In the summer of 1803 Gauss again came in contact with a councilor of the Superior Court of Appeals named von Ende, who was making determinations of latitude and longitude in Brunswick and vicinity. Von Ende, who died in 1816 in Cologne, had a small observatory in Celle. In August, 1803, von Zach was on the Brocken and was giving almost daily signals for the determination of longitude. In Brunswick, Helmstedt, and Wolfenbüttel the work was turned over to Gauss and von Ende. From solar observations Gauss determined the time with a chronometer belonging to the Duke of Gotha.
On August 27, 1803, Gauss journeyed from Brunswick to Brocken, met von Zach there, and then accompanied him to Gotha in order to confer with von Ende and Professor Bürg of Vienna. For three months he was a busy guest of the Seeberg observatory; on December 7 he and von Zach returned to Brunswick, where for ten days von Zach was his guest.
Gauss’ interest in this work was evidenced by the fact that he calculated observations made by Baron Friedrich Ferdinand Carl von Müffling (1775–1851), who in 1821 became chief of the general staff of the Prussian army. The latter had already participated in the trigonometric survey of Westphalia, made for military purposes in 1797–1802 by Colonel Lecoq. Lindenau now joined this project. With his sextant Gauss took part in this triangulation, and his calculation gave him a number of places from Brocken to Franconia which he thought were accurate enough for the needs of a map. In August, 1804, Gauss joined Olbers in the spa Rehburg for several days. The two friends measured some angles between Hanover, Brocken, and Minden, from whose approximately known geographic location Gauss calculated the latitude and longitude of Georgplatz, a small knoll in Bad Rehburg.
In the years 1803–1805 the French Colonel Epailly was directing surveys in the occupied zone, and his triangles, especially in the southern part of the Electorate of Hanover, were later of great use to Gauss in the measurement of an arc of meridian for setting up a network of triangles, although their configuration had an unfavorable influence. In 1805 Epailly went to Brunswick to make an observation from the tower of St. Andrew’s Church; Gauss used the opportunity to study his work and to familiarize himself with the instruments. Epailly became ill in Brunswick.
Soon after settling in Göttingen in November, 1807, Gauss secured from Harding a ten-inch Troughton sextant. In 1808 he published his Methodum peculiarem elevationem poll determinandi explicat D. Carolus Fridericus Gauss. With this method, the sextant and a clock would give good results. It required less preparatory calculation than a method he had previously published. In August, 1810, Gauss journeyed to Münden. One purpose of this trip was to find the effect of traveling on the running of a chronometer. On this occasion he determined on Freytagswerder at the confluence of the Werra and Fulda the latitude and difference in longitude from Göttingen. On February 19 and 20, 1812, Gauss observed star occultations for the determination of the longitude of Göttingen. In the same year on the Hanstein he determined the angles between Göttingen, Brocken, and Boineburg (two Müffling points).
On June 8, 1816, Schumacher wrote Gauss that the King of Denmark had appropriated the necessary funds for the measurement of an arc of meridian from Skagen to Lauenburg and a measurement of longitude from Copenhagen to the west coast of Jutland. He asked Gauss for an improvement in the usual methods and inquired whether he would be interested in extending the survey through Germany to the Bavarian triangles. Gauss replied on July 5, 1816, with enthusiasm and expressed hope of future activity in this field in Hanover, also offering his aid in calculating the principal triangles. He saw an advantage for Schumacher in the fact that Denmark had already been trigonometrically surveyed, although he felt there would be a difficulty inherent in Denmark’s flat terrain.
Gauss felt that such operations would help to clear up questions connected with the irregularity of the earth’s figure by furnishing additional empirical data. He was especially interested in making fundamental determinations in this field and also in the astronomical constants. His earlier interest in geodesy can be traced in part to a desire for accomplishment, in gratitude to the Duke of Brunswick. He was later greatly interested in von Zach’s survey of recently acquired lands authorized by the King of Prussia and realized that after the elaborate results in France and England a survey would be relatively unimportant unless carried out on a large scale. In 1816 Gauss received a copy of Epailly’s triangles.
At this time Gauss was considering as a prize question the problem of so projecting a given surface on another given surface that the projection would be similar to the original in the smallest parts. A special case occurs when the first surface is a sphere and the second a plane. The stereographic and mercator projections are particular solutions. The general solution, to include all particular solutions, would cover every kind of surface. In the Göttingen Royal Society of Sciences, Gauss had the opportunity of proposing a prize question only once every twelve years. He saw a chance of generalizing this theory and giving a geometric description of a large part of Europe by means of his geodetic measurements if others would also communicate their results to him. Thus his first idea of conformal projection can be dated early in the year 1815.
On the occasion of Gauss’ visit to Lindenau on the Seeberg in the fall of 1812, geodetic questions were discussed. His paper of 1813 on the theory of the attraction of homogeneous ellipsoids was a fruit of this visit. On September 25, 1814, the astronomer Encke, Gauss’ pupil, accompanied him to Seeberg for another visit with Lindenau. The fertility of Gauss’ thinking during this decade is comparable to that of his youth, 1790–1800. Encke recorded that he was as though intoxicated by the glorious theories which Gauss enunciated in the evening after dinner.
