Bessel could not fully agree with Olbers on this point, for he wrote to Gauss on May 28, 1837: “You have never recognized the duty of advancing the present knowledge of subjects through prompt publication of one part of your researches adapted to the whole; you are living for posterity. Where would the mathematical sciences be now, not only in your dwelling, but in all Europe, if you had expressed everything that you could express!”
Gauss always wished his presentation to be as compact as possible and found that such writing demanded much more time than the opposite type of writing. Schumacher refused to give up in his attempt to change Gauss’ viewpoint and in 1833 proposed a new seal with the words multa nec immatura. In 1836 he desired that Gauss use the words nec pluribus impar, but admitted he was afraid his friend would not use it.
A generation later, Kronecker, who was a leading mathematician of his day, had this to say about Gauss’ style:
The manner of presentation in the Disquisitiones [arithmeticae], as in Gaussian works generally, is the Euclidean. He sets up theorems and proves them, in which he industriously erases every trace of the train of thought which led him to his results. In this dogmatic form is certainly to be sought the reason for the fact that his work was not understood for so long and that the efforts and researches of Lejeune Dirichlet were needed to bring it to full effect and appreciation among posterity.62
Gauss was very slow in passing judgment on others. He expressed this viewpoint on February 12, 1837, in a letter to Schumacher: “Only when I have time to process something for myself and to transform in succum et sanguinem, can I permit a judgment.”
He felt that he would have to carry much of his work to the grave, because it would be incomplete and he was unwilling to turn over such results to another. And that is exactly what happened. The editing of his Collected Works extended from 1863 to 1934. He insisted on purity in determining mathematical concepts and missed it among most of his contemporaries. Gauss admitted in a letter to Bessel in 1826 that he had sometimes spent months of effort in vain on a problem.
Gauss never placed a high value on his method of least squares because he did not appraise things from the utilitarian viewpoint. He felt that people who had to do a great deal of calculating would have hit upon the same artifice, and made a bet that Tobias Mayer must have used the same method. Later, in looking through the papers of Tobias Mayer, he realized he would have lost that bet.
He was pleased by Abel’s work on transcendental functions because he felt that it relieved him of about one-third of the labor he had planned on that subject. Gauss praised the elegance and conciseness of Abel’s work, and stated that many of his formulas were like copies of his own. He recognized Abel’s independence by declaring that he had never communicated anything on the subject to anyone.
When Arago’s works were translated into German a change was made where he discussed the number of men who were qualified to pass final judgment on exact researches. The original stated “eight or ten,” but the translation by Hankel gave the figure at “seven or eight.” Gauss was quite pleased and attributed the change to Alexander von Humboldt. Actually it was done on the advice of Dirichlet.
Gauss gave an interesting sidelight on his methods of writing in a letter to Schumacher on October 12, 1840: “What is to be printed is usually written more than once. On the other hand I very rarely make a rough draft of letters, I believe that in my whole life I have made in advance a rough draft of scarcely a dozen letters.”
Archimedes was the man of antiquity whom Gauss esteemed most highly. He imagined him as a worthy old man of noble appearance. The only thing for which he could not pardon him was the fact that in his “sand calculating” he did not discover the arithmetic of position or decimal system of numbers. Gauss said, “How could he overlook that, and at what a pinnacle science would now be if Archimedes had made that discovery.”
On February 12, 1841, Gauss wrote to Schumacher the secret of how his memory worked: “My memory has the weakness (and has always had it) that everything read, unless at the moment of reading it is connected with something directly interesting, soon disappears without trace.”
Not long after this, Schumacher tried to pry out of Gauss the secret of his unique power in numerical calculating. After three weeks he got an answer from Gauss (January 6, 1842):
My occupation with higher arithmetic for almost fifty years now participates in the aptitude in numerical calculating ascribed to me, to the extent that of themselves many types of number relations have involuntarily stuck in my memory, relations which often occur in calculating. For example, such products as 13 × 29 = 377, 19 × 53 = 1,007 and the like, I look at directly, without taking thought; and in others which can be deduced immediately from such ones there is so little thought that I am scarcely conscious of it. Moreover, I have never purposely cultivated in any way skill in calculating, otherwise it could have been carried much further without doubt; I place no value on it, except in so far as it is the means but not the purpose.
