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Even so, Clairaut was unable to calculate from gravitational theory a value for the progression of the lunar apogee greater than 50percentt of the observed value; therefore, he supposed in 1747 (with Euler) that Newton's inverse-square law was but the first term of a series and, hence, an approximation not valid for distances as small as that between Earth and the Moon.
If this were true, then the longest lunar eclipse of the century should occur very close to lunar apogee; the slow speed should outweigh the relatively smaller size of Earth's umbral shadow.
Newton never found a way of deriving the precession of the lunar apogee from the theory of gravity, and consequently he never succeeded with a complete, gravity-derived account of the lunar orbit.[53] The mathematical treatment of the three lunar inequalities nevertheless did provide added support for his theory of gravity.
Newton's Theory of the Moon's Motion, published 1702 as part of David Gregory's Astronomiae Physicae & Geometricae Elementa, employed an epicycle-like device that Jeremiah Horrocks had devised for representing the precession of the lunar apogee and the inequality known as the "evection".
Still, it is worth noting that the conclusion Clairaut first drew from the factor of 2 discrepancy in the motion of the lunar apogee was not that Newton's theory of gravity was false, but that the inverse-square needed to be supplemented by a 1/r4 term — a response fully in keeping with Newton's fourth rule of reasoning.
The theory of gravity had too many loose ends, the most glaring of which was a factor of 2 discrepancy in the mean motion of the lunar apogee, a discrepancy that undercut the claim that the Moon is held in orbit by an inverse-square force.
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During the total lunar eclipse of July 27 , 2018 the smallest (apogee) full Moon appears smaller than the Sun at all times of the year.
The longest lunar eclipses possible correspond to the smallest apogee full Moons of all.
In 1798, the Institut de France offered a prize for the determination of the mean places of the apogee and ascending node of the lunar orbit, by means of at least 500 observations.
At this time the Moon will be more than 30,000 miles closer than its furthest point (its apogee) and it will appear almost 14% bigger in the sky – perfect timing for the lunar eclipse.
This is completely different from where apogee and perigee are on the Moon's elliptical orbit, but these also migrate throughout the course of a lunar year; the Moon's orbit is perturbed by the other masses in the Solar System, like the Sun, in its orbit around Earth.
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