Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work

by G. Waldo Dunnington

  The Disquisitiones were to have included an eighth section; at first a complete theory of congruences was planned, but later Gauss appears to have proposed to continue the work by a more complete discussion of the theory of circle division. Manuscript drafts on each of these subjects were found among his papers; the first of them is especially interesting, as it treats of the general theory of congruences from a point of view closely allied to that later taken by Evariste Galois, Serret, and Richard Dedekind. This draft appears to belong to the years 1797 and 1798.

  Euler had enriched the theory of numbers with a multitude of results, relating to Diophantine problems, to the theory of the residues of powers, and to binary quadratic forms; Lagrange had given the character of a general theory to some, at least, of these results by his discovery of the reduction of quadratic forms and of the true principles of the solution of indeterminate equations of the second degree. Legendre, with many additions of his own, had endeavored to arrange as many as possible of these scattered fragments of the science into a systematic whole in his Essai sur la théorie des nombres (1799). But the Disquisitiones were already in press, and in the Essai what was new to others was already known to Gauss.

  The remarkable interpretation of the arithmetical theory of positive binary and ternary quadratic forms is found in Gauss’ review of the Works of L. Seeber (1831).23 The two important memoirs on the theory of biquadratic residues appeared in 1825 and 1831. In the second of these two papers Gauss gives a theorem of biquadratic reciprocity between any two prime numbers, no less important than the quadratic law: “If p1 and p2 are two primary prime numbers, the biquadratic character of p1 with regard to p2 is the same as that of p2 with regard to p1.” This theorem itself and the introduction of imaginary integers upon which it depends are memorable in the history of higher arithmetic for the variety of the researches to which they have given rise.

  V

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  Astronomy and Matrimony

  The discovery of Ceres (Ferdinandea) made by Joseph Piazzi (1746–1826) in Palermo on New Year’s Day, 1801, did not become known through the German newspapers until May, and the first reasonably accurate reports about it were given by the Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde in its June number. The editor was Baron Franz Xavier G. von Zach (1754–1832), lieutenant colonel and director of the Seeberg observatory near Gotha, and this journal served as the collecting point for important new geographical and astronomical reports. Piazzi had dispatched on January 24 letters to Bode, director of the Berlin observatory, to Oriani in Milan, and to Lalande in Paris reporting that he had discovered a very small comet without tail and envelope. In February Lalande informed von Zach about it, without accurately indicating the place in the heavens, so that von Zach awaited further information. The letters of Piazzi to Oriani and Bode did not reach their destinations until April; the one to Bode was seventy-one days on the way. Meanwhile Piazzi had been able to pursue the object only up to February 11. In his letters Piazzi gave only two observed locations, those of January 1 and 23, rounded off in whole minutes, and only noted that from January 10 to 11 the receding motion went over into right motion; also he added in his letter to Oriani that he supposed that it was a planet, while he spoke in the letters to Bode and Lalande only of a comet.

  Both Bode and Oriani immediately gave the new report to von Zach, who promptly brought out in the June number of the Monatliche Correspondenz an exhaustive article “on a long supposed, now probably discovered, new major planet of our solar system between Mars and Jupiter.” Bode communicated the discovery to the Royal Prussian Academy of Sciences, and saw to its publication in several newspapers.

  On the occasion of a “little astronomical trip to Celle, Bremen, and Lilienthal” about which he published an exhaustive diary in the issues of the Monatliche Correspondenz in 1800–1801, von Zach had decided, along with five other astronomers (Schröder, Harding, Olbers, and probably von Ende and Gildemeister) who met in Lilienthal, to found an “exclusive society of twenty-four practical astronomers scattered over Europe” who were to plan the study of the supposed planet between Mars and Jupiter by simultaneous correction of the star catalogues. Piazzi was also among the twenty-four, but had not yet received the invitation to take part in the society. Bode, as well as Oriani, held fast to the belief that the new object was a planet, moving between Mars and Jupiter, and even von Zach agreed with this view, which would lead them beyond the superficial calculation of a circular orbit. He therefore tried a somewhat sharper computation of a circular orbit which exhibited a noteworthy similarity to the comet of 1770; the semi-major axis is the only element which shows similarity in both orbits. He also questioned whether both objects were not perhaps identical; certain doubts as to the nature of the new planet were, therefore, continually arising. Piazzi spoke of it in a later letter as only a comet, and even the Paris astronomers appear to have valued the discovery lightly.

  Meanwhile Bode had written Piazzi requesting an accurate record of his observations. He received no satisfactory answer, and an extensive correspondence on the subject of the discovery sprang up between Bode, von Zach, and Olbers (who had learned of it from the newspapers), and also Burckhardt in Paris. Just at this time Lalande arrived in Paris, and received a letter from Piazzi which gave the observations more accurately, but with the request not to publish them prematurely. Lalande however shared them with Burckhardt on the same conditions, and Burckhardt with the German astronomers.

