CONIC SECTION, the intersection of a plane with a cone (q.v.) . In Greek geometry and for centuries after the cone was regarded as a solid; hence any section was looked upon as a sur face bounded by a curve. Later geometers feeling that conical properties belong to cones as surfaces, the bounded space within being relatively indifferent, define a cone as a surface. Hence the plane section becomes not an area but a plane-curve, though ques tions of areal content of course remain in place. The study of conics seems to have issued from Plato's Academy, with its in scription, "Let none unversed in geometry enter." Plato himself was abreast but hardly ahead of the mathematics of his day, such as was afterwards enshrined in Euclid's thirteen books of Elements (c.
30o B.c.), but the interest and enthusiasm he aroused accomplished wonders.
But such curves did not exist in his day and it became necessary to invent them, as possessing geometric properties sym bolized now in these equations. This he is said to have done (as appears probable from the epigram of Eratosthenes, c. 23o B.C., "Do not . . . cut the cone in triads of Menaechmus") by imagining a right circular cone cut by a plane perpendicular to the edge or generatrix, and in one of three curves (bounding the plane section) according as the cone was right-, acute- or obtuse-angled at the vertex. In the first and simplest case the plane, being parallel to the opposite edge, could never meet it. The section then ex tended indefinitely, yielding a curve, now named parabola (q.v.), always the same in shape though varying in size. In the second case the plane cut the opposite edge, the section was closed and finite, and was later named the ellipse (q.v.). In the third case the plane, diverging more and more from the opposite element of the cone, never meets it (except in the opposite nappe, as recognized by Apollonius), the curve is now known as the hyperbola (q.v.). By help of this triad (figs. 1, 2, 3), says Eutocius of Ascalon (c. A.D. 560), its inventor found two solutions for the cube problem. Both the ellipse and the hyperbola change shape on varying the vertical angle, which in the parabola cannot vary. As the plane nears the vertex, the ellipse shrinks to a point, but the parabola and the hyperbola straighten out, each into a pair of right lines, divergent in the hyperbola but coincident in the parabola.
Through more than a century much progress was made by such writers as Aristaeus, Euclid and Archimedes. Euclid's four books on conics, now lost but mentioned by Pappus (fl. c. 300) and Proclus (c. 412-485), were probably summed up in the first four books of the conics of Apollonius. Archimedes (287-212 B.c.) also may have written such a treatise, now lost. His extant works contain weighty contributions, such as the quadrature of the parabola, the first of its kind, improving Eudoxus's method of ex haustion, and inserted as a tract between the two books on mass centres and the equilibrium of plane areas. Also in his books on Conoids and Spheroids he effected the cubature of these solids by like means (of compression), slicing each into thin parallel lam inae, then forming two series of thin cylinders with bases respec tively the larger and smaller bases of these layers, between which two series the volume sought must lie. As the strata become ever thinner, the two series close down upon the volume, lying always between,—a rigorous method suggesting integration. He also found the area of the ellipse (props. 5, 6 in the same work), but not of the hyperbola. The crowning peak in Greek mathematics, though not so broad-based as Archimedes, is Apollonius, born (c. 262 B.c.) in Perga, Pamphylia, a student in the Euclid-school at Alexandria and the author of eight books (387 propositions) on Conics, monumentally edited in two folios by Edmund Halley (Oxford, 171o), with commentaries. Books I.–IV. resume the science as known down to Conon of Samos, almost a contempo rary, and books V.–VIII. contain higher developments by the author himself.
According to Geminus of Rhodes (c. 7o B.c.), cited by Eutocius, Apollonius first showed that all conics are sections of any circular cone, right or oblique; also, as Pappus tells us, he gave the names parabola (application or equality, literally, casting alongside), ellipse (defect), and hyperbola (excess), to express certain facts in the comparison of areas. He assumed the axis of the cone oblique to its circular base, also a principal plane through the ver tex,and upright on that base, cutting the cone in the axial triangle, its sides being elements of the cone, its base a diameter of the circle, and the diameter perpendicular thereto being conjugate. His section planes were all at right angles to this axial triangle, but at varying angles to the base (e.g., rotating round the conjugate as axis), each section being thus symmetric as to the principal plane. He showed that all earlier types of conics (right-, acute and obtuse-angled cones) result as follows : the right-angled cone gives a parabola when the rotating plane becomes parallel to a cone-element ; the ellipse when it turns away from parallelism and from the axis ; and hyperbola when it turns away oppositely toward the axis,—in which case it would cut the opposite con gruent nappe traced by the same generatrix g prolonged backward (figs. 1-3).
