The Three First Sections and Part of the Seventh Section of Newton's Principia: With a Preface Recommending a Geometrical Course of Mathematical Reading, and an Introduction on the Atomic Constitution of Matter, and the Laws of Motion. By George Leigh Cooke ...

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John Henry Parker, 1850 - Dynamics of a particle - 163 pages
 

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Page 5 - Verily, verily, I say unto thee, We speak that we do know, and testify that we have seen ; and ye receive not our witness. If I have told you earthly things, and ye believe not, how shall ye believe, if I tell you of heavenly things?
Page 39 - Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal.
Page 95 - 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 39 - LEMMA I QUANTITIES, AND THE RATIOS OF QUANTITIES, WHICH IN ANY FINITE TIME CONVERGE CONTINUALLY TO EQUALITY, AND BEFORE THE END OF THAT TIME APPROACH NEARER THE ONE TO THE OTHER THAN BY ANY GIVEN DIFFERENCE, BECOME ULTIMATELY EQUAL.
Page 98 - Moreover, by means of the preceding Proposition and its Corollaries, we may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by Cor. 9 of this Prop.).
Page 114 - L place. Through the point S draw the chord PV, and the diameter VA of the circle: join AP, and draw QT perpendicular to SP, which produced, may meet the tangent PR in Z; and lastly, through the point Q, draw LR parallel to SP, meeting the circle in L, and the tangent PZ in R. And, because of the similar triangles ZQR, ZTP, VPA, we shall have RP2, that is, QRL to QT2 as AY2 to PV2.
Page 46 - WHEN THEIR BREADTHS ARE DIMINISHED IN INFINITUM, THE ULTIMATE RATIOS OF THE PARALLELOGRAMS IN ONE FIGURE TO THOSE IN THE OTHER, EACH TO EACH RESPECTIVELY, ARE THE SAME; I SAY, THAT THOSE TWO FIGURES AacE, PprT, ARE TO ONE ANOTHER IN THAT SAME RATIO.
Page 85 - DE, EF, of equal triangles; and these bases are reciprocally as the perpendiculars let fall upon them. COR. 2. If the chords AB, BC of two arcs, successively described in equal times by the same body, in spaces void of resistance, are completed into a parallelogram ABCV, and the diagonal BV of this parallelogram, in the position which it ultimately acquires when those arcs are diminished in infmitum, is produced both ways, it will pass through the centre of force COR.
Page 106 - SP; the centripetal force will be reciprocally as the solid xQ,it {{ t]^e solid be taken of that magnitude which it ultimately acquires when the points P and Q coincide. For QR is equal to the versed sine of double the arc QP, whose middle is P : and double the triangle SQP, or SP...
Page 45 - Much more will the rectilinear figure comprehended under the chords of the evanescent arcs ab, be, cd, &tc., ultimately coincide with the curvilinear figure. COR. 3. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs. COR. 4. And therefore these ultimate figures (as to their perimeters E) are not rectilinear, but curvilinear limits of rectilinear figures.

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