File:Principia Circular Orbit Propositions 7 and 8.jpg

English: In the Principia Book 1 Proposition 7 Newton determines the acceleration directed to a fixed point required to make a particle move in a circular arc, where the point is not the centre of the circle. The method is the same as that used to prove that the acceleration required to make a body move in an elliptical orbit directed to one of the foci is inversely proportional to the square of the distance of the body from that focus. In Proposition 8 he considers the case where the fixed point is infinitely distant from the circle, so the acceleration acts in a fixed direction at all points on the circle. In the Scholium to Proposition 8, he extends this by transforming the circle successively to an ellipse, parabola and hyperbola.