CONIC SECTION, or briefly Conic, a curve in which a plane intersects a cone. In ancient geometry the name was restricted to the three particular forms now designated the ellipse, parabola and hyperbola, and this sense is still retained in general works. But in modern geometry, especially in the analytical and projective methods, the “principle of continuity” renders advisable the inclusion of the other forms of the section of a cone, viz. the circle, and two lines (and also two points, the reciprocal of two lines) under the general title conic. The definition of conics as sections of a cone was employed by the Greek geometers as the fundamental principle of their researches in this subject; but the subsequent development of geometrical methods has brought to light many other means for defining these curves. One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in plano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio. This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse. In the case of the circle, the centre is the focus, and the line at infinity the directrix; we therefore see that a circle is a conic of zero eccentricity.
In projective geometry it is convenient to define a conic section as the projection of a circle. The particular conic into which the circle is projected depends upon the relation of the “vanishing line” to the circle; if it intersects it in real points, then the projection is a hyperbola, if in imaginary points an ellipse, and if it touches the circle, the projection is a parabola. These results may be put in another way, viz. the line at infinity intersects the hyperbola in real points, the ellipse in imaginary points, and the parabola in coincident real points. A conic may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the conic is an ellipse, if on the circle a parabola, and if within the circle a hyperbola. In analytical geometry the conic is represented by an algebraic equation of the second degree, and the species of conic is solely determined by means of certain relations between the coefficients. Confocal conics are conics having the same foci. If one of the foci be at infinity, the conics are confocal parabolas, which may also be regarded as parabolas having a common focus and axis. An important property of confocal systems is that only two confocals can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.
The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola. In this article we shall consider the historical development of the geometry of conics, and refer the reader to the article Geometry: Analytical and Projective, for the special methods of investigation.
History.—The invention of the conic sections is to be assigned to the school of geometers founded by Plato at Athens about the 4th century B.C. Under the guidance and inspiration of this philosopher much attention was given to the geometry of solids, and it is probable that while investigating the cone, Menaechmus, an associate of Plato, pupil of Eudoxus, and brother of Dinostratus (the inventor of the quadratrix), discovered and investigated the various curves made by truncating a cone. Menaechmus discussed three species of cones (distinguished by the magnitude of the vertical angle as obtuse-angled, right-angled and acute angled), and the only section he treated was that made by a plane perpendicular to a generator of the cone; according to the species of the cone, he obtained the curves now known as the hyperbola, parabola and ellipse. That he made considerable progress in the study of these curves is evidenced by Eutocius, who flourished about the 6th century A.D., and who assigns to Menaechmus two solutions of the problem of duplicating the cube by means of intersecting conics. On the authority of the two great commentators Pappus and Proclus, Euclid wrote four books on conics, but the originals are now lost, and all we have is chiefly to be found in the works of Apollonius of Perga. Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a geometrical and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces. He probably wrote a book on conics, but it is now lost. In his extant Conoids and Spheroids he defines a conoid to be the solid formed by the revolution of the parabola and hyperbola about its axis, and a spheroid to be formed similarly from the ellipse; these solids he discussed with great acumen, and effected their cubature by his famous “method of exhaustions.”
But the greatest Greek writer on the conic sections was Apollonius of Perga, and it is to his Conic Sections that we are indebted for a review of the early history of this subject. Of the eight books which made up his original treatise, only seven are certainly known, the first four in the original Greek, the next three are found in Arabic translations, and the eighth was restored by Edmund Halley in 1710 from certain introductory lemmas of Pappus. The first four books, of which the first three are dedicated to Eudemus, a pupil of Aristotle and author of the original Eudemian Summary, contain little that is original, and are principally based on the earlier works of Menaechmus, Aristaeus (probably a senior contemporary of Euclid, flourishing about a century later than Menaechmus), Euclid and Archimedes. The remaining books are strikingly original and are to be regarded as embracing Apollonius’s own researches.
