"椭圆"、"抛物线"和"双曲线"The names "ellipse," "parabola," and "hyperbola"

83° The names "ellipse," "parabola," and "hyperbola." [The following is adapted, with permission, from the article, by Howard Eves, of the same title that appeared in the Historically Speaking section of The Mathematics Teacher, April, 1960, pp. 280-281.]

Prior to Apollonius of Perga, the Greeks obtained the conic sections from three types of cones of revolution, according as the vertex angle of the cone was less than, equal to, or greater than a right angle. By cutting each of three such cones by planes perpendicular to a generator of the cone, an ellipse, a parabola, and a hyperbola respectively result. It follows that only one branch of a hyperbola was considered. Apollonius, on the other hand, in Book I of his great treatise Conic Sections, obtains all the conic sections in the now familiar way from one right or oblique double cone.

The names "ellipse," "parabola," and "hyperbola" were supplied by Apollonius, and were borrowed from the early Pythagorean terminology of application of areas. When the Pythagoreans applied a rectangle to a line segment (that is, placed the base of the rectangle along the line segment, with one end of the base coinciding with one end of the segment), they said they had a case of "ellipsis," "parabole," or "hyperbole" according as the base of the applied rectangle fell short of the line segment, exactly coincided with it, or exceeded it. Now let AB (see Figure 11) be the principal axis of a conic, P any point on the conic, and Q the foot of the perpendicular from P on AB. At A, which is a vertex of the conic, draw a perpendicular to AB and mark off on it a distance AR equal to what we now call the latus rectum, or parameter p, of the conic (that is, equal to the length of the chord which passes through a focus of the conic and is perpendicular to the principal axis of the conic). Apply, to segment AR, a rectangle having AQ for one side and an area equal to (PQ)². According as the application falls short of, coincides with, or exceeds the segment AR, Apollonius calls the conic an ellipse, a parabola, or a hyperbola. In other words, if we consider the curve referred to a Cartesian coordinate system having its x and y axes along AB and AR respectively and if we designate the coordinates of P by x and y, then the curve is an ellipse if y² < px, a parabola if y² = px, and a hyperbola if y² > px. Actually, in the cases of the ellipse and hyperbola,

y² = px ∓ px²/d,

where d is the length of the diameter through vertex A. Apollonius derives the bulk of the geometry of the conic sections from the geometrical equivalents of these Cartesian equations.

Now an ellipse, being a closed curve lying in the finite part of the plane, has no points in common with the so-called line at infinity in the plane. The parabola, on the other hand, is tangent to the line at infinity and thus has one and only one point in common with that line, and a hyperbola intersects the line at infinity in two distinct points. Because of these relations of the three types of conics with the line at infinity, the adjectives elliptic, parabolic, and hyperbolic have been employed in certain parts of mathematics. Thus, in 1871, Felix Klein called the non-Euclidean geometry of Lobachevsky and Bolyai hyperbolic geometry, that of Riemann he called elliptic geometry, while the name parabolic geometry was reserved for Euclidean geometry. Simplifying historical origins a little, the reason for applying these three epithets to the three geometries is essentially that in the Lobachevsky–Bolyai non-Euclidean geometry there exist two distinct lines through a point P and parallel to a line l not through P, in the Riemann non-Euclidean geometry there are no lines through P parallel to l, in Euclidean geometry there is one and only one line through P parallel to l.

The three adjectives elliptic, parabolic, and hyperbolic are also encountered in projective geometry, and for a similar reason. In projective geometry one studies, among other things, mappings of a line upon itself defined analytically by a symmetrical equation of the form

Axx' + B(x + x') + C = 0.

Here A, B, C are real constants, and x and x' are the coordinates of corresponding points under the mapping. Such a mapping of a line upon itself is called an involution, and of interest in the study of an involution are those points, called double points, which map into themselves. To find the double points of the above involution one merely sets x' = x, obtaining the quadratic equation

Ax² + 2Bx + C = 0.

The double points of the involution are the real solutions of this quadratic equation. Since the quadratic equation has two distinct real solutions if B² − AC > 0, one and only one real solution if B² − AC = 0, and no real solutions if B² − AC < 0, the involution has come to be called hyperbolic, parabolic, and elliptic in the three cases respectively. Thus a hyperbolic involution has two distinct double points, a parabolic involution has one and only one double point, and an elliptic involution has no double points.

Other uses of the adjective elliptic in mathematics occur in the following connections: elliptic cones and cylinders, elliptic coordinates, elliptic functions, elliptic integrals, elliptic paraboloids, elliptic partial differential equations, elliptic points on a surface, and elliptic Riemann surfaces. Similarly we have hyperbolic cylinders, hyperbolic functions, hyperbolic logarithms, hyperbolic paraboloids, hyperbolic partial