The Great Inequality of Jupiter and Saturn, Part 3: The Mathematicians

This is the third part of a post broken up into multiple parts, because of its length. This is my eighth post on geoheliocentrism, and the fourth post specifically focusing on Simon Shack’s TYCHOS model.¹ The previous posts were:

• Resonances in TYCHOS, Simon Shack’s Neo-Tychonian Model of Our Stellar System

• Are the Mayan Long and Short Periods for Mars Correct?

• Hamlet's Mill Advances at 1.6 km/h

In this post, we are focusing on the Great Inequality, in which seemingly, in the span of a single human lifetime, it was observed that the orbits of Jupiter and Saturn had sped up or slowed down. The first two parts of this post were:

• The Great Inequality of Jupiter and Saturn, Part 1: The English Astronomers:

  This problem was first noted by Johannes Kepler (1571-1630) after the publishing in 1627 of his Rudolphine Tables, and the English astronomers of the 17th century attempted, without much success, to reconcile the observed movements of Jupiter and Saturn with Kepler’s tables. Ultimately, Edmond Halley (1656-1742) printed in 1719—but only published posthumously in 1752—revised astronomical tables, with “an anomalous acceleration in the mean motion of Jupiter and an anomalous deceleration in the mean motion of Saturn”.

• The Great Inequality of Jupiter and Saturn, Part 2: The French Astronomers:

  This problem was picked up by Maraldi I (Giacomo Filippo Maraldi, 1665-1729), who first began looking at this problem in 1704. In his 1718 paper, he stated clearly that, since ancient times, Jupiter has been accelerating and Saturn decelerating, but not uniformly. In 1746, both Cassini II (Jacques Cassini, 1677-1756) and Pierre Charles Le Monnier (1715-1799) published papers attempting to explain the anomalies. After the reading of Le Monnier’s papers, the Académie Royale announced a prize for a mathematician resolving the anomaly.

The man associated with the mathematical resolution of the Great Inequality is Pierre-Simon Laplace (1749-1827), who, after making a presentation to the Académie Royale in November 1785, submitted three papers²³⁴ for the 1784 (published in 1787), 1785 and 1786 (published in 1788) proceedings. These would be the last proceedings of the Académie Royale, which was replaced following the French Revolution.

Pierre-Simon Laplace

This third part will focus on key developments leading to Laplace’s achievement, achieved approximately one century after Isaac Newton (1643-1727) published his Principia in 1687. As I did for the first part, for historical perspective, I will be using as guide the very lengthy paper by Curtis Wilson (1921-2012), “The great inequality of Jupiter and Saturn: from Kepler to Laplace.”⁵

It should be kept in mind that gravity was not explained in Newton’s Principia using the calculus, but, rather, geometry. It would take several decades before gravity was understood in terms of the calculus. For an introduction to this topic, see my post The Work of Leonhard Euler and the Bernoulli Family Ensured the Success of Newtonian Mechanics.

The three main actors in this development were arguably the three most important mathematicians of the 18th century: Leonhard Euler (1707-1783), Joseph-Louis Lagrange (1736-1813) and Laplace. Other contributors were Alexis Claude Clairaut (1713-1765) and Jean le Rond d’Alembert (1717-1783).

It is clear that I can only give a very high-level overview of the developments, since I am covering several decades worth of lengthy publications, each including pages and pages of detailed derivations. And Substack posts are limited in length.

Before gravitation could be presented in analytical terms, some basic analytical concepts had to be developed. This work would fall on Euler, who introduced the trigonometric series in 1729. These are of the form

a₀/2 + ∑_n=1..∞ (aₙ cos nθ + bₙ sin nθ)

where the aₙ and bₙ are constants. According to Wilson,

Without the extensive employment of such series, the celestial mechanics of Lagrange and Laplace would be inconceivable. [Wilson, pp.89-90]

With the introduction of these series, physics and mathematics would take a new turn:

The specific problems in the context of which these symbolic conceptions arose are left behind, and a new and apparently autonomous field of study emerges: the transformations and interrelations of algebraic and transcendental functions. It is within this field of study that the conception of the trigonometric series arises, as a means of approximating some other function, defined mathematically or physically. [Wilson, p.90, my emphasis]

In 1747, Euler made two significant steps towards the problem of resolving the motions of the planets. First, on 8 June 1747, he presented to the Berlin Academy the differential equations for motion. Second, three days later, he finished his submission for the 1748 prize—which he won—the Académie Royale had offered for resolving the motions of Jupiter and Saturn. Although he won both that prize and that of 1752, his results are unsatisfactory, as he had computed an acceleration for both Jupiter’s and Saturn’s orbits. This was around the same time as the discussions around the Principle of Least Action; see my post Pierre Louis Maupertuis and the Principle of Least Action.