In the autumn of 1816 Gauss moved into the new observatory. His spare time was claimed by research in theoretical astronomy, non-Euclidean geometry, and the theory of numbers; terrestrial magnetism and the theory of surfaces were already beginning to occupy his attention. It is understandable that he did not immediately agree to join Schumacher’s survey project. Gauss lacked practical experience in negotiating with the authorities on such a matter and in gaining the cooperation of experts; in addition he did not have trained assistants. Such a survey was expensive, and he disliked to request an appropriation at the same time as he was seeking funds to equip the new observatory. Even salaries were not always paid promptly. At this juncture Schumacher applied directly to Baron Karl Friedrich Alexander von Arnswaldt (1768–1845) in Hanover, curator of the University of Göttingen from 1816 to 1838. Von Arnswaldt proved to be very cooperative and at once inquired about all details, including the matter of assistants. In September, 1816, when von Arnswaldt was returning from a water cure in Wiesbaden, he stopped in Göttingen and requested Gauss to hand in a memorandum on the proposed survey. In early December Gauss received from Schumacher information on the cost of the Danish survey, including the duration of the measurements, the personnel, and related topics.
By the end of July, 1817, actual operations had begun in Denmark, and in November the triangulation of 1½ degrees of the arc of meridian from Hamburg to the Island of Alsen was complete. In 1818 when Schumacher was ready to measure his southernmost triangles, he again requested Gauss to participate in the connecting measurements, if possible in September, and sent him a sketch of the triangles already measured and of the planned connection. Gauss received this letter too late to discuss the plan with the curator. He had not yet received an answer to his memorandum on participation in the survey. Again Schumacher took the necessary steps and announced to Gauss at once that the continuation of the survey through Hanover was not only approved, but also that von Arnswaldt would do everything to promote it. In September, 1818, Gauss announced that the premier had commissioned him to undertake the survey in Lüneburg. He asked Schumacher about the most suitable tower in Lüneburg. His health had not been as good as usual that summer, but it soon improved and he looked forward with pleasure to the journey. He bought a coach and took along the twelve-inch Borda circle as well as the
eight-inch theodolite in order to see how the results would agree. Shumacher sent his assistant. Captain Caroc, who was busy with a survey in Lauenburg at the time. On this account Ursin (Georg Frederik Krüger) was ordered to assist Gauss. Later Schumacher himself came to Lüneburg with an eight-inch Repsold theodolite. The tower of St. Michael’s Church in Lüneburg was used to measure the two angles Hamburg-Hohenhorn and Hohenhorn-Lauenburg. This fine, big tower was of solid stone, and its centering was easy and sure.
During these observations Gauss had his first stimulus for the invention of the heliotrope; this was the reflection of the sun from a window in St. Michael’s tower in Hamburg, a disturbance in his observing. On advice of H. C. Albers, a geodesist, Gauss stayed in Lüneburg at the Schütting inn on the market place, some distance from St. Michael’s Church. On October 9, 1818, he determined on the bastion before a Lüneburg gate the angles between the various towers and the position of Schumacher’s theodolite. Gauss sent a report to Hanover on the Lüneburg measurements and requested a larger theodolite, but did not receive an answer.
Schumacher journeyed to London in April, 1819, to receive Ramsden’s zenith sector at Woolwich observatory. He used the opportunity to interest Sir Joseph Banks, president of the Royal Society and a man of unlimited influence, in the Hanoverian survey. He also stimulated the Danish envoy in London, Privy Councilor von Bourke, to negotiate on the survey with Count Münster, minister of Hanoverian affairs in London. Count Münster was rather surprised that Gauss had not applied directly to him, and requested an immediate memorandum on the costs. Gauss therefore sent Münster a document on the meaning and usefulness of the Hanoverian survey, and on advice of Schumacher informed von Arnswaldt of this letter, at the same time requesting the latter to approve participation in the observations in Lauenburg. On the first of June Schumacher personally conferred with von Arnswaldt and succeeded in getting a ministerial directive authorizing Gauss to go to Lauenburg and to receive all necessary advances of money. By the end of June the sector was set up in Lauenburg and Gauss participated in the observations, from which he returned home very suddenly on July 18, weakened by the summer heat. So great was the haste that he left behind various articles of clothing.
Early in 1820 Gauss again felt ready to make observations; he knew that eventually these data would be published and therefore wished to have a competent scientific witness present for the most important operations as an aid to accuracy. Soon afterward King Frederick VI of Denmark ordered his envoy in London to request of the Hanoverian government Gauss’ presence in the Danish survey, including the observations in Lauenburg. Bessel attached little importance to Gauss’ geodetic work and thought it required too much of his time. Such activity, he thought, should be carried on by one of lower mathematical stature. Gauss replied to Bessel:
All the measurements in the world are not worth one theorem by which the science of eternal truths is genuinely advanced. However, you are not to judge on the absolute, but rather on the relative value. Such a value is without doubt possessed by the measurements by which my triangle system is to be connected with that of Krayenhoff, and thereby with the French and English. And however low you estimate this work, in my eyes it is higher than those occupations which are interrupted by it. I am indeed far removed here from being master of my time. I must divide it between teaching (to which I have always had an antipathy, which is increased, though not caused, by the feeling of throwing my time away, an everpresent concomitant of this activity) and practical astronomical work. I have always enjoyed it so much; you will agree with me, that, when one does without all real help in numerous petty affairs, the feeling of losing one’s time can only be removed when one is conscious of pursuing a great important purpose. But you have made that difficult for the rest of us, since you have anticipated us and supplied most [astronomical] desires in such a model manner. Nothing remains for the rest of us except now and then to glean.
