Gauss was undoubtedly the first man to free himself from the fetters of Euclidean tradition, but the evidence does not show that he had direct influence on Bolyai and Lobachevsky, either by stimulation or furnishing of basic ideas. There is no doubt that Taurinus was independent of Gauss, even though he had a certain stimulus from him through his uncle Schweikart. Wolfgang Bolyai tried to steer his son away from the subject of parallels with the warning that even Gauss had failed to solve the problem. In later years Johann accused his father of having given his ideas to Gauss, who then claimed them as his own. In addition, it is perfectly clear that Bolyai and Lobachevsky were entirely independent of each other.
Possible influence of Gauss on Lobachevsky needs to be examined more closely. Bartels, friend and teacher of Gauss, spent nine years in Switzerland, returned to Brunswick 1805–1807, and in the latter year became professor of mathematics in Kasan. Bartels was the teacher of Lobachevsky, and introduced him to the work of Gauss in the theory of numbers, but there is no evidence that he gave him Gauss’ ideas on non-Euclidean geometry, simply because he did not know them. Gauss and Bartels were together only until 1807 and never saw each other again after that date. They exchanged letters of a personal nature in 1808 and again in 1821, but these letters contained nothing about mathematics. Of course, doubt of the correctness of the Euclidean axiom was in the air at that time, and we must assume that Bartels shared the common view—not due specifically to his friendship with Gauss. In later years Otto Struve, a grandson of Bartels, said that the latter regarded Lobachevsky as one of his foremost and most talented pupils in Kasan. The work on non-Euclidean geometry he regarded more as interesting intellectual speculations rather than as a work advancing science. Struve did not remember that Bartels ever spoke of similar ideas of Gauss’.
The strange fact is that in the course of a little more than a decade four men independently liberated themselves from Euclidean tradition. First, Gauss and Schweikart, almost simultaneously, then, again almost simultaneously, Lobachevsky and Bolyai. Even more miraculous is the fact that all four found only one of the two possible solutions. At that time none of them seems to have thought of the possibility of a consistent system of geometry in which the sum of the angles of a triangle is greater than two right angles. Much later Gauss recognized the possibility, and the idea was fully developed by his pupil Riemann, who on June 10, 1854, read before the philosophical faculty of Göttingen his famous trial lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. Of the three themes submitted it was Gauss’ choice which fixed upon this one, and he was in the audience.
There is a certain expression involving the equation of a surface and the coordinate of a point called the “Gauss curvature” which will be constant on all surfaces such as the sphere and the plane. It is also true, however, of certain other surfaces, at least if the dimensions of the figure are sufficiently small. Among the axioms of geometry which do not depend on the parallel postulate are those which secure the free mobility of a figure in space. This means that space is the same in nature throughout and that any figure can be carried along it without tearing or crumpling or stretching. Among the surfaces of constant Gauss curvature is that which is formed by revolving the tractrix about the X-axis. This surface is everywhere saddle shaped. The geometry of this surface is non-Euclidean and belongs to a type different from that of the sphere, being essentially Lobachevskian. The tractrix space is not a complete Lobachevskian plane, but a portion of it, with its edges joined to themselves by the same process by which a cylinder can be formed from a strip of Euclidean plane.
In a note dated about 1827 Gauss called the curved surface of constant negative measure of curvature generated by rotation of the tractrix (pseudo sphere) the “counterpart of the sphere.” The formulas set up by him lead to the theorem that in the pseudo sphere (and only in it) surfaces of rotation are congruent to each other and that, preserving this property, one can move a geodetic triangle on the pseudo-sphere just as one can move a spherical triangle on a sphere. It is thus worthy of note that Gauss studied surfaces of constant negative curvature. It is not known whether he originated the expression “counterpart of the sphere.”
