When Jensen finished the portrait, Therese accompanied him on August 13 on the trip home as far as Hamburg, to visit her brother Joseph and his bride, who were living in Stade at that time.
XVII
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Milestones on the Highways and Byways
Writers on Gauss have frequently commented on the fact that he published relatively little in comparison with what he had discovered. The first reason for this phenomenon lay in the fact that the: magnitude of the mathematical genius revealed in Gauss was rooted in the union of creative and critical power. This characteristic can be recognized in his doctoral dissertation as well as in his Disquisitiones arithmeticae. In his later publications criticism of the accomplishments of others recedes, but Gaussian rigor remains as a distinguishing characteristic.
This rigor is recognized externally in the form of presentation. Gauss always strove to give his research the form of completed works of art, and he never published a work until he had attained the desired form. He used to say that after a structure was completed one should no longer be able to see the scaffolding. This principle was exemplified in the letter seal which he used, a tree with very little fruit and the inscription: Pauca sed matura. In letters to Schumacher, Bessel, and Encke, Gauss explained why he would not and could not deviate from this mode of presentation.
When he again took up his theoretical work in the winter of 1825–1826, after completion of geodetic measurements in the field, he complained thus to Schumacher in a letter dated November 21, 1825: “The desire which I have always had in my works, to give them such a completion ut nihil amplius desiderari possit, makes them extraordinarily difficult for me.”
Schumacher answered him on December 2, 1825:
In reference to your works and the principle ut nihil amplius desiderari possit, I would almost like to wish, and to wish for the good of science, that you did not hold to it so strictly. We would then have more of the infinite richness of your ideas than now, and to me the subject matter seems more important than the most complete form of which the matter is capable. But I write my opinion with trepidation, since you have certainly considered for a long time the pro and con.
On February 12, 1826, Gauss further elaborated his viewpoint:
I was somewhat astonished at your utterance, as though my mistake consists of neglecting the subject matter too much in favor of completed form. During my whole scientific life I have always had the feeling of exactly the opposite, that is, I feel that the form could have been more completed, and that bits of carelessness have remained behind in it. For you do not wish to interpret it, as though I would accomplish more for science, if I were satisfied with furnishing individual building stones, tiles, and so forth, instead of a structure, be it a temple or a hut, since the building to a certain extent is only the form of the bricks. But I do not like to erect a building in which main parts are missing, even if I devote little to the outer ornaments. In no case, however, if you were otherwise right in your objection, does it fit my complaints about the present works, where it is not a matter of what I call subject matter; and likewise I can definitely assure you, that, though I also like to give a pleasing form, this claims comparatively little time and strength or has taken little in earlier works.
When Gauss sent Schumacher a little paper on the heliotrope for the Astronomische Nachrichten, he added (November 28, 1826): “This time I have certainly not deserved the criticism, as though I had conceded too much to form at the cost of subject matter, but rather the opposite.
Schumacher now felt called upon to explain his opinion fully (December 2, 1826): “I believed that another can do this filing just as well, and in that I can have made a mistake; where I have not made a mistake is the claim that you cannot turn over the inventing to another. Every year of your life increases the hints of new ideas understandable only to you. Shall all this be lost?”
Gauss was cold to all such pleas. On August 18, 1832, he wrote to his pupil and friend Encke: “I know that some of my friends wish that I might work less in this spirit, but that will never happen; I can have no real joy in the incomplete, and a work in which I have no joy is only torture to me. May each one work in that spirit which promises him most.”
On January 15, 1827, he wrote Schumacher that he was far advanced in working out his memoir on curved surfaces:
I find many difficulties in it, but that which one could rightfully call filing or form is in no wise what delays considerably (if I except the inflexibility of the Latin language), rather it is the inner concatenation of truths in their coherence, and such a work is not successful until the reader no longer recognizes the great effort which occurred in the execution. Therefore I cannot deny that I have no really clear idea of how I could carry out my work of such kind differently than I am accustomed, without, as I have already expressed myself, furnishing building stones instead of a building. Sometimes I have tried merely to bring to the public hints about this or that subject; either they are heeded by nobody or, as, for example, in some utterances in a book review, G. G. Anz, 1816, p. 619,58 they were afterward besmirched with mud. Therefore, in so far as the discussion is about important subjects, something essentially complete or nothing at all.
Such hints as those mentioned above are numerous in the works of Gauss’ youth, and are not entirely missing in his later works. The letter from Gauss to Encke on August 18, 1832, already quoted, shows how careful he was with his hints:
It has always been my conscientiously followed principle not to make such hints, which attentive readers find in every one of my writings in large quantity, until I have mastered the subject for myself (for example, see my Disquis. Arithmet, page 593, art. 335).
