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The 18Th-Century Battle Over Lunar Motion The 18th-century battle over lunar motion Siegfried Bodenmann In a dispute with more than just scientific import, Alexis Clairaut, Leonhard Euler, and Jean le Rond d’Alembert each employed their own strategies to establish that they were the first to understand a puzzling feature of the Moon’s orbit. Siegfried Bodenmann ([email protected]) is a coeditor of Leonhard Euler’s French correspondence for the Euler Commis- sion of the Swiss Academy of Sciences in Basel, Switzerland. About two years after Sputnik 1 began emitting radio sig- the direction of Pierre Louis Moreau de Maupertuis. Despite nals, the Soviet Union’s Luna 2 became the first manmade ob- the plague of flies, the cold winter, and all the dangers of the ject to reach the Moon. The intentional crash landing of the undertaking—including the perilous transport of astronom- spacecraft on 13 September 1959 once again demonstrated ical instruments through rapids and dense woods—Clairaut Soviet technological superiority over the US. That event can helped prove that Earth is flattened at the poles by measuring be regarded as the trigger of the so-called race to the Moon, a degree of the meridian. As historian Mary Terrall explores initiated by President John F. Kennedy in May 1961. As the in her imposing book The Man Who Flattened the Earth: Mau- much-broadcast commemoration of Apollo 11 and the first pertuis and the Sciences in the Enlightenment (University of manned landing on the Moon reminded us, the Americans Chicago Press, 2002), Clairaut and Maupertuis thus showed won the competition. that Isaac Newton was right about Earth’s oblateness; in con- Every story has a prologue. The one preceding Neil Arm- trast, René Descartes’s followers had deduced from strong’s historic steps shows that a captivating plot from the Descartes’s vortex theory of planetary motion and other past needn’t include a cold war or spies. It tells of three men works that the terrestrial globe was flattened at the equator. who competed to develop a predictive theory that would ac- On this autumn day, Clairaut carries under his arm a trea- curately describe the motion of the Moon and there- tise that criticizes Newton’s law of attraction. If Clairaut fore furnish a key tool that, much later, would is well aware of the importance of his assertions, help the Soviets and the Americans antici- he might still not expect them to start a pate the position of their celestial tar- prominent controversy with two of the get. The mid-18th-century contro- most renowned mathematicians of versy concerning lunar theory his time: Leonhard Euler and Jean involved strong personal and le Rond d’Alembert. political interests and shows Clairaut couldn’t have that even in mathematics, chosen a better day to pre - truth isn’t always the result sent his paper, as that of a solely objective and Wednesday in 1747 is the logical demonstration official start of the aca- but can be a construct demic year. On that day, born of tactics, strategy, the assembly of the and intrigue. Academy of Sciences in Paris is open to the pub- The principal actor lic, and the meeting The curtain rises on our room is almost always historical drama. The full. Those special cir- setting is Paris, capital cumstances, in addition of France and French to the implications of science. The calendar the talk, might explain reads 15 November 1747, the excitement generated and one of the protago- by Clairaut’s paper in the nists of the story about to course of the following unfold is leaving his apart- months. Historians cannot ment on Berry Street. The 34- year-old mathematician Alexis Figure 1. Alexis Claude Clairaut Claude Clairaut (figure 1) can al- (1713–65). This engraving, circa ready look back at an impressive 1770, is by Charles-Nicolas Cochin and list of achievements, one of which oc- Louis Jacques Cathelin, based on a drawing curred during the expedition to Lapland by Louis de Carmontelle. that had taken place 10 years earlier under © 2010 American Institute of Physics, S-0031-9228-1001-010-0 January 2010 Physics Today 27 a b Apogee Perigee t rbi ’s o Moon One year later Figure 2. The Moon’s orbital motion. (a) The apogee and perigee are key markers of the lunar orbit. Collectively, the two points are called apsides. (b) The orbit and apsides rotate at a rate of about 40° per year. The eccentricity of the orbit is exag- gerated in the illustrations. reconstruct exactly how many people came to the meeting; ground, and the orbit of the Moon are no longer one and the the secretary of the academy, Jean-Paul Grandjean de Fouchy, same. That is one of the chief criticisms offered by George- didn’t establish a list of all attendees as he would have done Louis Leclerc, comte de Buffon, author of the famous