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.