The regular observer was Karl Ludwig Harding (1766–1834), who was called to Göttingen in 1805 as assistant professor of astronomy and official observer of the observatory and in 1812 was promoted to the rank of full professor. In addition, he taught elementary astronomy and navigation. In the above-mentioned diary the observations of the sun and fixed stars are his. Harding mentioned in an article published in August, 1810, that the Göttingen observatory was not properly equipped with instruments. To supplement this the King of Westphalia granted Harding four thousand francs for his celestial atlas, and from March to June, 1811, one finds observations by Gauss, since Harding was at that time in Paris. The planets, the moon, Vesta, and Ceres were frequently observed by Gauss. In addition, various observations from 1809 to 1818 were carried out by sixteen of Gauss’ pupils.
The principal clock was the Shelton, a gift of George III, king of Great Britain and elector of Hanover. It had been in use since 1770. It is now in the east meridian hall and is still running. Gauss praised it in 1816 and 1818, but by 1819 he was dissatisfied with it.
Gauss always liked to use the sextant and spent many hours in determining the elevation and azimuth of terrestrial objects. In October, 1810, he journeyed to Gotha for a chronometric determination of longitude. During the period 1808–1812 he published five shorter memoirs in the field of spherical astronomy.
In the early days most of the high-quality astronomical instruments were of English make, and practically all of the Göttingen instruments were imported from England. However, there was a mechanic in Stuttgart named Baumann, who had studied for a long time under Ramsden in England, and soon Gauss gave him an order. But Germany had another instrument producer, also a pupil of Ramsden, who was becoming well known. This was an artillery captain, Georg von Reichenbach (1772–1826), in Munich. He was doing the best work of this kind to be found in Germany, and Gauss now began to purchase instruments from him. The results were highly pleasing; the first instrument arrived on November 26, 1812, and it is noticeable in the diary that Gauss’ interest in observing was immediately stimulated. His efforts were directed, not to accumulating a mass of observations, but to making a small number of extremely accurate ones. From this time on almost all the diary entries are in Gauss’ handwriting. Apparently Harding’s observations were no longer entered in the diary. Gauss’ astronomical work of this period is indicated in his correspondence with Bessel, one of the leading observers of that time. Their letters go into great detail; Bessel had considerable influence on Gauss in these matters and stimulated his interest in the theory of dioptrics in later years. On May 23, 1814, the observatory received a fine achromatic heliometer produced by Joseph Fraunhofer (1787–1826) in Munich. The observatory already possessed an older heliometer.
In 1810 a special fund was set up for the various departments of Göttingen University; 1,750 francs were earmarked annually for the observatory. Current expenditures, including the allowance for Gauss’ rent, absorbed about half of that amount. By application to the governing board of the university Gauss got funds for a stand to support the new heliometer, which arrived in April, 1815. After it was mounted, he tried some measurements of the diameters of Venus, Mars, and the ring of Saturn, as well as of the comet of 1815. In May, 1817, he sent the heliometer back to Fraunhofer for improvements. It was finally returned in November, 1817, but he never made real use of it.
In the winter of 1815–1816 the Göttingen observatory received a collection of astronomical instruments which it had purchased from the estate of Johann Hieronymous Schröter (1745–1816), founder of the Lilienthal observatory. These instruments had been purchased with the stipulation that Schröter could use them as long as he lived. When the Kingdom of Westphalia annexed Lilienthal, Schröter wished to disregard the sale and to ship the instruments to France. The French wanted to have some of the instruments, but realized a second sale would be illegal. Gauss felt so strongly about the matter that he expressed himself only orally!
On November 25, 1811, Laplace wrote to Gauss saying that he would do what he could. Meanwhile, nothing happened until Lilienthal was captured and burned on December 12, 1812, by the French. Schröter had to flee and never recovered from this blow. Fortunately the instruments were rescued. After the restoration of the Kingdom of Hanover Schröter arranged their transfer to Göttingen before his death.
Repsold had constructed a meridian circle which interested Gauss as early as 1810. Finally it was purchased for Göttingen and arrived on Friday, April 10, 1818. Incidentally Gauss visited the Mathematical Society in Hamburg in 1818 and became an honorary member of it. Repsold’s sons developed his workshop into a well-known astronomical instrument firm.