These distinctive areal properties are expressed in Cartesian co ordinates, the rectangular axes being the diameter and the tangent at the vertex, by the equation = px — -x2 for the ellipse ; = px for the parabola; and for the hyperbola. Either form, the ellipse or hyperbola, would hold for the parabola, since p/d = o for an infinite d; the parabola thus appears as a common limit of the ellipse and the hyperbola. Such equations were of course unknown to Apollonius, who employed their equivalents in their picturesque but cumbrous forms of areal relation.
Greek mathematics culminated in Apollonius. Little further ad vance was possible without new methods and higher points of view. Much later, the Arabs and other Muslims absorbed the classic science greedily; it was the Persian poet Omar Khayyam (c. 1044-1123/4) , one of the most prominent of mediaeval mathe maticians, with his remarkable classification and systematic study of equations, which he emphasized, who blazed the way to the modern union of analysis and geometry. In his Algebra he con sidered the cubic as soluble only by the intersection of conics, and the biquadratic not at all.
Johann Kepler (1571-1630) was the first to proclaim the reg nance of the conic in the sky. Apollonius had rightly ranked it among things worth study on their own account (Bk. V., Pref.); Kepler placed the sun in a world-focus, with planets rolling round it in ellipses, as confirmed and rationalized in Newton's Law of Inverse Squares. But Kepler's main advance in pure conic doctrine lay in enunciating (1604) as "analogy" the principle or law of continuity, Leibniz's lex continuationis, which supplies the parabola with a "blind focus" and with a vertex at infinity, all diameters being parallel, therewith preparing a path for projec tive geometry (see PROJECTIVE GEOMETRY). He also found ir(a+b) as the approximate length of the ellipse, a and b being the semi-axes.
The so-called Desargue's Theorem declared that a chord cuts a conic and an inscribed quadrangle in six points in involution (OP.
OP = ; also if two triangles have their corresponding vertices on three com punctal lines, then their corresponding sides meet in three collinear points (and conversely),—the basis of homology (Pon celot) (fig. 5).
Only Pascal seemed able to keep step with Desargues. He attended mathematical meetings with his father and before 164o composed a book on conics. Leibniz wrote to Pascal's nephew (Aug. 3o, 1676) vainly urging its publication; it perished, save for a small introductory fragment, so that the most that is known about it is only from Leibniz's letter. Pascal avows the leadership of Desargues, stating without proof, as the first lemma to the latter's theorem, his own about the hexagram inscribed in a circle, that the three inter sections of its three pairs of opposite sides are collinear. The reciprocal, Bianchon's Theorem, was first published in 1806. Since both regarded the cone solely without any axial triangle, and viewed any conic as a shadow or projection on a plane of any circle of the cone from its vertex, such a property of the circle-hexagram would in their minds pass over into the cor responding Hexagramma mysticum of the conic. From this prop osition Pere Mersenne (1588-1648) declares that Pascal deduced 400 corollaries.
To Rene Descartes (1596-1650) the algebraization and conse quent transfiguration of geometry, especially of conics, is com monly ascribed, although his Geometrie (1637) reads more like a geometrization of algebra (see ANALYTIC GEOMETRY). Pierre de Fermat (16o1-65), keener and deeper in mathematical insight as well as earlier in his inventions, nevertheless was later in publi cation. Their chief achievement was to introduce motion into Greek static conceptions, by using a pair of variables (x, y) to represent a moving point tracing a curve. An equation connecting the variables defined the motion of the point. Thus all is life and motion, "mouvement continu.” Equations symbolize the classic TO7r0t (10Ci), and conversely loci depict equations. All conics are grouped in a single equation of the second degree, ax2-1-2hxy+ by2+2gx+ 2fy +1 = o, each par ticular conic being determined by a set of values (a, h, b, f, g) con stant for any one curve but varying from conic to conic, while individual points are fixed by special value-pairs (x, y), each pair satisfying the curve's equation. Thus the pair x=3, y=4 satisfies the equation x2-1-y2= 25; hence the point (3, 4) is on the curve, a circle of radius 5 about the origin 0 (see COORDrNATES).
