## Theoria Motus Lunae Exhibens Omnes Eius Inaequalitates.

St. Petersburg: Academiae Imperialis Scientiarum, 1753.

First edition of one of Euler’s rarest books, containing his first lunar theory. “The observed motions of the planets, particularly of Jupiter and Saturn, as well as the moon, were evidently different from the calculated motions based on Newton’s theory of gravitation. Thus, the calculations of Clairaut and d’Alembert (1745) gave the value of eighteen years for the period of revolution of the lunar perigee, whereas observations showed this value to be nine years. This caused doubts about the validity of Newton’s system as a whole. For a long time Euler joined these scientists in thinking that the law of gravitation needed some corrections. In 1749 Clairaut established that the difference between theory and observation was due to the fact that he and others solving the corresponding differential equation had restricted themselves to the first approximation. When he calculated the second approximation, it was satisfactorily in accordance with the observed data. Euler did not at once agree. To put his doubts at rest, he advised the St. Petersburg Academy to announce a competition on the subject. Euler soon determined that Clairaut was right, and on Euler’s recommendation his composition received the prize of the Academy (1752). Euler was still not completely satisfied, however. In 1751 he had written his own *Theoria motus lunae exhibens omnes ejus inaequalitates* (published in 1753), in which he elaborated an original method of approximate solution to the three-body problem, the so-called first Euler lunar theory. In the appendix he described another method which was the earliest form of the general method of variation of elements. Euler’s numerical results also conformed to Newton’s theory of gravitation. The first Euler lunar theory had an important practical consequence: Johann Tobias Mayer, an astronomer from Göttingen, compiled, according to its formulas, lunar tables (1755) that enabled the calculation of the position of the moon and thus the longitude of a ship with an exactness previously unknown in navigation. The British Parliament had announced as early as 1714 a large cash prize for the method of determination of longitude at sea with error not to exceed half a degree, and smaller prizes for less exact methods. The prize was not awarded until 1765; £3,000 went to Mayer’s widow and £300 to Euler for his preliminary theoretical work. Simultaneously a large prize was awarded to John Harrison for his construction of a more nearly perfect chronometer. Lunar tables were included in all nautical almanacs after 1767, and the method was used for about a century” (DSB).ABPC/RBH lists only two copies in the last 20 years (Sotheby’s, 2004, £7,800 – Macclesfield copy; Zisska & Schauer, 2013, EUR 4,153). The Honeyman copy realised £1,400 in 1979.

“The description of the Moon’s motion was one of the most important problems in the history of positional astronomy. From antiquity, e.g. Ptolemy, until the middle of the seventeenth century geometric and kinematic theories were developed to predict solar and lunar eclipses and to construct calendars based on the observed motions of the Sun and the Moon. The determination of the Moon’s position with an accuracy of less than one arc minute became an official challenge for eighteenth century scientists by the ‘Longitude Act’ of 1714, initiating the search for a method to find the longitude at sea and, thus, to solve the most important problem of navigation. One of the most important astronomical solutions proposed was the method of lunar distances: from topocentric measurements of angular distances between the limb of the Moon’s disk and nearby stars and its elevations at the local time of the observer one obtained geocentric angular distances, which then were compared with predicted and calculated values for the meridian of London (or Paris) tabulated in lunar tables. The differences between observed and tabulated distances represent the time differences between these locations and thus yield the longitudes of the observer. The quality of this method depends crucially on the correctness and precision of the lunar tables involved in this procedure and consequently on the quality of the lunar theory used to construct them …

“From Euler’s studies of Newton’s *Principia* as reflected in his entries on mechanics written in his first ‘Basel notebook’ and his ‘travel diary’, we may assume that he was acquainted with Newton’s lunar theory. In particular, he must have been confronted with Section IX ‘De Motu Corporum in Orbibus mobilibus, deque motu Apsidum’ (The motion of bodies in mobile orbits, and the motion of the apsides), of book I in Newton’s *Principia*. Therein he would have read Newton’s famous statement, that ‘the [advance of the] apsis of the moon is about twice as swift,’ thus admitting the discrepancy by a factor of 2 between theory and observation. In addition, from Euler’s ‘Catalogus Librorum meorum’ (Euler’s own catalogue of his library, written between 1747 and 1749) we know that he possessed Gregory’s *Astronomiae* [*physicae & geometricae elementa*, 1702], which contains Newton’s lunar theory, from which he adopted at least numerical values of orbital parameters and from which he most probably learned the state-of-the-art on lunar theory … But there is strong evidence, that Euler studied the basic phenomena related to lunar theory from a treatise of the seventeenth century which became a standard textbook at the time, namely Book III (*De lunæ motibus*) of Boulliau’s *Astronomia philolaica* (1645) …

