Felix Klein stated80 that some Gaussian manuscripts reveal a knowledge of the fundamental ideas of quaternions, the discovery (in 1843) of the great Irish mathematician. Sir William Rowan Hamilton, P. G. Tait declared81 that Klein was mistaken and that the Gaussian restricted forms of linear and vector operators do not constitute an invention of quaternions.
As a matter of fact, Hamilton had expressed himself on the subject in a letter to Augustus De Morgan on January 6, 1852:
In fact, with all my very high admiration . . . for Gauss, I have some private reasons for believing, I might say knowing, that he did not anticipate the quaternions. In fact, if I don’t forget the year, I met a particular friend, and (as I was told) pupil of Gauss, Baron von Waltershausen, . . . at the second Cambridge meeting of the British Association in 1845, just after Herschel had spoken of my quaternions and your triple algebra, in his speech from the throne. The said Baron soon afterwards called on me here (Dublin), . . . he informed me that his friend and (in one sense) master, Gauss, had long wished to frame a sort of triple algebra; but that his notion had been, that the third dimension of space was to be symbolically denoted by some new transcendental, as imaginary, with respect to −1, as that was with respect to 1. Now you see, as I saw then, that this was in fundamental contradiction to my plan of treating all dimensions of space with absolute impartiality, no one more real than another.82
A tradition states that Laplace, upon being asked who was the foremost mathematician in Germany, replied, “Bartels,”83 whereupon the questioner wanted to know why he didn’t name Gauss. Then Laplace said: “Oh, Gauss is the greatest mathematician in the world.”
When Niels Henrik Abel (1802–1829) of Norway, one of the most important mathematicians of the nineteenth century, went to Germany in 1825 he had originally intended to visit Gauss. He was not well known at the time. A copy of his proof of the impossibility of solving the general equation of the fifth degree had been sent to Gauss, and Abel thought Gauss did not do enough to put him before the public. After that he had no use for Gauss and was extremely critical of him. He did not go to Göttingen; it is regrettable that the two did not meet. Abel had wanted to use the splendid university library in Göttingen. Too late. Gauss realized what he had missed, for he wrote to Schumacher on May 19, 1829: “Abel’s death, which I have not seen announced in any newspaper, is a very great loss for science. Should anything about the life circumstances of this highly distinguished mind be printed, and come into your hands, I beg you to communicate it to me. I would also like to have his portrait if it were to be had anywhere.”
Aurel Edmund Voss (1845–1931), professor of mathematics in Munich, commented, in a lecture he delivered for the Gauss Centenary in 1877, on the isolated position which Gauss seemed to his contemporaries to assume. Abel and Jacobi were acutely conscious of this and ascribed it to pride. But they were wrong. It was merely the isolation of a pioneer. Posterity knows more about such matters than do contemporaries. On at least one occasion Gauss expressed himself clearly on this point. His personal view is found in a letter to Bolyai on April 20, 1848:
It is true, my life has been adorned with much that the world considers worthy of envy. But believe me, dear Bolyai, the austere sides of life, at least of mine, which move through it like a red thread, and before which one is more defenseless in old age, are not balanced to the hundredth part by the joyous. I will gladly admit that the same fate, which has become so difficult for me to bear, and still is, would have been much easier for many another person, but the mental make-up belongs to our ego, the creator of our existence has given it to us and we can do little to change it. On the other hand I find in this consciousness of the nothingness of life, which the greater part of humanity must in any case express on approaching the goal (death), that it offers the strongest guarantee for the succession of a more beautiful metamorphosis. My dear friend, let us console ourselves with this and thereby seek to gain the necessary equanimity, in order to tarry with it until the end. Fortem facit vicina libertas sanem, says Seneca.
In reviewing an essay Laplace had published in the Connaissance des temps (1816), Gauss wrote Olbers on December 31, 1831: “The essay . . . is in my judgment quite unworthy of this great geometer. I find two different, very gross blunders in it. I had always imagined that among geometers of the first rank the calculation was always only the dress in which they present that which they created not by calculation, but by meditation about the subject itself.”
The monarch expressed himself on Lagrange in a letter to Schumacher on January 29, 1829: “The reproof hits Lagrange, like almost all analysts of modern times, of not always keeping the subject actively in mind during the game of signs.”
Gauss expressed himself in general terms about mathematicians in France and England in a letter to Schumacher on August 17, 1836: “For my part . . . I joyously accept all true scientific progress which is made beyond the Rhine and beyond the channel, but when they commit stupidities over there . . . then there is nothing further to be done, except to take no notice at all of it.”
In 1837 Humboldt had to decide whether to attend a convention of scientists in Prague or the Göttingen University Jubilee, since they met at the same time. As indicated in Chapter XVI, he spent about a week as Gauss’ house guest. The reason for his decision is found in a letter he wrote to Gauss on July 27, 1837: “Several hours with you, dear friend, are dearer than all the sectional meetings of the so-called scientists, who move in such great masses and so gastronomically that there has never been enough scientific intercourse for me. At the end I have always asked myself like the mathematician at the end of the opera, et dites-moi franchement ce que cela prouve.”