XVI
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Trials and Triumphs: Experiencing Conflict
The University of Göttingen very early began preparations to celebrate its centennial jubilee. It was decided to erect a new building and to dedicate it as part of the celebration. This large building in classical Greek style, known as the “Aula,” is on the Wilhelmsplatz. In the small park in front of it stands a monument of King Wilhelm IV of Hanover, to whose memory the Aula was dedicated; it was unveiled as part of the jubilee. The Aula has always been used as the administration building of the university, and it contains a beautiful auditorium adorned with oil portraits of men important in the history of the university.
The jubilee was celebrated from September 12 to 20, 1837; large numbers of alumni, officials, and delegations from other universities were present. The curators of the university and the Ministers von Arnswaldt and von Stralenheim were escorted into the city by an armed guard. On September 17 all the church bells chimed in the festival. The prorector Bergmann at the salute of cannons turned over to the student body a special jubilee flag of the university.
A choir sang in the steeple of St. John’s Church as King Ernst August, who had been officially received at the Weender Tor by the city council and a delegation of citizens, arrived. In the parade, which marched past the King, were students, the faculty, the pastors of the city, the curators, ambassadors and deputies of foreign courts, and representatives of many universities. The parade then moved to Wilhelmsplatz, where the Aula was dedicated and the monument unveiled. Karl Theodor Albert Liebner (1803–1871), professor of theology, gave the jubilee sermon, and the service was closed by a great Te Deum. After a large banquet the day was closed by a concert in St. John’s Church.
September 18 was the day of the actual academic celebration. Early in the morning Minister von Stralenheim handed to the prorector a golden key and thus turned over to the university the new Aula. At the same time he gave Bergmann a medallion and chain of pure gold, and decreed that each successor of the prorector should wear them. In the Aula, Carl Otfried Müller, Göttingen’s great classical scholar, gave the principal address of the occasion in eloquent Latin.50 After a gala dinner the citizens of Göttingen put on an imposing torchlight parade in the evening in honor of the curators of the university, followed by fireworks.
A large number of the leading scholars of Germany were present; the most important one was Alexander von Humboldt, who was Gauss’ house guest during this visit to his alma mater. When Carl Bolsche saw Gauss walking out of the Aula arm in arm with Humboldt, he was conscious of the great impression made on the youth. He wrote that they said to each other:
“Such an age—what good fortune and what a call for everyone to use his talent in his own way as honestly and as restlessly.” After I heard him [Gauss] mentioned and as often as I saw him in Göttingen, a feeling of edifying reverence and love overcame me. True greatness, without realizing it or desiring it, ennobling the youth!
On September 19, honorary doctorates were conferred in the Aula by the deans of the several faculties. Gauss regretted that Sophie Germain was no longer alive to receive one of them. In the afternoon he gave before an open session of the Royal Society of Sciences a lecture on his invention of the bifilar magnetometer. Gauss undertook the lecture at the special request of Humboldt, who was present at the session. He began his lecture with these words:
Our society approaches the celebration of the Georgia Augusta, as the daughter appears at the golden wedding anniversary of her mother, not in order to express her feelings in flowery language, but to share the joy of the house and to present a modest gift. According to native custom the daughter brings a simple work of her hands finished in the evening hours, or a fruit ripened in her own garden. But the feelings of the daughter o
n the day of honor of her dear mother, to whom she owes existence, care, and prosperity, the feelings of grateful, joyous emotion, are too much a part of her being to have need of words. Indeed, the day of honor of the mother is also the day of honor of the daughter.
While I have the honor, at this festive moment and before such a brilliant assemblage, of opening with a lecture the first session of our society in the new rooms in this sense, I am quite conscious of how very much I must count on a benevolent indulgence in more than one respect. A lecture from the area of the rigorous sciences, in itself slightly compatible, and in any case in my hands not adorned with eloquence, can in the most favorable instance arouse sympathy only among those who are more intimate with similar efforts. It will be more gratefully recognized if such persons as are more remote from these sciences do not deny their honoring attention to a lecture from which I cannot separate several developments appearing perhaps dry to them, without becoming superficial or even unintelligible.
