In the winter of 1811 Gauss was teaching one course for one Student whom he called “most simple.” The previous winter he had reported to Schumacher that he had several very capable students in two courses. The story is told that Gauss once saw a student staggering through the streets of Göttingen. When the student saw the professor approaching, he attempted to straighten himself out, but did not succeed and cut quite a figure. Gauss scrutinized him and threatened by pointing his finger at him, smiling as he said: “My young friend, I wish that science would intoxicate you as much as our good Göttingen beer!”
Concerning the gifted students Gauss expressed himself very clearly in a letter to Schumacher on October 2, 1808:
In my opinion instruction is very purposeless for such individuals who do not want merely to collect a mass of knowledge, but are mainly interested in exercising (training) their own powers. One doesn’t need to grasp such a one by the hand and lead him to the goal, but only from time to time give him suggestions, in order that he may reach it himself in the shortest way.
On May 21, 1843, Gauss wrote to Weber about some of the pitfalls which can trap even one of the greatest mathematicians, and frequently do:
In the last two months I have been very busy with my own mathematical speculations, which have cost me much time, without my having reached my original goal. Again and again I was enticed by the frequently intersecting prospects from one direction to the other, sometimes even by will-o’-the-wisps, as is not rare in mathematical speculations.
Gauss held at least one basic view in common with Goethe; the following passage in a letter to Bolyai on September 2, 1808, might well have been uttered by Faust:
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never satisfied man is so strange—if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms again for others.
Gauss was not by any means indifferent to the practical applications of his research and to the material value of such applications. But he insisted that the evaluation of scientific work should not be based on such standards. He once said that science should be the friend of practicality but not its slave. In the introduction to his work on magnetism he wrote: “Science, even though advancing material interests, cannot be limited by these, but demands equal effort for all elements of its research.”171
Gauss used to maintain that mathematics is far more a science for the eye rather than the ear. The eye surveys something at a glance and comprehends it, owing to the splendid language of signs which have been perfected in mathematics, as in no other science, especially with reference to the operations to be carried out. On the long road of translation of signs into words, this will never be presented to the ear so precisely that it can be made sufficiently accessible to the listener, in order to be pictured by those less familiar with the subject. Thus Gauss was uncommonly painstaking in the choice of designations, definitions, and even in the choice of individual letters. This attitude is shown in the following passage of a letter to Bessel on November 21, 1811:
The sin2ϕ is annoying to me every time, although even Laplaces uses it; if one fears that sin ϕ2 could be ambiguous (which occurs perhaps never or very rarely if one were speaking of sin [ϕ2]). Well now, let one write (sin ϕ)2, but not sin sin 2ϕ, which by analogy ought to mean sin (sin ϕ).
Gauss felt that the expression in the case of equal roots that “an equation of the nth degree has n roots” is a “façon de parler” which one may tolerate. He illustrated this point in a letter to Schumacher on April 30, 1830 (his fifty-third birthday):
In verbis simus faciles; on the other hand I always demand that the mathematician always remain conscious of the things, whereby of course nothing unsuitable can ever be built on the phrase. Our modern mathematics has, with respect to language, a completely different character from that of ancient mathematics; every moment one permits such modes of expression as must be understood cum grano salis.
A letter to Schumacher on September 17, 1808, expressed Gauss’ high hopes in the field of functions. He was a young man then, and unfortunately all of these hopes were never realized. The passage runs thus:
We now know how to operate with circular and logarithmic functions, just as with one times one, but the splendid gold mine which the interior of higher functions contains is still almost terra incognita. Formerly I did much work on the subject and some day shall publish my own major work on it, of which I have already given a hint in my Disquisitiones arithmeticae, p. 593. One is astonished at the superabundant richness in new, highly interesting truths and relations which such functions offer (where among other things belong also those with which the rectification of the ellipse and hyperbola is connected).
In 1811 Gauss stated that there are no true controversies in mathematics. In a letter to Schumacher on October 4, 1849, he expressed more fully his views on this subject: “I have a great antipathy against being drawn into any sort of polemic, an antipathy which is increased with every year for reasons similar to those which Goethe mentions in a letter to Frau von Wolzogen.”