One day a small circle of friends sat in Gauss’ home discussing the intelligence of animals. Georg Heinrich Bode (1802–1846), young professor of classical philology, had many wonderful things to tell on this subject, in particular about his travels in America and about the wisdom of a parrot that he had brought along from Northampton, Massachusetts. It was so wise that Bode had named it “Socrates.” At first Gauss listened silently to Bode’s praise. But when the young scholar maintained that his “Socrates” could even answer questions asked in Greek, Gauss remarked smilingly that he had not taught “Hansi,” his chaffinch from the Harz mountains, Greek, but that he had succeeded in teaching him several bits of his native Brunswick dialect, which the little animal knew how to use cleverly. A short time previously he had held out to him a cigar and his pipe and asked: “What shall I smoke, Hansi?” Whereupon the sly little bird after a short meditation promptly answered with “Piep.”63
Georg Julius Ribbentrop (1798–1874), who first served in the University of Göttingen library and later for many years as professor of law, was a confirmed bachelor and an eccentric; he was widely known as the perfect example of the absent-minded professor. One evening he was invited to Gauss’ home for dinner. After the meal a severe electrical storm came up, and it turned into a long-lasting cloudburst. In those days the observatory was some distance beyond the gates of the city. Therese Gauss, at that time her father’s hostess, invited the guest to stay overnight on account of the long walk home. Ribbentrop agreed, but soon disappeared before the others missed him. After some time the doorbell rang. There stood Ribbentrop in front of his astonished hosts, drenched. He had hurried home to get his necessary sleeping articles for the overnight invitation.
Gauss had promised to show Ribbentrop the eclipse of the moon visible on November 24, 1836. On the appointed evening the rain was pouring down, so that Gauss assumed his astronomical guest would not appear. He was so much the more surprised when the latter, drenched, suddenly stood before him.
“But, my dear colleague,” Gauss greeted him, “in this weather our planned observations of the sky might come to naught.”64
“Not at all,” opined Ribbentrop remonstrating, as he triumphantly waved his big umbrella. “My landlady has seen to it that this time I did not forget my umbrella.”
Gauss’ factotum, the optician J. H. Teipel, was an industrious and able artisan, who along with his upright, honest character possessed fine, native wit. He was proud of the fact that after he had learned his trade assisting Gauss at the Göttingen observatory, he was allowed to call himself “university optician.” He had to endure many malicious references to this title, but in his dry, clever way always had a ready comeback. One day a wag asked Teipel the difference between opticus (optician) and optimist. He got this answer: “Approximately the same as between Gustav and Gasthof.”
Teipel was not only the artisan assistant of Gauss; he also undertook to show visitors through the observatory, who as laymen
wanted “to look at the stars” and were satisfied with a popular explanation of the starry heavens. The following comic scene occurred once: In observing planets Teipel was once asked by a lady about the distance of Venus from the earth, whereupon he explained: “That I cannot tell you, Madame. For numbers in the world of stars Hofrat Gauss is there. I am here merely calling attention to the beauties in the sky.”
Eugene Gauss used to tell that his father first thought of the heliotrope while walking with him and noticing the light of the setting sun reflected from a window of a distant house. Gauss’ own version was slightly different. The heliotrope was his favorite invention. He emphasized that he was led to it not by an accident, but by mature meditation. It was true that from the steeple of St. Michael’s in Lüneburg he had seen the windowpane of a Hamburg steeple flashing in the sun, an incident which merely strengthened his conviction of its practicality. Gauss liked to tell how, when the heliotrope was first used, a crowd of curious spectators gathered and let out a shout of joy at the first appearance of the distant light.
