Carl Friedrich Gauss Quotes (50 quotes)

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Disquisitiones Arithmeticae

Carl Friedrich Gauss

Carl Friedrich Gauss: 50 quotes 7 likes

Famous Carl Friedrich Gauss Quotes

“One day he said: For the soul there is a satisfaction of a higher type; the material is not at all necessary. Whether I apply mathematics to a couple of clods of dirt, which we call planets, or to purely arithmetical problems, it s just the same; the latter have only a higher charm for me.”

Carl Friedrich Gauss

Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 348

“In researches in which an infinity of directions of straight lines in space is concerned, it is advantageous to represent these directions by means of those points upon a fixed sphere”

Carl Friedrich Gauss

"Gauss's Abstract of the Disquisitiones Generales circa Superficies Curvas presented to the Royal Society of Gottingen" (1827) Tr. James Caddall Morehead & Adam Miller Hiltebeitel in General Investigations of Curved Surfaces of 1827 and 1825 http://books.google.com/books?id=SYJsAAAAMAAJ& (1902)
Context: In researches in which an infinity of directions of straight lines in space is concerned, it is advantageous to represent these directions by means of those points upon a fixed sphere, which are the end points of the radii drawn parallel to the lines. The centre and the radius of this auxiliary sphere are here quite arbitrary. The radius may be taken equal to unity. This procedure agrees fundamentally with that which is constantly employed in astronomy, where all directions are referred to a fictitious celestial sphere of infinite radius. Spherical trigonometry and certain other theorems, to which the author has added a new one of frequent application, then serve for the solution of the problems which the comparison of the various directions involved can present.

“The centre and the radius of this auxiliary sphere are here quite arbitrary.”

Carl Friedrich Gauss

“The principle that the sum of the squares of the differences between the observed and computed quantities must be a minimum may, in the following manner, be considered independently of the calculus of probabilities.”

Carl Friedrich Gauss

Theoria motus corporum coelestium in sectionibus conicis solem ambientum (1809) Tr. Charles Henry Davis as Theory of the Motion of the Heavenly Bodies moving about the Sun in Conic Sections http://books.google.com/books?id=cspWAAAAMAAJ& (1857)
Context: The principle that the sum of the squares of the differences between the observed and computed quantities must be a minimum may, in the following manner, be considered independently of the calculus of probabilities. When the number of unknown quantities is equal to the number of the observed quantities depending on them, the former may be so determined as exactly to satisfy the latter. But when the number of the former is less than that of the latter, an absolutely exact agreement cannot be obtained, unless the observations possess absolute accuracy. In this case care must be taken to establish the best possible agreement, or to diminish as far as practicable the differences. This idea, however, from its nature, involves something vague. For, although a system of values for the unknown quantities which makes all the differences respectively less than another system, is without doubt to be preferred to the latter, still the choice between two systems, one of which presents a better agreement in some observations, the other in others, is left in a measure to our judgment, and innumerable different principles can be proposed by which the former condition is satisfied. Denoting the differences between observation and calculation by A, A’, A’’, etc., the first condition will be satisfied not only if AA + A’ A’ + A’’ A’’ + etc., is a minimum (which is our principle) but also if A4 + A’4 + A’’4 + etc., or A6 + A’6 + A’’6 + etc., or in general, if the sum of any of the powers with an even exponent becomes a minimum. But of all these principles ours is the most simple; by the others we should be led into the most complicated calculations.

“Spherical trigonometry and certain other theorems, to which the author has added a new one of frequent application, then serve for the solution of the problems which the comparison of the various directions involved can present.”

Carl Friedrich Gauss

“All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced.”

Carl Friedrich Gauss

March 14, 1824. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 360

Carl Friedrich Gauss Quotes about mathematics

“Mathematics is the queen of the sciences.”

Carl Friedrich Gauss

As quoted in Gauss zum Gedächtniss (1856) by Wolfgang Sartorius von Waltershausen; Variants: Mathematics is the queen of sciences and 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 the first rank.
Mathematics is the queen of the sciences and number theory is the queen of mathematics. [Die Mathematik ist die Königin der Wissenschaften und die Zahlentheorie ist die Königin der Mathematik.]