When the publication of the correspondence between Gauss and Bessel84 and the correspondence between Bessel and Olbers was being planned in 1848. Gauss wrote Schumacher on December 23, 1848, that he knew well how many of the compliments in almost all Bessel’s letters ought to be omitted. This occurred because Bessel liked to say something pleasant to people, or something he presumed they would like to hear. Gauss felt that if the letters were published during his own lifetime, such passages should be cut out. When the Bessel-Olbers correspondence appeared. Gauss felt that there were certain passages in Olbers’ letters that should have been omitted, since they indicated a lack of mature judgment and were more of a private chat on matters of science, contemporary scientists, and related subjects.
Moritz Cantor (1829–1920), later world-famous as a historian of mathematics and professor at the University of Heidelberg, delivered a lecture there on November 14, 1899, almost exactly a half century after his arrival in Göttingen late one October evening in 1849. Gauss was the subject of the lecture, at the close of which Cantor gave intimate and charming reminiscences of the great master:
I intended to enroll for the winter courses under Stern, Weber, Listing, and Gauss. I heard the former, but Gauss was not lecturing. It was the same in the summer of 1850, Gauss was not lecturing. In the winter 1850–1851 Gauss taught the announced course on the method of least squares, and I attended it. As far as I know, it was the last course which he taught.85 Later, mostly just as previously, he always had an excuse for not conducting his announced courses. It seems to me as though I were looking into the office in which the course was held. We listeners were sitting around a large table covered with books. Gauss was sitting in an armchair at one narrow side of the table and beside him on an easel there was a moderately large wooden blackboard on which he calculated with chalk. Gauss wore the black velvet house-cap . . . and when he got up, he continually had his left hand in his trouser pocket. . . . The use of ink was excluded by the small amount of space available for each listener at the table, but Gauss disliked even the taking of notes with pencil. Once when we wanted to take notes, he said: “Dispense with writing here, and pay better attention.” It is easy to comprehend that even with the strictest attention and the best memory one was not able later to finish any accurate notebook. One was
able to reproduce only especially ingenious, separate deductions. Moreover such turns of speech and interpolated remarks as had nothing to do with science, stuck in one’s memory, but they characterized the speaker. . . . On another occasion the table was covered with logarithm tables. Gauss explained their differences according to color of paper, formation of numbers, whether of equal size or whether they projected above or below the line; he spoke of the number of decimal places in the tables, of their calculation and rose to the remark, uttered very seriously: “You have no idea, how much poetry is contained in the calculation of a logarithm table.” In other utterances one noted the wag. . . . Laplace also wrote besides his great scientific work on the calculus of probabilities an Essai philosophique sur les probabilités, which found the quickest sale. Gauss had laid the three first editions on the table and showed us in the first edition a statement that the conqueror only harms his own country instead of helping it, which is missing in the second edition and returns in the following ones. The first edition appeared while Napoleon was on Elba, the second during the hundred days, further editions followed in measured intervals. Gauss even knew how to put in linguistic remarks, one of which I will not hold back. He declaimed against the phrase möglichst gut. Not the possibility is increased, but the goodness, therefore one would have to say bestmöglich.
This course on the method of least squares just described by Cantor had nine students in it. Of the nine, only five paid the required fee, and Gauss received the handsome sum of twenty-five thalers for giving the course. Later on, two additional students paid their fees. The names of the students were: A. Ritter, M. Cantor, R. Dedekind, Lieutenant von Uslar, Chr. Menges, A. Valett, L. Hildebrand, G. Wagener, J. C. Lion. Most of these names are forgotten today.
August Ritter (1826–1908) was born in Lüneburg and worked for the Leipzig-Dresden railroad as a draftsman until 1850. In that year he went to Göttingen where he studied until 1853 and took, his doctor’s degree. Later he spent some time as an engineer in Rome and Naples, and in 1856 was called to teach engineering and mechanics at the Technical Institute in Hanover. From 1869 until his retirement in 1899 he was a professor in Aachen. His last years were spent in Lüneburg.
It would be interesting to know more about the others. Cantor made an international reputation for himself. By far the most important person among the students was Richard Dedekind (1831–1916), one of the creators of the modern theory of algebraic numbers. In 1901 Dedekind set down his own reminiscences of the course in great detail. Part of them have special biographical interest for us:
As a native of Brunswick I early heard Gauss mentioned, and I gladly believed in his greatness without knowing of what it consisted. It made a deeper impression on me when I first heard of his geometric representation of imaginary or, as one still said at that time, impossible quantities. At that time as a student at the Collegium Carolineum (the present-day Institute of Technology) I had penetrated a little into higher mathematics, and soon afterward, when Gauss celebrated the golden jubilee of his doctorate in 1849, our faculty sent to him a congratulation composed by the brilliant philologist Petri, in which the passage, he has made possible the impossible, especially attracted my attention. At Easter, 1850, I went to Göttingen, and there my understanding increased somewhat when I was introduced to the elements of number theory in the seminar by a short, but very interesting course by Stern.86 On my way to or from the observatory, where I took a course of the excellent Professor Goldschmidt on popular astronomy, I occasionally met Gauss and rejoiced at the sight of his stately, awe-inspiring appearance, and very often I saw him at close range at his usual place in the Literary Museum, which he visited regularly in order to read newspapers.