multi- for a typical assembly. But a transcript of Clairaut’s talk, in- volume Natural History. But as science historian John Green- serted in the academy records, allows one to closely follow berg points out in a March 1986 Isis article, Buffon “defended his thoughts concerning the inverse-square law and the lunar the inverse-square law as sacred on metaphysical grounds. orbit. He harped on the matter ad nauseum but contributed nothing Because the Moon is attracted to both Earth and the Sun, to the mathematical work that led to the achievement that mathematicians calculating its orbit must confront the fa- eluded Newton.” In fact, a thorough reading of Clairaut’s mously difficult three-body problem. Thanks to the Leibniz- paper shows that the French mathematician does not contest ian calculus and its improvement by the brothers Jakob and the universality of the gravity law. Rather, he demonstrates Johann Bernoulli, by 1740 the problem could be formulated that the term he adds to correct Newton’s formula makes a as a system of four differential equations. But those only huge difference for astronomical objects, like the Moon and allow an approximate solution. One of the numerous irregu- Earth, attracting at a small distance, but it nearly vanishes for larities in the orbit of the Moon around Earth is the motion objects, like Earth and the Sun, attracting at a great distance. of the apsides—that is, the apogee, or point at which the He thus interprets Newton’s inverse-square law as a special Moon is farthest from Earth, and the perigee, at which it is case in astronomy and replaces it with what he thinks is a nearest. Those two points rotate around Earth with a period new, universally valid principle. of nine years, as illustrated in figure 2. Even the great Newton Dispersed via letters and periodicals, Clairaut’s thesis could account for only half of the motion of the apsides. soon reaches other 18th-century science centers such as Lon- That’s why Clairaut takes on the lunar orbit and tries to ex- don, Berlin, and Geneva. In January 1748, for example, the plain it with the inverse-square law. But after calculating over Jes uit Journal de Trévoux publishes a review of Clairaut’s paper and over again, he is forced to the conclusion that the law of along with a defense of his Newtonian views—the review gravitation cannot account for the observations—he keeps supposedly initiated by the French scholar himself as a public finding an 18-year period for a full revolution of the lunar ap- answer to Buffon’s attacks. sides around Earth. Clairaut then experiments with adding a 1/r3 term to Newton’s attraction law to explain the apsidal Two other protagonists motion. The drama’s first supporting role is played by Swiss mathe- To be sure, it isn’t the first time that a modification of the matician Euler (figure 3). By 1748, then in Berlin, he has al- attraction law has been proposed. For example, Gabrielle ready been informed of Clairaut’s new term by Clairaut him- Émilie Le Tonnelier de Breteuil, marquise du Châtelet, had self, who sent letters dated 3 September and 11 September already postulated a similar addition in her Institutions de 1747. The French scholar, though, did not send any mathe- physique of 1740. However, she was concerned with terrestrial matical justifications. events such as the motion of sap in plants or of fluids in cap- Euler’s answer to Clairaut might have offended its recip- illary tubes, not with astronomical phenomena. Moreover, ient a bit. Euler agrees that the Newtonian attraction law according to Swiss mathematician Gabriel Cramer and other doesn’t seem to work for the Moon but claims that he already contemporaries, Clairaut, who was a mentor to the marquise, pointed out the insufficiency of Newton’s law in a yet unpub- may have suggested the idea to her. In any event, Clairaut’s lished treatise on Saturn’s motion. He furthermore asserts revolutionary and unorthodox move is to question Newton’s that he already recognized the problem in his own studies of law in both the astronomical and terrestrial domains. the motion of the Moon’s apsides. It is hard to say for sure For some readers at the time, it seems clear that Clairaut’s when Euler reached that realization. Numerous documents additional term, specifically introduced to explain the motion confirm that he had indeed reflected on apsidal motion, even of the Moon’s apsides, implies that the laws that rule the mo- as far back as 1725. Moreover, in a paper read before the tion of Earth around the Sun, the fall of an apple to the Berlin Academy of Science in June 1747, Euler explicitly men- 28 January 2010 Physics Today www.physicstoday.org with Clairaut and Euler, who both had begun to work on the subject well before, d’Alembert is exceedingly cautious.
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