quotes of carl friedrich gauss

Carl Friedrich Gauss

I mean the word proof not in the sense of the lawyers, who settwo half proofs equal to a whole one, but in the sense of a math-ematician, where proof = 0, and it is demanded for proof thatevery doubt becomes impossible.

Johann Carl Friedrich Gauss (30 April 1777 23February 1855) was a German mathematician, as-tronomer and physicist.

1 Quotes But in our opinion truths of this kind should bedrawn from notions rather than from notations.

About the proof of Wilsons theorem. Disqui-sitiones Arithmeticae (1801) Article 76

The problem of distinguishing prime numbers fromcomposite numbers and of resolving the latter intotheir prime factors is known to be one of the mostimportant and useful in arithmetic. It has engagedthe industry and wisdom of ancient and modern ge-ometers to such an extent that it would be superu-ous to discuss the problem at length. Further, thedignity of the science itself seems to require that ev-

The enchanting charms of this sublime science reveal themselvesin all their beauty only to those who have the courage to go deeplyinto it she must without doubt have the noblest courage, quiteextraordinary talents and superior genius.

Mathematics is the queen of sciences and number theory is thequeen of mathematics.

ery possible means be explored for the solution of aproblem so elegant and so celebrated.

Problema, numeros primos a compositisdignoscendi, hosque in factores suos primosresolvendi, ad gravissima ac utilissima totiusarithmeticae pertinere, et geometrarum tumveterum tum recentiorum industriam ac

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I have had my results for a long time: but I do not yet know howI am to arrive at them.

sagacitatem occupavisse, tam notum est, utde hac re copiose loqui superuum foret. [P]raetereaque scientiae dignitas requirerevidetur, ut omnia subsidia ad solutionemproblematis tam elegantis ac celebris seduloexcolantur.

Disquisitiones Arithmeticae (1801): Article329

The enchanting charms of this sublime sciencereveal themselves in all their beauty only to thosewho have the courage to go deeply into it. Butwhen a person of that sex, that, because of ourmoresand our prejudices, has to encounter innitely moreobstacles and diculties than men in familiarizingherself with these thorny research problems, never-theless succeeds in surmounting these obstacles andpenetrating their most obscure parts, she must with-out doubt have the noblest courage, quite extraordi-nary talents and superior genius.

Letter to Sophie Germain (30 April 1807)([...]; les charmes enchanteurs de cette sub-lime science ne se dclent dans toute leurbeaut qu' ceux qui ont le courage del'approfondir. Mais lorsqu'une personne dece sexe, qui, par nos meurs [sic] et parnos prjugs, doit rencontrer inniment plusd'obstacles et de dicults, que les hommes, se familiariser avec ces recherches pineuses,sait nanmoins franchir ces entraves et pntrerce qu'elles ont de plus cach, il faut sans doute,

There are problems to whose solution I would attach an innitelygreater importance than to those of mathematics, for exampletouching ethics, or our relation to God, or concerning our destinyand our future; but their solution lies wholly beyond us and com-pletely outside the province of science.

qu'elle ait le plus noble courage, des talents tout fait extraordinaires, le gnie superieur.)

It is not knowledge, but the act of learning, notpossession but the act of getting there, whichgrants the greatest enjoyment. [Wahrlich es istnicht das Wissen, sondern das Lernen, nicht dasBesitzen sondern das Erwerben, nicht das Da-Seyn,sondern das Hinkommen, was den grssten Genussgewhrt.] When I have claried and exhausted asubject, then I turn away from it, in order to gointo darkness again. The never-satised man is sostrange; if he has completed a structure, then it isnot in order to dwell in it peacefully, but in order tobegin another. I imagine the world conqueror mustfeel thus, who, after one kingdom is scarcely con-quered, stretches out his arms for others.

