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Who said: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
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Home > Dictionary of Science Quotations > Scientist Names Index G > Carl Friedrich Gauss Quotes

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Carl Friedrich Gauss
(30 Apr 1777 - 23 Feb 1855)

German mathematician who transformed nearly all areas of mathematics, and contributed much to other areas of science.


Science Quotes by Carl Friedrich Gauss (51 quotes)

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... we must first base such words as “between” upon clear concepts, a thing which is quite feasible but which I have not seen done.
— Carl Friedrich Gauss
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 83.
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...durch planmässiges Tattonieren.
(... through systematic, palpable experimentation.)
Response, when asked how he came upon his theorems.
— Carl Friedrich Gauss
Quoted in A.L. Mackay, Dictionary of Scientific Quotations (1994).
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..und Juwele wägt man nicht mit der Krämerwaage
... and jewels are not weighed on a grocery scale.
Comment on Dirichlet's publication as being not prolific, but profound.
— Carl Friedrich Gauss
Letter to Alexander von Humbolt (9 Jul 1845). In Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, 267. By
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Pauca sed matura.
Few, but ripe.
— Carl Friedrich Gauss
His motto. He would limit his publications to work he regarded as complete and perfect.
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A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
— Carl Friedrich Gauss
Quoted in H. Eves, Mathematical Circles (1977) .
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As is well known the principle of virtual velocities transforms all statics into a mathematical assignment, and by D'Alembert's principle for dynamics, the latter is again reduced to statics. Although it is is very much in order that in gradual training of science and in the instruction of the individual the easier precedes the more difficult, the simple precedes the more complicated, the special precedes the general, yet the min, once it has arrived at the higher standpoint, demands the reverse process whereby all statics appears only as a very special case of mechanics.
— Carl Friedrich Gauss
Collected Works (1877), Vol. 5, 25-26. Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 412.
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Ask her to wait a moment. I am almost done.
When told, while working, that his wife was dying.
— Carl Friedrich Gauss
Quoted in E.T. Bell, Men of Mathematics, (1937).
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Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns.
— Carl Friedrich Gauss
In Letter to Bolyai (30 Jun 1803), in Franz Schmidt and Paul Stäckel, Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai, (1899), Letter XXIII , 55.
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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.
— Carl Friedrich Gauss
From his memoir 'Erdmagnetismus und Magnetometer' in Collected Works (1877), Vol. 5, 315-316. Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 411.
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Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
— Carl Friedrich Gauss
Quoted in H. Eves, Mathematical Circles Squared, (1972).
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For three days now this angel, almost too heavenly for earth has been my fiancée … Life stands before me like an eternal spring with new and brilliant colours. Upon his engagement to Johanne Osthof of Brunswick; they married 9 Oct 1805.
— Carl Friedrich Gauss
Letter to Farkas Wolfgang Bolyai (1804). Quoted in Stephen Hawking, God Created the Integers: The Mathematical Breakthroughs (2005), 567.
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God does arithmetic.
— Carl Friedrich Gauss
Attributed. Quoted in A.L. Mackay, A Dictionary of Scientific Quotations (1991), 100.
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I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.
— Carl Friedrich Gauss
Quoted in J Koenderink, Solid Shape (1990).
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I am coming more and more to the conviction that the necessity of our geometry cannot be proved, at least neither by, nor for, the human intelligence … One would have to rank geometry not with arithmetic, which stands a priori, but approximately with mechanics.
— Carl Friedrich Gauss
From Letter (28 Apr 1817) to Olbers, as quoted in Guy Waldo Dunnington, Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work (1955), 180.
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I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.
— Carl Friedrich Gauss
Letter to Friedrich Bessel (4 Dec 1808). In Gauss-Bessel Briefwechsel (1880), 107. In Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 158.
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I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
— Carl Friedrich Gauss
From Letter (Mar 1816), replying to Olbers’ attempt to entice him to work on Fermat’s Theorem. In James R. Newman (ed.) The World of Mathematics (1956), Vol. 1, 312-313.
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I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, for a multitude of such theorems can easily be set up, which one could neither prove nor disprove. But I have been stimulated by it to bring our 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 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.
In reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. The hope Gauss expressed for his success was never realised.
— Carl Friedrich Gauss
Letter to Heinrich Olbers (21 Mar 1816). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 413.
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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 [one who has superficial knowledge of the subject], 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.
— Carl Friedrich Gauss
Letter to Heinrich Olbers (26 Oct 1802). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 414.
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I have had my results for a long time: but I do not yet know how I am to arrive at them.
— Carl Friedrich Gauss
Quoted in A. Arber, The Mind and the Eye (1954).
