other, in two columns. There was a reclamation as to priority by Poncelet in the Bulletin Universel reprinted with remarks by Gergonne (Gerg., t. xix., 1827), and followed by a short paper by Gergonne, “Rectifications de quelques théorèmes, &c.,” which is important as first introducing the word class. We find in it explicitly the two correlative definitions :—-“a plane curve is said to be of the with degree (order) when it has with a line m real or ideal intersections,” and “a plane curve is said to be of the »nth class when from any point of its’ plane there can be drawn to it m real or ideal tangents.”

It may be remarked that in Poncelet’s memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, i or say the class of the curve, is in general and at most = m(m - l),and that he mentions that this number is subject to reduction when the curve has double points or cusps.

The theorem of duality as regards plane figures may be thus stated :—two...