Although the motions of the planets were discussed by the Greeks they believed that the planets revolved round the Earth so are of little interest to us in this article although the method of epicycles is an early application of Fourier series.

The first to propose a system of planetary paths which would set the scene for major advances was Copernicus who in *De revolutionibus orbium coelestium* (1543), argued that the planets and the Earth moved round the Sun. Although a major breakthrough, Copernicus proposed circular paths for the planets and accurate astronomical observations soon began to show that his proposal was not strictly accurate.

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 paths to unprecedented accuracy.

Kepler showed that a planet moves round the Sun in an elliptical path 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 path. Both these laws were first formulated for the planet Mars, and published in *Astronomia Nova* (1609).

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 paths, 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*.

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*^{4} 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 Académie des Sciences 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 Académie des Sciences 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 Académie des Sciences 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.

The Académie des Sciences Prize of 1772 for work on the orbit of the Moon was jointly won by Lagrange and Euler. Lagrange submitted *Essai sur le problème des trois corps * in which he showed that Euler's restricted three body solution held for the general three body problem. He also found another solution where the three bodies were at the vertices of an equilateral triangle. Lagrange considers his solutions do not apply to the solar system but we now know the both the Earth and Jupiter have asteroids sharing their orbits in the equilateral triangle solution configuration discovered by Lagrange. For Jupiter these bodies are called Trojan planets, the first to be discovered being Achilles in 1908. The Trojan planets move 60 in front and 60 behind Jupiter at what are now called the Lagrangian points.

However all this work on the orbits of bodies in the solar system failed to keep pace with observations which always seemed one step ahead, giving further and yet further problems for the theorists to explain. Laplace, from 1774 onwards, became an important contributor to the attempt of the theoreticians to explain the observations of the observers.

Lagrange introduced the method of variation of the arbitrary constants in a paper in 1776 stating that the method was of interest in celestial mechanics and, in special cases, had been already been used by Euler, Laplace and himself. Lagrange published further major papers in 1783 and 1784 on the theory of perturbations of orbits using methods of variations of the arbitrary constants and, in 1785, applied his theory to the orbits of Jupiter and Saturn.

An important development occurred on 13 March 1781 when the astronomer William Herschel (father of John Herschel) observing in his private observatory in Bath, England found

... a curious either nebulous star or perhaps a comet.

Almost immediately it was realised that it was a planet and within a year of its discovery it was shown to have an almost circular orbit. The name Uranus was eventually adopted although William Herschel himself proposed Georgium Sidus (perhaps in the hope of more funds from King George!) while in France it was known as Herschel until the middle of the following century.

Laplace read a memoir to the Académie des Sciences on 23 November 1785 in which he gave a theoretical explanation of all the remaining major discrepancies between theory and observation of all the planets and their moons excluding Uranus. He also addressed the question of the stability of the solar system for the first time. This work was to culminate in the publication of *Mécanique céleste * (1799) in which, among many other important results, he claimed to prove the stability of the solar system.

The remaining observations not explained by theory at the end of the 18 Century concerned the motion of the Moon. Laplace's work of 1787, that of Adams of 1854 and later Delaunay's work described below eventually provided solutions. Observations of Uranus in the early years of the 19th Century showed there were problems with its orbit and by 1830 Uranus had departed by 15" from the best fitting ellipse.

The next body to be discovered in the solar system was the minor planet Ceres, discovered in 1801. In 1766 J D Titus and in 1772 J E Bode had noted that

(1+4)/10, (3+4)/10, (6+4)/10, (12+4)/10, (24+4)/10, (48+4)/10, (96+4)/10

gave the distances of the 6 known planets from the Sun (taking the Earth's distance to be 1) except there was no planet at distance 2.8. The discovery of Uranus at distance 19.2 was close to the next term of the sequence 19.6.

A search was made for a planet at distance 2.8 and on 1 January 1801 G Piazzi discovered such a body. On 11 February Piazzi fell ill and ended his observations. The new planet, unobserved by other astronomers, passed behind the Sun and was lost. However Gauss in a brilliant piece of work was able to compute an orbit from the small number of observations. In fact Gauss' s method requires only 3 observations and is still essentially that used today in calculating orbits. Ceres, so named by Piazzi, was found to be where Gauss predicted by Olbers. Its distance from the Sun fitted exactly the 2.8 prediction of the Titus-Bode law.

Johann Encke, a student of Gauss, computed (using Gauss's method) an elliptical orbit for the comet of 1818. It had the shortest known period of 3.3 years. The period showed a periodic decrease which Encke could not explain by perturbations by other planets.

Work on the general three body problem during the 19th Century had begun to take two distinct lines. One was the developing of highly complicated methods of approximating the motions of the bodies. The other line was to produce a sophisticated theory to transform and integrate the equations of motion. The first of these lines was celestial mechanics while the second was rational or analytic mechanics. Both the theory of perturbations and the theory of variations of the arbitrary constants were of major mathematical significance as well as contributing greatly to the understanding of planetary orbits.

Papers published by Hamilton in 1834 and 1835 made major contributions to the mechanics of orbiting bodies. as did the significant paper published by Jacobi in 1843 where he reduced the problem of two actual planets orbiting a sun to the motion of two theoretical point masses. As a first approximation the theoretical point masses orbited the centre of gravity of the original system in ellipses. He then used a method, first discovered by Lagrange, to compute the perturbations. Bertrand extended Jacobi's work in 1852.

In 1836 Liouville studied planetary theory, the three body problem and the motion of the minor planets Ceres and Vesta. Many mathematicians around this period devoted much of their time to these problems. Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".