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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.
— Carl Friedrich Gauss
Letter to Heinrich Schumacher (17 Sep 1808). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 416.
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I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.
— Carl Friedrich Gauss
Quoted in G. Simmons, Calculus Gems (1992).
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If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
— Carl Friedrich Gauss
As quoted, without citation, in Eric Temple Bell, Men of Mathematics (1945), 254.
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In general I would be cautious against … plays of fancy and would not make way for their reception into scientific astronomy, which must have quite a 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 the stuff. My thoughts about the inhabitants of celestial bodies, for example, belong in that category. For my part, I am (contrary to the usual opinion) convinced … 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 break themselves off, in so far as the materials are not of a sort entirely heterogeneous with those on earth.
— Carl Friedrich Gauss
Letter to Heinrich Schumacher (7 Nov 1847). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 411.
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In general the position as regards all such new calculi is this That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able—without the unconscious inspiration of genius which no one can command—to solve the respective problems, yea, to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange’s calculus of variations, with my calculus of congruences, and with Möbius’s calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.
— Carl Friedrich Gauss
Letter (15 May 1843) to Schumacher, collected in Carl Friedrich Gauss Werke (1866), Vol. 8, 298, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914), 197-198. From the original German, “Überhaupt verhält es sich mit allen solchen neuen Calculs so, dass man durch sie nichts leisten kann, was nicht auch ohne sie zu leisten wäre; der Vortheil ist aber der, dass, wenn ein solcher Calcul dem innersten Wesen vielfach vorkommender Bedürfnisse correspondirt, jeder, der sich ihn ganz angeeignet hat, auch ohne die gleichsam unbewussten Inspirationen des Genies, die niemand erzwingen kann, die dahin gehörigen Aufgaben lösen, ja selbst in so verwickelten Fällen gleichsam mechanisch lösen kann, wo ohne eine solche Hülfe auch das Genie ohnmächtig wird. So ist es mit der Erfindung der Buchstabenrechnung überhaupt; so mit der Differentialrechnung gewesen; so ist es auch (wenn auch in partielleren Sphären) mit Lagranges Variationsrechnung, mit meiner Congruenzenrechnung und mit Möbius' Calcul. Es werden durch solche Conceptionen unzählige Aufgaben, die sonst vereinzelt stehen, und jedesmal neue Efforts (kleinere oder grössere) des Erfindungsgeistes erfordern, gleichsam zu einem organischen Reiche.”
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In mathematics there are no true controversies. (1811)
— Carl Friedrich Gauss
This quote is usually seen without any specific source citation. The sense of it is given, not in quotation marks, as “In 1811 Gauss stated that there are no true controversies in mathematics,” in G. Waldo Dunnington, Jeremy Gray and Fritz-Egbert Dohse, Carl Friedrich Gauss: Titan of Science (2003), 418. If you know the primary source, please contact Webmaster.
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In my opinion instruction is very purposeless for such individuals who do no 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.
— Carl Friedrich Gauss
Letter to Heinrich Schumacher (2 Oct 1808). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 416.
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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 interesting prospects from one direction to the other, sometimes even by will-o'-the-wisps, as is not rare in mathematic speculations.
— Carl Friedrich Gauss
Letter to Ernst Weber (21 May 1843). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 416.
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It is always noteworthy that all those who seriously study this science [the theory of numbers] conceive a sort of passion for it.
— Carl Friedrich Gauss
Letter to Jonos Boyai (2 Sep 1808). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 413.
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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 for others.
— Carl Friedrich Gauss
Letter to Farkas Wolfgang Bolyai (2 Sep 1808). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 416.
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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 such idle chatter that when a certain exclusive occupation is often connected with certain specific shortcomings, it is on the other hand always free of certain other shortcomings.
— Carl Friedrich Gauss
Letter to Heinrich Schumacher (1-5 Jan 1845). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 414.
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It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.
— Carl Friedrich Gauss
In Gauss-Schumacher Briefwechsel, Bd. 4, (1862), 387.
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Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.
— Carl Friedrich Gauss
Quoted in E.T. Bell, The Development of Mathematics (1945).
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Mathematics is concerned only with the enumeration and comparison of relations.
— Carl Friedrich Gauss
Quoted in E. T. Bell, The Development of Mathematics (1945).
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Mathematics is the queen of the sciences and arithmetic [number theory] is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations, she is entitled to first rank.
— Carl Friedrich Gauss
I>Sartorius von Waltershausen: Gauss zum Gedächtniss (1856), 79. Quoted in Robert Edouard Moritz, Memorabilia Mathematica (1914), 271.
