Physical Sciences and Mathematics | Open Access Articles

Seventeen Letters From Euler To Johann I Bernoulli, 1727-1740, Leonhard Euler

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Solutio Problematum Quorundam Astronomicorum, Leonhard Euler

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Solutio Problematis Arithmetici De Inveniendo Numero, Qui Per Datos Numeros Divisus Relinquat Data Residua, Leonhard Euler

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Euler proves the Chinese Remainder Theorem by constructing an algorithm to find the smallest number which, divided by given numbers, leaves given remainders. He begins by solving the case in which two relatively prime divisors with corresponding remainders are given and proposes that by repeating his algorithm, he can solve similar problems with any number of constraints. Euler then discusses scenarios in which divisors are not relatively prime, and ends the paper with an application of his algorithm to a classic problem: dating events in Roman indictions.

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Additamentum Ad Dissertationem De Infinitis Curvis Eiusdem Generis, Leonhard Euler

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De Infinitis Curvis Eiusdem Generis Seu Methodus Inveniendi Aequationes Pro Infinitis Curvis Eiusdem Generis, Leonhard Euler

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De Progressionibus Harmonicis Observationes, Leonhard Euler

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Euler gives the now-named Euler-Mascheroni constant, accurate to five decimal places, and examines several series related to log(n).

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De Minimis Oscillationibus Corporum Tam Rigidorum Quam Flexibilium. Methodus Nova Et Facilis., Leonhard Euler

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Dissertatio De Igne In Qua Ejus Natura Et Proprietates Explicantur, Leonhard Euler

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Euler argues that fire is the result of the bursting of tiny glassy balls of highly compressed air in the pores of bodies, so that "heat consists in a certain motion of the smallest particles of a body." Thus, all the phenomena associated with heat and fire can be deduced from the laws of mechanics without supposing any "occult qualities." He also says that light is the elastic vibration of the ether that is initiated by the explosions of little balls; hence, light is propagated by the same laws as sound.

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Tentamen Novae Theoriae Musicae, Leonhard Euler

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Problematis Isoperimetrici In Latissimo Sensu Accepti Solutio Generalis, Leonhard Euler

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Quomodo Data Quacunque Curva Inveniri Oporteat Aliam Quae Cum Data Quodammodo Iuncta Ad Tautochronismum Producendum Sit Idonea, Leonhard Euler

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De Solutione Problematum Diophanteorum Per Numeros Integros, Leonhard Euler

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Euler searches for integer solutions to axx+bx+c=yy and considers some applications to figurate numbers.

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Constructio Aequationis Differentialis AxN Dx = Dy + Y2 Dx, Leonhard Euler

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De Progressionibus Transcendentibus Seu Quarum Termini Generales Algebraice Dari Nequeunt, Leonhard Euler

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De Summatione Innumerabilium Progressionum, Leonhard Euler

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This paper concerns the sum of reciprocal squares, which equals π²/6. Euler does not yet have the tools to find this value directly, but instead approximates it as 1.644934. He says this follows from E25 and E19, and also refers us forward to E736. Then Euler brings in the harmonic series: letting f(x) denote the xth partial sum of the harmonic series, he approximates it as an integral and defines his constant γ as the limit of f(x) – log(x).

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De Communicatione Motus In Collisione Corporum, Leonhard Euler

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Solutio Singularis Casus Circa Tautochronismum, Leonhard Euler

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Observationes De Theoremate Quodam Fermatiano Aliisque Ad Numeros Primos Spectantibus, Leonhard Euler

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Euler shows that the fifth Fermat number, 2^2⁵ +1 = 4,294,967,297, is not prime because it is divisible by 641, though he does not give any clues about how he discovered this fact. He also tacks on a few "theorems" but says that he does not yet know how to prove them.

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De Curvis Rectificabilibus Algebraicis Atque Traiectoriis Reciprocis Algebraicis, Leonhard Euler

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De Formis Radicum Aequationum Cuiusque Ordinis Coniectatio, Leonhard Euler

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For an equation of degree n, Euler wants to define a "resolvent equation" of degree n-1 whose roots are related to the roots of the original equation. Thus, by solving the resolvent one can solve the original equation. In sections 2 to 7 he works this out for quadratic, cubic, and biquadratic equations. In section 8 Euler says that he wants to try the same approach for solving the quintic equation and general nth degree equations. In the rest of the paper he tries to figure out in what cases resolvents will work.

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Specimen De Constructione Aequationum Differentialium Sine Indeterminatarum Separatione, Leonhard Euler

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In this paper, Euler investigates a differential equation that he encountered in finding the arc length of an ellipse. This differential equation cannot be solved by separation of variables, as is indicated in the title of the article. Euler first develops a formula for the arc length of an ellipse by cleverly manipulating a binomial series, then shows that this formula satisfies the desired differential equation. Integrating factors make a brief appearance.

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De Indorum Anno Solari Astronomico, Leonhard Euler

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This work is an appendix following two other appendices in a book by Euler's friend and St. Petersbrg Academy colleague T. S. Bayer, Historia regni Graecorum Bactriani (History of the Bactrian kingdom of the Greeks). The original appendices were written by a Danish missionary in Tranquebar, C. T. Walther ("The Indian Doctrine of Time," pp. 145-190), and by Bayer himself, based on his correspondence with Walther and other Tranquebar missionaries ("Supplement to the Indian Doctrine of Time," pp. 191-200). Euler's contribution appears immediately after these.