The first book, which is almost entirely concerned with the construction of the three conic sections, contains one of the most brilliant of all the discoveries of Apollonius. Prior to his time, a right cone of a definite vertical angle was required for the generation of any particular conic; Apollonius showed that the sections could all be produced from one and the same cone, which may be either right or oblique, by simply varying the inclination of the cutting plane. The importance of this generalization cannot be overestimated; it is of more than historical interest, for it remains the basis upon which certain authorities introduce the study of these curves. To comprehend more exactly the discovery of Apollonius, imagine an oblique cone on a circular base, of which the line joining the vertex to the centre of the base is the axis. The section made by a plane containing the axis and perpendicular to the base is a triangle contained by two generating lines of the cone and a diameter of the basal circle. Apollonius considered sections of the cone made by planes at any inclination to the plane of the circular base and perpendicular to the triangle containing the axis. The points in which the cutting plane intersects the sides of the triangle are the vertices of the curve; and the line joining these points is a diameter which Apollonius named the latus transversum. He discriminated the three species of conics as follows:—At one of the two vertices erect a perpendicular (latus rectum) of a certain length (which is determined below), and join the extremity of this line to the other vertex. At any point on the latus transversum erect an ordinate. Then the square of the ordinate intercepted between the diameter and the curve is equal to the rectangle contained by the portion of the diameter between the first vertex and the foot of the ordinate, and the segment of the ordinate intercepted between the diameter and the line joining the extremity of the latus rectum to the second vertex. This property is true for all conics, and it served as the basis of most of the constructions and propositions given by Apollonius. The conics are distinguished by the ratio between the latus rectum (which was originally called the latus erectum, and now often referred to as the parameter) and the segment of the ordinate intercepted between the diameter and the line joining the second vertex with the extremity of the latus rectum. When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the parabola; if the inclination of the plane be greater than that of the side, we obtain the hyperbola. In modern notation, if we denote the ordinate by 𝑦, the distance of the foot of the ordinate from the vertex (the abscissa) by 𝑥, and the latus rectum by 𝑝, these relations may be expressed as 𝑦2<𝑝𝑥 for the ellipse, 𝑦2＝𝑝𝑥 for the parabola, and 𝑦2>𝑝𝑥 for the hyperbola. Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle. The word parabola was used by Archimedes, who was prior to Apollonius; but this may be an interpolation.
We may now summarize the contents of the Conics of Apollonius. The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact. The fifth book contains properties of normals and their envelopes, thus embracing the germs of the theory of evolutes, and also maxima and minima problems, such as to draw the longest and shortest lines from a given point to a conic; the sixth book is concerned with the similarity of conics; the seventh with complementary chords and conjugate diameters; the eighth book, according to the restoration of Edmund Halley, continues the subject of the preceding book. His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola. He also considered the two branches of a hyperbola, calling the second branch the “opposite” hyperbola, and shows the relation which existed between many metrical properties of the ellipse and hyperbola. The focus of the parabola was discovered by Pappus, who also introduced the notion of the directrix.
The Conics of Apollonius was translated into Arabic by Tobit ben Korra in the 9th century, and this edition was followed by Halley in 1710. Although the Arabs were in full possession of the store of knowledge of the geometry of conics which the Greeks had accumulated, they did little to increase it; the only advance made consisted in the application of describing intersecting conics so as to solve algebraic equations. The great pioneer in this field was Omar Khayyám, who flourished in the 11th century. These discoveries were unknown in western Europe for many centuries, and were re-invented and developed by many European mathematicians. In 1522 there was published an original work on conics by Johann Werner of Nuremburg. This work, the earliest published in Christian Europe, treats the conic sections in relation to the original cone, the procedure differing from that of the Greek geometers. Werner was followed by Franciscus Maurolycus of Messina, who adopted the same method, and added considerably to the discoveries of Apollonius. Claude Mydorge (1585–1647), a French geometer and friend of Descartes, published a work De sectionibus conicis in which he greatly simplified the cumbrous proofs of Apollonius, whose method of treatment he followed.
Johann Kepler (1571–1630) made many important discoveries in the geometry of conics. Of supreme importance is the fertile conception of the planets revolving about the sun in elliptic orbits. On this is based the great structure of celestial mechanics and the theory of universal gravitation; and in the elucidation of problems more directly concerned with astronomy, Kepler, Sir Isaac Newton and others discovered many properties of the conic sections (see Mechanics). Kepler’s greatest contribution to geometry lies in his formulation of the “principle of continuity” which enabled him to show that a parabola has a “caecus (or blind) focus” at infinity, and that all lines through this focus are parallel (see Geometrical Continuity). This assumption (which differentiates ancient from modern geometry) has been developed into one of the most potent methods of geometrical investigation (see Geometry: Projective). We may also notice Kepler’s approximate value for the circumference of an ellipse (if the semi-axes be 𝑎 and 𝑏, the approximate circumference is π(𝑎+𝑏)).