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My young friend, I wish that science would intoxicate you as much as our good Göttingen beer! Upon seeing a student staggering down a street.
— Carl Friedrich Gauss
Attributed. Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 416.
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Sophie Germain proved to the world that even a woman can accomplish something in the most rigorous and abstract of sciences and for that reason would well have deserved an honorary degree.
— Carl Friedrich Gauss
Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 68.
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That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
— Carl Friedrich Gauss
Theoria Residiorum Biquadraticorum, Commentario secunda', Werke (1863), Vol. 2. Quoted in Robert Edouard Moritz, Memorabilia Mathematica (1914), 282.
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The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounters infinitely more obstacles that an man in familiarizing herself with complicated problems, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, without doubt she must have the noblest courage, quite extraordinary talents and superior genius.
— Carl Friedrich Gauss
in a letter to Sophie Germain (c.April 1807)
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The knowledge whose content makes up astronomy is the gain from more than 2,000 years’ work on one of the most abundant objects of human knowledge, in which the foremost minds of all times have summoned up all the resources of genius and diligence.
— Carl Friedrich Gauss
From 'Inaugural Lecture on Astronomy', collected in G. Waldo Dunnington (ed.), Carl Friedrich Gauss: Inaugural Lecture on Astronomy and Papers on the Foundations of Mathematics (1937), 49.
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The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.
— Carl Friedrich Gauss
Disquisitiones Arithmeticae (1801), Article 329
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Theory attracts practice as the magnet attracts iron.
— Carl Friedrich Gauss
Attributed. Quoted in Speaking of Science: Notable Quotes on Science, Engineering, and the Environment (2000), 53.
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There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
— Carl Friedrich Gauss
Quoted in J.R. Newman, The World of Mathematics (1956), 314.
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There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein.
— Carl Friedrich Gauss
Attributed
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Thou, nature, art my goddess; to thy laws my services are bound...
— Carl Friedrich Gauss
His second motto (from King Lear by Shakespeare).
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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.
— Carl Friedrich Gauss
Letter to Friedrich Bessel (4 Dec 1808). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 415.
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To what heights would science now be raised if Archimedes had made that discovery [of decimal number notation]!
— Carl Friedrich Gauss
As quoted, without citation, in Eric Temple Bell, Men of Mathematics (1937), 256.
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We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.
— Carl Friedrich Gauss
Letter to Friedrich Bessel (1830).
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When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false.
— Carl Friedrich Gauss
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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 sacrifice to my gods [mathematical research]. For example, I hope to get the editting of the census, the birth and death lists in local districts, not as a job, but for my pleasure and satisfaction...
— Carl Friedrich Gauss
Letter to Heinrich Olbers (26 Oct 1802). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 415.
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You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
— Carl Friedrich Gauss
Quoted in G. Simmons, Calculus Gems (1992).
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…as our friend Zach has often noted, in our days those who do the best for astronomy are not the salaried university professors, but so-called dillettanti, physicians, jurists, and so forth.Lamenting the fragmentary time left to a professor has remaining after fulfilling his teaching duties.
— Carl Friedrich Gauss
Letter to Heinrich Olbers (26 Oct 1802). Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 415.
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Quotes by others about Carl Friedrich Gauss (26)

The importance of C.F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming indeed; also his achievement of the system of absolute measurement in the field of electromagnetism. In my opinion it is impossible to achieve a coherent objective picture of the world on the basis of concepts which are taken more or less from inner psychological experience.
Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 350.
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The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.
Quoted in G. Waldo Dunnington, Carl Friedrich Gauss: Titan of Science (2004), 350.
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In describing the honourable mission I charged him with, M. Pernety informed me that he made my name known to you. This leads me to confess that I am not as completely unknown to you as you might believe, but that fearing the ridicule attached to a female scientist, I have previously taken the name of M. LeBlanc in communicating to you those notes that, no doubt, do not deserve the indulgence with which you have responded.
Explaining her use of a male psuedonym.
Letter to Carl Friedrich Gauss (1807)
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[About Gauss’ mathematical writing style] He is like the fox, who effaces his tracks in the sand with his tail.
In G. F. Simmons, Calculus Gems (1992), 177.
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For other great mathematicians or philosophers, he [Gauss] used the epithets magnus, or clarus, or clarissimus; for Newton alone he kept the prefix summus.
In History of Mathematics (3rd Ed., 1901), 362.
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[Gauss calculated the elements of the planet Ceres] and his analysis proved him to be the first of theoretical astronomers no less than the greatest of “arithmeticians.”
