That the second focal ray PF' is really a diameter appears on transforming to oblique axes PF' and PT; for the trans formed equation readily reduces to Hence to any value of x correspond two equal and opposite values of y, i.e., the second focal ray PF' bisects all chords parallel to the tangent PT. The constant 4p' is easily shown to be the focal chord among these parallels, and p'=x' p=focal ray of the new origin P(x',y').
By an improved method of exhaustion, suggesting modern in tegration, Archimedes (287-212 B.c.) proved the half-segment OSP =i0SPU ; hence the whole segment POP'=iPP'U'U. So too for any oblique segment, since the form of the equation is the same and the areal elements parallel to the tangent are all sloped at the same angle (a) to the axis. Less simple, though readily proved, are such properties as that the circle circumscrib ing the triangle of any three tangents to the parabola passes through the focus F. More important, in the parabola as in all conics the radius (p) of curvature equals the cubed normal di vided by the squared half-parameter; i.e., PC= p or p : : 02. The locus of this centre of curvature C is the semi-cubical or cuspidal parabola remarkable as the first curve rectified (by W. Neil, 1657, and H. Van Haureat, 1659), unless slightly antedated by the cycloid (fig. I).
The parabola is perhaps best known as the path of a projectile through an unresisting medium under constant like-directed accel eration. However, air resists and the acceleration of gravity is neither quite constant nor fixed in direction; hence, especially in long high flights, the trajectory deviates much from a parabola, which is uniform (since e= I), the curves differing only in size, with F the centre of similitude for coaxial confocals. The parab ola appears in many other physical problems and connections.
taut against ED, traces a para bola (fig. 2). To find points of a parabola, given the focus F and the directrix DR, draw the axis through F upright on DR at D, and bisect FD; this determines the vertex 0, and OF= p. About F as centre draw any circle cutting the axis at T and N; from N lay off (toward F) NS= 2p and through S draw a circle chord PP' upright on the axis; then P and P' are points of P.
To find F and DR of a parabola, draw two mutually perpen dicular pairs of parallel chords, and also the bisectors (or diame ters) of the pairs meeting the parabola at P and Q. Draw the tangents at P and Q (parallel to the bisected chords) ; being at right angles, they meet on the directrix, as at R; draw DR per pendicular to the diameters. Through P and Q draw chords sloped to the tangents as the tangents are to the diameters; these chords meet in the focus F, the diameter through F is the axis, the segment FD is 2p, its mid-point 0 is the vertex.
The name parabola is applied loosely to many other curves, especially of the equational form ytn+"= amx", besides the Apol lonian or quadratic = 4Px. For example, the cubic = the biquadratic the semicubic parabola all having been subjects of early study (Descartes, Newton, et al.).
There is also the cubic parabola
called the cartesian parabola from its use by Descartes to solve sextic equa tions by its intersections with a circle. There are also the which vary widely with varying a
See T. H. Eagles, Plane Curves (1885).