Practical Astronomy (privatissime)
Summer 1819
The Use of the Calculus of Probabilities in Applied Mathematics
Theoretical Astronomy
Winter 1819
Theory of Astronomy
Theory of the Movement of Comets
Practical Astronomy (privatissime)
Summer 1820
Theoretical Astronomy
Principal Theories of Astronomical Calculation
Winter 1820
Theory of the Movement of Planets and Comets
Practical Astronomy (privatissime)
Summer 1821
(on leave of absence for geodetic survey)
Winter 1821
Theory of the Movement of Comets
Practical Astronomy (privatissime)
Summer 1822
(on leave of absence for geodetic survey)
Winter 1822
Theory of the Movement of Comets
Practical Astronomy (privatissime)
Summer 1823
(on leave of absence for geodetic survey)
Winter 1823
Use of the Calculus of Probabilities in Applied Mathematics
Practical Astronomy (privatissime)
Summer 1824
(on leave of absence for geodetic survey)
Winter 1824
(not announced in catalogue)
Summer 1825
(on leave of absence for geodetic survey)
Winter 1825
Theory of the Motion of Comets
Practical Astronomy (privatissime)
Summer 1826
Theory of the Motion of Celestial Bodies
Practical Astronomy (privatissime)
Winter 1826
Use of the Calculus of Probabilities in Applied Mathematics
Practical Astronomy (privatissime)
Summer 1827
General Theory of Curved Surfaces
Practical Astronomy (privatissime)
Winter 1827
Use of Calculus of Probabilities in Applied Mathematics
Summer 1828
Theory of the Movement of Comets
Winter 1828
Instruments, Observations, and Calculations Used in Higher Geodesy
Practical Astronomy (privatissime)
Summer 1829
Theory of the Movements of Comets
Instruments, Observations, and Calculations Used in Higher Geodesy
Winter 1829
Use of Calculus Probabilities in Applied Mathematics
Practical Astronomy (privatissime)
Summer 1830
Theory of the Movements of Comets
Instruments, Observations, and Calculations Used in Higher Geodesy
Winter 1830
Calculation of Perturbations of Planets and Comets
Practical Astronomy (privatissime)
Summer 1831
Theory of the Movements of Planets and Comets
Instruments of Geodesy
Winter 1831
Use of Calculus of Probabilities in Applied Mathematics, Especially Astronomy, Geodesy, and Crystallography
Practical Astronomy (privatissime)
Summer 1832
Theory of Motion of Planets and Comets Instruments of Geodesy
Winter 1832
Theory and Practice Observing Magnetic Phenomena
Practical Astronomy (privatissime)
Summer 1833
Theory of Numerical Equations
Instruments of Geodesy
Winter 1833
Use of Calculus of Probabilities
Practical Astronomy
Summer 1834
Practical Astronomy
Winter 1834
Same as Winter, 1833–1834
Summer 1835
Practical Astronomy
Winter 1835
Method of Least Squares and Its Application to Astronomy, Higher Geodesy, and Natural Science
Practical Astronomy
Summer 1836
Theory of Observation of Magnetic Phenomena
Practical Astronomy
Winter 1836
Method of Least Squares
Practical Astronomy
Summer 1837
Theory of Observation of Magnetic Phenomena
Practical Astronomy
Winter 1837
Method of Least Squares
Practical Astronomy
Summer 1838
Theory of Observation of Terrestrial Magnetic Phenomena
Practical Astronomy
Winter 1838
Method of Least Squares
Practical Astronomy
Summer 1839
Same as 1838
Winter 1839
Same as Winter, 1836–1837
Summer 1840
Special Topics in Dynamics
Practical Astronomy
Winter 1840
Practical Astronomy
Summer 1841
Practical Astronomy with Preliminary
Explanation of Principles of Dioptrics (privatissime)
Winter 1841
Same as winter, 1835–1836
Summer 1842
Practical Astronomy
Winter 1842
Same as winter, 1835–1836
Summer 1843
Practical Astronomy
Winter 1843
Same as winter, 1835–1836
Summer 1844
Instruments, Measurements, and Calculations of Higher Geodesy
Practical Astronomy
Winter 1844
Same as winter, 1835–1836
Summer 1845
Same as summer, 1844
Winter 1845
Same as winter, 1835–1836
Summer 1846
Same as summer, 1844
Winter 1846
Same as winter, 1835–1836
Summer 1847
Same as summer, 1844
Winter 1847
Same as winter, 1835–1836
Summer 1848
Same as summer, 1844
Winter 1848
Same as winter, 1835–1836
Summer 1849
Same as summer, 1844
Winter 1849
Same as winter 1835–1836
Summer 1850 to Winter 1854
Same as winter, 1835–1836
Appendix I — Doctrines, Opinions, Theories, and Views
In his memoir Erdmagnetismus und Magnetometer Gauss gave a definition on which he based his work; at least he showed the principle on which he operated: “By explanation the scientist understands nothing except the reduction to the least and simplest basic laws possible, beyond which he cannot go, but must plainly demand them; from them’ however he deduces the phenomena absolutely completely as necessary.”166
How clearly he adhered to this austere ideal is shown in the following passage of a letter to Schumacher dated November 7, 1847:
