If in any of the Conick sections about the Focus therof then he demonstrates that the vis centripeta or tendency towards that focus is in all places reciprocally as the square of the distance therfrom, and that according to the velocity of the impressed motion, the Curve described is an Hyperbola, if the body moved be swift to a certain degree then a parabola if slower an Ellipse, or Circle in one case from this some tendency or gravitation it follows likewise that the squares of the times of the periodical revolutions are as the cubes of the Radii or Transverse Axes of the Ellipses. All which being found to agree with the phenomena of the Celestiall motions, as descovered by the great sagacity and industry of Kepler Our Author extends him self upon the consequences of this <s>gravity \s> sort of vis centripeta, showing how to <s>discover <\s> find the conick section which a bodie shall describe when cast with any velocity in a given line, supposing the quantity of the said force known and laying down severall neat constructions to determine the orbs either from the focus given and two points or Tangents: or without it by 5 points or Tangents or any number of points and Tangents making together five. Then he shows how from the time given to find the point in the Orb answering therto: which he performs accuratly in the parabola and by concise approximations <s>shows <\s> comes as near as he pleases in the Ellipse and Hyperbola: all which are problems of the highest concern in Astronomy. Next he lays down the rates of the perpendicular descent of bodies towards their center, particularly in the case where the tendency shewn is reciprocally as the square of the distance, <s>therfrom <\s> and generally in all other cases, supposing a generall quadrature of Curve lines: upon which supposition likewise he delivers a generall method of discovering the Orbs described by a body moving in such a tendency towards a Center, encreasing or decreasing in any given relation to the distance from the Center and then with great subtilty he determines in all cases the motion of the Apsides (or of the points of greatest distance from the Center of all these Curves in such orbs as are nearly Circular shewing the Apsides fixt if the tendency be reciprocally as the square of the distance; direct in motion if in any ratio between the square and the Cube, and retrograde if under the square <s>if reciprocally as the cube <\s> which motion he determines exactly from the rule of the increase or decrease of the Vis Centripeta.
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Manuscript details
 Author
 Edmond Halley
 Reference
 CLP/21/13
 Series
 Cl.P
 Date
 1687
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Cite as
Of Newton's Principia , 1687. From The Royal Society, CLP/21/13
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