“Although Isaac Newton established the principle of universal gravitation in his Principia (1687) and even published a ‘Theory of the Moon’s Motion’ in 1702, the first lunar theories based on the law of gravitation and treated as a so-called ‘three-body-problem’ including the application of equations of motion formulated in three dimensions were only developed in the 1740s by Leonhard Euler, Alexis-Claude Clairaut and Jean le Rond d’Alembert. The reason for this delay was the lack of adequate mathematical methods and physical principles to cope with the analytical treatment of this three-body-problem. It was Euler who first formulated general equations of motion of the three-body-problem for the Earth–Sun–Moon system and introduced trigonometric series to integrate them using the method of undetermined coefficients. Using his (unpublished) ‘embryonic’ lunar theory of 1743/1744, Euler constructed and published between 1744 and 1750 numerous lunar tables incorporating perturbational effects of the Sun. But only in 1753 his ‘first’ lunar theory was published, one year after Clairaut’s. They solved the problem of the apsidal motion of the Moon’s orbit, which was actually first noted already by Newton but which re-appeared in 1747 when neither Euler nor Clairaut could derive this motion from the inverse square law of gravitation in accordance with observational data” (Verdun).

“In the *Theoria motus lunae*, one more of his landmark books, Euler was taking the lead in transforming theoretical astronomy into a modern mathematical science. In a letter of 10 April 1751 to Clairaut, Euler wrote, ‘I have the satisfaction of writing to you that I am now altogether clear concerning the motion of lunar apogee snd that I find it, as do you, entirely in agreement with the inverse-square law (Newton’s theory).’ In a letter of 29 June, he praised Clairaut’s new method profusely: ‘The more I consider this happy discovery, the more important it seems to me, and in my opinion it is the greatest discovery in the theory of astronomy, without which it would be absolutely impossible ever to succeed in knowing the perturbations that the planets cause in each other.’

“More than a century after planetary and solar tables had become Keplerian following the Rudolphine Tables of 1627, they were transformed as Newtonian tables, taking into account perturbations of each body caused by others in universal gravitation. Euler, Clairaut and D’Alembert provided crucial analysis to compute perturbations in the aphelia and nodes of planetary and solar motion produced among them. Since Euler found the classical expansion method of perturbations ineffective, he proposed a variant; he devised a single-step procedure with five ordinary differential equations and applied trigonometric series to describe the motion of planets. All computations used the same mean motion arguments. Euler’s procedures gave closer approximations than Clairaut’s for the intermediate orbit and the three-body problem. Since some effects were exceedingly small, Euler could work from the series for them to make good approximations. His text and Mayer’s writings, which resembled Euler’s, were two of the first to confirm that lunar apses follow Newton’s inverse-square law of attraction. Having two computational methods achieve the same result, so Euler believed, placed this finding beyond doubt. In his seventy-five page appendix he added another computational method, an initial form of the calculus of variations, with new formulas to sharpen calculations …

“[Euler] had completed most of the main body of the text [of *Theoria motus lunae*] in 1751 [in Berlin]. Euler’s comments on this text and on his next important book of the early 1750s, the forthcoming *Institutiones calculi differentialis*, indicate problems in printing, including the representation of new curves, and disagreements about costs. In April Euler expressed delight that President Razumovskij [1728-1803, President of the St. Petersburg Imperial Academy of Sciences 1746-98] thought highly enough of the *Theoria motus lunae* to offer to send payment for the costs of the printer and bookbinder. That month Schumacher [Johann Daniel Schumacher (1690-1761), Secretary of the St. Petersburg Imperial Academy of Sciences] replied that Razumovskij considered the *Theoria *extraordinary, advancing two hundred rubles to pay for it and requesting a French translation; in July Euler gave the actual cost of 633 Reichsthaler, and the Russian Imperial Academy paid for its publication” (Calinger, pp. 376-8)

Verdun, ‘Leonhard Euler’s early lunar theories 1725–1752,’ *Archive for History of Exact Sciences* 67 (2013), pp. 235–303.

4to (276 x 221 mm), pp. viii, 347, [1], with one folding engraved plate. Contemporary half-calf and marbled boards, spine gilt with red lettering-piece. A fine copy.

Item #5125

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Price:
$19,500.00
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