Around the same time, and independently, Clairaut and d’Alembert also developed differential equations of motion. Without such equations, Newton’s gravity could never have been studied from an analytical point of view.

The problems to be solved were not simply mathematical, as not all agreed on the underlying physical principles. Euler, for example, was not a fully convinced Newtonian:

Up to 1750 Euler remained fundamentally sceptical as to the correctness of the inverse-square law of universal gravitation, and vigorously defended the view that all forces arise from the impenetrability of matter, and are thus forces of contact. Only with Clairaut’s proof that the inverse-square law could account for the motion of the lunar apogee did Euler come to agree that all further work on planetary theory must assume this law to be exact. And even after so agreeing, Euler remained convinced that the aether responsible for transmitting light through the interplanetary spaces must introduce frictional forces, and so cause the planets to fall gradually toward the Sun, with accelerating mean motions. The inverse-square law might be necessary for the explanation of the phenomena, but Euler did not believe it to be sufficient. Euler was fortified in this conviction by the phenomenon of the secular acceleration of the Moon, which until 1787 seemed to be inexplicable otherwise than by the assumption of frictional forces. But if the inverse-square law were fundamentally insufficient, then so also were the Keplerian rules that followed from it in the solution of the two-body problem. [Wilson, pp.74-75, my emphasis]

One could argue that Euler was channelling Paolo Sarpi (1552-1623) and Galileo Galilei (1564-1642). See one of my very first posts, Paolo Sarpi on Local Motion. In addition, Euler felt that the aether was making the celestial bodies slow down:

In his Opuscula varii argumenti of 1746, Euler had shown that a resisting aether of very low density would cause a planet or satellite to fall with extreme slowness toward the central body, and hence to have an ever decreasing period, or ever increasing mean angular motion about the central body. During these years as later, it was Euler’s conviction that light was produced by waves in an aether, and that the matter of this aether could not fail to have inertia and therefore to exercise resistance against any body moving through it. [Wilson, p.135, my emphasis]

Nor was Laplace consistently Newtonian:

Laplace, who appears to have shared neither Euler’s theological convictions nor his belief in a luminiferous aether, in 1774 put forward a modification of Newton’s theory, involving a finite speed of transmission of the gravitational force, and a difference in the resulting action depending on the speed and direction of motion of the body acted on; and these assumptions led to the same result as did the assumption of a resisting medium, namely that the mean motions of planets and satellites would undergo slow, secular accelerations. The apparent deceleration of Saturn, on the other hand, could not be accounted for in the same way, since it implied an increase in what we would now call potential energy; here the best Laplace could do was to invoke a hypothetical interaction between Saturn and one or more comets, as possibly having produced the effect. He was still putting forward these hypotheses as probable a decade later, in 1784. [Wilson, p.21, my emphasis]

Back to the mathematics. It turned out that the differential equations of motion introduced by Euler—when applied to the motion of a planet perturbing the motion of another planet—referred to the inverse cube of the distance between the perturbing planet and the perturbed planet. The result was therefore an irrational expression, impossible to integrate except by integration. Euler resolved the situation by introducing trigonometric series—easy to integrate, term by term—to approximate that expression. This still led to the task of computing the initial coefficients of the terms of the series; the initial work was done by Euler, with successive refinements, leading to faster convergence of the series, by Lagrange and Laplace.

But there was a problem. The solution of the differential equations for the radius vector—the distance from the Sun to a planet—turned out to contain terms proportional to the time. These terms were called at the time «arcs de cercles» [“arcs of circles”]. What does this mean? Occurrences of t—for “time”—occurred independently, and not as part of arguments of the sine or cosine functions. There were two possibilities: the first was that these occurrences were artifacts of the approximation process, and the second that the solar system was unstable. Should the second case apply, this would mean that the Great Inequality were not some long-term oscillatory process.