Gauss planned to purchase from Reichenbach a second meridian circle. This plan led to his trip to Bavaria in the Easter vacation of 1816. He left Göttingen on April 18 and was absent five weeks. The letters written to Minna at this time are extremely interesting. Gauss was not a frequent traveler, and it is probable that he enjoyed this trip more than any other he ever took. The exquisite Bavarian Alps made a strong appeal to him. Twelve days were spent in Munich and Benediktbeuern, where Fraunhofer’s optical work was carried on. His companions on this trip were his ten-year-old son Joseph and Dr. P. Tittel, one of his pupils. Gauss had hesitated to leave Minna at this time, since it was only several weeks before the birth of his daughter Therese.
The primary purpose of this trip was to make the personal acquaintance of Fraunhofer and Reichenbach, as well as Joseph von Utzschneider (1763–1840), partner of Fraunhofer in the optical company, and Traugott Lebrecht Ertel (1778–1858), owner of Reichenbach’s firm, and to discuss with them the ordering of two new meridian instruments of the best construction.
Gauss journeyed via Gotha, where he spent several days with his friend Bernhard August von Lindenau (1799–1854), premier of Saxony and director of the Seeberg observatory. Lindenau furnished his coach to Gauss, who was not satisfied with the slowness of his Göttingen coachman. At Berchtesgaden he visited the salt mines. Benediktbeuern was some hours distant beyond Munich, and Gauss found the optical shop set up in an old abbey belonging to Utzschneider. In addition, these men had as assistant a Munich mechanic named Liebherr. The workers were at the time busy with meridian instruments for Warsaw, Ofen, and Turin.
On April 26 Gauss wrote Minna from Munich:
Yesterday evening about eight o’clock we arrived here in good shape. . . . Now first of all a little account of the trip. On Sunday, early, we left Seeberg, in Lindenau’s coach and with horses we hired, to drive through the Thuringian Forest where the roads were still completely covered with ice. In summer this region must be romantically beautiful. These horses brought us to Meiningen, where we immediately took post-horses for the all-night drive over the fine Bavarian highway, reaching Würzburg next morning rather weary. Here we refreshed ourselves with a midday meal. As night approached we continued on our way, again to drive all night and into the next day as far as Augsburg, where we spent the night. Thursday morning we looked around a little and by midday were again on our way. Those last eight and a half miles over an incomparably beautiful road took seven and a half hours. And so we came yesterday to beautiful Munich.
I was rather exhausted by the journey, but rest has set everything right again, and today I feel as well as in Göttingen. We are lodged in a very good inn. Early this morning Reichenbach came to see me, having already learned of my arrival. I have spent the greater part of the day with him and have accepted his very friendly and urgent invitation to stay at his home. Tomorrow we will move there, also Tittel. . . . I have also become acquainted with Utzschneider, and on Tuesday we will go with him to his estate in Benedictbeuern on the Tyrolean border. Reichenbach is a very gracious man who overwhelms me with kindness; his house is in the suburbs, has an extremely pleasing location and bears the stamp of great affluence.
A second letter is headed “Reichenhall, 36 hours beyond Munich, Sunday evening. May 11, 1816,” and gives an
account of the return journey:
At last I am on the way home. After twelve very pleasant days in Munich, in which I include the trip to Benedictbeuern, I have come this far with Reichenbach, who had to come here on business. After seeing something of this vicinity and of the nearby Berchtesgaden as well as the extremely interesting salt mines and the incomparably beautiful region, tomorrow will find me on my way back to Göttingen. We do not return to Munich, but take the nearer way to Regensburg and Nürnberg, then Gotha for a couple of days. . . . I write this at midnight with my eyes almost closing, since today we made the excursion to Berchtesgaden, and there in the underground salt mines we were constantly on the move.
The tour of inspection also took Gauss to the Munich observatory and that of Father Placidus Heinrich in Regensburg. The entire trip lasted exactly thirty-six days, of which eight were required for the trip to Munich via Mühlhausen, Gotha, Meiningen, and Augsburg. Sixteen days were required for the return trip, because Gauss detoured via Reichenhall, Landshut, Regensburg, and Nürnberg.
