The Principia : Mathematical Principles of Natural Philosophy
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falling by the force of its own gravity. The first of these spaces is equal to the versed sine of the arc described by the moon during the same time, inasmuch as this versed sine measures the departure of the moon from the tangent caused by centripetal force and thus can be calculated if the moon’s periodic time and its distance from the center of the earth are both given. The second space is found by experiments with pendulums, as Huygens has shown. Therefore, the result of the calculation will
moon was occupying three hours earlier, H the place on the earth situated perpendicularly beneath L, h the place opposite H, K and k places 90 degrees distant from H and h, CH and Ch the greatest heights of the sea (measured from the center of the earth), and CK and Ck the least heights. If an ellipse is described with axes Hh, Kk, and then if by the revolution of this ellipse about the major axis Hh a spheroid HPKhpk is described, this spheroid will represent the figure of the sea very nearly,
terrestrial atmosphere, shining with the light of the sun, by its thickness of only a few miles obscures and utterly extinguishes the light not only of all the stars but also of the moon itself; yet the smallest stars are known to shine, without any loss in their brightness, through the immense thickness of the tails, which are likewise illuminated by the light of the sun. Nor is the brightness of most cometary tails generally greater than that of our air reflecting the light of the sun in a
is decreased as the square of that ratio, and so the sum of these forces is decreased in a less than squared ratio of the distance PT, and therefore (by prop. 45, corol. 1) causes the auge, or upper apsis, to regress. But in conjunction and opposition the force whereby body P is urged toward body T is the difference between the force by which body T attracts body P and the force KL; and that difference, because the force KL is increased very nearly in the ratio of the distance PT, decreases in a
opposition of themselves and body S than in the quadratures. The nodes of this ring, or its intersections with the plane of the orbit of body S or T, will be at rest in the syzygies, but outside the syzygies they will move backward [or in antecedentia], and do so most swiftly in the quadratures and more slowly in other places. The inclination of the ring will also vary, and its axis will oscillate in each revolution; and when a revolution has been completed, it will return to its original