What do I have for such work, on which I myself could place a higher value, except fleeting hours of leisure? A character different from mine, less sensitive to unpleasant impressions, or I myself, if many other things were different from what they actually are, would perhaps be able to gain more from such hours of leisure than I, in general, can. As things are, I must not reject an undertaking, which, although connected with a thousand complaints and perhaps acting as an irritant to my strength, is really useful, which of course could be carried out by others, while I myself under more favorable circumstances would do something better, but which, if I do not take it on myself, would definitely not be carried out; finally, I must not conceal from you a matter which somewhat balances the inequality which exists between my salary—the same in 1824 as was scheduled in 1810 under Jérôme—and the needs of a large family. [Gauss wanted Bessel to treat this as confidential.]
In the spring of 1820, as a result of Count Münster’s letter from London, the King approved the continuation of the survey through the Kingdom of Hanover. This cabinet order of George IV, king of Great Britain and Hanover, went to the ministry on May 9, 1820, and the ministry notified Gauss on June 30, 1820. On November 1, 1820, he reported to the ministry on his participation in the survey at Braak, twelve kilometers northeast of Hamburg, which lasted from September 12 to October 25. On the return trip Gauss found out through von Arnswaldt that Count Münster had been negotiating with the Duke of Wellington (as Great Master of the Ordnance) on the question of transferring the Ramsden zenith sector, which Gauss valued highly for this work. Actually, it was returned to London in 1827 by Gauss’ assistant. Captain Müller. The Danish survey as planned was not completed in Schumacher’s lifetime. There was a long delay from 1824 to 1838; his death occurred in 1850.
By the close of 1820 Gauss had begun many preparations for the survey. There were numerous difficulties which often put him in a bad temper. The Göttingen observatory was chosen as a beginning point; he found a good view on a hill five kilometers north, just beyond Weende, the closest village. Difficulty was encountered in getting Epailly’s data on ninety-four triangles surveyed in 1804–1805, which by a detour connected southern Hanover with Hamburg. Epailly’s points were difficult to locate. The Prussian general staff furnished results of von Müffling’s survey of Hessen, Thuringia, and Brandenburg, which was planned to connect with the French, Bavarian, and Austrian network. Another irritation was the delay in getting the proper instruments. Two artillery officers. Captain G. W. Müller and Lieutenant F. Hartmann of the city of Hanover, were appointed his assistants.
First reconnaissance and preliminary measurements in the neighborhood of Göttingen began in the middle of April, 1821. A French signal tower had once stood on the mountain Hohenhagen, 508 meters above sea level, not far from Dransfeld, a small village fifteen kilometers distant. Because of the fine view this spot proved to be suitable as a main point of triangulation, Müller and Hartmann found the Hils near Ammensen to be a good point for an open view northward to the city of Hanover and even farther, Hartmann was assigned the building of a signal tower on the Hohenhagen, and Müller had the same task on the Hils, as well as the reconnaissance of Kruksberg at Lichtenberg in the Brunswick area.
A unified reconnaissance of the survey area was not made and evidently was not planned. Later this caused Gauss many unpleasant hours, although one does not have to look far for the reasons why it was not made: poor travel facilities, slow means of communication, mediocre maps, poor accommodations in villages, expense, and technical difficulties in building reconnaissance structures. Gauss had to familiarize his assistants with the scientific instruments and often performed himself what he did not entrust to them. The principal reason for lack of reconnaissance is found in Gauss’ desire to see some results of the survey and to be able to present them to the proper authorities. His method of procedure in theoretical work sometimes worked to a disadvantage in practical geodetic work.
Gauss’ invention of the heliotrope was the circumstance which stimulated him to accelerate the surve
y. At first he convinced himself on a photometric basis that sunlight reflected by a small mirror can be seen at considerable distances. Next he experimented with a mirror attached to a theodolite. In the Astronomical Yearbook for 1825 he wrote: “With a somewhat favorable condition of the atmosphere there are no longer any limits for the sides of a triangle, except such as are set by the earth’s curvature.”
In connection with the heliotrope the Göttingen Almanach attributed to Gauss the statement that this discovery would be greater than that of America, if with this instrument we could communicate with our neighbors on the moon. Gauss did write: “With 100 mirrors united, each of 16 square feet area, one would be able to send good heliotrope light to the moon. It is a shame that we can’t send out such an apparatus with a detachment of 100 people and several astronomers to give us signals for determination of longitude.”
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 15