Of Newton's Principia

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                                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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Edmond Halley
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Cite as

Of Newton's Principia , 1687. From The Royal Society, CLP/21/13



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