The First Three Sections of Newton's Principia: With Copious Notes and Illustrations, and a Great Variety of Deductions and Problems. Designed for the Use of Students |
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Common terms and phrases
accelerating force angle arc described body revolving centre of force centrifugal force centripetal force chord of curvature circle conic section curve curvilinear area diameter diminished duplicate ratio dx² earth ellipse evanescent finite focus force of gravity given Hence hyperbola indefinitely greater indefinitely less indefinitely small inversely latus rectum Lemma limit mean distance motion Note to Prop orbit parabola parallel parallelograms perpendicular polygon Prob proportional PROPOSITION Q T² quantity radii reciprocally revolve round right line round its axis SCHOLIUM subduplicate ratio subtense supposed tangent triangles ultimate ratio vanish velocity
Popular passages
Page 45 - In a parabola, the velocity of a body at any distance from the focus is to the velocity of a body revolving in a circle, at the same distance...
Page 26 - From the same demonstration it likewise follows that the arc which a body, uniformly revolving in a circle by means of a given centripetal force, describes in any time is a mean proportional between the diameter of the circle and the space which the same body falling by the same given force would descend through in the same given time.
Page 10 - BF parallel to the tangent, always cutting any right line AF passing through A in F, this line BF will be ultimately in the ratio of equality with the evanescent arc ACB; because, completing the parallelogram AFBD, it is always in a ratio of equality with AD. COR. 2. And if through B and A more right lines are drawn, as BE, BD, AF, AG, cutting the tangent AD and its parallel BF; the ultimate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the...
Page 5 - If you deny it, suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is against the supposition.
Page 18 - I should happen to consider quantities as made up of particles, or should use little curve lines for right ones, I would not be understood to mean indivisibles, but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas.
Page 20 - EF, &c., they will all lie in the same plane; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE.
Page 26 - Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed); and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres.
Page 23 - Proposition is that the force which results out of all tends to the point S. But if any force acts perpetually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion: but will neither augment nor diminish the quantity of the described surface, and is therefore to be neglected in the composition of forces. PROPOSITION III.
Page 43 - ... time. Subduct from both sides the subduplicate ratio of the latus rectum, and there will remain the sesquiplicate ratio of the greater axis, equal to the ratio of the periodic time. QED COR. Therefore the periodic times in ellipses are the same as in circles whose diameters are equal to the greater axes of the ellipses.
Page 41 - For the focus, the point of contact, and the position of the tangent, being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and velocity of the body being given; and two orbits, touching one the other, cannot be described by the same centripetal force and the same velocity.