In general I would be cautious against . . . plays of fancy and would not make way for their reception into scientific astronomy, which must have a quite different character. Laplace’s cosmogenic hypotheses belong in that class. Indeed, I do not deny that I sometimes amuse myself in a similar manner, only I would never publish such stuff. My thoughts about the inhabitants of celestial bodies, for example, belong in that category. For my p
art I am (contrary to the usual opinion) convinced (which in such things one calls conviction) that the larger the cosmic body, the smaller are the inhabitants and other products. For example, on the sun trees, which in the same ratio would be larger than ours, as the sun exceeds the earth in magnitude, would not be able to exist, for on account of the much greater weight on the surface of the sun, all branches would of themselves break off, in so far as the materials are not of a sort entirely heterogenous with those of the earth.
In a memoir on mechanics Gauss gave another example of how closely he followed the principle of exact definition:
As is well known the principle of virtual velocities transforms all statics into a mathematical assignment, and by Dalembert’s principle for dynamics the latter is again reduced to statics.
Although it is very much in order that in the gradual training of science and in the instruction of the individual the easier precedes the more difficult, the simpler precedes the more complicated, the special precedes the general, yet the mind, once it has arrived at the higher standpoint, demands the reverse process whereby all statics appears only as a very special case of mechanics.167
Gauss gave his views on certain extensions of mathematics in a letter to Schumacher dated May 15, 1843:
All . . . new systems of notation are such that one can accomplish nothing by means of them which would also not be accomplished without them; but the advantage is that when such a system of notation corresponds to the innermost essence of frequently occurring needs, each one who has entirely made it his own, even without the equally unconscious inspirations of the genius, which nobody can conquer, can solve the problems belonging in that category, indeed can mechanically solve them, just as mechanically in cases so complicated that without such an aid even the genius becomes powerless. Thus it is with the invention of calculating by letters in general; thus it was with the differential calculus; thus it is also (even though in partial spheres) with Lagrange’s calculus of variations, with my calculus of congruences, and with Möbius’ (barycentric) calculus. Through such conceptions innumerable problems which are otherwise isolated and every time demand new (minor or major) efforts of the spirit of invention, become equally an organic realm.
A letter to Schumacher on September 1, 1850, contains Gauss’ ideas on convergent and divergent series:
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 by which the most complicated arguments can be reduced to a certain mechanism. Thereby science has won infinitely in richness, in beauty, and in solidity, but, as the business is usually carried on, has lost just as much. How often that lever is applied only mechanically, although the authorization for it in most cases implies certain tacit hypotheses. I demand that in all use of the system of notation, in all uses of a concept one shall remain conscious of the original conditions, and never regard as (one’s) property any products of the mechanism beyond the clear authorization. But the usual course is that one claims for the analysis a character of generality and expects of the other of the results thus produced, not yet recognized as proved, that it shall prove the opposite. But one may make this demand only of the one who for his part maintains that a result is wrong, but not of the one who recognizes as unproved a result which rests on a mechanism whose original essential conditions do not tally in the present case. Thus it is very often with divergent series. Series have a clear meaning when they converge; this clarity of meaning disappears with this condition, and nothing is essentially changed whether one uses the word sum or value. . . . Instead of the above comparison of a machine take that of paper money. This can be used advantageously for great works, but the usage is sound, if I am not mistaken, in being able to convert it at any moment into hard cash.
As a footnote to the above remarks we should add here a sentence found in a letter to Bessel dated May 5, 1812: “As soon as a series ceases being convergent, its sum as sum has no meaning.”
