Theoria motus corporum coelestium in sectionibus conicis solem ambientium.

Hamburg: Friedrich Perthes and I.H. Besser, 1809.

First edition of the work which, “with the Disquisitiones [Arithmeticae, 1801], established his reputation as a mathematical and scientific genius of the first order” (DSB). “The Theoria motus will always be classed among those great works the appearance of which forms an epoch in the history of the science to which they refer. The processes detailed in it are no less remarkable for originality and completeness than for the concise and elegant form in which the author has exhibited them. Indeed, it may be considered as the textbook from which have been chiefly derived those powerful and refined methods of investigation which characterize German astronomy and its representatives of the nineteenth century, Bessel, Hansen, Struve, Encke, and Gerling … It was forty years before the methods of the Theoria motus became the common possession of all astronomers” (Dunnington, Carl Friedrich Gauss: Titan of Science (2004), p. 91). “In this work Gauss systematically developed the method of orbit calculation from three observations he had devised in 1801 to locate the planetoid Ceres, the earliest discovered of the ‘asteroids,’ which had been spotted and lost by G. Piazzi in January 1801. Gauss predicted where the planetoid would be found next, using improved numerical methods based on least squares, and a more accurate orbit theory based on ellipse rather than the usual circular approximation. Gauss’s calculations, completed in 1801, enabled the astronomer Heinrich W. M. Olbers to find Ceres in the predicted position, a remarkable feat that cemented Gauss’ reputation as mathematical and scientific genius. Gauss found the reputation his astronomical work gained for him so attractive that he decide upon a career in astronomy, becoming director of the Göttingen Observatory in 1807” (Norman). As well as providing a tool for astronomers, Gauss’s method of orbit computation also offered a way of reducing inaccuracy of calculations arising from measurement error – the method of least squares, “the automobile of modern statistical analysis” and the origin of “the most famous priority dispute in the history of statistics” (Stigler). The French mathematician Adrien-Marie Legendre had published the method of least squares, though without justification, in 1805 in his Nouvelles méthodes pour la détermination des orbites des comète, but Gauss states in the present work that he had the method since 1795.

“On 1 January 1801, Giuseppe Piazzi in Palermo discovered a comet or planet in the constellation of Taurus, detectable only telescopically. He observed it through 11 February, when illness interrupted his observations. He informed three astronomers of his discovery, and in May sent his detailed observations to J.J. Lalande in Paris, asking that publication be postponed.

“Since the 1770s two astronomers, J.E. Bode of Berlin and Franz Xaver von Zach (1754– 1832) of Gotha, had entertained the notion of a missing planet between Mars and Jupiter. A numerical series due to J.D. Titius, publicized by Bode in 1772, gave approximate mean solar distances of the known planets, but predicted a planet in this ‘gap’. It received surprising corroboration in 1781 with the discovery of Uranus, a planet whose nearly circular orbit had a radius close to the next term after Saturn in the series. In autumn 1800 Zach and other German astronomers formed a society to promote systematic search for the missing planet.

“In spring 1801 the question arose: might Piazzi’s ‘comet’ be the quarry sought? It must be re-discovered! From June onward, Zach’s monthly reports in a periodical which he published, the Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde (hereafter, ‘MC’) gave an ongoing account of the search.

“The July issue reported the efforts of J.C. Burckhardt, in Paris, to put an orbit to Piazzi’s observations. Parabolic orbits, Burckhardt found, were unsatisfactory; circular orbits could accommodate more of the data. He proposed an approximate elliptical orbit, but in agreement with P.S. Laplace (1749–1827), held that an accurate orbit determination would require more observations.

“Through late summer and autumn, cloudy weather prevented a systematic search. In the September issue, Zach published Piazzi’s revised observations. Gauss, a subscriber to the MC, set about determining an orbit.

“The November issue of the MC contained a review of Piazzi’s memoir on his discovery. Finding parabolic trajectories hopeless, he had derived two circular orbits with radii 2.7067 and 2.68626 astronomical units. From the second of these Zach computed an ephemeris for November and December. Piazzi named the planet Ceres Ferdinandea, thus honoring Sicily’s ruler.

“Zach now received Gauss’s results, and to them devoted his entire report in the December issue. Gauss had computed four different elliptical orbits, each based on a different trio of observations; the four sets of elements were in near agreement with each other and with the 19 observations Piazzi had considered undoubtful. Gauss put the planet in January 1801 about a quadrant past aphelion; and assigned it a considerably higher eccentricity than had Burckhardt, so that in December 1801 the planet would be 6° or 7° farther east than any of the other proposed orbits implied. He gave positions for Ceres at 6-day intervals from 25 November to 31 December.

“The weather continued unpropitious. As Zach reported in the January 1802 issue of the MC, in the early morning hours of 7–8 December he clocked a star very close to Gauss’s prediction for Ceres, but bad weather on the following nights prevented verification.