  On the basis of these more accurate observations Burckhardt computed an ellipse; his attempts to represent the observations by a parabola failed. The corresponding elements are in the July issue of the Monatliche Correspondenz, in which von Zach gave in all monthly periodicals “Continued Reports about a New Major Planet.”

  The complete observations of Piazzi from January 1 to February 11 were finally published in the September issue of the Monatliche Correspondenz, after Piazzi had sent them with several corrections to Bode, Lalande, and Oriani, and thus they reached Gauss.

  In the October issue of the Monatliche Correspondenz von Zach recorded that from about the middle of August until the close of September attempts were made by almost all astronomers to find the planet as it again emerged from the rays of the sun, but in vain; bad weather was also prevalent at this time. The elliptic orbit calculated by Burckhardt was unreliable, not so much because the observed portion of the orbit was quite small (which was then considered by astronomers as the chief difficulty) as because it deviated from an arbitrary attraction in the region of perihelion. The problem of determining a completely unknown planetary orbit from observations had arisen previously only in the case of Uranus, where a circular orbit could be reckoned more accurately through reference to Bode’s discovery of the much earlier observations of Flamsteed (1690) and Tobias Mayer (1756). Olbers had also begun the calculation of an elliptical orbit, but with little hope of success, since he considered the preliminary calculation of a circular orbit to be fundamental. He gave the elements of such an orbit as follows:

  If the new planet had gone through its aphelion before January 1, then its heliocentric rapidity is always increasing, and even its geocentric lengths must be greater in August and September than according to the circle hypothesis. But if it went through its perihelion in February, then it later decreased the heliocentric velocity and its geocentric lengths must be smaller in August and September than according to the circle hypothesis. Since it is not known now which of the two cases obtains, it is safer, in future searching for the star, to hold as basic those cases from the circle hypothesis which cannot deviate very much from the true ones and to keep the mean of both possible cases.

  Olbers, just as Burckhardt, falsely assumed that the planet stood at the time of its discovery not far from perihelion or aphelion, while Gauss later showed that it was about halfway between.

  Piazzi wrote a brief essay in which he made careful communications about the first discovery and the later ob
servations. He also included in this his own calculations of one of the circular orbits as well as those sent to him by Oriani and the orbits computed by the other astronomers. To this paper is attached a corrected list of his observations. In the November issue of the Monatliche Correspondenz, von Zach gave an exhaustive review of this work “which probably could not come into the German bookstores either soon or easily.” He printed opposite the observations that had appeared in the September issue the corrected observations, which contain a correction of the right ascension for February 11 of about 15 minutes.

  With respect to Olbers’ proposal to base the preliminary calculation for rediscovery of the planet on a circular orbit, von Zach computed an ephemeris for November and December “in order to perform a little service for all astronomers and lovers of the subject who want to busy themselves with the search for the star.”

  During the interim Gauss, who received the Monatliche Correspondenz in Brunswick, had silently given himself over to the problem; the interest in the new planet caused him to lay aside temporarily his purely mathematical researches and his theory of the moon. In his diary notes, entries 119 and 120 show that he was working out the method in 1801 (September and October). His earliest notes on Ceres date from the first part of November, 1801, but lack clearness.

  One can see that Gauss, as soon as he took up this work, created new practical methods for orbit determination. He would not limit himself to a certain hypothesis by experiment, but systematically sought the orbit which would fit the observations as well as possible; if the Piazzi observations embrace only forty-one days, then there must be an ellipse which fits them, and which approximates the predicted places for the rediscovery. It was a matter, therefore, of finding an ellipse that was free from all arbitrary assumptions. On the first pages of his manual for November, 1801, we find this problem completely solved, although in a less complete form than in the Theoria motus. In a brief manuscript, “Summary Survey of the Methods Applied in the Determination of the Orbits of Both New Planets,” Gauss collected his earliest methods, and sent this to Olbers on August 6, 1802; it was returned to him in November, 1805. Shortly after the appearance of the Theoria motus, von Lindenau got it, supposedly on a visit with Gauss, and published it with Gauss’ consent in the Monatliche Correspondenz for September, 1809.

  The determination of the orbit of a celestial body from three given observations is an almost impossible task; an explicit solution is not achievable, because the observed places and the relations to the determining elements of the orbit are very involved. The solution of the problem is indicated by approximations, and hence the setting up of an almost unlimited number of methods is possible; these are distinguished by more or less important factors.

  It is to be expected that Gauss thoroughly investigated the field to which these methods apply; but it will also be understood that he was not able at the time to carry through this work on his first orbital calculations because of the need to speed calculation of the single case now pressing. It was important that the orbit of the new planet should be known as soon and as accurately as possible so that the planet might be rediscovered in the heavens. Thus Gauss explained that the first orbital calculations rest on an important new fundamental thought to which the arbitrariness of older methods no longer applies. But in this particular achievement one does not find the consummate perfection and delicate elaboration of the methods of the Theoria motus.

  This first fundamental thought consists in the setting up of an equation between the distances of the planet from the sun and from the earth in the mean observation; it is too involved to be given here. Gauss wrote of it: “This formula is the most important part of the entire method and its main foundation.” He tells us that he came on his fundamental formula in a very bizarre way.