“The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it.”

Carl Friedrich Gauss

As quoted by Louise Grinstein, Sally I. Lipsey, Encyclopedia of Mathematics Education (2001) p. 235.

“If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.”

Carl Friedrich Gauss

The World of Mathematics (1956) Edited by J. R. Newman

“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

As quoted in The World of Mathematics (1956) Edited by J. R. Newman

“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.”

Carl Friedrich Gauss

Gauss-Schumacher Briefwechsel (1862)
Context: 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.

“I scarcely believe that in psychology data are present which can be mathematically evaluated. But one cannot know this with certainty, without having made the experiment. God alone is in possession of the mathematical bases of psychic phenomena.”

Carl Friedrich Gauss

As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 306

Carl Friedrich Gauss Quotes about time

“I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance… The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer J. Bolyai a genius of the first rank.”

Carl Friedrich Gauss

Letter to Gerling (1832)

“It is beyond doubt that the happiness which love can bestow on its chosen souls is the highest that can fall to mortal's lot. But when I imagine myself in the place of the man who, after twenty happy years, now in one moment loses his all, I am moved almost to say that he is the wretchedest of mortals, and that it is better never to have known such happy days. So it is on this miserable earth: 'the purest joy finds its grave in the abyss of time.”

Carl Friedrich Gauss

What are we without the hope of a better future?
As quoted in Kneller, Karl Alois, Kettle, Thomas Michael, 1911. "Christianity and the leaders of modern science; a contribution to the history of culture in the nineteenth century" https://archive.org/stream/christianitylead00kneluoft#page/44/mode/2up, Freiburg im Breisgau, p. 44-45

“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

The Mind and the Eye (1954) by A. Arber

“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

As quoted in Calculus Gems (1992) by George F. Simmons

“The perturbations which the motions of planets suffer from the influence other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time”

Carl Friedrich Gauss

Theoria motus corporum coelestium... (1809) Tr. Charles Henry Davis as Theory of the Motion of the Heavenly Bodies moving about the Sun in Conic Sections (1857)
Context: The perturbations which the motions of planets suffer from the influence other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion.

“I believe you are more believing in the Bible than I. I am not, and, you are much happier than I. I must say that so often in earlier times when I saw people of the lower classes, simple manual laborers who could believe so rightly with their hearts, I always envied them, and now tell me how does one begin this?”

Carl Friedrich Gauss

A reply to Rudolf Wagner's on his religious views as quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 305.

Carl Friedrich Gauss: Trending quotes

“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 Wilhelm Bessel (1830)

Reality

“But of all these principles ours is the most simple; by the others we should be led into the most complicated calculations.”

Carl Friedrich Gauss

“It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.”

Carl Friedrich Gauss

Letter to Farkas Bolyai (2 September 1808)
Context: It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. [Wahrlich es ist nicht das Wissen, sondern das Lernen, nicht das Besitzen sondern das Erwerben, nicht das Da-Seyn, sondern das Hinkommen, was den grössten Genuss gewährt. ] 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.

Knowledge , Learning

Carl Friedrich Gauss Quotes

“The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.”

Carl Friedrich Gauss

Letter to Sophie Germain (30 April 1807) ([...]; les charmes enchanteurs de cette sublime science ne se décèlent dans toute leur beauté qu'à ceux qui ont le courage de l'approfondir. Mais lorsqu'une personne de ce sexe, qui, par nos meurs [sic] et par nos préjugés, doit rencontrer infiniment plus d'obstacles et de difficultés, que les hommes, à se familiariser avec ces recherches épineuses, sait néanmoins franchir ces entraves et pénétrer ce qu'elles ont de plus caché, il faut sans doute, qu'elle ait le plus noble courage, des talents tout à fait extraordinaires, le génie superieur.)
Context: The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it. But when a person of that sex, that, because of our mores and our prejudices, has to encounter infinitely more obstacles and difficulties than men in familiarizing herself with these thorny research problems, nevertheless succeeds in surmounting these obstacles and penetrating their most obscure parts, she must without doubt have the noblest courage, quite extraordinary talents and superior genius.

