## Nouvelles méthodes pour la détermination des orbites des comètes; avec un supplément contenant divers perfectionnemens de ces methods et leur application auc deux Comètes de 1805.

Paris: Firmin Didot, 1806.

First edition, second issue, but the first to contain the Supplement, of the invention of 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). “In 1805 Legendre published the work by which he is chiefly known in the history of statistics, *Nouvelles méthodes pour la détermination des orbites des comètes*. At eighty pages this work made a slim book, but it gained a fifty-five page supplement (and a reprinted title page [i.e., the offered issue]) in January of 1806 ... For stark clarity of exposition the presentation is unsurpassed; it must be counted as one of the clearest and most elegant introductions of a new statistical method in the history of statistics. In fact, statisticians in the succeeding century and three-quarters have found so little to improve upon that … the explanation of the method could almost be from an elementary text of the present” (*ibid*.). “The great advances in mathematical astronomy made during the early years of the nineteenth century were due in no small part to the development of the method of least squares. The same method is the foundation for the calculus of errors of observation now occupying a place of great importance in the scientific study of social, economic, biological, and psychological problems. Gauss says in his work on the *Theory of Motions of the Heavenly Bodies* (1809) that he had made use of this principle since 1795 but that it was first published by Legendre. The first statement of the method appeared as an appendix entitled “Sur la méthode des moindres quarrés” in Legendre’s Nouvelles méthodes pour la détermination des orbites des comètes, Paris 1805” (Wolberg, ‘The Method of Least Squares,’ in *Designing Quantitative Experiments, *2010). This is a rare book on the market, ABPC/RBH listing just four copies of either issue since 1941.

“[By] 1800 the principle of combining observational equations had evolved, through work of [Tobias] Mayer and [Pierre-Simon] Laplace, to produce a convenient ad hoc procedure for quite general situations. We have also seen how the idea of starting with a mathematical criterion had led, in work of [Roger] Boscovich and Laplace, to an elegant solution suitable for simple linear relationships involving only two unknowns. The first of these approaches developed through problems in astronomy; the second was (at least in these early years) exclusively employed in connection with attempts to determine the figure of the earth. These two lines came together in the work of a man who, like Laplace, was an excellent mathematician working on problems in both arenas − Adrien Marie Legendre.

“Legendre came to deal with empirical problems in astronomy and geodesy at a time when [methods] had been developed separately in the two fields. It was also a time when a half-century's successful use of these methods had seen a change in the view scientists took of them − from Euler’s early belief that combination of observations made under different conditions would be detrimental to the later view of Laplace that such combination was essential to the comparison of theory and experience. Legendre brought a fresh view to these problems; and it was Legendre, and not Laplace, who took the next important step.

“Legendre did not hit upon the idea of least squares in his first exposure to observational data. From 1792 on he was associated with the French commission charged with measuring the length of a meridian quadrant (the distance from the equator to the North Pole) through Paris. One of the major projects initiated by the National Convention in the early years after the French Revolution had been the decision in 1792 to change the ancient system of measurement by introducing the metric system as a new order, toppling existing standards of measurement in an action symbolic of the French Revolution itself. The basis of the new system was to be the meter, defined to be 1/10,000,000 of a meridian quadrant. It remained for French science to come up with a new determination of the length of this arc. In keeping with the nationalism that inspired the enterprise, the determination was to be based only on new measurements made on French lands. To this end an arc of nearly 10', extending from Montjouy (near Barcelona) in the south to Dunkirk in the north, was measured in 1795. By 1799 the complex task of reducing the multitude of angular measurements to arc lengths had been completed by J. B. J. Delambre and P. F. A. Mechain … In early 1799, Delambre published an extensive discussion of the theoretical results underlying the reduction of the raw data on this arc. This volume is prefaced by a short memoir by Legendre that is dated 9 Nivose, an VII (30 December 1798) and indicates that Legendre did not have the method of least squares at that time …