An official report on this trip was given by Gauss to the university board of curators on June 5, 1816, just four days before Therese’s birth. In it he made his recommendations on the purchase of instruments. Success crowned his efforts; he received the Reichenbach passage instrument in November, 1818, and the meridian circle in August, 1819.
The Repsold meridian circle arrived, as mentioned above, on April 10, 1818. The following day Repsold himself arrived and was a house guest of Gauss until the evening of the nineteenth. During his visit the Shelton clock was cleaned and regulated. On May 1, 1818, Gauss began a new diary entitled “Diary of Observations at the Repsold Meridian Circle.” This volume lay unnoticed at the observatory until after 1927. From May 1 to 31 Gauss observed twenty-one days—mostly principal stars, on some days the sun. Occasionally he complained about weather conditions as well as the fact that, in one direction, fruit trees in adjacent yards and, in another, the wooded Hainberg obscured his view. He suggested the possibility of chopping down some of the trees. On the first of June, 1818. Repsold, returning home on his trip, again visited Gauss and made further improvements on the instrument.
Gauss set up a rather extensive plan of observation which in large part he completed, but did not publish his results except for occasional observations of planets. In the catalogue of the Göttingen observatory is Gauss’ notebook entitled “Calculations and Notes Concerning the Observations at the Meridian Circles.” On the first pages of this book is an unfinished list of 316 stars, predominantly circumpolar. His observations fall almost without exception in the time from noon or early afternoon to the late evening hours and include daily ten to fifteen stars. From the end of September to the end of October, 1818, Gauss was in Lüneburg to make measurements on St. Michael’s Church tower as a connecting point with the Danish triangulation. From his return until the summer of 1819 he was observing as many as thirty stars daily, as well as Jupiter, the sun, and the comet of 1819.
The Reichenbach passage instrument was set up in September, 1818, and Gauss began to devote all his attention to it. He stopped using the Repsold circle and began to use more and more the Reichenbach meridian circle, which was set up on October, 1819. It is noteworthy that Gauss seldom permitted Harding or one of the pupils to use these instruments. When he handled the Reichenbach meridian circle, Gauss wore gloves, according to the story told by Encke, his pupil. Encke and Nicolai, another pupil, were allowed to take down notes when Gauss was observing. He made use of a town mechanic who rejoiced in the name of Philipp Rumpf.
In the years 1819–1822 at Gauss’ suggestion the right ascensions of the moon and several moon stars were observed by Nicolai in Mannheim, Soldner in Munich, and Encke in Seeberg for the determination of differences in longitude. The observations were continued for many years, although Gauss withdrew in July, 1820.
On the first of May, 1820, he obtained a Liebherr clock, which had been sorely needed, as evidenced by the fact that he had borrowed one from Repsold. Even the Liebherr clock soon caused trouble by running fast, and Rumpf was away for several months! Fortunately a Hardy clock was presented to the observatory in 1826 by the Duke of Sussex.
Gauss was made happy in August, 1819, by a four-and-a-half-day visit from his old friend Olbers. At the moment he was especially vexed at Heidelbach, Harding’s father-in-law, who owned the orchard just south of the observatory. The owner would not allow the fruit trees to be cut down and refused compensation for these obstructions to observation. In June, 1819, Gauss was in Lauenburg for the setting up of a Ramsden zenith sector.
One should not imagine that Gauss’ work of this period was confined to practical astronomy. He was considering magic squares, quadratic residuals, and a new method of determining integrals by approximation (1814), founded on Newton’s method for that purpose. There was one memoir of more than common interest, devoted to the demonstration of a very remarkable proposition in planetary theory that the secular variations which the elements of the orbit of a planet would experience from another planet which disturbs it are the same as if the mass of the disturbing planet were distributed into an elliptic ring coincident with its orbit, in such a manner that equal masses of the ring would correspond to portions of the orbit described in equal times.
