Newton's Principia, Sections I, II, III, with Notes and Illustrations; Also a Collection of Problems (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1863 Excerpt: ... the Moon's orbit relative to the Earth to be nearly circular, and let ABCD be this orbit, E the Earth. 1. The areas described by the radii drawn from the Moon to the Earth are nearly proportional to the times of describing; hence the resultant force on the Moon tends nearly to E. 2. If ES the line joining the centers of the Earth and Sun meets the Moon's relative orbit about the Earth in A, C, and DEB be perpendicular to ES, the description of areas is accelerated as the Moon moves from D to A and from B to C, and retarded from A to B and from C to D; hence the direction of the resultant force on the Moon in the positions Mx, M2, MB, Mt, is in the directions of the arrows slightly inclined to the radii drawn to E. From these observed facts, we see that when the force, under the action of which E moves, is applied to the Moon in the contrary direction, the remaining force tends in the directions of the arrows. By the supposition that the Earth and Moon are acted on by forces tending to the Sun, whose distance compared with EM is very great, and that the differences of the forces on these bodies are not very great, the circumstances of the description of areas in the motion of the Moon are accounted for. Prop. IV. Theorem IV. The centripetal forces of equal bodies, which describe different circles with uniform velocity, tend to the centers of the circles, and are to each other as the squares of arcs described in the same time, divided by the radii of the circles. The bodies move uniformly, therefore the arcs described are proportional to the times of describing them; and the sectors of circles are proportional to the arcs on which they stand, therefore the areas described by radii drawn to the centers are proportional to the times of describing them; hence, ...