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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.

-Carl Friedrich Gauss

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Mathematics Is The Queen

Carl Friedrich Gauss

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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.

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I Protest Against The Use Of Infinite Magnitude ..., Which Is Never Permissible In Mathematics.

When I Have Clarified And Exhausted A Subject, Then I Turn Away From It, In Order To Go Into Darkness Again.

If Others Would But Reflect On Mathematical Truths As Deeply And As Continuously As I Have, They Would Make My Discoveries.

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.

Sin2 φ Is Odious To Me, Even Though Laplace Made Use Of It; Should It Be Feared That Sin2 φ Might Become Ambiguous, Which Would Perhaps Never Occur, Or At Most Very Rarely When Speaking Of Sin(φ2), Well Then, Let Us Write (sin φ)2, But Not Sin2 φ, Which By Analogy Should Signify Sin (sin φ)

The Enchanting Charms Of This Sublime Science Reveal Only To Those Who Have The Courage To Go Deeply Into It.

[on Sophie Germain] When A Person Of The Sex Which, According To Our Customs And Prejudices, Must Encounter Infinitely More Difficulties Than Men... Succeeds Nevertheless In Surmounting These Obstacles And Penetrating The Most Obscure Parts Of [number Theory], Then Without Doubt She Must Have The Noblest Courage, Quite Extraordinary Talents And Superior Genius.

In Mathematics There Are No True Controversies.

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.

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.

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.

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.

The Infinite Is Only A Manner Of Speaking.

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.

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.

When A Philosopher Says Something That Is True Then It Is Trivial. When He Says Something That Is Not Trivial Then It Is False.

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.

I Have The Vagary Of Taking A Lively Interest In Mathematical Subjects Only Where I May Anticipate Ingenious Association Of Ideas And Results Recommending Themselves By Elegance Or Generality.

In 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.

Theory Attracts Practice As The Magnet Attracts Iron.

Less Depends Upon The Choice Of Words Than Upon This, That Their Introduction Shall Be Justified By Pregnant Theorems.

No Contradictions Will Arise As Long As Finite Man Does Not Mistake The Infinite For Something Fixed, As Long As He Is Not Led By An Acquired Habit Of Mind To Regard The Infinite As Something Bounded.

The Total Number Of Dirichlet's Publications Is Not Large: Jewels Are Not Weighed On A Grocery Scale.

I Believe You Are More Believing In The Bible Than I. I Am Not, And, You Are Much Happier Than I.

I Have Had My Results For A Long Time: But I Do Not Yet Know How I Am To Arrive At Them.

Ask Her To Wait A Moment - I Am Almost Done.

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.

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.

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.

Mathematicians Stand On Each Other's Shoulders.

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.

Life Stands Before Me Like An Eternal Spring With New And Brilliant Clothes.

Arc, Amplitude, And Curvature Sustain A Similar Relation To Each Other As Time, Motion, And Velocity, Or As Volume, Mass, And Density.

There Have Been Only Three Epoch-making Mathematicians, Archimedes, Newton, And Eisenstein.

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.

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.

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.

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.

Mathematics Is Concerned Only With The Enumeration And Comparison Of Relations.

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.

The Higher Arithmetic Presents Us With An Inexhaustible Store Of Interesting Truths - Of Truths, Too, Which Are Not Isolated, But Stand In A Close Internal Connection, And Between Which, As Our Knowledge Increases, We Are Continually Discovering New And Sometimes Wholly Unexpected Ties.

Mathematics Is The Queen Of Science, And Arithmetic The Queen Of Mathematics.

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.

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.

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.

...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.

It Is Always Noteworthy That All Those Who Seriously Study This Science [the Theory Of Numbers] Conceive A Sort Of Passion For It.

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.

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.

You Have No Idea, How Much Poetry There Is In The Calculation Of A Table Of Logarithms!

Mathematics Is The Queen Of The Sciences.

It Is Not Knowledge, But The Act Of Learning, Not Possession But The Act Of Getting There, Which Grants The Greatest Enjoyment.

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.

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.

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.

Does The Pursuit Of Truth Give You As Much Pleasure As Before? Surely It Is Not The Knowing But The Learning, Not The Possessing But The Acquiring, Not The Being-there But The Getting There That Afford The Greatest Satisfaction. If I Have Exhausted Something, I Leave It In Order To Go Again Into The Dark. Thus Is That Insatiable Man So Strange: When He Has Completed A Structure It Is Not In Order To Dwell In It Comfortably, But To Start Another.

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.

Complete Knowledge Of The Nature Of An Analytic Function Must Also Include Insight Into Its Behavior For Imaginary Values Of The Arguments. Often The Latter Is Indispensable Even For A Proper Appreciation Of The Behavior Of The Function For Real Arguments. It Is Therefore Essential That The Original Determination Of The Function Concept Be Broadened To A Domain Of Magnitudes Which Includes Both The Real And The Imaginary Quantities, On An Equal Footing, Under The Single Designation Complex Numbers.

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.

Mathematical Discoveries, Like Springtime Violets In The Woods, Have Their Season Which No Human Can Hasten Or Retard.

His Second Motto: Thou, Nature, Art My Goddess; To Thy Laws My Services Are Bound.

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.

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