In the course of the last investigation Gauss arrived at some elliptic integrals, the evaluation of which he was enabled to effect by means of a transformation which is included in one of the series immortalized by the name of C. G. J. Jacobi (1804–1851), the brilliant German-Jewish mathematician. Somewhat later Gauss wrote to Schumacher (1828) that he had anticipated Abel and Jacobi by a quarter of a century, but that his principle of not publishing anything unless it was a completed work of art (form and content), had robbed him of priority rights in these difficult theories.
In the years with which this chapter is concerned Gauss enunciated a number of theorems on the so-called arithmetico-geometric mean. As a matter of fact, he had been working on elliptic and lemniscate integrals as early as his student days. Posthumous fragments of the year 1808 show that the 1828 letter to Schumacher was correct in its statement. The use of i for √-1 goes back to early work of Gauss (1801), but he did not lend his authority to placing the imaginary on a firm basis until the appearance of a memoir in 1831. The names “complex” and “lateral number” are credited to him, but others gave the first geometric interpretation of the imaginary.
Gauss’ name has been incorrectly given to a table of addition and subtraction logarithms which he produced and published in 1812. However, he scrupulously referred to Z. Leonelli, whose idea it was and who actually published such a table in 1802. The so-called Gaussian logarithms are arranged to give the logarithms of the sum and difference of numbers whose logarithms are given. Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers, the numbers themselves being unknown, but their logarithms being known, wherefore they are frequently called addition and subtraction logarithms.
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Geodesy and Bereavement: The Transitional Decade, 1822−1832
Gauss’ interest in geodesy must be traced back to the year 1794 when he discovered the method of least squares. In it he found a method of logically combining magnitudes which involve accidental errors. For Gauss the method of least squares was at once one of the bridges which led from pure to applied mathematics. He visualized his highest ideal in activity in both directions. In a letter to Olbers he expressed it thus: “The most refined geometer and the perfect astronomer—these are two separate titles which I highly esteem with all my heart, and which I worship with passionate warmth whenever they are united.”
Originally Gauss did not attach great importance to the method of least squares; he felt it was so natural that it must have been used by many who were engaged in numerical calculations. Frequently he said he would be willing to bet that the elder Tobias Mayer had used it in his calculat
ions. Later he discovered by examining Mayer’s papers that he would have lost the bet. Legendre anticipated Gauss in the matter of publication, and Gauss showed Legendre’s study to his fellow students, including Bolyai. The interesting thing is that Gauss discovered the method at such an early age. Yet he did not underestimate the practical importance of its use. In June, 1798, he adapted it to the principles of the calculus of probabilities, and as early as 1802 used it in astronomical calculations. Legendre used the name méthode des moindres carrés, and by adopting this nomenclature Gauss showed that he did not feel hurt because he was anticipated. His first publication on the subject occurred in Book II, Section III, of the Theoria motus.
On February 15, 1821, Gauss presented to the Royal Society of Göttingen his memoir Theoria combinationis observationum erroribus minimis obnoxiae, pars prior, in which he gave what he felt to be the only proper connection of the method of least squares with the calculus of probabilities. He stated that he started from the same viewpoint as Laplace, but that he used a different mode of development. On February 2, 1823, he transmitted to the Royal Society pars posterior of the above memoir. His Supplementum theoriae combinationis observationum erroribus minimis obnoxiae, presented to the Royal Society on September 16, 1826, was a direct result of his practical work in geodesy.35
Very little is known about Gauss’ first experience in practical geodetic work. During his student years he had no experience in observing, other than a few exercises with the mural quadrant and the reflecting sextant. A letter written in 1797 shows that he did have a certain interest in observing, and realized that he had no training in drawing, architecture, or mechanical arts. In a letter dated February 21, 1802, the astronomer von Zach, speaking from his own experience, wrote that he regarded Gauss’ nearsightedness as a hindrance, and alluded to danger to the eyes from solar and sextant observations. However, he let Gauss borrow a sextant, a clock, and a telescope. With the little telescope Gauss began to practice occasionally—primarily as a pastime, but also in preparation for his future vocation. In years 1803–1805 he began practicing with the sextant in an area around Broitzen, five kilometers from Brunswick. At this time he had visions of one day engaging in a large triangulation; an essay of Beigel on the trigonometric survey of Bavaria attracted his attention.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 14