term to the gravitational law to explain the observed motion of the perihelion, the point in the orbit of the Moon where it is closest to the Earth.
The first to propose a system of planetary orbits which would set the scene for major advances was Copernicus who in De revolutionibus orbium coelestium (1543), argued that the planets and the Earth were in orbit round the Sun. Although a major breakthrough, Copernicus proposed circular orbits for the planets and accurate astronomical observations soon began to show that his proposal was not strictly accurate.
You can see a diagram from De revolutionibus orbium coelestium showing Copernicus's solar system.
In 1600 Kepler became assistant to Tycho Brahe who was making accurate observations of the planets. After Brahe died in 1601 Kepler continued the work, calculating planetary orbits to unprecedented accuracy.
Kepler showed that a planet moves round the Sun in an elliptical orbit which has the Sun in one of its two foci. He also showed that a line joining the planet to the Sun sweeps out equal areas in equal times as the planet describes its orbit. Both these laws were first formulated for the planet Mars, and published in Astronomia Nova (1609).
You can see a diagram from Astronomia Nova showing Kepler's elliptical orbit for Mars..
However scientists certainly did not accept Kepler's first two laws with enthusiasm. The first was given a cool reception and was certainly thought to require further work to confirm it. The second of Kepler's laws suffered an even worse fate in being essentially ignored by scientists for around 80 years.
Kepler's third law, that the squares of the periods of planets are proportional to the cubes of the mean radii of their orbits, appeared in Harmonice mundi (1619) and, perhaps surprisingly in view of the above comments, was widely accepted right from the time of its publication.
In 1679 Hooke wrote a letter to Newton. In the letter he explained how he considered planetary motion to be the result of a central force continuously diverting the planet from its path in a straight line. Newton did not answer this directly but explained his own idea that the rotation of the Earth could be proved from the fact that an object dropped from the top of a tower should have a greater tangential velocity than one dropped near the foot of the tower.
Newton provided a sketch of the path that the particle would follow, quite incorrectly showing it spiral towards the centre of the Earth. Hooke replied that his theory of planetary motion would lead to the path of the particle being an ellipse so that the particle, were it not for the fact that the Earth was in the way, would return to its original position after traversing the ellipse.
Newton, not one to like being corrected, had to admit that his original sketch was incorrect but he "corrected" Hooke's sketch on the assumption that gravity was constant. Hooke replied to Newton that his own theory involved an inverse square law for gravitational attraction. Many years later Hooke was to claim priority for proposing the inverse square law of gravitation and used this letter to Newton to support his claim.
It is worth emphasising that there is a major step to be made from an inverse square law of force to explain planetary motion and a universal law of gravitation. Certainly the motion of the Moon round the Earth was not seen to necessarily be part of the same laws which govern the motion of the planets round the Sun.
Fifty years after these events Newton was to record his own recollections of these events which, although interesting, do not really agree with the known historical facts! [I preserve Newton's old English.]
In the same year I began to think of gravity extending to ye orb of the Moon and (having found out how to estimate the force with wch globe revolving within a sphere presses the surface of a sphere) from Kepler's rule of the periodical times of the Planets being in sesquialternate proportion to their distances from the centres of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must reciprocally as the squares of their distances from the centres about wch they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth, and found them answer pretty nearly. All this was in the two plague years of 1665-1666...In 1684 Wren, Hooke and Halley discussed, at the Royal Society, whether the elliptical shape of planetary orbits was a consequence of an inverse square law of force depending on the distance from the Sun. Halley wrote that
Mr Hook said that he had it, but that he would conceale it for some time so that others, triing and failing might know how to value it, when he should make it publick.Later in the same year in August, Halley visited Newton in Cambridge and asked him what orbit a body would follow under an inverse square law of force
Sr Isaac replied immediately that it would be an Ellipsis, the Doctor struck with joy and amasement asked him how he knew it, why, said he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it, and then to send it him.Despite the claims by Newton in the above quote, he had in fact proved this result in 1680 as a direct result of the letters from Hooke. Newton indeed reworked his proof and sent a nine page paper De motu corporum in gyrum (On the motion of bodies in an orbit) to Halley. It did not state the law of universal gravitation nor Newton's three laws of motion. All this was to develop over the next couple of years to become the basis for the Principia .
Halley was largely responsible for ensuring that the Principia was published. He received Newton's complete manuscript by April of 1687 but there were many problems not the least being that Newton tried to prevent the publication of the Book III when Hooke claimed priority with the inverse square law of force.
In the Principia the problem of two attracting bodies with an inverse square law of force is completely solved (in Propositions 1-17, 57-60 in Book I). Newton argues that an inverse square law must give produce elliptical, parabolic or hyperbolic orbits.
A bright comet had appeared on 14 November 1680. It remained visible until 5 December 1680 when it moved too close to the Sun to be observed. It reappeared two weeks later moving away from the Sun along almost the same path along which it had approached. Newton found good agreement between its orbit and a parabola. He uses the orbit of this comet, and comets in general, to support his inverse square law of gravitation in the Principia .
You can see Newton's diagram of the orbit of the comet of 1680 from the Principia .