Beauty , Courage , Science

“I am almost amazed that you consider a professional philosopher capable of no confusion in concepts and definitions. Such things are nowhere more at home than among philosophers who are not mathematicians, and Wolff was no mathematician, even though he made cheap compen- diums. Look around among the philosophers of today, among Schelling, Hegel, Nees von Esenbeck, and their like; doesn t your hair stand on end at their definitions? Read in the history of ancient philosophy what kinds of definitions the men of that day, Plato and others, gave (I except Aristotle). But even in Kant it is often not much better; in my opinion his distinction between analytic and synthetic theorems is such a one that either peters out in a triviality or is false.”

Carl Friedrich Gauss

As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 362

Men , Home , History , Reading

“Ask her to wait a moment — I am almost done.”

Carl Friedrich Gauss

When told, while working, that his wife was dying, as attributed in Men of Mathematics (1937) by E. T. Bell

Waiting

“If the object of all human investigation were but to produce in cognition a reflection of the world as it exists, of what value would be all its labor and pains, which could result only in vain repetition, in an imitation within the soul of that which exists without it?”

Carl Friedrich Gauss

As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 365

Pain , World , Soul

“Dark are the paths which a higher hand allows us to traverse here… let us hold fast to the faith that a finer, more sublime solution of the enigmas of earthly life will be present, will become part of us.”

Carl Friedrich Gauss

In his letter to Schumacher on February 9, 1823. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 361

Life , Faith , Parting

“But in our opinion truths of this kind should be drawn from notions rather than from notations.”

Carl Friedrich Gauss book Disquisitiones Arithmeticae

About the proof of Wilson's theorem. Disquisitiones Arithmeticae (1801) Article 76

Truth

“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

A reply to Olbers' 1816 attempt to entice him to work on Fermat's Theorem. As quoted in The World of Mathematics (1956) Edited by J. R. Newman

“There is in this world a joy of the intellect, which finds satisfaction in science, and a joy of the heart, which manifests itself above all in the aid men give one another against the troubles and trials of life. But for the Supreme Being to have created existences, and stationed them in various spheres in order to taste these joys for some 80 or 90 years — that were surely a miserable plan…. Whether the soul were to live for 80 years or for 80 million years, if it were doomed in the end to perish, such an existence would only be a respite. In the end it would drop out of being. We are thus impelled to the conclusion to which so many things point, although they do not amount to a coercive scientific proof, that besides this material world there exists another purely spiritual order of things, with activities as various, as the present, and that this world of spirit we shall one day inherit.”

Carl Friedrich Gauss

As quoted in Kneller, Karl Alois, Kettle, Thomas Michael, 1911. "Christianity and the leaders of modern science; a contribution to the history of culture in the nineteenth century" https://archive.org/stream/christianitylead00kneluoft#page/48/mode/2up, Freiburg im Breisgau, p. 48-49

Life , Men , World , Soul

“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

As quoted in Solid Shape (1990) by Jan J. Koenderink

“The history of the apple is too absurd. Whether the apple fell or not, how can any one believe that such a discovery could in that way be accelerated or retarded? Undoubtedly, the occurrence was something of this sort. There comes to Newton a stupid, importunate man, who asks him how he hit upon his great discovery. When Newton had convinced himself what a noodle he had to do with, and wanted to get rid of the man, he told him that an apple fell on his nose; and this made the matter quite clear to the man, and he went away satisfied.”

Carl Friedrich Gauss

As quoted by Robert Chambers, "Sir Isaac Newton and the Apple," The Book of Days (1832) Vol. 2 https://books.google.com/books?id=K0UJAAAAIAAJ, p. 757.

