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Cosine, do leave off throwing your hyperbolas and parabolas at one's head. I want to know the only interesting thing about the business. We shall follow one or other of your curves. Very well. But where will they take us to?" "Nowhere," answered Nicholl. "How nowhere?" "Evidently they are unfinished curves, prolonged indefinitely!" "Ah, savants!
Consequently, anything coming to the Sun from a great distance must have one exact amount of proper motion to produce a parabola: all other amounts would give hyperbolas or ellipses. And if there are no hyperbolic orbits, then it is infinity to one that all the orbits are elliptical. This is just what they would be if comets had the genesis above supposed.
There's nothing more to fear from your hyperbolas or parabolas or any other of your open curves!" "Nothing more, certainly, from an open curve, but every thing from a closed one." "A closed curve! What is it called? And what is the trouble?"
For it can similarly be demonstrated, by taking any other point in the curve, such as G, that the excess of AG over AD, namely VG, is to the excess of BD over DG, namely DP, in this same ratio of 3 to 2. And following this principle Mr. Des Cartes constructed these curves in his Geometric; and he easily recognized that in the case of parallel rays, these curves became Hyperbolas and Ellipses.
As we have seen, Newton's contemporaries had perceived how circular motions could be explained; that was a special case, but Newton furnished the solution of the general problem, containing all special cases of motion in circles, ellipses, parabolas, hyperbolas that is, in all the conic sections.
Why should Denis? He is not nearly as much of a fool as many people look. And I would not listen to Keith. He moves among hyperbolas." Mr. Heard felt slightly relieved. What a sensible fellow this was; so matter-of-fact and sure of his ground. The ideal scholar. Sirocco did not exist for him. He stood aloof from human passions and infirmities.
"Is it of the slightest possible importance which of the two curves controls the Projectile?" "Not the slightest except in the Eye of Science!" "Then let the Eye of Science and her parabolas and hyperbolas, and conjugates, and asymptotes, and the rest of the confounded nonsensical farrago, all go to pot! What's the use of bothering your heads about them here!
The ellipse can be generated ideally out of the definition, and would have been generated, like asymptotes and hyperbolas, even if never illustrated in nature at all. Lemmas from a foreign tongue have only served to disclose a great fecundity in the native one, and the legitimate word that the context required has supplanted the casual stranger that may first have ushered it into the mind.
And finally I shall treat of the various shapes of transparent and reflecting bodies by which rays are collected at a point or are turned aside in various ways. From this it will be seen with what facility, following our new Theory, we find not only the Ellipses, Hyperbolas, and other curves which Mr.
Ellipses and hyperbolas have countless varieties of forms, but there is only one form of parabola; or, to speak literally, all parabolas are similar, while there are infinitely numerous dissimilar ellipses and dissimilar hyperbolas.
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