Laplace found a workaround in which the «arcs de cercles» could be avoided:

Laplace will write again, seeking a formal generality, on the subject of eliminating “arcs de cercle” from the solutions of differential equations, when these equations are themselves free of such “arcs”. And Lagrange will respond by showing that Laplace’s procedure is subject to certain limitations, and by presenting a different procedure that avoids these limitations and is also free of certain features of Laplace’s memoir that Lagrange finds “metaphysically unsatisfying”. Lagrange’s metaphysical discontent with the Laplacian techniques presumably stems here, as it stems elsewhere, from his dislike of any hint of the infinitesimal. [Wilson, pp.129-130, my emphasis]

The highlighted sentence reflects a distrust of any kind of indivisible or infinitesimal in the world of mathematics, since their introduction in Western Europe during the 17th century. For more on this topic, including Lagrange’s position, see my post The War Against the Infinitesimals.

Lagrange and Laplace each wrote several papers in which the Great Inequality appears, and each, with mutual respect, carefully read the work of the other, and adjusted his work accordingly. But for neither was this topic the main focus. Compare the titles of their first papers:

• Lagrange: «Solution de différents problèmes de calcul intégral»⁶ [Solution of several problems of the integral calculus], first published in 1762-1765 in Miscellanea Taurinensia, the Turin-based journal created by Lagrange with some colleagues;

• Laplace: «Sur le principe de la gravitation universelle et sur les inégalités séculaires des planètes qui en dépendent»⁷ [On the principle of universal gravitation and the secular inequalities of the planets that depend upon it], dated 1773 and published in 1776.

As we can see, Lagrange was focusing on the development of the integral calculus, and only the last section, 60 of 200 pages, targeted a particular use case: «Application de la solution précédente à la théorie de Jupiter and de Saturne» [Application of the previous solution to the theory of Jupiter and Saturn]. Laplace, on the other hand, was focusing on the development of Newton’s theory of universal gravitation. It seems to me that Lagrange was focusing on mathematical elegance, while Laplace was acting “like an engineer”, looking for a solution “that worked”.

Here is the overall assumption under which they were working:

In the view of planetary motion that guides the work of Laplace and Lagrange throughout the 1770’s and 1780’s, the fundamental motion of each planet is the Keplerian motion, following the elliptical orbit in accordance with Kepler’s areal rule. But owing to their mutual action, the planets are subject to inequalities that perturb the Keplerian motion, and these inequalities are seen as of two kinds. Some of them, labelled “periodic”, depend on the positions of the planets, either with respect to each other or with respect to their aphelia; these inequalities prove to be small in relation to the equation of center, and in general, reestablish themselves within relatively short periods, usually not longer than a small number of years. In treating of these inequalities, Laplace views the orbital elements as fixed and unchanging, and the planet as merely executing an oscillatory dance of small ambit about a mean position which moves forward in strict accordance with the Keplerian rules. The other inequalities, called “secular”, alter the elements of the orbits by nearly insensible nuances at each revolution, but, accumulating unceasingly, they end by changing entirely, over the course of centuries, the nature and position of the orbits. But even the secular variations, as Lagrange first argues and Laplace comes to accept as established doctrine, are periodic; their periods, however, are very long, measured in tens of thousands of years rather than a few years or decades, and their formulas do not involve the positions of the planets. [Wilson, pp.166-167, my emphasis]

But the inequalities all affect each other, and there are constraints that needed to be respected:

In accordance with this formula, the secular variations of the eccentricities and inclinations that Lagrange and Laplace have educed imply some exchange of angular momentum between the planets, but the gains and losses of angular momentum deriving from these variations turn out to balance one another, without the mean solar distances being assumed to change. But if one does suppose an exchange of angular momentum involving change in the mean solar distances and not the eccentricities or inclinations, then the relation between the changes in mean motion derivable from this assumption… will be contradicted by the observations. Indeed, there can be no exchange of angular momentum involving a change in the mean solar distances, without changes in eccentricity or inclination or both being involved as well. This is because a second requirement has to be met, the conservation of forces vives or, as we now say, energy, and this puts a restriction on the ways in which angular momentum can be exchanged. Laplace in 1774 gives no indication of being aware of the necessity of premising conservation of forces vives as well as angular momentum. We believe it is from Lagrange that he will learn to take energy conservation into account. [Wilson, pp.167-168, my emphasis]