At the beginning of the following winter semester I considered myself mature enough to hear his lectures on the method of least squares, and so, armed with the lecture attendance book and not without heart palpitation, I stepped into his living room, where I found him sitting at a desk. My announcement seemed to gladden him very little, I had also heard that he did not like to decide to conduct courses; after he had entered his name in the book, he said after a short silence: “Perhaps you know that it is always very doubtful whether my lectures materialize; where do you live? at the barber Vogel’s? Well, that’s a piece of good luck, for he is also my barber, I shall notify you through him.”
Several days later Vogel, a character known throughout the city, quite filled with the importance of his mission, entered my room in order to tell that several other students had announced themselves and that Privy Court Councillor Gauss would conduct the course.
There were nine of us students, of whom I gradually became more closely acquainted with A. Ritter and Moritz Cantor; we all came very regularly, rarely was one of us absent, although the way to the observatory was sometimes unpleasant in winter.87 The auditorium, separated from Gauss’ office by an anteroom, was rather small. We sat at a table, whose long sides offered a comfortable place for three, but not for four, persons. Opposite the door at the upper end sat Gauss at a moderate distance from the table, and when we were all present, then the two of us who came last had to move up quite close to him and take their notebooks on their lap. Gauss wore a lightweight black cap, a rather long brown coat, grey trousers; usually he sat in a comfortable attitude, looking down, slightly stooped, with hands folded above his lap. He spoke quite freely, very clearly, simply and plainly; but when he wanted to emphasize a new viewpoint, in which he used an especially characteristic word, then he suddenly lifted his head, turned to one of his neighbors and gazed at him with his beautiful, penetrating blue eyes during the emphatic speech. That was unforgettable. His language was almost free of dialect, only sometimes came sounds like our Brunswick dialect; in counting, for example, in which he was not ashamed to use his fingers, he did not say eins, zwei, drei, but eine, zweie, dreie, and so forth, as one can even now hear among us at the market place. If he proceeded from an explanation of principles to the development of mathematical formulas, then he got up, and in stately, very upright posture he wrote on a blackboard beside him in his peculiarly beautiful handwriting, in which he always succeeded through economy and purposeful arrangement in making do with the rather small space. For numerical examples, on whose careful completion he placed special value, he brought along the requisite data on little slips of paper.
On January 24, 1851, Gauss closed his presentation of the first part of his course, through which he familiarized us with the essence of the method of least squares. There followed now an extremely clear development of the fundamental concepts and principal theorems of the calculus of probabilities, explained by original examples, which served as an introduction to the second and third mode of establishing the method, which I must not go into here. I can only say that we followed with ever increasing interest this distinguished lecture, in which several examples from the theory of definite integrals were also treated. But it also seemed to us, as if Gauss himself, who previously had shown little inclination to conduct the course, sensed some joy in his teaching activity during it. Thus the close came on March 13, Gauss got up, all of us with him, and he dismissed us with the friendly farewell words: “It only remains for me to thank you for the great regularity and attention with which you have followed my lecture, probably to be called rather dry.” A half century has now passed since then, but this allegedly dry lecture is unforgettable in memory as one of the finest which I have ever heard.
Throughout his later years Gauss was continually sought out by German and foreign scientists for conferences and personal meetings. A sample of this is found in a letter he wrote to his daughter Minna on July 16, 1839:
This summer I am still expecting many visits here. One or two Englishmen will come probably in the next weeks, in order to discuss magnetic observations with me, which are to be instituted on the expedition of two English ships to the south polar regions. At about the same time Kupffer88 of Petersburg (a former student of mine) will come, also principally
on account of observations to be made in the Russian Empire. In a similar connection Hansteen of Norway has announced himself for the end of August. In September comes Dirichlet of Berlin, a man whom I esteem very highly, and who is especially friendly with Dirichlet [sic!].89
As I hear, Listing will also arrive very shortly, although at first only for several days, in order to journey then to his native city of Frankfurt, until he begins his professorship here.
Although all these visits from outside are in themselves dear to me, yet it will be strenuous for me if they fall on hot days.
On July 15, 1838, Gauss mentions in a letter to Olbers a visit of Sir John Herschel: “Mr. Herschel arrived here yesterday evening. He had journeyed by stage-coach directly from Harburg to Hanover, and came here after he had been there 2 or 3 days. Perhaps I can persuade him to return via Bremen.”
Christopher Hansteen (1784–1873), mentioned in the above letter to Minna, was professor of astronomy and director of the observatory in Oslo. He early became interested in terrestrial magnetism and published several works in the field, having been one of the first who participated in the observations of the Magnetic Association. Hansteen stayed two weeks with Gauss and practiced the use of the two magnetic instruments which Gauss had invented. He ordered reproductions of them for the magnetic observatory which was being built in Oslo. Kupffer also ordered both instruments for St. Petersburg and three or four places in eastern Russia.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 30