Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work

by G. Waldo Dunnington

. every one is the equal rational function (viz., the gth power) of the above. With this insight the original viewpoint was won, and from it arose the entire Gaussian theory of circle-division equations.

  When Gauss went to Göttingen, J. Wildt (1770–1844) was studying mathematics; he published in 1795 a paper with the title “Theses quae de lineis parallelis respondent.” Although Gauss did not become very intimate with Wildt and Kästner, he corresponded with Seyffer until the latter’s death. It was in Seyffer’s home that Gauss and Bolyai met accidentally, as the latter’s son Johann related. Wolfgang spoke freely of the ease with which he could handle mathematics. Shortly afterward, while out walking, he met Gauss on the old town fortification. They approached each other. Wolfgang mentioned, among other things, his definition of the straight line, and several ways of proving Euclid’s eleventh axiom. Gauss, surprised and delighted, broke in with the laconic words: “You are a genius; you are my friend!”

  Gauss gave to Bolyai the tablet on which he had made the discovery of the 17-gon, as a souvenir; also a pipe. In 1797 the two made a trip to Brunswick by foot to visit Gauss’ parents. When Gauss was not in the room, Frau Dorothea asked Bolyai whether Gauss would amount to anything. Bolyai replied: “The first mathematician in Europe,” and she burst into tears.

  On September 28, 1798, Gauss returned to Brunswick, while Bolyai remained until June 5, 1799. Their letters of this period are extremely interesting from a personal side. Bolyai tells of their last meeting in very touching terms:

  When he left a year earlier, and desired that we see each other once more, he wrote that I was to set the time and place [outside of Göttingen]. I decided on Clausthal; and we appeared punctually at the same time, on foot. The professor of astronomy [Seyffer], who was with Napoleon at Austerlitz and then became his engineer colonel, and others accompanied me by foot to the next village. On leave-taking I wept like a child; I went back against my will, but finally gained control of myself. From the last hill, from which Göttingen was still visible, I looked back once again. That picture of the farewell has remained impressed on me forever.

  I accompanied him [Gauss] in the morning, Saturday, May 25, 1799, to the peak of a small mountain toward Brunswick. That feeling of seeing each other for the last time is indescribable. Even a word about tears is ineffective. The Book of the Future is closed. Then we parted with a dying, farewell handshake, almost without words, with the distinction that he, led by angels of the temple of fame and glory, returns to Brunswick, and I, much less worthy, yet calm with a good conscience, go to Göttingen, although pursued afterward as a martyr of truth by an army of the many less worthy ones.

  The two friends never saw each other again, but later we shall come to their correspondence on scientific and personal matters. On Thursday, May 23, 1799, Ide, to whom Bolyai had just related the episode of his meeting with Gauss in Klausthal, wrote to Gauss:

  If I reckon properly, then our Bolyai arrives [in Göttingen again] at the right time to be able to take some part in the pleasures of the rifle meeting. (Saturday, I believe, the king is led around the house three times with noisy playing and to the sound of bells; afterward follow firecrackers and a riotous parade.) He will in all probability attend this, but only as a philosopher, who on such occasions finds material with which to institute observations on human follies. This is so much his rule of habit, as I have discovered from several cases, that it is difficult for him to miss any of these worldly affairs, not that he wants to enjoy them along with the others, but in order to strengthen his tranquillity of mind. Recently there was a turbulent student uprising again, at which we were accidentally spectators. I went home about ten o’clock because the affair lasted so late into the night, but couldn’t persuade him to go along, not because he would have still been willing to be in the action (for we were both empty-handed), but in order to be able to philosophize even further on the idleness of the deed, which he had done the whole time over and over while I was by him.

  Unfortunately there is extant no picture of Gauss during his student days. The first we have of him (which is reproduced in this book) was made by Schwarz in 1803.

