# Carl Friedrich Gauss

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.

## 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 … she must without doubt have the noblest courage, quite extraordinary talents and superior genius.
Mathematics is the queen of sciences and number theory is the queen of mathematics.
I have had my results for a long time: but I do not yet know how I am to arrive at them.
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.
• But in our opinion truths of this kind should be drawn from notions rather than from notations.
• About the proof of Wilson's theorem. Disquisitiones Arithmeticae (1801) Article 76
• 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.
• 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
• 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.
• 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.)
• 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.
• 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.
• Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.
• "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 (1902)
• Arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density.
• 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.
• 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
• 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.
• 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.
• 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.
• Mathematics is the queen of the sciences.
• 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 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.
• 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.
• "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'. What are we without the hope of a better future?
• May the dream which we call life be for you a happy dream, a foretaste of that true life which we shall inherit in our real home, when the awakened spirit shall labour no longer under the grievous bondage of the flesh, the fetters of space, the whips of earthly pain, and the sting of our paltry needs and desires. Let us carry our burdens to the end, stoutly and uncomplainingly, never losing sight of that higher goal. Glad then shall we be to lay down our weary lives, and to see the dropping of the curtain."
• 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.
• 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.
• 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)
• 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.
• As quoted by Robert Chambers, "Sir Isaac Newton and the Apple," The Book of Days (1832) Vol. 2, p. 757.
• 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.
• 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.
• Gauss-Schumacher Briefwechsel (1862)
• Ask her to wait a moment — I am almost done.
• When told, while working, that his wife was dying, as attributed in Men of Mathematics (1937) by E. T. Bell
• I have had my results for a long time: but I do not yet know how I am to arrive at them.
• The Mind and the Eye (1954) by A. Arber
• If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.
• The World of Mathematics (1956) Edited by J. R. Newman
• 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.
• 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 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.
• As quoted in The World of Mathematics (1956) Edited by J. R. Newman
• 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.
• Mathematical Circles Squared (1972) by Howard W. Eves
• 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?
• 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.
• 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.
• As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 306
• You say that faith is a gift; this is perhaps the most correct thing that can be said about it.
• As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 305.
• Yes! The world would be nonsense, the whole creation an absurdity without immortality.
• As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 357
• All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced.
• March 14, 1824. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 360
• Even though much error and hypocrisy may often be mixed in such pietistic tendencies, nevertheless I recognize with all my heart the business of a missionary as a highly honorable one in so far as it leads to civilization the still semisavage part of earth s inhabitants. May my son try it for several years.
• Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 359
• 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: Titan of Science (1955) by Guy Waldo Dunnington. p. 349
• 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: Titan of Science (1955) by Guy Waldo Dunnington. p. 348
• 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.
• On higher arithmetic. Mathematical Circles Adieu (1977) by Howard W. Eves
• 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.
• As quoted in Solid Shape (1990) by Jan J. Koenderink
• 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.
• As quoted in Calculus Gems (1992) by George F. Simmons
• 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.
• 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
• The austere sides of life, at least of mine, which move through it like a red thread, and which one faces more and more defenselessly in old age, are not balanced to the hundredth part by the pleasurable. I will gladly admit that the same fates which have been so hard for me to bear, and still are, would have been much easier for many another person, but the mental constitution belongs to our ego, which the Creator of our existence has given us, and we can change little in it.
• As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 360
• 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.
• As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 362

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

• 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
• 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.
• 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
• 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.
• In a letter dated April 25, 1825. As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 361
• 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?
• As quoted in Carl Friedrich Gauss: Titan of Science (1955) by Guy Waldo Dunnington. p. 365

Gauss considered the three dimensions of space as specific peculiarities of the human soul. ~ Wolfgang Sartorius von Waltershausen
• Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticæ he observed that the principles which he used in the division of the circle were applicable to many other functions, besides the circular, and particularly to the transcendents dependent on the integral ${\displaystyle \displaystyle \int {\frac {\,dx}{\sqrt {1-x^{4}}}}}$. From this Jacobi concluded that Gauss had thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity. The papers in the collected works of Gauss confirm this conclusion.
• Not only could nobody but Gauss have produced it, but it would never have occurred to anyone but Gauss that such a formula was possible.
• Albert Einstein, on the formula developed by Gauss for finding the date of Passover, as quoted in The Calculated Confusion of Calendars (1976) by W. A. Schocken
• There is no doubt... that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgement has caused mathematicians to equate elegance and conciseness at the cost of intelligibility. Gauss himself wrote elegant, but highly compact, carefully polished papers with no hint of the motivation, meaning, or details of the steps. When criticized, he said that no architect leaves the scaffolding after completing the building. But the fact is that even excellent mathematicians found the reading of Gauss's papers very difficult, and the same is true of many other mathematicians.
• Morris Kline, Mathematics and the Physical World (1959) Ch. 5: Numbers Known and Unknown, pp. 58-59.
• Toward the ends of their lives, Euler, D'Alembert, and Lagrange agreed that the realm of mathematical ideas had been practically exhausted and that no new great minds were appearing on the mathematical horizons. Of course, these men had grown old and their vision was already dimmed, for Laplace, Legendre, and Fourier were in young manhood. In one respect, however, these elder statesmen were correct... their immediate successors continued to explore and polish the very same ideas which the mid-eighteenth century had pursued.
But history shows that the human mind is fertile, ingenious, and creative beyond all possible anticipations. ...even the richest vein of thought is ultimately exhausted, and then, indeed, a period of stagnation may ensue. Inevitably, however, there arise new conceptions and new periods of feverish and rewarding research. Euler and his contemporaries failed to reckon with history. ...
The man who was to change the course of mathematics was but six years old when Euler and D'Alembert died in 1783... Gauss is commonly ranked with Archimedes and Newton. ...all three of these men were as much devoted to physical research as to mathematics.
• Morris Kline, Mathematics and the Physical World (1959) Ch. 26: Non-Euclidean Geometries, pp. 443-445.
• It is to Gauss, to the Magnetic Union, and to magnetic observers in general, that we owe our deliverance from that absurd method of estimating forces by a variable standard which prevailed so long even among men of science. It was Gauss who first based the practical measurement of magnetic force (and therefore of every other force) on those long established principles, which, though they are embodied in every dynamical equation, have been so generally set aside, that these very equations, though correctly given... are usually explained... by assuming, in addition to the variable standard of force, a variable, and therefore illegal, standard of mass.
• If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds.
• According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically.

### The Music of the Primes (2003)

For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians. ~ Marcus du Sautoy
Quotations about some of the work of Gauss from a book about Prime Numbers by Marcus du Sautoy, professor of mathematics at Oxford University
• Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians.
• Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist.
• The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'
• Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built.