Letter to Farkas Bolyai (2 September 1808)

In researches inwhich an innity of directions of

3straight lines in space is concerned, it is advan-tageous to represent these directions by meansof those points upon a xed sphere, which arethe end points of the radii drawn parallel to thelines. The centre and the radius of this aux-iliary sphere are here quite arbitrary. The ra-dius may be taken equal to unity. This procedureagrees fundamentally with that which is constantlyemployed in astronomy, where all directions are re-ferred to a ctitious celestial sphere of innite ra-dius. Spherical trigonometry and certain othertheorems, to which the author has added a newone of frequent application, then serve for thesolution of the problems which the comparisonof the various directions involved can present.

Gausss Abstract of the Disquisitiones Gen-erales circa Supercies Curvas presented tothe Royal Society of Gottingen (1827) Tr.James Caddall Morehead & Adam MillerHiltebeitel in General Investigations of CurvedSurfaces of 1827 and 1825 (1902)

Less depends upon the choice of words thanupon this, that their introduction shall be jus-tied by pregnant theorems.


Arc, amplitude, and curvature sustain a similarrelation to each other as time, motion, and veloc-ity, or as volume, mass, and density.


I mean the word proof not in the sense of thelawyers, who set two half proofs equal to a wholeone, but in the sense of a mathematician, where proof = 0, and it is demanded for proof thatevery doubt becomes impossible.

In a letter to Heinrich Wilhelm MatthiasOlbers (14 May 1826), defending Cheva-lier d'Angos against presumption of guilt (byJohann Franz Encke and others), of havingfalsely claimed to have discovered a comet in1784; as quoted in Calculus Gems (1992) byGeorge F. Simmons

We must admit with humility that, while number ispurely a product of our minds, space has a realityoutside our minds, so that we cannot completely pre-scribe its properties a priori.

Letter to Friedrich Wilhelm Bessel (1830)

To praise it would amount to praising myself. Forthe entire content of the work coincides almostexactly with my own meditations which have occu-pied my mind for the past thirty or thirty-ve years.

Letter to Farkas Bolyai, on his son JnosBolyai's 1832 publishings on non-Euclideangeometry.

I will add that I have recently received fromHungarya little paper on non-Euclidean geometry in which Irediscover all my own ideas and results worked outwith great elegance... The writer is a very youngAustrian ocer, the son of one of my early friends,with whom I often discussed the subject in 1798, al-though my ideas were at that time far removed fromthe development and maturity which they have re-ceived through the original reections of this youngman. I consider the young geometer J. Bolyai agenius of the rst rank.

Letter to Gerling (1832)

Mathematics is the queen of the sciences. As quoted inGauss zumGedchtniss (1856) byWolfgang Sartorius von Waltershausen; Vari-ants: Mathematics is the queen of sciencesand number theory is the queen of math-ematics. She often condescends to renderservice to astronomy and other natural sci-ences, but in all relations she is entitled tothe rst rank.Mathematics is the queen of the sciences andnumber theory is the queen of mathemat-ics. [Die Mathematik ist die Knigin der Wis-senschaften und die Zahlentheorie ist die Kni-gin der Mathematik.]

The function just found cannot, it is true, expressrigorously the probabilities of the errors: for sincethe possible errors are in all cases conned withincertain limits, the probability of errors exceedingthose limits ought always to be zero, while our for-mula always gives some value. However, this de-fect, which every analytical function must, from itsnature, labor under, is of no importance in prac-tice, because the value of our function decreases sorapidly... that it can safely be considered as vanish-ing. Besides, the nature of the subject never ad-mits of assigning with absolute rigor the limitsof error.

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Theoria motus corporum coelestium in sec-tionibus conicis solem ambientum (1809) Tr.Charles Henry Davis as Theory of the Motionof the Heavenly Bodies moving about the Sunin Conic Sections (1857)

There is in this world a joy of the intellect, whichnds satisfaction in science, and a joy of the heart,which manifests itself above all in the aid men giveone another against the troubles and trials of life.But for the Supreme Being to have created exis-tences, and stationed them in various spheres in or-der to taste these joys for some 80 or 90 years that were surely a miserable plan.... Whether thesoul were to live for 80 years or for 80 million years,if it were doomed in the end to perish, such an ex-istence would only be a respite. In the end it woulddrop out of being. We are thus impelled to the con-clusion to which so many things point, although theydo not amount to a coercive scientic proof, that be-sides this material world there exists another purelyspiritual order of things, with activities as various,as the present, and that this world of spirit we shallone day inherit.