  IV

  —

  The Young Man

  When Gauss returned to his parents in Brunswick on September 28, 1798, at the close of the summer semester, the future was still uncertain. He hoped that the Duke would not withdraw his assistance. Zimmermann did his best to get a promise for him. Shortly after Gauss’ arrival Zimmermann had inquired in writing if the Duke would be willing to talk to his protégé. Months passed without an answer, and no opportunity presented itself. It was not likely that Gauss himself would be able to hasten the decision with a request for an audience, because Carl Wilhelm Ferdinand had suffered from considerable abuse of this indulgent custom and now seldom admitted anyone unless personally summoned. Gauss felt keenly the fact that he could not discuss his case personally with the Duke. He wrote Bolyai on September 30, 1798:

  Brunswick.

  dear bolyai:

  I reached here last Tuesday. On the second day of my trip I had to travel a good while in the rain. This, together with the fact that I left Monday with a half-empty stomach and spent the entire night traveling in the open, though not indeed on foot, brought on a little indisposition from which I have slightly recovered. Now the dear native air has already completely cured me again. Except for Zimmermann, I haven’t yet seen any of my older friends. I am planning to visit the duke in a few days. Thus I know very little that is certain about my future fate: if I meanwhile infer your sentiments from my own, then even this item will not be uninteresting to you. Of my duke I have reason to hope that he will still continue his assistance until I receive a regular job. I lost a very lucrative occupation. A Russian envoy16 is staying here; I was to have instructed his two young, very spirited daughters in mathematics and astronomy. But because I arrived too late, a French immigrant has already taken over the assignment.

  But another affair, very attractive to me, is awaiting me.

  Major General von Stamford,17 whom I have often mentioned to you as an excellent man, intelligent connoisseur, and warm friend of mathematics, desires to go through certain parts of it with me. What parts and on what basis I do not know, because I haven’t visited him yet. I think that this will be adequate for my subsistence, and that almost my entire time will be my own. That is the most important news that I can write you now. My future landlord seems to be a good, honest man: I have seen my room only at night: it appears quite comfortable and suitable to me. In about eight days . . . when my riding trousers are done . . . I plan to take a trip to Helmstedt. Then I hope to be able to write you of my other circumstances. And so, until then, farewell, good Bolyai,

  gauss

  P.S. I have used the Latin letters on account of your eyes and because you once told me that you would rather have them than the Gothik [i.e., cursive].

  My address: Charles Frederic Gauss, Candid. en philosophie (or even nothing at all) at Mr. Schröder’s on Wendenstrasse. I need not say that all tidings from you will be welcome to me.

  Remember me to my acquaintances there: Ide, Simonis, Eichhorn, Seyffer, Lichtenberg, Kaestner, Persoon, or whomsoever you see.

  Because you are in danger of traveling with Murhard,18 I consider it my duty to tell you of his behavior here, which may show you that it is not without danger to join with such a man on a trip. Save for this reason I would not even mention such a name in a letter to you, even as I do not make mention here of a single one of any of his other windy chatterings.

  Here in Brunswick he hires a man who drives his (Murhard’s) carriage to Helmstedt with his own horses. When they get there he asks him (without paying him) if he doesn’t want to come again two days later in order to drive him back to Brunswick again, when he is then to receive double pay. The coachman agrees. When he comes again in two days Murhard has been gone a long time. (En passant: When I went through Nordheim Monday night, Murhard came directly into the taver
n; he appeared very disturbed and said that he had just traveled from Helmstedt to Goslar, and planned to go to Göttingen the next day. What became of him there you probably know already.)

  He goes with Secr. Brückmann,19 whose father’s famous mineralogical cabinet he had inspected, into a loan library and takes out several books. Thereupon, when he is to leave something as security, as is the custom, he calls on his companion, Mr. Brückmann, to identify him. But M. didn’t return the books, and Mr. Brückmann (who doesn’t know M.) has to pay for them.

  On November 29, the following observations occur in another letter to Bolyai:

  I am living for the most part now on credit, because my financial prospects are all shattered. . . . I have been in Helmstedt and found a very good reception with Pfaff as well as with the custodian of the library. Pfaff came up to my expectations. He shows the unmistakable sign of the genius, of not leaving a matter until he has dug it out as far as possible. With great kindness he offered me the use of his library and I am going to write him in a few days to request various books.