“As he reported in the February 1802 issue, early on 1 January Zach discovered the planet some 6°east of its December position, and through January he followed its motion, which agreed closely with Gauss’s orbital elements. Wilhelm Olbers (1758–1840) also re-discovered the planet, reporting the fact to the newspapers, where Gauss read about it. Gauss’s ellipse, exclaimed Zach, was astonishingly exact. ‘Without the ingenious efforts and calculations of Dr. Gauss, we should probably not have found Ceres again; the greater and more beautiful part of the achievement belongs to him’” (Landmark Writings in Western Mathematics 1640-1940, pp. 317-8).

According to Kepler, the orbit of a celestial body is a conic section with focus at the centre of the Sun. To specify its orbit five parameters, or elements, are required, namely: two parameters determining the position of the plane of the body’s orbit relative to the Earth’s orbit; the relative scale of the orbit; the eccentricity of the orbit or perihelial distance, the shortest distance from the orbit to the centre of the Sun; and the relative ‘tilt’ of the main axis of the orbit. In addition to these five parameters, a single time when the object was at a particular point in the orbit is needed, so that its location at a given time can be computed. Gauss had a total of 22 observations made by Piazzi over 41 days. The data from these observations consisted of a specific moment in time together with two angles defining the direction in which the object had been seen relative to an astronomical system of reference defined by the sphere of fixed stars. In principle, each of these observations defined a line in space, starting from the location of Piazzi’s location at the moment of observation and directed along the direction defined by the two angles. Gauss had to make corrections for various effects such as the rotation of the Earth’s axis, the motion of the Earth’s orbit around the sun, and possible errors in Piazzi’s observations or in their transcription. Gauss began by determining a rough approximation to the unknown orbit, and he then refined it to a higher degree of precision. Gauss initially used only three of Piazzi’s 22 observations, those from January 1, January 21, and February 11. The observations showed an apparent retrograde motion from January 1 to January 11, around which time Ceres reversed to a forward motion. Gauss chose one of the unknown distances, the one corresponding to the intermediate position of the there observations, as the target of his efforts. After obtaining that important value, he determined the distances of the first and third observations, and from those the corresponding spatial positions of Ceres. From the spatial positions Gauss calculated a first approximation of the elements of the orbit. Using this approximate orbital calculation, he could then revise the initial calculation of the distances to obtain a more precise orbit, and so on, until all the values in the calculation became coherent with each other and with the three selected observations. Subsequent refinements in his calculation adjusted the initial parameters to fit all of Piazzi’s observations more smoothly.

Gauss sent a manuscript summarizing his methods in a letter to Olbers dated 6 August, 1802, just seven months after the discovery of Ceres; it was published only seven years later in the September 1809 issue of MC. By this time Gauss had so refined his methods of orbit calculation that he writes in the preface to Theoria motus that “Scarcely any trace of resemblance remains between the method by which the orbit of Ceres was first computed and the form given in this work.”

“In 1809, the bookseller Perthes of Hamburg published Gauss’s Theoria motus corporum coelestium in sectionibus conicis solem ambientium. The book … contains the sum of Gauss’s work in theoretical astronomy but it does not always describe the actual methods Gauss used in his research. Like Disquisitiones arithmeticae, Theoria motus was published in Latin; Gauss had written it in German but had to translate it because Perthes thought it would sell better. The subject matter of Theoria motus is the determination of the elliptic and hyperbolic orbits of planets and comets from a minimum of observations and without any superfluous or unfounded assumption … Theoria motus is systematic to the point of being pedantic; it consists of two books, one with preliminary material and one with the solution of the general problem. The work is the first rigorous account of Gauss’s methods for calculating the orbits of celestial bodies, directly deduced from Kepler’s laws. Up to Gauss’s time, astronomers used ad hoc methods which varied from case to case, despite the fact that the theoretical foundations had been clear for more than 100 years. Gauss’s essential contribution consisted in a combination of thorough theoretical knowledge, the unusual algebraic facility with which he handled the considerable complications which occur in a direct development of these equations, and his practical astronomical experience” (Bühler, Gauss).

