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parabola Geom.|pəˈræbələ|
[a. 16th c. L. parabola (also parabolē), a. Gr. παραβολή juxtaposition, application, spec. in Geometry, the ‘application’ of a given area to a given straight line, hence also, the curve described below: for derivation and other senses, cf. parable. In F. parabole. See note below.]
One of the conic sections; the plane curve formed by the intersection of a cone with a plane parallel to a side of the cone; also definable as the locus of a point whose distance from a given point (the focus) is equal to its distance from a given straight line (the directrix).
Sometimes distinguished from parabolas of the higher kind (see b) as the Apollonian parabola or quadratic parabola. It is approximately the path of a projectile under the influence of gravity.
[1544Archimedis Opera 142 (heading) Archimedis qvadratvra parabolæ, id est portionis contentæ a linea recta & sectione rectanguli coni. 1558Commandinus Archimedis Opera 18 b, (heading) Archimedis qvadratvra paraboles.] 1579Digges Stratiot. 188, I demaunde whether then this Eleipsis shal not make an Angle with the Parabola Section equal to the distaunce betweene the grade of Randon proponed, and the grade of vttermost Randon. 1656[see parabolaster]. 1668Phil. Trans. III. 876 The Spindle made of the same Parabola by rotation about its Base. 1696Whiston Th. Earth i. (1722) 14 The Orbits describ'd will be one of the other Conick Sections, either Parabola's or Hyperbola's. 1706W. Jones Syn. Palmar. Matheseos 246 'Tis evident the Parabola has but one Focus. 1788Chambers Cycl. (ed. Rees), Parabola, osculatory, in Geometry, is used particularly for that parabola which not only osculates or measures the curvature of any curve at a given point, but also measures the variation of the curvature at the point. 1828Hutton Course Math. II. 136 The Area or Space of a Parabola, is equal to Two-Thirds of its Circumscribing Parallelogram. 1832Nat. Philos. II. Introd. Mech. p. xviii. (U.K.S.), The curve-line which a ball describes, if the resistance of the air be not taken into consideration, is called in geometry a parabola. 1868Lockyer Elem. Astron. xxiii. (1870) 124 The orbit of a comet is generally best represented by what is called a parabola; that is, an infinitely long ellipse. 1881C. Taylor Anc. & Mod. Geom. 82 The parabola was so called from the equality of the square of the ordinate of any point upon it to the rectangle contained by its abscissa and the latus rectum... It is reported by Proclus in his Commentaries on the first book of Euclid..that the terms parabola, hyperbola, and ellipse had been used by the Pythagoreans to express the equality or inequality of areas, and were subsequently transferred to the conic curves.
b. Extended to curves of higher degrees resembling a parabola in running off to infinity without approaching to an asymptote, or having the line at infinity as a tangent, and denoted by equations analogous to that of the common parabola.
campaniform parabola or bell-shaped parabola: a name formerly given to cubic parabolas without cusp or node. Cartesian p.: a cubic curve denoted by the equation xy = ax3 + bx2 + cx + d, having four infinite branches, two parabolic and two hyperbolic. cubic parabola or cubical p.: a parabola of the third degree. double p.: a parabola having the line at infinity for a double tangent. helicoid p.: see helicoid. Neilian p.: the semicubical parabola (ax2 = y3), rectified by William Neil in 1657. semicubical p.: see semicubical.
1664Phil. Trans. I. 15 A Method for the Quadrature of Parabola's of all degrees. 1727–41Chambers Cycl. s.v., Parabola's of the higher kinds are algebraic curves, defined by am - 1x = ym... Some call these Paraboloids. 1765Croker Dict. Arts, Cartesian Parabola. 1795Hutton Math. Dict. II. 192 A bell-form Parabola, with a conjugate point.
[Note. To the earlier Greek geometers, including Archimedes, b.c. 287–212, who investigated only sections perpendicular to the surface of the cone, the parabola was known as ὀρθογωνίου κώνου τοµή = sectio rectanguli coni ‘the [perpendicular] section of a right-angled cone’. The use of παραβολή, ‘application’, in this sense is due to Apollonius of Perga, c 210 b.c., and, with him, referred to the fact that a rectangle on the abscissa, having an area equal to the square on the ordinate, can be ‘applied’ to the latus rectum, without either excess (as in the hyperbola), or deficiency (as in the ellipse). (See C. Taylor Anct. & Mod. Geom. 195; T. L. Heath Apollonius of Perga, Introd. lxxx.) But an explanation of the name, from the much more obvious property of the parallelism of the section to a side of the cone, is given by Eutokius of Ascalon c a.d. 550, and is frequent in later writers.] |