“The occasion for Legendre’s reconsideration of observational equations, and for the appearance of the method of least squares, was the preparation in 1805 of a memoir on the determination of cometary orbits. The memoir is a scant seventy-one pages (excluding the appendix); and, aside from a few brief remarks at the end of the preface, the method of least squares makes no appearance before page 64. Even this first mention of least squares seems to be an afterthought because, after presenting an arbitrary solution to five linear equations in four unknowns (one that assumed that two equations held exactly and two of the unknowns were zero), Legendre wrote that the resulting errors were of a size “quite tolerable in the theory of comets. But it is possible to reduce them further by seeking the minimum of the sum of the squares of the quantities E', E", E"'” (*Nouvelles méthodes*, p. 64). He then reworked the solution in line with this principle. It seems plausible that Legendre hit on the method of least squares while his memoir was in the later stages of preparation, a guess that is consistent with the fact that the method is not employed earlier in the memoir, despite several opportunities. It is clear, however, that Legendre immediately realized the method’s potential and that it was not merely applications to the orbits of comets he had in mind. On pages 68 and 69 he explained the method in more detail (with the word minimum making five italicized appearances, an emphasis reflecting his apparent excitement), and the memoir is followed on pages 72-80 by the elegant appendix. The example that concludes the appendix reveals Legendre’s depth of understanding of his method (notwithstanding the lack of a formal probabilistic framework). It also suggests that it was because Legendre saw these problems of the orbits of comets as similar to those he had encountered in geodesy that he was inspired to introduce his principle and was able to abstract it from the particular problem he faced. Indeed, the example he chose to discuss was not just given as an illustration, it was a serious return to what must have been the most expensive set of data in France − the 1795 measurements of the French meridian arc from Montjouy to Dunkirk” (Stigler).

The appendix is dated 6 March 1805. Legendre states the principle of his method of “distributing the errors among the equations” on pp. 72-3: “Of all the principles that can be proposed for this purpose, I think there is none more general, more exact, or easier to apply, than that which we have used in this work; it consists of making the sum of the squares of the errors a minimum. By this method, a kind of equilibrium is established among the errors which, since it prevents the extremes from dominating, is appropriate for revealing the state of the system which most nearly approaches the truth.”

“The clarity of the exposition no doubt contributed to the fact that the method met with almost immediate success. Before the year 1805 was over it had appeared in another book, Puissant’s *Traite de geodesie*; and in August of the following year it was presented to a German audience by von Lindenau in von Zach’s astronomical journal, *Monatliche **Correspondenz*. Ten years after Legendre's 1805 appendix, the method of least squares was a standard tool in astronomy and geodesy in France, Italy, and Prussia. By 1825 the same was true in England. The rapid geographic diffusion of the method and its quick acceptance in these two fields, almost to the exclusion of other methods, is a success story that has few parallels in the history of scientific method” (*ibid*.).

An unintended consequence of Legendre’s publication was a protracted priority dispute with Carl Friedrich Gauss, who claimed in 1809 that he had been using the method since 1795. In *Theoria motus corporum coelestium* (1809, Section 186), “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).

*Nouvelles méthodes* was first issued in 1805, and reissued in January 1806 with a reset title-page and a 55-page supplement. In the supplement Legendre makes some improvements to the methods he had introduced for determining the orbital elements of comets. When these methods had been applied to the observations by Alexis Bouvard of a comet that appeared in 1805 (now called 2P/Encke), Legendre had found that the elements were “not sufficiently exact”. He traced the problem to a coefficient in one of his equations which, if it happened to be very small (as it was for the observations relating to Bouvard’s comet), could lead to large errors in the calculation of the orbital elements. In the supplement he introduced a modification of his method which avoided this problem. In the second part of the supplement he applied his new method to Bouvard’s observations of a second comet that had appeared in 1805. Bouvard is best known for his prediction of the existence of a planet beyond Uranus, but he died before he could complete his investigations and the discovery of Neptune was made by Adams and Le Verrier.

“Legendre (born 18 September 1752, died 10 January 1833) was a mathematician of great breadth and originality. He was three years Laplace’s junior and succeeded Laplace successively as professor of mathematics at the École Militaire and the École Normale. Legendre’s best-known mathematical work was on elliptic integrals (he pioneered this area forty years before Abel and Jacobi), number theory (he discovered the law of quadratic reciprocity), and geometry (his *Éléments de géométrie *was among the most successful of such texts of the nineteenth century). In addition, he wrote important memoirs on the theory of gravitational attraction. He was a member of two French commissions, one that in 1787 geodetically joined the observatories at Paris and Greenwich and one that in 1795 measured the meridian arc from Barcelona to Dunkirk, the arc upon which the length of the meter was based. It is at the nexus of these latter works in theoretical and practical astronomy and geodesy that the method of least squares appeared” (Stigler).

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).

4to (266 x 212 mm), pp. [2] viii, 80 and 1 engraved plate; 1-55 (supplement), uncut in original pink plain wrappers, very light sporring to a few leaves, overall a very fine and untouched copy.

Item #5112

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Price:
$4,000.00
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