“It was Gauss in his Theoria motus who first connected probability theory to the method of least squares … The Theoria motus, which was written to explain how to calculate planetary positions, came into being because the methods available to the astronomers of the eighteenth century were not adequate to determine the orbit of the planet Ceres … Against this background it was natural for Gauss to concern himself with the problem of how to use redundant observations. It seems clear that the more observations available the more accurately will the orbit be known. Gauss said on the subject: ‘But in such a case, if it is proposed to aim at the greatest precision, we shall take care to collect and employ the greatest possible number of accurate places. Then, of course, more data will exist than are required: but all these data will be liable to errors, however small, so that it will generally be impossible to satisfy all perfectly. Now as no reason exists, why, from among those data, we should consider any six as absolutely exact, but since we must assume, rather, upon the principles of probability, that greater or less errors are equally possible in all, promiscuously; since, moreover, generally speaking, small errors oftener occur than large ones; it is evident, that an orbit which, while it satisfies precisely six data, deviates more or less from the others, must be regarded as less consistent with the principles of the calculus of probabilities, than one which, at the same time that it differs a little from those six data, presents so much the better an agreement with the rest. The investigation of an orbit having, strictly speaking, the maximum probability, will depend upon a knowledge of the law according to which the probability of errors decreases as the errors increase in magnitude: but that depends upon so many vague and doubtful considerations — physiological included — which cannot be subjected to calculation, that it is scarcely, and indeed less than scarcely, possible to assign properly a law of this kind, in any case of practical astronomy. Nevertheless, an investigation of the connection between this law and the most probable orbit, which we will undertake in its utmost generality, is not to be regarded as by any means a barren speculation’ (Goldstine, A History of Numerical Analysis from the 16th through the 19th Century, pp. 212-3).

In Section 186 of the present work, “Gauss writes: “Our principle, which we have made use of since the year 1795, has lately been published by Legendre in the work Nouvelles méthodes pour la détermination des orbites des comètes, Paris, 1806, where several other properties of this principle have been explained, which, for the sake of brevity, we here omit.”

“The Theoria motus was originally written in German and completed in the autumn of 1806. In July 1806 Gauss had for some weeks at his disposal a copy of Legendre’s book before it was sent to Olbers for reviewing. It was not until 1807 that Gauss finally found a publisher, who, however, required that the manuscript should be translated into Latin. Printing began in 1807 and the book was published in 1809. Gauss had thus ample time to elaborate on the formulation of the relation of his version of the method of least squares to that of Legendre, if he had wished so.

“Gauss’s use of the expression “our principle” naturally angered Legendre who expressed his feelings in a letter to Gauss dated May 31, 1809. The original is in the Gauss archives at Gottingen; it contains the following statement: “It was with pleasure that I saw that in the course of your meditations you had hit on the same method which I had called Méthode des moindres quarrés in my memoir on comets. The idea for this method did not call for an effort of genius; however, when I observe how imperfect and full of difficulties were the methods which had been employed previously with the same end in view, especially that of M. La Place, which you are justified in attacking, I confess to you that I do attach some value to this little find. I will therefore not conceal from you, Sir, that I felt some regret to see that in citing my memoir p. 221 you say principium nostrum quojam inde ab anno 1795 usi sumus etc. There is no discovery that one cannot claim for oneself by saying that one had found the same thing some years previously; but if one does not supply the evidence by citing the place where one has published it, this assertion becomes pointless and serves only to do a disservice to the true author of the discovery.”

“It therefore became important for Gauss to get his claim of having used the method of least squares since 1795 corroborated. He wrote to Olbers in 1809 asking whether Olbers still remembered their discussions in 1803 and 1804 when Gauss had explained the method to him. In 1812 he again wrote to Olbers saying “Perhaps you will find an opportunity sometime, to attest publicly that I already stated the essential ideas to you at our first personal meeting in 1803.” In an 1816 paper Olbers attested that he remembered being told the basic principle in 1803.

“In 1811 Laplace brought the matter of priority before Gauss, who answered that “I have used the method of least squares since the year 1795 and I find in my papers, that the month of June 1798 is the time when I reconciled it with the principle of the calculus of probabilities.” [In his Théorie analytique des probabilités (1812)] Laplace writes that Legendre was the first to publish the method, but that we owe Gauss the justice to observe that he had the same idea several years before, that he had used it regularly, and that he had communicated it to several astronomers” (Hald, pp. 394-5).

“The heat of the dispute never reached that of the Newton − Leibniz controversy, but it reached dramatic levels nonetheless. Legendre appended a semi-anonymous attack on Gauss to the 1820 version of his Nouvelles méthodes pour la détermination des orbites des comètes, and Gauss solicited reluctant testimony from friends that he had told them of the method before 1805. A recent study of this and further evidence suggests that, although Gauss may well have been telling the truth about his prior use of the method, he was unsuccessful in whatever attempts he made to communicate it before 1805. In addition, there is no indication that he saw its great general potential before he learned of Legendre’s work. Legendre’s 1805 appendix, on the other hand, although it fell far short of Gauss's work in development, was a dramatic and clear proclamation of a general method by a man who had no doubt about its importance” (Stigler).

Dibner 114n; Norman 879; Sparrow, Milestones of Science 81; PMM 257n. Hald, A History of Mathematical Statistics from 1750 to 1930, 1998. Stigler, A History of Statistics, 1986 (see pp. 12-15, 55-61 & 145-6).

Large 4to (296 x 236 mm), pp. [i-iii], iv-xi, [1], [1], 2-227, [1, errata], 1-20 (tables) and one engraved plate (occasional minor stains). Recent calf-backed marbled boards.

Item #5365

Price: $6,500.00