Hints of this kind -had already been put forth in Muslim works, and in France by Francois Viete (154o—i6o3), but not the notion of the moving point (x, y). This mutual depiction of value-pair (x, y) and moving point P, by its effecting a union of algebra and geometry, marks the birth of modern mathematics.
A prompt reaction to the stimulus of Descartes's Geometrie was seen in, John Wallis's Tractatus on conics, spreading the "new method," quite ignoring the classics, and notable for introducing the sign oo : "Esto oo Nota nu meri infiniti" (1655). A similar reaction appears in the writings of De l'Hopital (17°7), although De La Hire still followed his master's method of projection (1685). Despite the wide and clear vision thus opened, the classic precedent retained a fascination, more or less determining later works, which introduced new syn thetic ideas even while following analytic paths. Thus Newton 0642-1727) conceived the conic as envelope of two sides of two constant angles rotating round fixed vertices, the other two sides meeting always on the same right line; and Colin Maclaurin (Geometrica organica, 171o) imagined the conic as a locus of the vertex of a triangle, each side fixed at one point, the other ver tices moving each on a fixed line. Michel Chasles 0793-1880) em ployed and extended both these conceptions.
Later advances in the doc trine of conics have been made mainly in connection with an alytic and especially the projec tive geometry and coordinates (qq.v.). Nevertheless the Greek synthetic spirit found extraor dinary reincarnation in the work of Jacob Steiner (1796-1863).
As a pupil of Pestalozzi, he naturally laid especial stress on in tuition and envisagement, while unjustly disliking all forms of analysis and eschewing its use as a reproach to geometry proper. But his insight and ingenuity were alike amazing and not only levelled the way for Von Staudt 0798-1867) and his followers in the geometry of position, but may also be said to have rounded out and filled in apparently the whole circuit of the theory of conic sections.
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Benaist, 19o3) ; W. Dette, Anal. Geom. der Kegelschnitte (1909) ; F. Enriques, L'Evolution des idees geometriques dans la pensee grecque (tr. M. Solovine, 1927) ; P. Ver Eecke, Les Coniques d' Apollonios de Perge (1924) ; F. Klein, Elementarmathematik vom hoh. Standpunkt aus (1925), and Vorl. d. Entwicklung d. Math. im Jahrh. (1926) ; B. Niewenglowski, Cours de geometric analytique (192s) ; W. L. Osgood and W. C. Graustein, Plane and Solid Analytic Geom. (1922) ; G. Salmon, A Treatise on Conic Sections 0848, 6th ed., 1879) ; A. Schoenfliess, Einfiihrung in die anal. Geom. der Ebene u. des Raumes (1925) ; J. Steiner, V orl. iiber synthetische Geom. (2nd ed., 1875-76) ; F. Wicke, Einfiihrung in d. Hohere Math., 2 VO1S., 1927; H. P. Hudson, Cremona Transformations (1927). On the history of the subject, F. Cajori, History of Mathematics (1919) ; M. Cantor, Vorl. d. Geschichte d. Mathematik, 4 vols., 1880-19o8, 1922-24; Encyclopedic des Sciences Math. pures et aPpliques (x913— 14) ; Fundamenta mathematica (192o-26) ; S. Giinther and H. Wieleitner, Geschichte der Mathematik (19o8-21) ; T. L. Heath, A History of Greek Mathematics (1921), and Apollonius of Perga (1896), Archimedes (1897), Aristarchus of Samos (192o) ; D. E. Smith, History of Mathematics, 2 VOIS. 0923-25) ; J. Tropfke, Geschichte d. Elementarmath. (2nd ed., 1g21 seq.); E. Duporcq, Premiers Principes de Geometrie moderne (1924). (W. Sm.)