  Greet all my acquaintances. But mention to no one what I have told you of my circumstances, and should anyone inquire about the matter, then say that you only know in general that I have good, although not entirely certain, prospects, which in fact is quite true. And visit me as soon as you can.

  gauss.

  An improvement in his circumstances soon came, as the following extracts from a letter of December 30, 1798, to Bolyai show:

  Several favorable changes have occurred in my situation since my last letter. Indeed I have not yet talked with the duke himself, but he has explained that I am to continue receiving the sum which I enjoyed in Göttingen (which amounts to 158 thalers annually and is now fairly adequate to my needs). He desires further that I become a doctor of philosophy, but I am going to postpone it until my Work is done, when I hope I can become one without costs and without the usual harlequinery.

  About the middle of December, 1799, Gauss returned to Helmstedt again in order to use its library. He was cordially received by the librarian Bruns, as well as by the professor of mathematics Johann Friedrich Pfaff (1765–1825), whose acquaintance Gauss had already made during his stay in Helmstedt in October, 1798. He rented a room in Pfaff’s home and furnished it himself, but studied so strenuously and incessantly that the others in the house saw him for only a few hours in the evening. He and Pfaff would then take walks “to the spring and to Harpke.” The topics of conversation were usually mathematical in nature. On such occasions it is believed that Gauss gave out much more than he received.

  After a brief introductory paragraph Gauss wrote to Bolyai on December 16, 1799:

  Do not allow me to spoil very many lines in explaining the cause of my somewhat belated answer [to your last letter]. The chief one is that I did not know for certain until October that I would not yet take my trip to Gotha as intended then; and I desired to be able to write you only when certain what sort of a place I have now temporarily substituted for it. That I can do now; it is Helmstedt, where I arrived a few days ago and from where I write you this letter. I now narrate from there in chronological order.

  You remember that I had already sent in a thesis to the philosophical faculty at Helmstedt, when we saw each other in Clausthal for the last time, as Candidate for the title of doctor. This affair has progressed since then, and the faculty conferred this title on me on July 16 without burdening me with most of the formalities heretofore customary. Our good prince has taken over the costs of it. The paper is printed and was already finished in August. I know no way, at once safe and convenient, of sending it to you, but you will probably be able to get it easily through the bookstore; hence I am writing the complete title. It is: Demonstratio nova theorematis, omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse, auctore Carolo Frederico Gauss, Helmstadii apud C. G. Fleckeisen, 1799, 5 folios in quarto, with a copperplate print. The title indicates quite definitely the chief purpose of the essay; only about a third of the whole, nevertheless, is used for this purpose, the remainder contains chiefly the history and a critique of works on the same subject by other mathematicians (viz. d’AIembert, Bougainville, Euler, de Foncenex, Lagrange, and the encyclopedists . . . which latter, however, will probably not be much pleased) besides many and varied comments on the shallowness which is so dominant in our present-day mathematics.

  Certainly this pamphlet will interest you at least as the first attempt of your friend. To my knowledge, published reviews of it haven’t yet appeared anywhere. Up until now I have distributed some thirty copies, partly to mathematicians, partly to those whom I owed a debt of courtesy. Until now I have lacked an opportunity to send one to France. Of private comments which have come to my knowledge only that of General von Tempelhoff20 in Berlin is especially important to me and gladdened me the more, because he is one of the best German mathematicians, and especially because my criticism touched him as the author of a compendium. From third hand I have found out that he thus passes opinion: (his own words) “that Gauss is an absolutely hopeless mathematician; he doesn’t yield a hand’s breadth of ground; he has fought bravely and well and holds the battlefield completely.” . . .

  Since I shall probably not enter the fetters of a position very soon and had too slight assistance in Brunswick, I made the decision to betake myself here to Helmstedt for a while, and I shall probably remain here until Easter; you can send your letters at your pleasure either here or to Brunswick, for I have arranged that all letters directed to me there are forwarded at once.