In the reference mentioned, Goethe said that “in advanced years when one must proceed so economically with one’s time, one becomes most annoyed on account of wasted days.’”172
In one of his valuable flashes of insight Gauss declared: “I have the result, only I do not yet know how to get to it.” In this utterance we see above all that he emphasizes a lightninglike intuition. He has possession of a thing, which is, however, not yet his own, and which can only become his own when he has found the way to it. From the point of view of elementary logic, this is certainly contradictory; but methodologically, by no means. Here it is a question of Erwirh es um es zu besitzen! This makes necessary a series of further intuitions along the road of invention and of construction.
Gauss repudiated certain proofs of earlier algebraists as being “not sufficiently rigorous,” and replaced them by more rigorous proofs. This means that even in mathematics, what appears to one investigator as flawless, strict, and evident, is found by another to have gaps and weaknesses. Absolute correctness belongs only to identities, tautologies, that are absolutely true in themselves, but cannot bear fruit. Thus at the foundation of every theorem and of every proof there is an incommensurable element of dogma, and in all of them taken together there is the dogma of infallibility that can never be proved nor disproved. That is Gauss’ view. Today there is common agreement on this point.
Coming to more personal matters, we find that Gauss in 1803, while he was still a bachelor, uttered the dictum that only he who is a father has the full right of citizenship on earth. After the birth of his six children he must certainly have felt that he was a full-fledged citizen. A year earlier he had said that marriage is like a lottery in which there are many blanks and few winning tickets, and continued by expressing the hope that he would not draw a blank. In 1812 Bessel reminded him that he united ideally interest in science and interest in a woman, adding that he hoped to imitate Gauss in this respect. Shortly afterward Gauss wrote these words to Bessel: “Certainly you too will find out that among all the goods of life happiness springs from a well-chosen marriage which is the greatest and purest that crowns all others.”
In 1807 when Gauss calculated the orbit of a planetoid and gave it the name of Vesta, Bessel congratulated him and stated that it was particularly pleasant because it showed to which goddess he offered sacrifices.
Bibliography
1. Publications of Gauss
(This list does not include book reviews and similar notices.)
1. Doctoral dissertation: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in
factores primi vel secundi gradus resolvi posse. Helmstedt: C. G. Fleckeisen, 1799.
2. “Berechnung des Osterfestes,” Monat. Corresp. (ed. Zach), II (1800), 121–130.
3. “Neigung der Bahn der Ceres,” Monat. Corresp. (ed. Zach), IV (1801), 649.
4. Disquisitiones arithmeticae. Leipzig: Gerhard Fleischer, Jr, 1801. French translation: Recherches arithmétiques, by A. C. M. Poulet-Delisle. Paris, 1807. German translation: Untersuchungen liber höhere Arithmetik, by H. Maser. Berlin: Julius Springer, 1889.
5. “Sur la division de la circonférence du cercle en parties égales,” Soc. philom. bull. (Paris), III (1802), 102–103.
6. “Berechnung des jüdischen Osterfestes,” Monat. Corresp. (ed. Zach), V (1802), 435–37.
7. “Vorschriften, um aus der geocentrischen Länge und Breite eines Himmelskörpers, dem Orte seines Knotens, der Neigung der Bahn, der Länge der Sonne und ihrem Abstande von der Erde abzuleiten: des Himmelskorpers heliocentrische Lange in der Bahn, wahren Abstand von der Sonne und wahren Abstand von der Erde,” Monat. Corresp. (ed. Zach), V (1802), 540–546.
8. “Erste Elemente der Pallas,” Monat. Corresp. (ed. Zach), V (1802).
9. “Störungsgleichungen für die Ceres,” Monat. Corresp. (ed. Zach), VI (1802), 387–389.
10. “Tafeln für die Störungen der Ceres,” Monat. Corresp. (ed. Zach), VII (1803), 259–275.
11. “Einige Bemerkungen zur Vereinfachung der Rechnung für die geocentrischen Oerter der Planeten,” Monat. Corresp. (ed. Zach), IX (1804), 385–400.
12. “Ueber die Grenzen der geocentrischen Oerter der Planeten,” Monat. Corresp. (ed. Zach), X (1804), 173–191.
13. “Erste Elemente der Juno,” Monat. Corresp. (ed. Zach), X (1804), 464–552.
14. “Theorematis arithmetici demonstratio nova,” Comment. (Göttingen), XVI (1804–1808), 69–74.
15. “Der Zodiacus der Juno,” Monat. Corresp. (ed. Zach), XI (1805), 225–228.
16. “Ueber den zweiten Cometen von 1805,” Monat. Corresp. (ed. Zach), XIV (1806).
