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Updated: May 23, 2025
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 demonstrable that the motion produced would be in a circle, or a parabola, or an hyperbola: and it has been surmised that in the case of some comets one of these is really the fact.
"'The squares of the times are as the cubes of the distance' means that all motion is the result of two forces acting upon prakriti, and that where the two forces are balanced, or equal, the result in motion is a circle or ellipse, the square of the Repulsion being equal to the cube of the Attraction to make them equal and produce a circle. In other cases they produce hyperbola and parabola.
Now for the high high bully old curve!" "The hyperbola," continued the Captain, not minding Ardan's antics, "the hyperbola is a curve of the second order, formed from the intersection of a cone by a plane parallel to its axis, or rather parallel to its two generatrices, constituting two separate branches, extending indefinitely in both directions."
I cannot do more than merely allude to the treatises on Conic Sections and on Maxima and Minima by Apollonius, who is said to have been the first to introduce the words ellipse and hyperbola. In like manner I must pass the astronomical observations of Alistyllus and Timocharis.
It is relatively like what is said of a straight line drawn about a hyperbola, always approaching but never touching it, and like what is said about squaring a circle. Hence it may be plain what is meant by the means by which divine providence acts in order that man may be man and be perfected in understanding, and that these means are called by the common term truths.
We all know those meetings now the fraternal handshake, the menagerie smell, the reek of tobacco, the indistinguishable hubbub of tongues, the frothy violence, the bottomless inanity of abstract dissensions, that have less concern with human realities than the curve of the hyperbola through space.
The conic section is submitted to algebra, an obstetrical appliance which brings forth the equation; and behold, entreated now in one direction, now in another, the womb of the formula gives birth to the ellipse, the hyperbola, the parabola, their foci, their radius vectors, their tangents, their normals, their conjugate axes, their asymptotes and the rest.
Never, never rest contented with any circle of ideas, but always be certain that a wider one is still possible. For my thought is like a hyperbola that continually widens ascending. For grief there is no known consolation. It is useless to fill our hearts with bubbles. A loved one gone is gone, and as to the future even if there is a future it is unknown.
"The projectile has the choice between two mathematical curves, and it will follow one or the other according to the speed with which it is animated, and which at this moment I cannot estimate." "Yes," said Nicholl, "it will follow either a parabola or a hyperbola." "Just so," replied Barbicane. "With a certain speed it will assume the parabola, and with a greater the hyperbola."
In these it is plain he used Hyperbola and Amplification, for he was not satisfied with comparing the clamor to the sound of the wind, but to the waves beating on a craggy shore, where the high sea makes the noise greater.
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