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CONGRUOUS—CONIC SECTION

plane; for protecting buildings against fire; inlaying and combining metals; unforgeable bank-note paper; a method of killing whales by means of rockets; improvements in the manufacture of gunpowder; stereotype plates; fireworks; gas meters, &c. The first friction matches made in England (1827) were named after him by their inventor, John Walker. He published a number of works, including three treatises on The Congreve Rocket System (1807, 1817 and 1821; the last was translated into German, Weimar, 1829); An Elementary Treatise on the Mounting of Naval Ordnance (1812); A Description of the Hydropneumatical Lock (1815); A New Principle of Steam-Engine (1819); Resumption of Cash Payments (1819); Systems of Currency (1819), &c.

See Colonel J. R. J. Jocelyn in Journal of the Royal Artillery, vol. 32, No. 11, and sources therein referred to. The account in the Dictionary of National Biography is very inaccurate.


CONGRUOUS (from Lat. congruere, to agree), that which corresponds to or agrees with anything; the derivation appears in “congruence,” a condition of such correspondence or agreement, a term used particularly in mathematics, e.g. for a doubly infinite system of lines (see Surface), and in the theory of numbers, for the relation of two numbers, which, on being divided by a third number, known as the modulus, leave the same remainder (see Number). The similar word “congruity” is a term of Scholastic theology in the doctrine of merit. God’s recompense for good works, if performed in a state of grace, is based on “condignity,” meritum de condigno; if before such a state is reached, it should be fit or “congruous” that God should recompense such works by conferring the “first grace,” meritum de congruo. The term is also used in theology, in reference to the controversy between the Jesuits and the Dominicans on the subject of grace, at the end of the 16th century (see Molina, Luis, and Suarez, Francisco).


CONIBOS, or Manoas, a tribe of South American Indians inhabiting the Pampa del Sacramento and the banks of the Ucayali, Peru. Spanish missionaries first visited them in 1683, and in 1685 some Franciscans who had founded a mission among them were massacred. A like fate befell a priest in 1695. They have since been converted and are now a peaceful people.


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 confocal can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.

The definitions given above refiect 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