As quoted in Kneller, Karl Alois, Kettle,Thomas Michael, 1911.Christianity and theleaders of modern science; a contribution tothe history of culture in the nineteenth century,Freiburg im Breisgau, p. 48-49

It is beyond doubt that the happiness which lovecan bestow on its chosen souls is the highest that canfall to mortals lot. But when I imagine myself in theplace of the man who, after twenty happy years, nowin one moment loses his all, I am moved almost tosay that he is the wretchedest of mortals, and that itis better never to have known such happy days. Soit is on this miserable earth: 'the purest joy nds itsgrave in the abyss of time'. What are we without thehope of a better future?


May the dream which we call life be for you a happydream, a foretaste of that true life which we shallinherit in our real home, when the awakened spiritshall labour no longer under the grievous bondage ofthe esh, the fetters of space, the whips of earthlypain, and the sting of our paltry needs and desires.Let us carry our burdens to the end, stoutly and un-complainingly, 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.

As quoted in Kneller, Karl Alois, Kettle,Thomas Michael, 1911.Christianity and theleaders of modern science; a contribution tothe history of culture in the nineteenth century,Freiburg im Breisgau, p. 46

Believe me,... the bitterness of life, or at least ofmine, which runs through it like a strand of red, andbecomes less and less endurable as I grow older, isnot compensated in the hundredth part by the joy oflife. I will freely admit that these burdens, which tome have been so grievous, would have been lighter tomany another; but our temperament is part of our-selves, given to us by the Creator with our very ex-istence, and we have very little power to change it. Ind, on the other hand, in this very consciousness ofthe vanity of life, which nearly all men must confessto as they draw near the end, my strongest assuranceof the approach of a more beautiful metamorphosis.In this, my dear friend, let us nd comfort, and en-deavour to call up calmness to bear life out to theend.


The perturbations which the motions of planetssuer from the inuence other planets, are sosmall and so slow that they only become sensi-ble after a long interval of time; within a shortertime, or even within one or several revolutions, ac-cording to circumstances, the motion would dierso little from motion exactly described, accord-ing to the laws of Kepler, in a perfect ellipse, thatobservations cannot show the dierence. As longas this is true, it not be worth while to undertake pre-maturely the computation of the perturbations, butit will be sucient to adapt to the observations whatwe may call an osculating conic section: but, af-terwards, when the planet has been observed fora longer time, the eect of the perturbations willshow itself in such a manner, that it will no longerbe possible to satisfy exactly all the observationsby a purely elliptic motion; then, accordingly, acomplete and permanent agreement cannot be ob-tained, unless the perturbations are properly con-nected with the elliptic motion.

Theoria motus corporum coelestium... (1809)Tr. Charles Henry Davis as Theory of the Mo-tion of the Heavenly Bodies moving about theSun in Conic Sections (1857)

The principle that the sum of the squares of thedierences between the observed and computed

5quantities must be a minimum may, in the fol-lowing manner, be considered independently ofthe calculus of probabilities. When the numberof unknown quantities is equal to the number of theobserved quantities depending on them, the formermay be so determined as exactly to satisfy the latter.But when the number of the former is less than thatof the latter, an absolutely exact agreement cannotbe obtained, unless the observations possess abso-lute accuracy. In this case care must be taken toestablish the best possible agreement, or to dimin-ish as far as practicable the dierences. This idea,however, from its nature, involves something vague.For, although a system of values for the unknownquantities which makes all the dierences respec-tively less than another system, is without doubt tobe preferred to the latter, still the choice betweentwo systems, one of which presents a better agree-ment in some observations, the other in others, isleft in a measure to our judgment, and innumerabledierent principles can be proposed by which theformer condition is satised. Denoting the dier-ences between observation and calculation by A, A,A, etc., the rst condition will be satised not onlyif AA + A A + A A + etc., is a minimum (whichis our principle) but also if A4 + A4 + A4 + etc., orA6 + A6 + A6 + etc., or in general, if the sum ofany of the powers with an even exponent becomes aminimum. But of all these principles ours is themost simple; by the others we should be led intothe most complicated calculations.