17. “Ephemeride für den Lauf der Ceres,” Monat. Corresp. (ed. Zach), XV (1807), 154–157.
18. “Beobachtungen der Pallas,” Monat. Corresp. (ed. Zach), XV (1807), 377–378.
19. “Erste und zweite Elemente der Vesta,” Monat. Corresp. (ed. Zach), XV (1807), 596–598.
20. “Allgemeine Tafeln für Aberration und Nutation,” Monat. Corresp. (ed. Zach), XVII (1808), 312–317.
21. “Beobachtungen der Juno, Vesta und Pallas,” Monat. Corresp. (ed. Zach), XVUI (1808), 83–86, 173–175, 182–188, 269–273.
22. “Aus zwei beobachteten Höhen zweier Sterne, deren Rectascensionen und Declinationen als gegeben angesehen werden, und den entsprechenden Zeiten der Uhr, die entweder nach Stemzeit geht oder deren Gang während der Beobachtungen als bekannt angenommen wird, den Stand der Uhr und die Polhöhe zu bestimmen,” Monat. Corresp. (ed. Zach), XVIII (1808), 277–293; XIX (1809).
23. “Summatio quarundam serierum singularium,” Comment. (Göttingen), I (1808–1811).
24. “Disquisitio de elementis ellipticis Palladis,” Comment. (Göttingen), I (1808–1811).
25. “Beobachtungen der neuen Planeten,” Monat. Corresp. (ed. Zach), XX (1809), 78–79.
26. Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Hamburg: Perthes and Besser, 1809. English translation by Charles Henry Davis. Boston: Little, Brown and Company, 1857. French translation by Edm. Dubois, 1860. German translation by Carl Haase, Hanover, 1865. There was also a Russian translation.
27. “Summarische Uebersicht der zur Bestimmung der Bahnen der beiden neuen Hauptplaneten angewandten Methoden,” Monat. Corresp. (ed. Zach), XX (1809), 197–224.
28. “Fortgesetzte Nachrichten von dem neuen Hauptplaneten Vesta,” Monat. Corresp. (ed. Zach), XIX (1809), 407–410, 504–515.
29. “Pallas- und Vesta-Beobachtungen,” Monat. Corresp. (ed. Zach), XXI (1810). 276–280.
30. “Bestimmung der grössten Ellipse, welche die vier Seiten eines gegebenen Vierecks berührt,” Monat. Corresp. (ed. Zach), XXII (1810), 112–121.
31. “Elemente der Pallas,” Monat. Corresp. (ed. Zach), XXII (1810) 400–403; XXIII (1811), 97–98; XXIV (1811), 449–465.
32. “Tafeln für die Mittagsverbesserung,” Monat. Corresp. (ed Zach), XXIII (1811), 401–409.
33. “Beobachtungen des Cometen,” Monat. Corresp. (ed. Zach) XXIV (1811), 180–182.
34. “Elemente des zweiten Cometen von 1811,” Monat. Corresp. (ed Zach), XXIV (1811).
35. “Observationes cometae secundi ann. 1813 in observatorio Got tingensi factae, adjectis nonnullis annotationibus circa calculum orbitarum parabolicarum,” Comment. (Göttingen), II (1811–1813); Nouv. Ann. math, XV (1856), 5–17.
36. “Disquisitiones generales circa seriem infinitam
Comment. (Göttingen), II (1811–1813). German translation by Heinrich Simon. Berlin: Julius Springer, 1888.
37. “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata,” Comment. (Göttingen), II (1811–1813). German translation by von Lindenau in Monat. Corresp. (ed. Zach), XXVIII (1813), pp. 37–57, 125–234. Also edited by A. Wangerin as No. 19 in Ostwald’s Klassiker der exakten Wissenschaften.
38. “Neue Methode aus der Höhe zweier Sterne die Zeit und die Polhöhe zu bestimmen,” (1809), Astron. Jahrh. (ed. Bode), 1812, pp. 129–143. German translation by Ludwig Harding of “Methodus peculiaris elevationem poli determinandi.” Göttingen, 1808.
39. “Ueber die Tafel für die Sonnen-Coordinaten in Beziehung auf den Aequator,” Monat. Corresp. (ed. Zach), XXV (1812), 23–36.
40. “Parabolische Elemente des zweiten Cometen von 1811,” Monat. Corresp. (ed. Zach), XXV (1812), 95–97.
41. “Sternbedeckungen,” Monat. Corresp. (ed. Zach), XXV (1812), 206–207.