Of the works Gauss published, none refers directly to the geometria situs, although this subject occupied his attention most of his life. In the last years of his life, about 1847 to 1855, he gave special attention to it and hoped for great things in this field, because he regarded it as almost untouched. Fifty years earlier he had written to Olbers that he was interested in Carnot’s Géométrie de position, which was then about to appear, and referred to the work of Euler and Vandermonde. In 1810 his friend Schumacher translated Carnot’s work.
It should be remembered that Carnot meant by the term “geometry of position” something different from geometria situs; he had in mind the application of negative numbers to geometry. Later, projective geometry was frequently designated as geometry of position as opposed to metric geometry. But these are not what Gauss had in mind by the term geometria situs. He referred to that branch of mathematics which today is called topology.
On October 30, 1825, Gauss reported to his friend Schumacher that he had made great progress in his work on the general theory of curved surfaces, and then continued: “One must pursue the tree to all its root fibers, and much of it costs me weeks of strenuous meditation. Much of it even belongs in the geometria situs, an almost unworked field.”
In a notebook under date of January 22, 1833, Gauss wrote: “Of the geometria situs, which Leibniz foresaw, and which two geometers65 have been permitted to glimpse weakly, after a century and a half we still know and have not much more than nothing. A principal problem of the border field of geometria situs and the geometria magnitudinis will be to count the entwinings of two closed or infinite lines.”
In 1834 the Göttingen Gelehrte Anzeigen contained a review by Moritz A. Stern (1807–1894), who was for many years professor of mathematics in Göttingen, of a book by the Dutch mathematician Uylenbrock dealing with the work of Huygens and other important seventeenth-century mathematicians. In the review, Stern mentioned having heard Gauss discuss his research in topology. It is regrettable that Gauss never found time to publish something on the subject. In a letter to his pupil Möbius (1790–1868) on August 13, 1849, Gauss thanked him for a copy of his paper on the forms of curves of the third order and urged him to investigate in a similar manner the form relationships of algebraic curves which occur in his own dissertation (1799).
One of the oldest notes by Gauss to be found among his papers is a sheet of paper with the date 1794. It bears the heading “A Collection of Knots” and contains thirteen neatly sketched views of knots with English names written beside them. This is probably an excerpt he made from an English book on knots. With it are two additional pieces of paper with sketches of knots. One is dated 1819; the other is much later, for it bears the notation: “Riedl, Beiträge zur Theorie des Sehnenwinkels, Wien 1827.” Notes of Gauss referring to the knotting together of closed curves are printed in his Collected Works (VIII, 271–285). In particular he found in a note dated December, 1844, the numerous forms which closed curves with four knots can exhibit. He was conscious of continual semantic difficulty in topology.
The above-mentioned remark of January 22, 1833, concerns the concatenation of two curves in space, in which at the close the definite integral formula for the number of knots is given. It has been alleged that Schnürlein, a pupil of Gauss, carried on intensive research on the application of higher analysis to topology, with his help, but no one has been able to verify this.
The determination of the mutual position of curves in the plane is the means which Gauss used in his dissertation (1799) for the deduction of the fundamental theorem of algebra. This viewpoint is even stronger in his last proof of the theorem (1849).
Under the heading of topology should be mentioned the research which Gauss did on the possible types of distribution of geocentric points of a planet on the zodiac (Collected Works, VI, 106), and the case mentioned here of a chainlike overlapping of two planetary orbits, as is often the case among the asteroids.
There can be no doubt that Gauss had some influence on the later work in topology carried on by his pupils Möbius, Listing, and Riemann. His influence probably lay in giving stimulation; and one need not detract from their work by saying that Gauss stimulated them to work in topology.