By around 1840 irregularities in the orbit of Uranus prompted many scientists to seek reasons them. Alexis Bouvard (a collector of planetary data) proposed that a planet might explain the irregularities and he wrote to the English Astronomer Royal Airy proposing this idea. Bessel also proposed this solution to the problem but died before completing his calculations. Delaunay, famed for his work on the orbit of the Moon, investigated the perturbations in a paper of 1842. Arago urged Le Verrier to work on the problem and on 1 June 1846 Le Verrier showed that the irregularities could be explained by an unknown planet and he determined the coordinates at which the planet would be found. The astronomer Galle in Berlin found the new planet on 26 September remarkably close to the position predicted by Le Verrier. The observations were confirmed on 29 September 1846 at the Paris observatory.

This was a remarkable achievement for Newton's theory of gravitation and of celestial mechanics. Le Verrier's personal triumph however was somewhat diminished when, on 15 October, a letter was published from the English astronomer Challis claiming that John Couch Adams of Cambridge University had made similar calculations to those of Le Verrier which he had completed in September 1845. His predicted position for the new planet had been almost as accurate as Le Verrier's but the English astronomers had been much less industrious in their search. John Herschel and Airy also supported Adams' claim. In fact Challis had, after a long delay, begun to search for the new planet on 29 July 1846. He observed it on 4 August but did not compare his observations with those of the previous night so only realised he had observed the planet after its discovery in Berlin about 7 weeks later. Arago was unimpressed by Adams' priority claims

Mr Adams does not have the right to appear in the history of the discovery of the planet Le Verrier either with a detailed citation or even with the faintest allusion. In the eyes of all impartial men, this discovery will remain one of the most magnificent triumphs of theoretical astronomy, one of the glories of the Académie and one of the most beautiful distinctions of our country.

The success of the mathematical analysis of both Le Verrier and Adams was somewhat fortunate. The orbits which they predicted were different and both not particularly good except around the 1840's. An argument over the naming of the new planet was, however, unfortunate. Arago was given the task of selecting a name by Le Verrier and Le Verrier made his wishes known in an unsubtle way by writing a paper on Herschel's planet, insisting that Uranus should be named after its discoverer. Encke, Gauss's student referred to above, suggested Neptune as a name. However Arago said

I commit myself never to call the new planet by any other name than Le Verrier. In this way, I think I will give an impeachable token of my love for science and follow the inspiration of a legitimate national sentiment.

The argument over a name led to Le Verrier resigning from the Bureau des Longitude and eventually Arago lost his battle over the name which became accepted as Neptune.

Delaunay, mentioned above for his work on the perturbations of Uranus, worked for 20 years on lunar theory. He treated it as a restricted three body problem and used transformations to produce infinite series solutions for the longitude, latitude and parallax for the Moon. The beginnings of his theory was published in 1847 and he had refined the theory until it was published in 2 volumes in 1860 and 1867 and was extremely accurate, its only drawback being the slow convergence of the infinite series.

Delaunay detected discrepancies between the observed motion of the Moon and his predictions. Le Verrier claimed that Delaunay's methods were in error but Delaunay claimed that the discrepancies were due to unknown factors. In 1865 Delaunay suggested that the discrepancies arose from a slowing of the Earth's rotation due to tidal friction, an explanation which is today believed to be correct.

Le Verrier had published an account of his theory of Mercury in 1859. He pointed out that there was a discrepancy of 38" per century between the predicted motion of the perihelion (the point of closest approach of the planet to the Sun) which was 527" per century and the observed value of 565" per century. In fact the actual discrepancy was 43" per century and this was pointed out by later by Simon Newcomb. Le Verrier was convinced that a planet or ring of material lay inside the orbit of Mercury but being close to the Sun had not been observed.

Le Verrier's search proved in vain and by 1896 Tisserand had concluded that no such perturbing body existed. Newcomb explained the discrepancy in the motion of the perihelion by assuming a minute departure from an inverse square law of gravitation. This was the first time that Newton's theory had been questioned for a long time. In fact this discrepancy in the motion of the perihelion of Mercury was to provide the proof that Newtonian theory had to give way to Einstein's theory of relativity. More details relating to the advance of Mercury's perihelion are contained in the article on general relativity.

G W Hill published an account of his lunar theory in 1878. Earlier approaches started with an elliptic orbit of the Moon round the Earth, assuming the Sun had no effect, then perturbing the orbit to take account of the gravitation of the Sun. Hill, on the other hand, started with circular orbits for the Sun and Moon about the Earth and went on to examine the perturbations caused by assuming elliptic orbits.

The final major step forward in the study of the three body problem which we shall consider was that of Poincaré. Bruns proved in 1887 that apart from the 10 classical integrals, 6 for the centre of gravity, 3 for angular momentum and one for energy, no others could exist. In 1889 Poincaré proved that for the restricted three body problem no integrals exist apart from the Jacobian. In 1890 Poincaré proved his famous recurrence theorem, namely that in any small region of phase space trajectories exist which pass through the region infinitely often. Poincaré published 3 volumes of *Les méthods nouvelle de la mécanique celeste * between 1892 and 1899. He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent. The stability proofs of Lagrange and Laplace became inconclusive after this result.

Poincaré introduced further topological methods in 1912 for the theory of stability of orbits in the three body problem. It fact Poincaré essentially invented topology in his attempt to answer stability questions in the three body problem. He conjectured that there are infinitely many periodic solutions of the restricted problem, the conjecture being later proved by Birkhoff. The stability of the orbits in the three body problem was also investigated by Levi-Civita, Birkhoff and others.

**Article by:** *J J O'Connor* and *E F Robertson*

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Orbits.html]