In the Principia Newton also deduced Kepler's third law. He looked briefly (in Propositions 65 and 66) at the problem of three bodies. However Newton later said that an exact solution for three bodies
exceeds, if I am not mistaken, the force of any human mind.It is important at this stage to examine the problems which now arose. Newton had completely solved the theoretical problem of the motion of two point masses under an inverse square law of attraction. For more than two point masses only approximations to the motion of the bodies could be found and this line of research led to a large effort by mathematicians to develop methods to attack this three body problem. However, the problem of the actual motion of the planets and moons in the solar system was highly complicated by other considerations.
Even if the Earth - Moon system were considered as a two body problem, theoretically solved in the Principia , the orbits would not be simple ellipses. Neither the Earth nor the Moon is a perfect sphere so does not behave as a point mass. This was to lead to the development of mechanics of rigid bodies, but even this would not give a completely accurate picture of the two body problem since tidal forces mean that neither the Earth nor Moon is rigid.
The observational data used by Newton in the Principia was provided by the Royal Greenwich Observatory. However modern scholars such as Richard Westfall claim that Newton sometimes adjusted his calculations to fit his theories. Certainly the observational evidence could not be used to prove the inverse square law of gravitation. Many problems relating observation to theory existed at the time of the Principia and more would arise.
Halley used Newton's method and found almost parabolic orbits for a number of comets. When he computed the orbits for three comets which had appeared in 1537, 1607 and one Halley observed himself in 1682, he found that the characteristics of the orbits were almost identical. Halley deduced they were the same comet and later was able to identify it with one which had appeared in 1456 and 1378. He computed an elliptical orbit for the comet and he noticed that Jupiter and Saturn were perturbing the orbit slightly between each return of the comet. Taking the perturbations into account Halley predicted the comet would return and reach perihelion (the point nearest the Sun) on 13 April 1759. He gave an error of one month on either side of this date. The comet was actually first seen again in December 1758 reaching perihelion on 12 March 1759.
In 1713 a second edition of the Principia , edited by Roger Cotes, appeared. Cotes wrote a preface defending the theory of gravitation given in the Principia . Cotes was himself to provide the next mathematical steps by finding the derivatives of the trigonometric functions, results published after his death.
Euler developed methods of integrating linear differential equations in 1739 and made known Cotes' work on trigonometric functions. He drew up lunar tables in 1744, clearly already studying gravitational attraction in the Earth, Moon, Sun system. Clairaut and d'Alembert were also studying perturbations of the Moon and, in 1747, Clairaut proposed adding a 1/r term to the gravitational law to explain the observed motion of the perihelion, the point in the orbit of the Moon where it is closest to the Earth.
However by the end of 1748 Clairaut had discovered that a more accurate application of the inverse square law came close to explaining the orbit. He published his version in 1752 and, two years later, d'Alembert published his calculations going to more terms in his approximation than Clairaut. In fact this work was of importance in having Newton's inverse square law of force accepted in Continental Europe.
The Earth's axis of rotation precesses, that is the direction of the axis of rotation itself rotates in a circle with a period of about 26000 years. Precession is caused by the gravitational attraction of the Sun on the equatorial bulge of the Earth, the bulge being predicted by Newton. Cassini made a measurement of an arc of longitude in 1712 but obtained a result which wrongly suggested that the Earth was elongated at the poles. In 1736 Maupertuis obtained the correct result verifying Newton's predictions. However, this illustrates the problems encountered by mathematicians at this time with basic data about bodies in the solar system, even the Earth, being highly inaccurate.
There is a small periodic effect called nutation superimposed on precession caused by the motion of the perihelion of the Moon. This superimposed effect has a period of 18.6 years and was first observed by Bradley in 1730 but not announced until 18 years later when he had observed the full cycle. D'Alembert quickly showed that Bradley's observed period was deducible from the inverse square law and Euler further clarified this with further work on the mechanics of rigid bodies during the 1750's.
The problem of the orbits of Jupiter and Saturn had troubled astronomers and mathematicians from Kepler's first theory of elliptical orbits. The Paris Academy offered Prizes for work on this topic in 1748, 1750 and 1752. In 1748 Euler's studies of the perturbation of Saturn's orbit won him the Prize. His work for the 1752 Prize, however, contains many mathematical errors and was not published until 17 years later. It did contain significant ideas, however, which were independently discovered since Euler's work was not known.
Lagrange won the Paris Academy Prize in 1764 for a work on the libration of the Moon. This is a periodic movement in the axis the Moon pointing towards the Earth which allows, over a period of time, more than 50% of the surface of the Moon to be seen. He also won the Paris Academy Prize of 1766 for work on the orbits of the moons of Jupiter where he gave a mathematical analysis to explain an observed inequality in the sequence of eclipses of the moons.
Euler, from 1760 onwards, seems to be the first to study the general problem of three bodies under mutual gravitation (rather than looking at bodies in the solar system) although at first he only considered the restricted three body problem when one of the bodies has negligible mass. When one body has negligible mass it is assumed that the motions of the other two can be solved as a two body problem, the body of negligible mass having no effect on the other two. Then the problem is to determine the motion of the third body attracted to the other two bodies which orbit each other. Even in this form the problem does not lead to exact solutions. Euler, however, found a particular solution with all three bodies in a straight line.
The first comet to have an elliptical orbit calculated which was far from a parabola was observed by Messier in 1769. The elliptical orbit was computed by Lexell who correctly realised that the small elliptical orbit had been produced by perturbations by Jupiter. The comet made no reappearance and again Lexell correctly deduced that Jupiter had changed the orbit so much that it was thrown far away from the Sun.
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