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Updated: May 23, 2025
He proved that if a body be projected with any velocity in free space, subject only to a central force, varying inversely as the square of the distance, the body must revolve in a curve which may be any one of the sections of a cone a circle, ellipse, parabola, or hyperbola; and he found that those comets of which he had observations move in parabolae round the Sun, and are thus subject to the universal law.
"It is very nearly," continued Nicholl, "the course described by a bomb launched from a mortar." "Perfect! And the hyperbola?" "The hyperbola, Michel, is a curve of the second order, produced by the intersection of a conic surface and a plane parallel to its axis, and constitutes two branches separated one from the other, both tending indefinitely in the two directions."
A single blow was enough to melt the wax over a certain zone; and this indicated clearly how much of the lateral faces had been raised by the shock to the temperature of melting wax. The form of this melted part could be made to differ considerably, but approximated to that of an equilateral hyperbola.
For about a century and a half, to the time of Apollonius, the beauty of this curve, and of its variations, the parabola and hyperbola, so fascinated the minds of Plato's followers, that Apollonius found theorems and problems relating to these figures sufficient to fill eight books with condensed truths concerning them.
Whether it was a parabola or a hyperbola, however, or whether it was disturbed or not, made very little difference as, in any case, the Projectile was bound to quit pretty soon the cone of the shadow, at a point directly opposite to where it had entered it.
For example, when we have proved with respect to the circle, that a straight line can not meet it in more than two points, and when the same thing has been successively proved of the ellipse, the parabola, and the hyperbola, it may be laid down as a universal property of the sections of the cone.
Suiting her action to the word, she picked up a red book and sent it in a gentle hyperbola toward the ceiling, where it sank into the lamp beside the first. It was a few minutes before either of them could do more than rock back and forth in helpless glee; but then by mutual agreement they took up the sport anew, this time in unison.
This is the play of Brahm; endlessly to dissever, in time and space, and to unite in consciousness, like the geometrician who discovers every ellipse, parabola, and hyperbola, in the cone where all inhere." A surgeon once remarked to the author that among his professional associates he had noticed an increasing awareness of the invisible.
Menaechmus, a pupil of Eudoxus, was the discoverer of the conic sections, two of which, the parabola and the hyperbola, he used for solving the problem of the two mean proportionals. If a:x=x:y=y:b, then x²=ay, y²=bx and xy=ab.
The combination of a centripetal with a projectile force, in the proportions which obtain in all the planets and satellites of our solar system, gives rise to an elliptical motion; but if the ratio of the two forces to each other were slightly altered, it is demonstrated that the motion produced would be in a circle, or a parabola, or an hyperbola; and it is thought that in the case of some comets one of these is probably the fact.
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