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Updated: May 23, 2025
Thus the properties of the ellipse, the hyperbola, and the parabola, continued to be studied by after mathematicians; but no use was made of this knowledge till nearly two thousand years later, when Kepler crowned the labours of Apollonius with the great discovery that the paths of the planets round the sun were conic sections.
Safe from that, I could say, 'Fate, do your worst! Besides, I don't believe in the infallibility of your ellipsic. It may prove just as unreliable as the hyperbola. And it is no harm to hope that it may!" From present appearances there was very little to justify Ardan's hope.
The parabola is the boundary of the projection of a sphere upon a plane, when the eye is just as far from the plane as the outer edge of the sphere is, and the hyperbola is a similar curve formed by bringing the eye still nearer to the plane. By these metamorphoses the circle loses much of its monotony, without losing much of its simplicity.
The outline of a sphere, projected by the beholder against any plane surface behind it, is a circle only when a perpendicular, let fall on the plane from the eye, passes through the centre of the sphere. In other positions the projection of the sphere becomes an ellipse, or one of its varieties, the parabola and hyperbola.
Among the unfamiliar faces that show themselves now and then at Lisconnel, some make no second appearance, never coming our way again, but passing out of our ken as utterly as if their route lay along a tangent, or the branch of an hyperbola, or other such unreverting line. We seldom, it is true, get proof positive, as in the case of the Dermodys, father and son, that they will no more return.
Some are elliptic, but, for the greater part round: some represent an hemisphere, others a segment, others an hyperbola. The glossopetrae are naturally of a conic figure, representing acute, obtuse, regular, and irregular cones.
The origin of comets has been long and eagerly inquired into, not altogether apart from the cheering guidance of ascertained facts. Sir William Herschel regarded them as fragments of nebulæ scattered débris of embryo worlds; and Laplace approved of and adopted the idea. But there was a difficulty. No comet has yet been observed to travel in a decided hyperbola.
"That if," cried Rhimeson, interrupting him, "he addresses a lady as cold, slow, and misty-hearted as himself, they may go on courting the whole course of their natural lives, like the assymptotes of a hyperbola, which approach nearer and nearer, ad infinitum, without the possibility of ever meeting."
The sun’s action, and the original impelling force, might have been in such a ratio to one another, that the reaction of the effect would have been such as to alter the causes more and more, without ever bringing them back to what they were at any former time. The planet would then have moved in a parabola, or an hyperbola, curves not returning into themselves.
The parabolic curve, as it is called, is a line partaking of the closeness of the ellipse on the one hand, and the openness of the hyperbola on the other. A parabola is an ellipse passing into a hyperbola; or, in other words, it is a part of an ellipse whose length, compared with its breadth, is too great to be estimated, and is consequently deemed to be endless for all practical purposes.
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