In the evening of September 19 a great dance took place in the riding academy, which had been specially equipped for the occasion. Over two thousand people took part in the dance. On the following morning the students took their flags into the Aula and closed the jubilee by singing the traditional “Gaudeamus igitur” on Wilhelmsplatz.
Gauss was not a friend of official celebrations. How did he react to all this gaiety and ceremony? The answer is found in a letter which he wrote to his friend Olbers on September 26, 1837:
After our jubilee ceremonies are surmounted—of which you get news through the press ad nauseam usque—I must give you a sign that I am still alive. Indeed, little would have been lacking for me to have met the same fate as Göschen,51 who died day before yesterday most probably as a sacrifice of the same [ceremonies]. Another local professor, Dissen,52 had died a few days earlier, but if in his case the jubilee had some effect, then it can only have been disturbances of mind, since he, confined to his room for many years, could not take any part in the ceremonies.
I myself had not been quite well previously and had decided to take no part in all the processions and the stay in church and Aula where the tasteless as well as oppressively heavy so-called official robe, a monk’s cloak, had to be put on. Not until early Sunday, when we were informed that the aforesaid processions and sermon presentation would occur before the King, did I decide not to remain behind. But even at the meeting in the library, where 100 persons were crowded together in close quarters, I was really sick in the deoxygenized air; then came the procession in the heavy-as-lead robe; then the stay in the terribly overcrowded church, where one could scarcely breathe, and an insipid sermon which went on and on. I almost fainted. Then a new procession, then standing an hour in the open air at the unveiling of the statue, then presentation. When I finally came home, I was melting away in sweat, and had to change linens and clothes quickly, in order to attend the banquet.
I believe that I escaped Göschen’s fate only by the fact that on the following day I avoided all participation and remained quietly at home, Just as on Tuesday [I avoided] participation in the honorary degree ceremony; finally by the fact that just after it through the solicitude of my good Weber all windows were opened, in order to let in fresh air again, so that when I arrived after 12 o’clock for the session of the Royal Society, I found enjoyable air and could hold my lecture.
Gerling had taken quarters in my home during the week. I saw Humboldt every day at my home, mostly several times every day. He departed last evening. I admired his vigor. He traveled night and day from Berlin here; from here to Hanover he again traveled during the night.
It is now as quiet again in Göttingen as it had been previously noisy. During the jubilee many a goblet of wine was emptied to your health.
At the time Gauss was quite concerned about his youngest son, Wilhelm, who in the fall of 1837 migrated to America with his bride, a niece of Bessel. His oldest son was engaged at this time to Sophie Erythropel of Stade, daughter of a physician, who died in October, 1837. (A sketch of all Gauss’ children is given in Appendix D.)
Humboldt on his return to Berlin spent two days in Hanover visiting ministers, ambassadors, and other members of the court. In a letter thanking Gauss for his recent hospitality he told of a meeting with the astronomer Caroline Herschel, who seemed to be especially glad to have some news of Gauss. Humboldt had an hour’s audience with King Ernst August, who said that he was pleased with everything he had seen in Göttingen during the jubilee. The King remarked that he had never seen better behaved young people.
Yet the sounds of the jubilee with all its good feeling, pride, and cheer had scarcely died away when a great catastrophe hit the university. It was to affect Gauss deeply and directly the rest of his life. It affected the university adversely for many years to come. It affected many prominent persons over a long period of years. And in the long run little was accomplished by the act which caused the catastrophe.