Theoria motus corporum coelestium in sec-tionibus conicis solem ambientum (1809) Tr.Charles Henry Davis as Theory of the Motionof the Heavenly Bodies moving about the Sunin Conic Sections (1857)

It may be true, that men, who are mere math-ematicians, have certain specic shortcomings,but that is not the fault of mathematics, for it isequally true of every other exclusive occupation.So there aremere philologists,mere jurists,mere sol-diers,meremerchants, etc. To such idle talk it mightfurther be added: that whenever a certain exclusiveoccupation is coupled with specic shortcomings, itis likewise almost certainly divorced from certainother shortcomings.

Gauss-Schumacher Briefwechsel (1862)

Ask her to wait a moment I am almost done. When told, while working, that his wife wasdying, as attributed in Men of Mathematics(1937) by E. T. Bell

I have had my results for a long time: but I donot yet know how I am to arrive at them.

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

If others would but reect on mathematicaltruths 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 Fermats Theorem as an isolatedproposition has very little interest for me, be-cause I could easily lay down a multitude of suchpropositions, which one could neither prove nordispose of.

A reply to Olbers 1816 attempt to entice himto work on Fermats Theorem. As quoted inThe World of Mathematics (1956) Edited by J.R. Newman

There are problems to whose solution I wouldattach an innitely greater importance than tothose of mathematics, for example touchingethics, or our relation to God, or concerningour destiny and our future; but their solutionlies wholly beyond us and completely outside theprovince of science.

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

Finally, two days ago, I succeeded not on ac-count of my hard eorts, but by the grace of theLord. Like a sudden ash of lightning, the rid-dle was solved. I am unable to say what was theconducting thread that connected what I previ-ously knew with what made my success possible.

Mathematical Circles Squared (1972) byHoward W. Eves

I believe you are more believing in the Bible thanI. I am not, and, you are much happier than I. Imust say that so often in earlier times when I sawpeople of the lower classes, simplemanual labor-ers who could believe so rightly with their hearts,I always envied them, and now tell me how doesone begin this?

A reply to Rudolf Wagners on his religiousviews as quoted in Carl Friedrich Gauss: Titanof Science (1955) by Guy Waldo Dunnington.p. 305.

I scarcely believe that in psychology data arepresent which can be mathematically evaluated.But one cannot know this with certainty, with-out having made the experiment. God alone isin possession of the mathematical bases of psy-chic phenomena.

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

You say that faith is a gift; this is perhaps themost correct thing that can be said about it.

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

Yes! The world would be nonsense, the wholecreation an absurdity without immortality.


All the measurements in the world do not bal-ance one theorem by which the science of eternaltruths is actually advanced.

March 14, 1824. As quoted in Carl FriedrichGauss: Titan of Science (1955) by Guy WaldoDunnington. p. 360

Even though much error and hypocrisy may oftenbe mixed in such pietistic tendencies, nevertheless Irecognize with all my heart the business of a mis-sionary as a highly honorable one in so far as it leadsto civilization the still semisavage part of earth s in-habitants. 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 muchevidence even though without rigorous scientic ba-sis, that besides this material world another, second,purely spiritual world order exists, with just as manydiversities as that in which we live-we are to par-ticipate in it.


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


A great part of its theories derives an additionalcharm from the peculiarity that important proposi-tions, with the impress of simplicity on them, are of-ten easily discovered by induction, and yet are of so

profound a character that we cannot nd the demon-strations till after many vain attempts; and even then,when we do succeed, it is often by some tediousand articial process, while the simple methods maylong remain concealed.