It was a noteworthy accident that Gauss, soon after he had complained to Schumacher about the failures of his hints, received through his friend on July 24 and August 14, 1827, the two letters of Jacobi with which his research on elliptic functions began, and that not long afterward he got acquainted with Abel’s researches which anticipated “probably a third” of his own. The influence which the famous passage in Article 335 of the Disquisitiones arithmeticae had on Abel and Jacobi is well known. Here Gauss sowed the seed which bore much fruit, and other hints did not fall on stony soil.
Almost a quarter of a century later the same point of debate between the two friends turned up again when Schumacher printed in his Astronomische Nachrichten a work of Jacobi on Kepler’s equation. On December 5, 1849, Schumacher replied to Gauss: “If I did not know how much time the final polishing of your work costs you, I would ask for your paper.”
Gauss then replied that he was inclined to use a part of his leisure time for working out a paper on the subject; but it would demand considerable time to execute the whole theory in a form satisfactory to himself. He continued (February 5, 1850):
You are entirely in error if you believe I mean by that only the last polishing in relation to language and elegance of presentation. These cost comparatively only an unimportant expenditure of time; what I mean is the inner completeness. In many of my works are such points of incidence which have cost me years of meditation, and in whose later presentation concentrated in a small space nobody noted the difficulty which must first be conquered.
Similar utterances are in the letter to Bessel dated February 28, 1839; Bessel supported in a letter to Gauss dated June 28, 1839, the same viewpoint expressed by Schumacher.
The completed presentation, in which Newton and Archimedes were Gauss’ models, was to be only the external sign of inner completeness. Here the word of Gaussian rigor gains its true meaning. In contrast to the habits of the eighteenth century. Gauss adopted what he called the rigor apud veteres consuetus or rigor antiquus. On September 1, 1850, he thus expressed his conviction to Schumacher:
It is the character of mathematics of modern times (in contrast to antiquity) that through our language of signs and nomenclature we possess a lever whereby the most complicated arguments are reduced to a certain
mechanism. Science has thereby gained infinitely, but in beauty and solidity, as the business is usually carried on, has lost so much. How often that lever is applied only mechanically, although the authorization for it in most cases implied certain silent hypotheses. I demand that in all use of calculation, in all uses of concepts, one is to remain always conscious of the original conditions, and never regard as property all products of the mechanism beyond the clear authorization. The usual course is, however, that one claims for analysis the character of generality and expects of the other person who does not recognize the results produced as proved that he should prove the opposite. But one must expect this of him who for his part maintains that a result is wrong, but not of him who recognizes a result as not proved, a result which rests on a mechanism whose original, essential conditions are not pertinent in the case under consideration.
A second reason for the discrepancy between Gauss’ richness of new thoughts and his relatively small number of publications during his lifetime lay in inhibitions which, owing to his mode of working, stood in the way of preparing to go to press.
In the above-mentioned letter of February 28, 1839, to Bessel Gauss stressed with unusual vigor of tone that he needed for working out his research “time, more time, much more time than you can imagine. And my time is often limited, very limited.” Such complaints about lack of time for theoretical research are continually repeated in his letters. Probably the happiest years of his life were those from 1799 to 1807 when he was living as a private scholar on the bounty of his duke. In his old age he thought of these years with emotion and gratitude. Thus on February 15, 1845, he wrote to Encke about young Gotthold Eisenstein, who was pursuing mathematical research supported by the King of Prussia:
He is still living in the happy time when he can yield completely to his talent, without finding it necessary to be disturbed by something foreign. I am vividly reminded of the—long past—years when I lived in similar circumstances. On the other hand just the purely mathematical speculations demand an unencroached upon and undivided time.
The duties of a professorship weighed heavily on Gauss, especially his position as director of the Göttingen observatory. This is reflected by his words in a letter of June 28, 1820, to Bessel: “As much as I love astronomy, I feel nevertheless the burdens of the life of a practical astronomer, often too much, but most painfully in the fact that I can hardly come to any coherent major theoretical work.”
In 1821 were added the geodetic measurements; the tedious and time-consuming field work ended in 1825, but for twenty more years much of Gauss’ time was occupied by purely mechanical calculating which should have been done by a person of lesser creative ability.
Then came the duty of holding lectures for his students. His real desire was to be an astronomer at some observatory and to give much of his time to pure research. The optimum arrangement, according to Gauss, would be for a professor to lecture to his students on the research in which he was engaged at the moment. He disliked administrative duties, red tape, and the “business” of being a professor.
Gauss performed his duties with his usual conscientiousness, but in a letter to Bessel on January 27, 1816, he called the giving of courses “a very burdensome, ungrateful business.” His complaints became very bitter when the burden of geodetic work was added. In 1824 when he was considering a call to Berlin, he cried out in a letter to Bessel (March 14, 1824):
I am so far removed here from being master of my time. I must divide it between giving courses (against which I have always had an antipathy, which, although it did not originate in, is increased by the ever-present feeling that I am throwing away my time) and practical astronomical work. What remains to me for such works on which I could place a higher value, except fleeting leisure hours? A character different from mine, less sensitive to unpleasant impressions, or I myself, if many things were different from what they are, would perhaps gain more from such leisure hours than I generally can.