  I am staying here at Professor Pfaff’s whom I esteem as an excellent geometer as well as a good man and my warm friend; a man of an innocent, childlike character, without any of the violent emotions which so dishonor a man and are widespread among scholars. Since I haven’t been here eight days, I can’t decide yet how well I shall be pleased otherwise; the place itself is terrible, the environs are praised; one must do without many comforts of life; the tone among the students as a whole is said to be rather rough; among the professors with whom I have become acquainted there are well-bred men.

  You write me of my portrait; I shall certainly send it to you, but for several reasons not now. Also you will prefer a later picture of me, when my looks will have changed, for my present features are, I hope, as fresh in your memory as yours are in mine.

  This letter will hardly reach you this year; tell me in your next one when you received it; the last day of December will at least be the last day which we call seventeen hundred (if micrological exegetes now postpone the end of the century one more year) and will be especially sacred to me. Note that when it is midnight here for us, midnight is already an hour past with you. On such festive occasions my mind passes into a loftier mood, into another spiritual world; the partitions of the room disappear, our filthy, paltry world with everything that appears so big to us, makes us so happy and so unhappy, disappears, and I am an immortal pure spirit united with all the good and noble who adorned our planet and whose bodies space or time separated from mine, and I enjoy the higher life of those greater joys which an impenetrable veil conceals from our eyes until death . . . do not cease to love your constant friend

  C. F. G.

  In 1797 Gauss discovered a new proof of Lagrange’s theorem, and at the close of this year two of his letters addressed to Zimmermann show that he was working out the Disquisitiones arithmeticae. These are probably the first reports of a work which was to put Gauss in the ranks of the greatest mathematicians of all time. The printing of it was begun in Brunswick by the printer Kircher, who had recently acquired ownership of a print shop as an adjunct of the public-school system under the superintendency of Joachim Heinrich Campe. In 1799 he sold it to Friedrich Vieweg, Campe’s son-in-law, in order that he might move back to Goslar, where he also had a printing shop, previously owned by Duncker, which he had gotten in 1783 by marriage. In Goslar he completed the pri
nting of the Disquisitiones, although the work took much longer than Gauss had anticipated. Of unusual interest is the part which Meyerhoff21 took in this book—the correction of the Latin.

  The above is striking enough, if one considers how little Gauss needed to mistrust his own proficiency in this respect. According to Moritz Cantor, Gauss wrote a classical Latin, giving rise to the expression that Cicero, if he could understand the mathematics of it, would have censured nothing in the Gaussian Latinity, except perhaps several customary incorrect modes of expression which Gauss used purposely. But it was Latin just the same and therefore attractive and stimulating to only a narrow circle of readers. Referring to Meyerhoff’s work. Gauss wrote:

  Of course I understand that it cannot be an especially attractive work for Mr. Meyerhoff, since he does not seem to be sufficiently acquainted with mathematics, in order to look on it just as reading. Thus the word algorithmus was unknown to him. Only on a single point must I take the liberty of disagreeing with him. I well know that si with the subjunctive is not good Latin; but modern mathematicians seem to have made for themselves the rule of constantly using the subjunctive in hypotheses and definitions; I do not remember an example of the opposite, and in Huyghens, who according to my notion writes the most elegant Latin and whom I purposely, therefore, have imitated, I find the subjunctive continually in these cases. I open at random and find Opera, p. 156, Quodsi fuerit; p. 157, Si sit, si fiat, si agitetur; p. 158, si suspendatur; pp. 188 seqq. are examples by the dozen. Therefore, since in this instance the desire to be a genuine Roman would be only purism (which as far as I am concerned would be less allowable, because I am well minded not to be so in any case) and the thing is not at all absurd in itself, I went with the current. I hope Mr. M. will not take offense at me. What was incomprehensible to him in the accedere possunt, p. 5. I have not been able to guess; I have therefore let it stand. The passage p. 7, which previously ran thus: Si numeri decadice expressi figurae singulae sine respectu loci quern occupant addantur, Mr. M. misunderstood, because he probably didn’t know that figurae means numbers; he took numeri for the nominative plural and figurae for the dative singular and on that account suggested to me that singulus is not wrong; but just for this reason a mathematician will probably not construe it incorrectly, chiefly because it doesn’t make sense; nevertheless I have now arranged the words somewhat differently by this time.