42. “Pallas-Beobachtungen,” Monat. Corresp. (ed. Zach), XXVI (1812). 199–203; XXVIII (1813), 197–198.
43. “Juno-Beobachtungen,” Monat. Corresp. (ed. Zach), XXVI (1812), 297–299.
44. “Tafel zur bequemern Berechnung des Logarithmen der Summe Oder Differenz zweier Grössen, welche selbst nur durch ihre Logarithmen gegeben sind,” Monat. Corresp. (ed. Zach), XXVI (1812). 498–528.
45. “Ueber Attraction der Sphäroiden,” Monat. Corresp. (ed. Zach), XXVII (1813), 421–431.
46. “Beobachtungen mit einem 12-zolligen Reichenbach’schen Kreise zur Bestimmung der Polhöhe der Göttinger Sternwarte,” Monat. Corresp. (ed. Zach), XXVII (1813), 481–484.
47. “Verzeichniss von Stern-Declinationen,” Monat. Corresp. (ed. Zach), XXVIII (1813), 97–99.
48. “Beobachtungen des zweiten Cometen vom Jahre 1813, angestellt auf der Sternwarte zu Göttingen, nebst einigen Bemerkungen über die Berechnung parabolischer Bahnen,” Monat. Corresp. (ed. Zach), XXVIII (1813), 501–513. “Observationes cometae secundi anni 1813.” German translation by Nicolai in Gött. gel. Anz, 1814, 25.
49. “Achte Opposition der Juno,” Monat. Corresp. (ed. Zach), XXVIII (1813), 574–578.
50. “Nachricht von dem Reichenbach’schen Repetitionskreise und dem Theodolithen,” Gött. gel. Anz, 1813.
51. “Beobachtungen usw. zur Bestimmung der Polhöhe der Göttinger Sternwarte,” Monat. Corresp. (ed. Zach), XXVII (1813).
52. “Methodus nova integralium valores per approximationem inveniendi,” (1814), Comment. (Göttingen), III (1814–1815), 39–76; Nouv. Ann. math, XV (1856), 109–129, 207–211, 315–321.
53. “Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse,” Comment. (Göttingen), III (1814–1815), 107–134, 135–142; Nouv. Ann. math, XV (1856), 134–139.
54. “Ejusdem theorematis demonstratio, tertia,” ibid.
55. “Eigenthümliche Darstellung der Pfaff’schen Integrationsmethode,” Gött. gel. Anz, 1813.
56. “Bestimmung der Genauigkeit der Beobachtungen,” Zeitschr. für Astr. (ed. Lindenau and Bohnenberger), (1816), 185–197.
57. “Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae,” Comment. (Göttingen), IV (1816–1818), 3–20.
58. “Determinatio attractionis, quam in punctum quodvis positionis datae exerceret planeta, si ejus massa per totam orbitam, ratione temporis, que singulae partes describuntur, uniformiter esset dispertita,” Comment. (Göttingen), IV (1816–1818), 21–48.
59. “Ueber die Differenz der Polhöhe, wenn sie aus Sonnenbeobachtungen oder Nordsternbeobachtungen mit dem Multiplications-Kreise abgeleitet wird,” Zeitschr. (ed. Lindenau), IV (1817), 119–131.
60. “Beobachtungen des Polarsterns in der untern Culmination auf der Göttinger neuen Sternwarte,” Zeitschr. (ed. Lindenau), IV (1817), 126–131.
61. “Ueber die achromatischen Doppelobjective besonders in Rücksicht der vollkommnern Aufhebung der Farbenzerstreuung,” Zeitschr. (ed. Lindenau), IV (1817), 345–351; Annal. (ed. Gilbert), LIX (1818), 188–195.
62. “Ueber einige Berichtigungen an Bordaischen Wiederholungskreisen,” Zeitschr. (ed. Lindenau), V (1818), 198–211.
63. “Nachricht von d. Repsold’schen Meridiankreise,” Gött. gel. Anz. (1818).
64. “Cometenbeobachtungen,” Zeitschr. (ed. Lindenau), V (1818), 276–277.
65. “Distances au zénith de quelques étoiles, observées à Göttingen, . . .” Monat. Corresp. (ed. Zach), II (1819), 53–61.
66. “Vom Reichenbach’schen Meridiankreise,” Gött. gel. Anz. (1820), p. 905. English translation by Herschel in the Memoirs of the Astronomical Society, Vol. I.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 47