In History of Mathematics (3rd Ed., 1901), 458.
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The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
In History of Mathematics (3rd Ed., 1901), 468.
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The mathematical giant [Gauss], who from his lofty heights embraces in one view the stars and the abysses …
Kurzer Grundriss eines Versuchs (1851), 44. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 158.
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The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.
In 'What Knowledge is of Most Worth?', Presidential address to the National Education Association, Denver, Colorado (9 Jul 1895). In Educational Review (Sep 1895), 10, 109.
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Gauss [replied], when asked how soon he expected to reach certain mathematical conclusions, “that he had them long ago, all he was worrying about was how to reach them.”
In Louis Pasteur, Free Lance of Science (1950), 365. Also excerpted in 'Mechanisms of Discovery', collected in I.S. Gordon and S. Sorkin (eds.) The Armchair Science Reader (1959), 336.
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Commenting on Archimedes, for whom he also had a boundless admiration, Gauss remarked that he could not understand how Archimedes failed to invent the decimal system of numeration or its equivalent (with some base other than 10). … This oversight Gauss regarded as the greatest calamity in the history of science.
In Men of Mathematics (1937), 256.
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Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.
Quoted in a space filler, without citation, in The Pentagon: A Mathematics Magazine for Students (Fall 1951), 11, No. 1, 12. Primary source needed (can you help).
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Gauss once said, “Mathematics is the queen of the sciences and number theory the queen of mathematics.” If this is true we may add that the Disquisitions is the Magna Charter of number theory.
In Allgemeine Deutsche Biographie (1878, 8, 435. As cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 158.
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The advantage which science gained by Gauss’ long-lingering method of publication is this: What he put into print is as true and important today as when first published; his publications are statutes, superior to other human statutes in this, that nowhere and never has a single error been detected in them. This justifies and makes intelligible the pride with which Gauss said in the evening of his life of the first larger work of his youth: “The Disquisitiones arithmeticae belong to history.”
In Allgemeine Deutsche Biographie (1878, 8, 435. As cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 158.
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In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months’ time, was achieved in three days by Euler with aid of improved methods of his own. … With still superior methods this same problem was solved by the illustrious Gauss in one hour.
In History of Mathematics (1897), 248.
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In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. “Pauca sed matura” was Gauss’s motto.
In Presidential Address British Association for the Advancement of Science, Section A, (1890), Nature, 42, 467. [The Latin motto translates as “Few, but ripe”. —Webmaster]
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It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty—balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service.
In Lectures on Science, Philosophy and Art (1908), 32-33.
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The greatest mathematicians, as Archimedes, Newton, and Gauss, always united theory and applications in equal measure.
In Elementarmathematik von höheren Standpunkte aus (1909), Bd. 2, 392.
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Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.
In Zahlentheorie, Teil 1 (1901), 43.
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Dirichlet was not satisfied to study Gauss’ Disquisitiones arithmetical once or several times, but continued throughout life to keep in close touch with the wealth of deep mathematical thoughts which it contains by perusing it again and again. For this reason the book was never placed on the shelf but had an abiding place on the table at which he worked. … Dirichlet was the first one, who not only fully understood this work, but made it also accessible to others.
In Dirichlet, Werke, Bd. 2, 315. As translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 159.
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Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence”.
In 'A Plea for the Mathematician', Nature, 1, 238 in Collected Mathematical Papers, Vol. 2, 655-56.
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This [the fact that the pursuit of mathematics brings into harmonious action all the faculties of the human mind] accounts for the extraordinary longevity of all the greatest masters of the Analytic art, the Dii Majores of the mathematical Pantheon. Leibnitz lived to the age of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; Plato, the supposed inventor of the conic sections, who made mathematics his study and delight, who called them the handles or aids to philosophy, the medicine of the soul, and is said never to have let a day go by without inventing some new theorems, lived to 82; Newton, the crown and glory of his race, to 85; Archimedes, the nearest akin, probably, to Newton in genius, was 75, and might have lived on to be 100, for aught we can guess to the contrary, when he was slain by the impatient and ill mannered sergeant, sent to bring him before the Roman general, in the full vigour of his faculties, and in the very act of working out a problem; Pythagoras, in whose school, I believe, the word mathematician (used, however, in a somewhat wider than its present sense) originated, the second founder of geometry, the inventor of the matchless theorem which goes by his name, the pre-cognizer of the undoubtedly mis-called Copernican theory, the discoverer of the regular solids and the musical canon who stands at the very apex of this pyramid of fame, (if we may credit the tradition) after spending 22 years studying in Egypt, and 12 in Babylon, opened school when 56 or 57 years old in Magna Græcia, married a young wife when past 60, and died, carrying on his work with energy unspent to the last, at the age of 99. The mathematician lives long and lives young; the wings of his soul do not early drop off, nor do its pores become clogged with the earthy particles blown from the dusty highways of vulgar life.