So by late 1774 or early 1775, significant progress had taken place, but Laplace was still convinced that gravitation was not the cause of the Great Inequality. During the next decade, Laplace would focus on other problems, such as the tides:

In the decade from the mid-1770’s to the mid-1780’s, Laplace, insofar as he dealt with questions of gravitational action, focussed on problems such as those of the tides and the precession of the equinoxes, that depend on the extended mass and non-spherical shape of planetary bodies. Having concluded that the apparent acceleration of Jupiter and the apparent deceleration of Saturn were not caused by gravitation, he had hypothesized tentatively that the acceleration of Jupiter could be due to the finite speed of propagation of the gravitational force, and that the deceleration of Saturn must be due to the action of comets. It would appear that he had decided that further investigation of the anomalies of these planets, at this time, was unlikely to be profitable. [Wilson, p.71, my emphasis]

Lagrange, on the other hand, kept at it, and ultimately introduced what Laplace would call the “perturbing function”.

Lagrange in these years, by contrast, continued to engage in systematic development of the general theory of planetary perturbations. He introduced at least three ideas that would prove of the greatest importance in Laplace’s final resolution of the anomalies. He defined the perturbing function, which was to become an essential feature of Laplace’s procedure for calculating higher-order perturbations. The perturbing function is a potential function; Lagrange uses it in the derivation of the various integrals of the equations of motion, including the energy or force vive integral. This integral, which had not entered into Laplace’s earlier work, will play a significant role in his drawing the conclusion contrary to his earlier view, that the anomalies in the motions of Jupiter and Saturn can be caused by gravitational action. Finally, it will be Lagrange who first remarks on the fact that near-commensurability of the mean motions of the planets implies long-term and potentially sizable inequalities—a suggestion that [Wilson] believe[d] led to Laplace’s renewed attack on the anomalies of Jupiter and Saturn in 1785. [Wilson, p.71, my emphasis]

[A potential function is a function whose value, scalar or vector, varies with the position in space.]

Among Lagrange’s successive works, reading Wilson’s paper, these two stand out:

• «Remarques générales sur le mouvement de plusieurs corps qui s’attirent mutuellement en raison inverse des carrés des distances»⁸ [General remarks on the motion of several bodies which attract one another in inverse proportion to the squares of the distances between them], published in 1779.

  In this memoir Lagrange shows how the perturbing function simplifies the derivation of the integrals for the center of gravity, angular momentum, and force vive of a system of gravitating bodies. [Wilson, p.206]

• «Théorie des variations périodiques (Première partie contenant les formules générales de ces variations)» [Theory of Periodic Variations (Part One: containing the general formulae for these variations)], published in 1785.

  In this memoir Lagrange mentioned, in passing, the circumstances under which a periodic inequality, even though proportional to the eccentricity or inclination, might prove to be sizable. The context was not the same as the one in which Laplace would later apply the idea so successfully. Nevertheless, in the total absence of other evidence as to why Laplace returned to the problem of Jupiter and Saturn in 1785, after a lapse of ten years, and at the same time relinquished the earlier hypotheses he had entertained on this subject, we are inclined to think Lagrange’s publication of 1785 on perturbations had a good deal to do with it; and the passage therein pointing out that inequalities otherwise expected to be small can become large because of a near-commensurability of the mean motions of perturbing and perturbed planets is the most relevant of all to Laplace’s discovery. It may also be worth mentioning that in the same memoir Lagrange derived the very formula that Laplace would later use in the actual calculation of the great inequality; however, Lagrange’s derivation differs from the one that Laplace will later give. [Wilson, p.215, my emphasis]

And this brings us to Laplace’s «Théorie de Jupiter et de Saturne» [Theory of Jupiter and Saturn], reference 3. Wilson provides a quick overview of the contributions. First, at the theoretical level, Laplace computed expressions involving higher powers of the basic parameters:

Of the new Laplacian procedures, the most important on the theoretical side concerned the calculation of perturbations proportional to the powers and products of the second and higher dimensions of the orbital eccentricities and inclinations. No planetary perturbations of this type had previously been calculated; the task was an exceedingly laborious one, most terms of this order were negligible, and previously there had been no procedure for determining in advance whether the calculation of a given perturbational term was likely to yield an empirically measurable result. Laplace, following a clue that he had undoubtedly gleaned from Lagrange, developed a rule for the purpose; and this rule became the basis of subsequent practice. Since many of the inequalities discovered by its means were quite as large as and even larger than the lower-order terms previously calculated, the result was a sharp increment in the accuracy of planetary prediction. [Wilson, p.23]

Second, in order to determine the constants in the expressions, he used an averaging technique based on actual observations made by astronomers:

A second important technique introduced into general use through Laplace’s example in the “Théorie de Jupiter et de Saturne” was the employment of equations of condition in the correction of orbital elements, and in the evaluation of coefficients of perturbational terms. The method is used to bring large numbers of observations to bear on the determination of constants so related to one another that the value given to one of them influences the values given to the others. It had been invented by Euler, and used successfully by Tobias Mayer and by Roger Boscovich. But it was Laplace in 1786 who first obtained general recognition for the method, with his application of it to the determination of the orbital elements of Jupiter and Saturn. Delambre, imitating Laplace’s example, made use of it in determining the constants of his solar theory, and in the refinement of the empirical constants of Laplace’s theories of Jupiter and Saturn, of the satellites of Jupiter, and of the planet Uranus. With the evident superiority of the resulting tables over all earlier tables, the employment of statistical procedures in astronomy became de rigueur. [Wilson, pp.23-24]

The techniques used in the «Théorie de Jupiter et de Saturne» turned out to be successful in resolving a number of issues, and astronomers successfully used Laplace’s ideas:

Laplace was an opportunist in mathematical astronomy—successfully, importantly so. He became the ‘organizer of victory’ for the science that had earlier been called “physical astronomy”, and which he in 1799 was to re-name “celestial mechanics”. His “Théorie de Jupiter et de Saturne” is the first in a series of victories leading beyond the earlier stalemate to a new level of precision in the agreement of planetary tables and observations. [Wilson, pp.23-24]

Still, for me, this success raises some issues. When we learned about the Ptolemaic and the Copernican systems, we were taught to mock the elaborate system of epicycles and deferents. Yet, if we look at the perturbation theory introduced by Newton, then successively developed by Euler, Lagrange, Laplace and others, aren’t we simply looking at another form of epicycles?

With the introduction of trigonometric series the mathematics became in a sense Ptolemaic, epicyclic, and therewith opaque to the force relations, the consequences of which it was designed to express. [Wilson, p.25]

In the next post, we will see how the motions of Jupiter and Saturn are understood in the TYCHOS system.

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1. Simon Shack. The TYCHOS: Our Geoaxial Binary System. 2023. https://book.tychos.space

2. M. de la Place. Mémoire sur les Inégalités séculaires des planètes et des satellites. Mémoires de Mathématique et de Physique, pp.1-50. In Histoire de l’Académie Royale des Sciences, Année 1784. Paris, 1787.

3. M. de la Place. Théorie de Jupiter et de Saturne. Mémoires de Mathématique et de Physique, pp.33-160. In Histoire de l’Académie Royale des Sciences, Année 1785. Paris, 1788.

4. M. de la Place. Suite de la Théorie de Jupiter et de Saturne. Mémoires de Mathématique et de Physique, pp.201-234. In Histoire de l’Académie Royale des Sciences, Année 1786. Paris, 1788.

5. Curtis Wilson. The great inequality of Jupiter and Saturn: from Kepler to Laplace. Archive for History of Exact Sciences 33(1-3):15-290, 1985.

6. Lagrange. Solution de différents problèmes de calcul intégral. Œuvres. Tome Premier, pp.469-668, 1867.

7. Laplace. Sur le principe de la gravitation universelle et sur les inégalités séculaires des planètes qui en dépendent. Œuvres. Tome Huitième, pp.201-275, 1891.

8. Lagrange. Remarques générales sur le mouvement de plusieurs corps qui s’attirent mutuellement en raison inverse des carrés des distances. Œuvres. Tome Quatrième, pp.399-418, 1869.