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

I am coming more and more to the conviction thatthe necessity of our geometry cannot be demon-strated, at least neither by, nor for, the human in-tellect. . . Geometry should be ranked, not witharithmetic, which is purely aprioristic, but with me-chanics.

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

You know that I write slowly. This is chieybecause I am never satised until I have saidas much as possible in a few words, and writ-ing briey takes far more time than writing atlength.

As quoted in Calculus Gems (1992) by GeorgeF. Simmons

In general the position as regards all such newcalculi is this - That one cannot accomplish bythem anything that could not be accomplishedwithout them. However, the advantage is, that,provided such a calculus corresponds to the in-most nature of frequent needs, anyone whomas-ters it thoroughly is able - without the uncon-scious inspiration of genius which no one cancommand - to solve the respective problems, yeato solve them mechanically in complicated casesin which, without such aid, even genius becomespowerless. Such is the case with the inventionof general algebra, with the dierential calculus,and in a more limited region with Lagrangescalculus of variations, with my calculus of con-gruences, and withMobiuss calculus. Such con-ceptions unite, as it were, into an organic wholecountless problems which otherwise would re-main isolated and require for their separate so-lution more or less application of inventive ge-nius.

As quoted in Gauss, Werke, Bd. 8, page 298 As quoted in Memorabilia Mathematica (orThe Philomaths Quotation-Book) (1914) byRobert Edouard Moritz, quotation #1215

As quoted in The First Systems of WeightedDierential and Integral Calculus (1980)by Jane Grossman, Michael Grossman, andRobert Katz, page ii

7 The austere sides of life, at least of mine, whichmove through it like a red thread, and which onefaces more andmore defenselessly in old age, are notbalanced to the hundredth part by the pleasurable. Iwill gladly admit that the same fates which have beenso hard for me to bear, and still are, would have beenmuch easier for many another person, but thementalconstitution belongs to our ego, which the Creator ofour existence has given us, and we can change littlein it.


I am almost amazed that you consider a profes-sional philosopher capable of no confusion in con-cepts and denitions. Such things are nowheremore at home than among philosophers who are notmathematicians, and Wol was no mathematician,even though he made cheap compen- diums. Lookaround among the philosophers of today, amongSchelling, Hegel, Nees von Esenbeck, and their like;doesn t your hair stand on end at their denitions?Read in the history of ancient philosophy what kindsof denitions the men of that day, Plato and others,gave (I except Aristotle). But even in Kant it is of-ten not much better; in my opinion his distinctionbetween analytic and synthetic theorems is such aone that either peters out in a triviality or is false.


One cannot reduce to concepts the distinction betweentwo systems of three straight lines each (directed lines, ofwhich the one system points forward, upward to the right,the other forward, upward to the left) but one can onlydemonstrate by holding to actually present spatial things.Two minds cannot reach agreement about it unless theirviews connect up with one and the same system presentin the real world

In a letter to Gerling on June 23, 1846. Asquoted in Carl Friedrich Gauss: Titan of Sci-ence (1955) by Guy Waldo Dunnington. p.364

Dark are the paths which a higher hand allowsus to traverse here... let us hold fast to the faiththat a ner, more sublime solution of the enig-mas of earthly life will be present, will becomepart of us.

In his letter to Schumacher on February 9,1823. As quoted in Carl Friedrich Gauss: Ti-tan of Science (1955) by GuyWaldo Dunning-ton. p. 361

In such apparent accidents which nally producesuch a decisive inuence on one s whole life, oneis inclined to recognize the tools of a higher hand.The great enigma of life never becomes clear to ushere below.

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

If the object of all human investigation were but toproduce in cognition a reection of the world as itexists, of what value would be all its labor and pains,which could result only in vain repetition, in an im-itation within the soul of that which exists withoutit?