  As Gauss explained in Article 2 of the “Summary Survey,” the first computation of the completely unknown orbit of a celestial body from three observations rests on the solution of two different problems; first, to find an approximate orbit in any sort of way; second, so to correct this orbit that it “satisfies” the observations as well as possible.

  If the orbit is even approximately known, then the first problem vanishes.

  His older calculations for the determination of Ceres’ orbit are (so far as preserved) dated November, 1801. Gauss numbered the systems of elements which he found in further correction of the orbit.

  Von Zach and many others had sought Ceres according to the Gaussian ephemeris in December, but under the most unfavorable weather and without success. In February, however, von Zach could finally announce in his Monatliche Correspondenz the lucky rediscovery of the planet. He had informed Gauss by letter on January 17, 1802. During the night of December 31 to January 1, von Zach made sure that a suspicious star observed by him was definitely Ceres. On January 1, Olbers also discovered the planetoid, its location agreeing exactly with the Gaussian ephemeris. Gauss seems to have first heard of Olbers’ rediscovery of Ceres through the newspapers. He wrote to him on January 18 in order to get Olbers’ observations; with this letter began the correspondence which shows how busy Gauss was with new corrections of the Ceres orbit. The correspondence ripened into a lifelong, intimate friendship between Gauss and Olbers.24

  The discovery of the planet Ceres introduced Gauss to the world as a theoretical astronomer of the highest order. He was able to calculate in one hour the orbit of a comet, for which task Euler, using the old methods, had used three full days. Through such close application Euler later lost the sight of one eye. “To be sure,” said Gauss, “I would probably have become blind also, if I had been willing to keep on calculating in this manner for three days.” The planet Uranus had been discovered twenty years before Ceres, when near opposition; this was a critical position, which at once gave a near approximation to the elements of its orbit. A stationary elongation of Ceres, though less fertile in its results, was sufficient to assign her such a place between Mars and Jupiter as was required to satisfy Bode’s singular law. Kepler had found a planet to be lacking, as he sought to round out one of that series of cosmic speculations which had guided him to the discovery of his laws. The complete determination, however, of the elements of a planet’s orbit from three geocentric longitudes and latitudes—or from four of the first and two of the second in those cases where the latitudes are evanescent or small —was still a new problem, already solved only in the case of comets moving in parabolic orbits. Newton, whom we can thank for its first solution, had pronounced it to be problema omnium longe difficillimum.

  Gauss devoted one memoir to the beautiful demonstration of a very remarkable proposition that the secular variations which the elements of the orbit of a planet would undergo from another planet which disturbs it are the same as if the mass of the disturbing planet were distributed into an elliptical ring coincident with its orbit, and in such a manner that equal masses of the ring would correspond to portions of the orbit described in equal times.

  In marked contrast to the publication of the Disquisitiones, this rediscovery of Ceres was a spectacular accomplishment. It was Piazzi’s discovery which gave Gauss the opportunity of revealing, in most impressive form, his remarkable mathematical superiority over all his contemporaries. This particular work also had the effect of greatly improving his personal affairs in a financial way, and made possible the establishment of his own home. Brunswick had no observatory, very likely not even a telescope (worthy of the name) within its walls. This young, little-known scholar, without the external aids and instruments of astronomers, was possessed of an inner vision so far-reaching and a mathematical genius so marvelously piercing that he was able from calculations at his desk to locate the missing orbit of the lost asteroid so accurately that now the work of retracing and rediscovery by men equipped with the telescope could no longer fail.

  Further observations of Ceres by Olbers led him in 1802 to the discovery of a second planet in the immediate neighborhood of Ceres, and, with the privilege acc
orded him as discoverer, he named it Pallas. In 1804 a third planetoid was discovered by Ludwig Harding, and named by him Juno. A natural consequence of Gauss’ calculation of the orbit of Ceres was that all calculations of these new celestial bodies now devolved upon him. In this he had no rival. Gauss devoted especial care and work to the Olbers planetoid, Pallas, making close investigation and calculations of its movements and perturbations, as a result of which he came to call it his favorite.

  The year 1801 was truly epoch-making for Gauss. In later years he recalled that so many new discoveries and important ideas came to him, as sources of scientific truths, that he could not control them. Many of these occupied his time during the remainder of his life, while others were lost in the press of other affairs. The works of Lagrange and Laplace, which appeared at the time, gave him material for investigation in the field of celestial mechanics. He clearly realized how these men had attained their discoveries by means of Newton’s creative ideas, and his admiration for Newton reached new heights. “Newton remains forever the master of all masters!” he exclaimed. The deeper he penetrated into mathematics, the more fully he was persuaded that its true meaning lies in its application to practical life and natural science. Thus we explain his turning to theoretical and practical astronomy. He had made use of the “astronomical instruments” at the Collegium Carolinum, but we do not know what they were. It was there that he may have gotten his first practice in observing.