Such passages in letters to intimate friends could be multiplied considerably. One more, found in a letter to Olbers on February 19, 1826, will suffice:
Independence, that is the great watchword for deep intellectual work. But when I have my head full of mental images hovering in the air, when the hour approaches that I must teach courses, then I cannot describe to you how exhausting for me the digression, the animation of heterogeneous ideas, is, and how hard for me things often become, which under other circumstances I would consider a miserable ABC work. Meanwhile, dear Olbers, I do not want to tire you out with complaints about things which are not to be changed; my whole position in life would have to be a different one if such adversities did not occur rather often.
From the above statements it is perfectly clear that the progress of Gauss’ theoretical work was hindered not merely by lack of leisure time, but also by his constitution of mind. Of course, this is to be tied up with the tragedies which struck Gauss several times in his domestic life.
The sensitiveness to unpleasant impressions, of which he spoke in his letter to Bessel on March 14, 1824, must have contributed to the fact that in his publications he avoided touching on subjects which could have led to debate. In his doctoral dissertation he was cautious about the use of imaginaries59 and their geometric interpretation, which he possessed before 1799 but suppressed and did not publish until 1831. The same delay occurred with respect to his work in non-Euclidean geometry.
This shyness was strengthened by his unpleasant experiences in 1816 when he reviewed the parallel theories of Schwab and Metternich and hinted at the impossibility of proving Euclid’s eleventh axiom. He had such attacks in mind when he wrote to Gerling on August 25, 1818: “I am glad that you have the courage to express yourself (in your textbook), as though you recognized the possibility that our theory of parallels, consequently all our geometry, might be false. But the wasps, whose nest you stir up, will fly around your head.”
In addition there was the low opinion which Gauss had of the great majority of mathematicians. As early as December 16, 1799, he wrote to Wolfgang Bolyai, who had sent him an attempt to prove the axiom of parallels:
Publish your work soon; certainly you will not reap for it the thanks of the general public (to which belongs many a one who is considered an able mathematician), for I am convinced more and more that the number of true geometers is extremely small and most of them can neither judge the difficulties in such works nor even understand them—but certainly the thanks of all those whose judgment can be really valuable to you alone.
When Bolyai sent Gauss in 1832 his son Johann’s work on non-Euclidean geometry, in which the problem of parallels was solved, he got this reply from Gauss (March 6, 1832): “Most people do not have the right sense of what is involved, and I have found only few people who received with special interest that which I communicated to them. In order to do that one must have felt quite vividly what is really missing, and most people are in the dark about that.”
In a letter to Gerling on June 25, 1815, Gauss expressed himself even more severely: “It seems to me that it is important in more than one respect to keep awake in pupils the sense of rigor, since most people are too inclined to pass over to a lax observance. Even our greatest mathematicians have mostly somewhat dull feelers in this respect.”
On September 29, 1837, Gauss expressed to his young friend Möbius his low opinion of the intelligence of the general public: “One must always consider that, where the readers for whom one is writing take no offense, it would perhaps not be beneficial to penetrate more deeply than is profitable for them.”
In 1831 even Gauss’ pupil and intimate friend Schumacher believed he had proved the axiom of parallels, and Gauss had great difficulty in convincing him of the weakness of his process. Bessel could not find the flaw in Schumacher’s work, but merely bowed to the authority of Gauss. Schumacher took no offense and recognized the importance of Gauss’ motto and method of Working when he wrote him on May 24, 1839: “The pauc
a sed datura are here,60 golden apples of the Hesperides, ripened under the sun of genius, one of which has more value than a shipload of Borstorf [apples].”
Bessel urged Gauss to hasten the publication of his results in these words (January 4, 1839): “Mr. von Boguslawski61 has told me that you have reached a point satisfactory to yourself in your research on terrestrial magnetism. I know the meaning of this word and therefore wish you good luck for the most complete success, also cherish the hope that you will no longer leave it in your exclusive possession.”
Concerning his writing on terrestrial magnetism and the magnetometer. Gauss wrote to Schumacher on July 9, 1835:
As you know, I work slowly, most slowly in such (semipopular) subjects; I am almost ashamed to say how long I have been writing on these few pages.
I would think that now a nonmathematical scholar (for they are often difficult as to ideas), no less than, for example, an intelligent carpenter who has had only a little schooling, or to whom one has explained abbreviations like square and such, would have to be able to comprehend the subject.
In 1825 Olbers wrote to Bessel and praised him for prompt publication of his results. At the same time he indulged in some criticism of Gauss, who, he thought, always wanted to pluck the finest fruits to which his path had led him, before showing them to others. Olbers considered this a weakness of character, since with his wealth of ideas Gauss had so much to give away.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 25