2 Quotes about Gauss

Gauss considered the three dimensions of space as specic pecu-liarities of the human soul. ~ Wolfgang Sartorius von Walter-shausen

Not only could nobody but Gauss have producedit, but it would never have occurred to anyonebut Gauss that such a formula was possible.

Albert Einstein, on the formula developed byGauss for nding the date of Passover, asquoted in The Calculated Confusion of Calen-dars (1976) by W. A. Schocken

It is to Gauss, to the Magnetic Union, and to mag-netic observers in general, that we owe our deliv-erance from that absurd method of estimatingforces by a variable standard which prevailed solong even among men of science. It was Gauss

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who rst based the practical measurement of mag-netic force (and therefore of every other force) onthose long established principles, which, though theyare embodied in every dynamical equation, havebeen so generally set aside, that these very equations,though correctly given... are usually explained... byassuming, in addition to the variable standard offorce, a variable, and therefore illegal, standard ofmass.

James Clerk Maxwell, Introductory Lectureon Experimental Physics, The Scientic Pa-pers of James Clerk Maxwell (1890) Vol.2

If explaining minds seems harder than explain-ing songs, we should remember that sometimesenlarging problems makes them simpler! Thetheory of the roots of equations seemed hard forcenturies within its little world of real numbers, butit suddenly seemed simple once Gauss exposed thelarger world of so-called complex numbers. Sim-ilarly, music should make more sense once seenthrough listeners minds.

Marvin Minsky, Music, Mind, and Meaning(1981)

According to his frequently expressed view, Gaussconsidered the three dimensions of space as spe-cic peculiarities of the human soul; people,which are unable to comprehend this, he designatedin his humorous mood by the name Botians. Wecould imagine ourselves, he said, as beings whichare conscious of but two dimensions; higher beingsmight look at us in a like manner, and continuingjokingly, he said that he had laid aside certainproblems which, when in a higher state of being,he hoped to investigate geometrically.

Wolfgang Sartorius von Waltershausen inGauss zum Gedchtniss (1856)

2.1 The Music of the Primes (2003)Quotations about some of the work of Gauss froma book about Prime Numbers by Marcus du Sautoy,professor of mathematics at Oxford University

Gauss liked to call [number theory] 'the Queen ofMathematics. For Gauss, the jewels in the crownwere the primes, numbers which had fascinated andteased generations of mathematicians.

Armed with his prime number tables, Gauss beganhis quest. As he looked at the proportion of num-bers that were prime, he found that when he countedhigher and higher a pattern started to emerge. De-spite the randomness of these numbers, a stunningregularity seemed to be looming out of the mist.

For Gauss, the jewels in the crown were the primes, numberswhich had fascinated and teased generations of mathematicians.~ Marcus du Sautoy

The revelation that the graph appears to climb sosmoothly, even though the primes themselves areso unpredictable, is one of the most miraculous inmathematics and represents one of the high pointsin the story of the primes. On the back page of hisbook of logarithms, Gauss recorded the discoveryof his formula for the number of primes up to N interms of the logarithm function. Yet despite theimportance of the discovery, Gauss told no onewhat he had found. The most the world heard ofhis revelation were the cryptic words, 'You haveno idea how much poetry there is in a table of log-arithms.'

Maybe we have become so hung up on looking atthe primes from Gausss and Riemanns perspectivethat what we are missing is simply a dierent wayto understand these enigmatic numbers. Gauss gavean estimate for the number of primes, Riemann pre-dicted that the guess is at worst the square root of No its mark, Littlewood showed that you can't dobetter than this. Maybe there is an alternative view-point that no one has found because we have becomeso culturally attached to the house that Gauss built.

3 External links MacTutor biography of Gauss

Carl Friedrich Gauss at Planet Math

Carl Frederick Gauss, site by Gauss great-great-great granddaughter, including a scanned letter writ-ten to his son, Eugene, and links to his genealogy.

Gauss and His Children, site for Gauss researchers

Gauss, general information

3.1 Obituaries 9

3.1 Obituaries MNRAS 16 (1856) 80

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