An important generalization of the conic sections was developed about the beginning of the 17th century by Girard Desargues and Blaise Pascal. Since all conics derived from a circular cone appear circular when viewed from the apex, they conceived the treatment of the conic sections as projections of a circle. From this conception all the properties of conics can be deduced. Desargues has a special claim to fame on account of his beautiful theorem on the involution of a quadrangle inscribed in a conic. Pascal discovered a striking property of a hexagon inscribed in a conic (the hexagrammum mysticum); from this theorem Pascal is said to have deduced over 400 corollaries, including most of the results obtained by earlier geometers. This subject is mathematically discussed in the article Geometry: Projective.
While Desargues and Pascal were founding modern synthetic geometry, René Descartes was developing the algebraic representation of geometric relations. The subject of analytical geometry which he virtually created enabled him to view the conic sections as algebraic equations of the second degree, the form of the section depending solely on the coefficients. This method rivals in elegance all other methods; problems are investigated by purely algebraic means, and generalizations discovered which elevate the method to a position of paramount importance. John Wallis, in addition to translating the Conics of Apollonius, published in 1655 an original work entitled De sectionibus conicis nova methodo expositis, in which he treated the curves by the Cartesian method, and derived their properties from the definition in plano, completely ignoring the connexion between the conic sections and a cone. The analytical method was also followed by G. F. A. de l’Hôpital in his Traité analytique des sections coniques (1707). A mathematical investigation of the conics by this method is given in the article Geometry: Analytical. Philippe de la Hire, a pupil of Desargues, wrote several works on the conic sections, of which the most important is his Sectiones Conicae (1685). His treatment is synthetic, and he follows his tutor and Pascal in deducing the properties of conics by projection from a circle.
A method of generating conics essentially the same as our modern method of homographic pencils was discussed by Jan de Witt in his Elementa linearum curvarum (1650); but he treated the curves by the Cartesian method, and not synthetically. Similar methods were devised by Sir Isaac Newton and Colin Maclaurin. In Newton’s method, two angles of constant magnitude are caused to revolve about their vertices which are fixed in position, in such a manner that the intersection of two limbs moves along a fixed straight line; then the two remaining limbs envelop a conic. Maclaurin’s method, published in his Geometria organica (1719), is based on the proposition that the locus of the vertex of a triangle, the sides of which pass through three fixed points, and the base angles move along two fixed lines, is a conic section. Both Newton’s and Maclaurin’s methods have been developed by Michel Chasles. In modern times the study of the conic sections has proceeded along the lines which we have indicated; for further details reference should be made to the article Geometry.
Authorities.—For the ancient geometry of conic sections, especially of Apollonius, reference should be made to T. L. Heath’s Apollonius of Perga (1886); more general accounts are given in James Gow, A Short History of Greek Mathematics (1884), and in H. G. Zeuthen, Die Lehre von dem Kegelschnitten in Alterthum (1886). Michel Chasles in his Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837, a third edition was published in 1889), gives a valuable account of both the ancient and modern geometry of conics; a German translation with the title Geschichte der Geometrie was published in 1839 by L. A. Sohncke. A copious list of early works on conic sections is given in Fred. W. A. Murhard, Bibliotheca mathematica (Leipzig, 1798). The history is also treated in general historical treatises (see Mathematics).
Geometrical constructions are treated in T. H. Eagles, Constructive Geometry of Plane Curves (1886); geometric investigations primarily based on the relation of the conic sections to a cone are given in Hugo Hamilton’s De Sectionibus Conicis (1758); this method of treatment has been largely replaced by considering the curves from their definition in plano, and then passing to their derivation from the cone and cylinder. This method is followed in most modern works. Of such text-books there is an ever-increasing number; here we may notice W. H. Besant, Geometrical Conic Sections; C. Smith, Geometrical Conics; W. H. Drew, Geometrical Treatise on Conic Sections. Reference may also be made to C. Taylor, An Introduction to Ancient and Modern Geometry of Conics (1881).
See also list of works under Geometry: Analytical and Projective.