On September 26, 1833, during a political uprising. King Wilhelm of Hanover (William IV of England) recognized and signed an amended constitution, by which certain rights were secured for the people. In June, 1837, he died, a weak but benevolent ruler who had enjoyed far-reaching sympathies among the populace of both countries. The English parliamentary reform of 1832 corresponded to the introduction of the Hanoverian constitution. William IV was succeeded by Ernst August, Duke of Cumberland (1771–1851), fifth son of George III of England. His first act53 was to revoke the liberal constitution adopted by his predecessor, and to restore the constitution of 1819. One of the provisions of the new constitution declared that an heir to the throne who suffered from any physical or moral defect was thereby debarred from the throne. The new King’s only son was blind, and probably his sole purpose in setting aside the constitution of 1833 was to make his son legal heir to the throne. Friedrich C. Dahlmann (1785–1860), professor of history and political science at Göttingen, made a motion in the academic senate that the university should undertake steps to call the King’s attention to the danger of his plan. This occurred just before the jubilee, and found no response. Few professors wanted to intervene in the matter. People in Göttingen thought they saw an evil omen in the fact that at the unveiling of the monument to Wilhelm IV during the jubilee, Ernst August turned his head the other way.
There were also financial reasons why Ernst August nullified the constitution of 1833. It secured the acknowledgment of the provincial diets, and the division of the chief diet or parliament into an upper and lower house; it gave the lower house the exclusive right of initiating legislation and handed over to the ministry the control of the finances. The new king never gave his official approval to the constitution of 1833, as prescribed. New elections were ordered according to the constitution of 1819. The king dismissed his cabinet ministers and immediately renamed them department ministers. His purpose was to release all his subjects from their oaths of loyalty to the constitution of 1833.
Leaders in both church and state, and the people at large, were vehemently opposed to the arbitrary and illegal course taken by the King. Wilhelm IV had granted concessions in answer to the demands emphasized and enforced by riots at Göttingen and elsewhere. Dahlmann had taught that a constitution can be changed only in a constitutional manner. Such a constitution, involving all national, moral, and intellectual forces, formed a symbol for the duty of the individual in the service of the fatherland. An oath of loyalty to the constitution was at least theoretically almost a religious matter. It was a matter of conscience. Such were the feelings of the high-minded men of the time.
The people protested all in vain. The King had a will of iron, and a furious temper withal. Opposition merely inflamed his passions and stiffened his resolution. The imperious old man swore he would leave the people no constitution at all if they refused to accept the one he offered. He proved himself to be bluff, brutal, and self-willed. He did not know the German language. He knew only “subjects,” not full-fledged citizens, and hated con
stitutional government. Ernst August was also rather unfriendly to research and science.
One can imagine the King’s indignation when he received a protest signed by seven Göttingen professors. They were Dahlmann, Ewald, Weber, the famous Grimm brothers, Wilhelm Eduard Albrecht (1800–1876), professor of law, and Georg Gottfried Gervinus (1805–1871), professor of history and literature. Dahlmann and Ewald were the prime movers in the protest, which was directed not to the board of curators of the university, but to the royal cabinet. The Seven declared that they felt bound by their oath of loyalty to the constitution of 1833. The King’s anger greatly increased when he found out that the document had immediately been made public far beyond the borders of Hanover. Curator von Arnswaldt and Cabinet Councilor Hoppenstedt tried to get the Seven to take back their protest. Opinion in Göttingen was divided.
Prorector Bergmann and the academic senate secured an audience with the King at his castle Rotenkirchen in nearby Soiling. Bergmann presented a document in which the dissemination of the protest was called “an unhappy event.” The government used this in a release to the press, which attempted to isolate the Seven and make it appear that the whole university was against them. Six other leading Göttingen professors three weeks later released to the press a statement in which they said that they could never criticize the action of their seven colleagues.
On December 12, 1837, the famous Göttingen Seven were fired from their jobs. Jacob Grimm, Dahlmann, and Gervinus were ordered to leave the Kingdom of Hanover within three days, since they had “confessed” having a part in circulating the protest. The others were allowed to stay in Göttingen as long as they “behaved” and kept quiet. The government ignored the declaration of the other six professors.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 23