The mathematician Georg Cantor called the following passage in a letter from Gauss to Schumacher (July 12, 1831) a horror infiniti and a sort of nearsightedness: “The use of an infinite magnitude (quantity) as a completed one is never permitted in mathematics. The infinite is only a façon de parler, while one really speaks of limits which certain ratios approach as closely as one desires, while others are permitted to increase without limitation.”
Writing to his friend Bolyai on September 2, 1808, Gauss touched on his favorite study: “It is always noteworthy that all those who seriously study this science [the theory of numbers] conceive a sort of passion for it.”
Certainly the reason for his preference just mentioned can be found in a letter of Gauss to Schumacher on September 17, 1808: “I have the vagary of taking a lively interest in mathematical subjects only where I may anticipate ingenious association of ideas and results recommending themselves by elegance or generality.”
In a letter to Olbers on March 21, 1816, Gauss gave vent to some views about the famous theorem of Fermat:
I confess that the Fermat theorem as an isolated one has little interest for me, for a multitude of such theorems can easily be set up, which one can neither prove nor disprove. But I have been stimulated by it to bring out again several old ideas for a great extension of the theory of numbers. Of course this theory belongs to the things where one cannot predict to what extent one will succeed in reaching obscurely hovering distant goals. A happy star must also rule, and my situation and so manifold distracting affairs of course do not permit me to pursue such meditations as in the happy years 1796–1798 when I created the principal topics of my Disquisitiones arithmeticae. But I am convinced that if good fortune should do more than I may expect, and make me successful in some advances in that theory, even the Fermat theorem will appear in it only as one of the least interesting corollaries.168
The following passage in a letter to Schumacher dated January 1–5, 1845, is Gauss’ answer to a charge frequently made against mathematicians:
It may be true that people who are merely mathematicians have certain specific shortcomings; however that is not the fault of mathematics, but is true of every exclusive occupation. Likewise a mere linguist, a mere jurist, a mere soldier, a mere merchant, and so forth. One could add to such idle chatter that when a certain exclusive occupation is often connected with certain specific shortcomings, it is on the other hand almost always free of certain other specific shortcomings.
It has often been alleged that Gauss actually did not like to teach, but this statement needs some qualification and clarification. The truth is that Gauss did not enjoy giving elementary instruction, and unfortunately he found most of his students poorly prepared for advanced work. He felt that the most desirable situation would be to lecture on research in which one was engaged at the time. The following passage in a letter to Olbers dated October 26, 1802, gives a rather full explanation of his views on this matter:
I have a true aversion to teaching. The perennial business of a professor of mathematics is only to teach the ABC of his science; most of the few pupils who go a step further, and usually, to keep the metaphor, remain in the process of gathering information, become only Halbwisser,169 for the rarer talents do not want to have themselves educated by lecture courses, but train themselves. And with this thankless work the professor loses his precious time. At the home of my excellent friend Pfaff, with whom I lived several months,170 I saw how few fragmentary hours for his own work he has left from the public and private lectures, the preparations for them and from other occupations connected with the office of a professor. Experience also seems to corroborate this. I know of no professor who really would have done much for science, other than the great Tobias Mayer, and in his time he rated as a bad professor. Likewise, as our friend Zach has often noted, in our days those who do the best for astronomy are not the salaried university teachers, but so-called dilettanti, physicians, jurists, and so forth.
A
nd in that attitude, if the colors should perhaps be somewhat too dark, I would infinitely prefer the latter, rather than the former, if I had the choice of only two. With a thousand joys I would accept a nonacademic job for which industriousness, accuracy, loyalty, and such are sufficient without specialized knowledge, and which would give a comfortable living and sufficient leisure, in order to be able to sacrifice to my gods. For example, I hope to get the editing of the census, the birth and death lists in local districts, not as a job, but for my pleasure and satisfaction, to make myself somewhat useful for the advantages which I enjoy here.
Writing to Bessel on December 4, 1808, Gauss told what his difficulties were:
To the distracting occupations belong especially my lecture courses which I am holding this winter for the first time, and which now cost much more of my time than I like. Meanwhile I hope that the second time this expenditure of time will be much less, otherwise I would never be able to reconcile myself to it, even practical (astronomical) work must give far more satisfaction than if one brings up to B a couple more mediocre heads which otherwise would have stopped at A!
Again to Bessel he wrote on January 7, 1810: “This winter I am teaching two courses for three listeners, of whom one is only moderately prepared, one scarcely moderately prepared, and the third lacks preparation as well as ability. Those are the onera of a mathematical profession.”
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 46