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Mechanica, Volume 2, Leonhard Euler

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Mechanica (this volume, along with E15) is Euler's outline of a program of studies embracing every branch of science, involving a systematic application of analysis. It laid the foundations of analytical mechanics, the result of Euler's consideration of the motion produced by forces acting on both free and constrained points. It was also the first published work in which the number e appeared. In this volume, Euler considers motion of a point-mass lying on a given curve or surface. He derives some differential equations of the geodesics governing the problem of free motion on a surface. In this way, …

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Mechanica, Volume 1, Leonhard Euler

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Mechanica (this volume, along with E16) is Euler's outline of a program of studies embracing every branch of science, involving a systematic application of analysis. It laid the foundations of analytical mechanics, the result of Euler's consideration of the motion produced by forces acting on both free and constrained points. It was also the first published work in which the number e appeared. This volume focuses on the kinematics and dynamics of a point-mass, introducing infinitely small bodies that can be considered to be points under certain assumptions. Euler focuses on single mass-points except for a few pages at …

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Curva Tautochrona In Fluido Resistentiam Faciente Secundum Quadrata Celeritatum, Leonhard Euler

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De Innumerabilibus Curvis Tautochronis In Vacuo, Leonhard Euler

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Solutio Problematis Astronomici Ex Datis Tribus Stellae Fixae Altitudinibus Et Temporum Differentiis Invenire Elevationem Poli Et Declinationem Stellae. Auct. L. Eulero, Leonhard Euler

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Constructio Aequationum Quarundam Differentialium, Quae Indeterminatarum Separationem Non Admittunt, Leonhard Euler

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Nova Methodus Innumerabiles Aequationes Differentiales Secundi Gradus Reducendi Ad Aequationes Differentiales Primi Gradus, Leonhard Euler

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Solutio Problematis De Invenienda Curva, Quam Format Lamina Utcunque Elastica In Singulis Punctis A Potentiis Quibuscunque Sollicitata, Leonhard Euler

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Full-Text Articles in Physical Sciences and Mathematics

Seventeen Letters From Euler To Johann I Bernoulli, 1727-1740, Leonhard Euler

All Works by Eneström Number

Solutio Problematum Quorundam Astronomicorum, Leonhard Euler

All Works by Eneström Number

Solutio Problematis Arithmetici De Inveniendo Numero, Qui Per Datos Numeros Divisus Relinquat Data Residua, Leonhard Euler

All Works by Eneström Number

Additamentum Ad Dissertationem De Infinitis Curvis Eiusdem Generis, Leonhard Euler

All Works by Eneström Number

De Infinitis Curvis Eiusdem Generis Seu Methodus Inveniendi Aequationes Pro Infinitis Curvis Eiusdem Generis, Leonhard Euler

All Works by Eneström Number

De Progressionibus Harmonicis Observationes, Leonhard Euler

All Works by Eneström Number

De Minimis Oscillationibus Corporum Tam Rigidorum Quam Flexibilium. Methodus Nova Et Facilis., Leonhard Euler

All Works by Eneström Number

Dissertatio De Igne In Qua Ejus Natura Et Proprietates Explicantur, Leonhard Euler

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Tentamen Novae Theoriae Musicae, Leonhard Euler

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Problematis Isoperimetrici In Latissimo Sensu Accepti Solutio Generalis, Leonhard Euler

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Quomodo Data Quacunque Curva Inveniri Oporteat Aliam Quae Cum Data Quodammodo Iuncta Ad Tautochronismum Producendum Sit Idonea, Leonhard Euler

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De Solutione Problematum Diophanteorum Per Numeros Integros, Leonhard Euler

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Constructio Aequationis Differentialis AxN Dx = Dy + Y2 Dx, Leonhard Euler

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De Progressionibus Transcendentibus Seu Quarum Termini Generales Algebraice Dari Nequeunt, Leonhard Euler

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De Summatione Innumerabilium Progressionum, Leonhard Euler

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De Communicatione Motus In Collisione Corporum, Leonhard Euler

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Solutio Singularis Casus Circa Tautochronismum, Leonhard Euler

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Observationes De Theoremate Quodam Fermatiano Aliisque Ad Numeros Primos Spectantibus, Leonhard Euler

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De Curvis Rectificabilibus Algebraicis Atque Traiectoriis Reciprocis Algebraicis, Leonhard Euler

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De Formis Radicum Aequationum Cuiusque Ordinis Coniectatio, Leonhard Euler

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Specimen De Constructione Aequationum Differentialium Sine Indeterminatarum Separatione, Leonhard Euler

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De Indorum Anno Solari Astronomico, Leonhard Euler

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Mechanica, Volume 2, Leonhard Euler

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Mechanica, Volume 1, Leonhard Euler

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Curva Tautochrona In Fluido Resistentiam Faciente Secundum Quadrata Celeritatum, Leonhard Euler

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De Innumerabilibus Curvis Tautochronis In Vacuo, Leonhard Euler

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Solutio Problematis Astronomici Ex Datis Tribus Stellae Fixae Altitudinibus Et Temporum Differentiis Invenire Elevationem Poli Et Declinationem Stellae. Auct. L. Eulero, Leonhard Euler

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Constructio Aequationum Quarundam Differentialium, Quae Indeterminatarum Separationem Non Admittunt, Leonhard Euler

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Nova Methodus Innumerabiles Aequationes Differentiales Secundi Gradus Reducendi Ad Aequationes Differentiales Primi Gradus, Leonhard Euler

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Solutio Problematis De Invenienda Curva, Quam Format Lamina Utcunque Elastica In Singulis Punctis A Potentiis Quibuscunque Sollicitata, Leonhard Euler

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