In the last decade of his life Gauss continued his observations at the instruments of the observatory, and the numbers of the Astronomische Nachrichten of that period contain frequent communications from him on various observations of eclipses and locations of planets and comets. His last observation communicated to this journal, which he himself carried out, was that of the solar eclipse of July 28, 1851, when he was seventy-four years old; his assistants Klinkerfues and Westphal participated in this. Gauss’ last communication to the periodical occurred in 1854. His diary of the Reichenbach circle from March 6 to June 21, 1846, shows twenty-one observation days. The diary closes with the latter date. From July 4, 1846, to June 27, 1851, he observed most of the previously observed fundamental stars—several of them ten to twenty times—and several other stars. Moreover he included the planets Venus and Mercury and observed the newly discovered planets Metis (Graham’s planet), Parthenope, Victoria, Iris, Flora, and Neptune. He published the latter observations; they are printed in Volume VI of the Collected Works. He also observed the solar eclipse of October 8, 1848.
Gauss’ love for astronomy was demonstrated by the fact that he gave four of his six children the first names of the discoverers of the asteroids. On May 6, 1807, Bessel wrote to Gauss: “I saw with pleasure that you have calculated the orbit of Vesta; also the name chosen by you is splendid, and therefore certainly also pleasant to all your friends because it shows them to which goddess you sacrifice.”
XVIII
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Senex Mirabilis
In 1848 the so-called “literary museum” was founded in Göttingen as a protection against the Revolution. It was in part a social organization, designed to bring students and professors closer together. This museum, later called the “union,” was set up at Hospitalstrasse 1 in the former home of Karl Otfried Müller, the professor of Greek who had died in 1840 at Athens while on a journey. As a means of educating the students many newspapers and large quantities of “unobjectionable” literature in every form were offered. When Göttingen got a warning from Hanover about the Jung Deutschland movement, officials proudly pointed to the literary museum. Only two professors in Göttingen were suspected of belonging to the movement. One was a former pupil of Gauss, Moritz A. Stern in mathematics, and the other was Theodor Benfey (1809–1881) in Oriental languages. This suspicion had arisen merely because they were Jews.
Gauss joined the club and in his last years almost the only physical recreation he took was a daily walk to the reading room between eleven and one o’clock. He rapidly scanned political, financial, literary, and scientific news, making a mental note or writing down things which especially interested him. Some of this was for statistical studies he made. It w
as his custom to collect recent issues of the newspapers he wanted to read, and to prevent their being taken away before he had read them, he arranged them chronologically on the seat of his chair and sat On them. Then he cautiously pulled out one after the other and passed them on after having read them. Students generally stood in awe of Gauss, and when one of them happened to be reading a newspaper which Gauss desired, he cast a questioning look at the student who then hurried to give him the paper.66
Personal and scientific correspondence with friends, relatives, and colleagues in all parts of the world took up much of Gauss’ time in his later years. He received occasional letters begging for financial aid, several from persons who claimed to be his kin. Yet, in spite of new correspondents, he was both relieved of letter writing and grieved at the death of his intimate friends: Olbers died in 1840, Bessel in 1846, and Schumacher in 1850, At one period he had written weekly to Schumacher. Bolyai, Encke, Gerling, and Humboldt survived him; hence his correspondence with them continued to the last. He noted on his calendar, several months in advance, the day in 1853 when Humboldt would reach the age attained by Newton and sent him special congratulations. On February 15, 1851, Gauss’ assistant Benjamin Goldschmidt died very suddenly at the age of forty-four. He had been observing the night before and had shown some visitors the Pleiades through the telescope. He was found dead in bed early the next morning. Goldschmidt was a man of benevolent character, and well informed in his field. His sudden death had a serious effect on the aged Gauss, who was sincerely attached to the young man and honored him for his ability. Goldschmidt was succeeded by E. F. W. Klinkerfues, a picturesque character of considerable talent, who remained at the observatory until his suicide there in 1884, Klinkerfues had early become an orphan, was a member of a large family, and had had a difficult life. Gauss had great sympathy for him.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 26