In Presidential Address to the British Association, Collected Mathematical Papers, Vol. 2 (1908), 658.
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Two extreme views have always been held as to the use of mathematics. To some, mathematics is only measuring and calculating instruments, and their interest ceases as soon as discussions arise which cannot benefit those who use the instruments for the purposes of application in mechanics, astronomy, physics, statistics, and other sciences. At the other extreme we have those who are animated exclusively by the love of pure science. To them pure mathematics, with the theory of numbers at the head, is the only real and genuine science, and the applications have only an interest in so far as they contain or suggest problems in pure mathematics.
Of the two greatest mathematicians of modern tunes, Newton and Gauss, the former can be considered as a representative of the first, the latter of the second class; neither of them was exclusively so, and Newton’s inventions in the science of pure mathematics were probably equal to Gauss’s work in applied mathematics. Newton’s reluctance to publish the method of fluxions invented and used by him may perhaps be attributed to the fact that he was not satisfied with the logical foundations of the Calculus; and Gauss is known to have abandoned his electro-dynamic speculations, as he could not find a satisfying physical basis. …
Newton’s greatest work, the Principia, laid the foundation of mathematical physics; Gauss’s greatest work, the Disquisitiones Arithmeticae, that of higher arithmetic as distinguished from algebra. Both works, written in the synthetic style of the ancients, are difficult, if not deterrent, in their form, neither of them leading the reader by easy steps to the results. It took twenty or more years before either of these works received due recognition; neither found favour at once before that great tribunal of mathematical thought, the Paris Academy of Sciences. …
The country of Newton is still pre-eminent for its culture of mathematical physics, that of Gauss for the most abstract work in mathematics.
In History of European Thought in the Nineteenth Century (1903), 630.
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Ultima se tangunt. How expressive, how nicely characterizing withal is mathematics! As the musician recognizes Mozart, Beethoven, Schubert in the first chords, so the mathematician would distinguish his Cauchy, Gauss, Jacobi, Helmholtz in a few pages.
In Ceremonial Speech (15 Nov 1887) celebrating the 301st anniversary of the Karl-Franzens-University Graz. Published as Gustav Robert Kirchhoff: Festrede zur Feier des 301. Gründungstages der Karl-Franzens-Universität zu Graz (1888), 29, as translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-book (1914), 186-187. From the original German, “Ultima se tangunt. Und wie ausdrucksfähig, wie fein charakterisirend ist dabei die Mathematik. Wie der Musiker bei den ersten Tacten Mozart, Beethoven, Schubert erkennt, so würde der Mathematiker nach wenig Seiten, seinen Cauchy, Gauss, Jacobi, Helmholtz unterscheiden.” [The Latin words translate as “the final touch”. —Webmaster]
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May not Music be described as the Mathematic of sense, Mathematic as Music of the reason? the soul of each the same! Thus the musician feels Mathematic, the mathematician thinks Music, Music the dream, Mathematic the working life each to receive its consummation from the other when the human intelligence, elevated to its perfect type, shall shine forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz!
In paper read 7 Apr 1864, printed in 'Algebraical Researches Containing a Disquisition On Newton’s Rule for the Discovery of Imaginary Roots', Philosophical Transactions of the Royal Society of London (1865), 154, 613, footnote. Also in Collected Mathematical Papers, Vol. 2, 419.
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Gauss was not the son of a mathematician; Handel’s father was a surgeon, of whose musical powers nothing is known; Titian was the son and also the nephew of a lawyer, while he and his brother, Francesco Vecellio, were the first painters in a family which produced a succession of seven other artists with diminishing talents. These facts do not, however, prove that the condition of the nerve-tracts and centres of the brain, which determine the specific talent, appeared for the first time in these men: the appropriate condition surely existed previously in their parents, although it did not achieve expression. They prove, as it seems to me, that a high degree of endowment in a special direction, which we call talent, cannot have arisen from the experience of previous generations, that is, by the exercise of the brain in the same specific direction.
In 'On Heredity', Essays upon Heredity and Kindred Biological Problems (1889), Vol. 1, 96.
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See also:
  • 30 Apr - short biography, births, deaths and events on date of Gauss's birth.
  • Carl Friedrich Gauss: Titan of Science, by G. Waldo Dunnington. - book suggestion.

Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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- 90 -
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- 60 -
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