Stupidity , Way , History

“Believe me,… the bitterness of life, or at least of mine, which runs through it like a strand of red, and becomes less and less endurable as I grow older, is not compensated in the hundredth part by the joy of life. I will freely admit that these burdens, which to me have been so grievous, would have been lighter to many another; but our temperament is part of ourselves, given to us by the Creator with our very existence, and we have very little power to change it. I find, on the other hand, in this very consciousness of the vanity of life, which nearly all men must confess to as they draw near the end, my strongest assurance of the approach of a more beautiful metamorphosis. In this, my dear friend, let us find comfort, and endeavour to call up calmness to bear life out to the end.”

Carl Friedrich Gauss

Life , Men , Beauty , Change

“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 ½ proof = 0, and it is demanded for proof that every doubt becomes impossible.”

Carl Friedrich Gauss

In a letter to Heinrich Wilhelm Matthias Olbers (14 May 1826), defending Chevalier d'Angos against presumption of guilt (by Johann Franz Encke and others), of having falsely claimed to have discovered a comet in 1784; as quoted in Calculus Gems (1992) by George F. Simmons

Sense

“To praise it would amount to praising myself. For the entire content of the work … coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.”

Carl Friedrich Gauss

Letter to Farkas Bolyai, on his son János Bolyai's 1832 publishings on non-Euclidean geometry.

Past

“Yes! The world would be nonsense, the whole creation an absurdity without immortality.”

Carl Friedrich Gauss

As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 357

World

“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

Mathematical Circles Squared (1972) by Howard W. Eves

Effort , Success

“In such apparent accidents which finally produce such a decisive influence on one s whole life, one is inclined to recognize the tools of a higher hand. The great enigma of life never becomes clear to us here below.”

Carl Friedrich Gauss

In a letter dated April 25, 1825. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 361

Life , Decision

“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 Mobius'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

As quoted in Gauss, Werke, Bd. 8, page 298
As quoted in Memorabilia Mathematica (or The Philomath's Quotation-Book) (1914) by Robert Edouard Moritz, quotation #1215
As quoted in The First Systems of Weighted Differential and Integral Calculus (1980) by Jane Grossman, Michael Grossman, and Robert Katz, page ii

Nature , Problem

“You say that faith is a gift; this is perhaps the most correct thing that can be said about it.”

Carl Friedrich Gauss

As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 305.

Faith

“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 book Disquisitiones Arithmeticae

Problema, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tam notum est, ut de hac re copiose loqui superfluum foret. … [P]raetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem problematis tam elegantis ac celebris sedulo excolantur.
Disquisitiones Arithmeticae (1801): Article 329

Wisdom , Problem , Science

“The function just found cannot, it is true, express rigorously the probabilities of the errors: for since the possible errors are in all cases confined within certain limits, the probability of errors exceeding those limits ought always to be zero, while our formula always gives some value. However, this defect, which every analytical function must, from its nature, labor under, is of no importance in practice, because the value of our function decreases so rapidly… that it can safely be considered as vanishing. Besides, the nature of the subject never admits of assigning with absolute rigor the limits of error.”

Carl Friedrich Gauss

Error , Nature

“One is forced to the view, for which there is so much evidence even though without rigorous scientific basis, that besides this material world another, second, purely spiritual world order exists, with just as many diversities as that in which we live-—we are to participate in it.”

Carl Friedrich Gauss

Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 349

World

“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 instance, +1, -1, √-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 question.”

Carl Friedrich Gauss

In Theoria residiorum biquadraticorum, Commentatio secunda; Werke, Bd. 2 (Goettingen, 1863), p.177. As quoted by Robert Edouard Moritz in Memorabilia mathematica: the philomath's quotation book (1914) p. 282.

Question

“Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.”

Carl Friedrich Gauss

Dependency

“One cannot reduce to concepts the distinction between two systems of three straight lines each (directed lines, of which the one system points forward, upward to the right, the other forward, upward to the left) but one can only demonstrate by holding to actually present spatial things. Two minds cannot reach agreement about it unless their views connect up with one and the same system present in the real world”

Carl Friedrich Gauss

In a letter to Gerling on June 23, 1846. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 364

World

“A great part of its 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

On higher arithmetic. Mathematical Circles Adieu (1977) by Howard W. Eves

Parting