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Démonstration De La Somme De Cette Suite 1 + 1/4 + 1/9 + 1/16 + ..., Leonhard Euler

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Leonhardus Eulerus Mathematicus Acutissimus Ad Auctorem, Leonhard Euler

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De Oscillationibus Fili Flexilis Quotcunque Pondusculis Onusti, Leonhard Euler

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De Constructione Aequationum Ope Motus Tractorii Aliisque Ad Methodum Tangentium Inversam Pertinentibus, Leonhard Euler

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Solutio Problematis Ad Geometriam Situs Pertinentis, Leonhard Euler

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Inventio Summae Cuiusque Seriei Ex Dato Termino Generali, Leonhard Euler

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Euler continues with the methods of E25 to attack ζ(2) for a second time. He starts with a Taylor series, builds a "Bernoulli polynomial" and uses it to evaluate 0ⁿ + 1ⁿ + 2ⁿ + 3ⁿ + ... + (x-1)ⁿ, (x = 1, 2, 3, ...) and gets the relationship (B+1)ⁿ⁺¹ – Bⁿ⁺¹ = 0 for Bernoulli numbers. He gets an infinite series approximation for the nth partial sum of the harmonic series.

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Solutio Problematum Rectivicationem Ellipsis Requirentium, Leonhard Euler

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Theorematum Quorundam Ad Numeros Primos Spectantium Demonstratio, Leonhard Euler

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Curvarum Maximi Minive Proprietate Gaudientium Inventio Nova Et Facilis, Leonhard Euler

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Investigatio Binarum Curvarum, Quarum Arcus Eidem Abscissae Respondentes Summam Algebraicam Constituant, Leonhard Euler

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Methodus Universalis Series Summandi Ulterius Promota, Leonhard Euler

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Inquisitio Physica In Causam Fluxus Ac Refluxus Maris, Leonhard Euler

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Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, Leonhard Euler

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This paper begins with an expression of the trapezoid rule for "formal mechanical quadrature." Euler also sums the first ten terms of ζ(2) to get 1.549768 and gives an expression for the error term. Then he determines the sum of the first million terms of the harmonic series to be 14.392669.

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Methodus Computandi Aequationem Meridiei, Leonhard Euler

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

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Additamentum Ad Dissertationem De Infinitis Curvis Eiusdem Generis, Leonhard Euler

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De Minimis Oscillationibus Corporum Tam Rigidorum Quam Flexibilium. Methodus Nova Et Facilis., 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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Solutio Problematum Quorundam Astronomicorum, 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 Linea Celerrimi Descensus In Medio Quocunque Resistente, Leonhard Euler

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De Motu Planetarum Et Orbitarum Determinatione, Leonhard Euler

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Orbitae Solaris Determinatio, Leonhard Euler

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De Summis Serierum Reciprocarum, Leonhard Euler

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Euler finds the values of ζ(2n), where ζ is now named as the Riemann zeta function. He introduces the Euler numbers E_n. All the odd-indexed Euler numbers are zero, E₀=1, E₂=-1, E₄=5, E₆=-61, E₈=1380, and they satisfy (E+1)ⁿ+(E-1)ⁿ = 0. (The Euler numbers are the coefficients of the series expansion of sech(x) and are related to Bernoulli numbers.) Among other things, this paper includes an infinite product formula for sin(x)/x.

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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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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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De Communicatione Motus In Collisione Corporum, Leonhard Euler

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De Curvis Rectificabilibus Algebraicis Atque Traiectoriis Reciprocis Algebraicis, Leonhard Euler

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Solutio Singularis Casus Circa Tautochronismum, Leonhard Euler

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Full-Text Articles in Entire DC Network

Démonstration De La Somme De Cette Suite 1 + 1/4 + 1/9 + 1/16 + ..., Leonhard Euler

All Works by Eneström Number

Leonhardus Eulerus Mathematicus Acutissimus Ad Auctorem, Leonhard Euler

All Works by Eneström Number

De Oscillationibus Fili Flexilis Quotcunque Pondusculis Onusti, Leonhard Euler

All Works by Eneström Number

De Constructione Aequationum Ope Motus Tractorii Aliisque Ad Methodum Tangentium Inversam Pertinentibus, Leonhard Euler

All Works by Eneström Number

Solutio Problematis Ad Geometriam Situs Pertinentis, Leonhard Euler

All Works by Eneström Number

Inventio Summae Cuiusque Seriei Ex Dato Termino Generali, Leonhard Euler

All Works by Eneström Number

Solutio Problematum Rectivicationem Ellipsis Requirentium, Leonhard Euler

All Works by Eneström Number

Theorematum Quorundam Ad Numeros Primos Spectantium Demonstratio, Leonhard Euler

All Works by Eneström Number

Curvarum Maximi Minive Proprietate Gaudientium Inventio Nova Et Facilis, Leonhard Euler

All Works by Eneström Number

Investigatio Binarum Curvarum, Quarum Arcus Eidem Abscissae Respondentes Summam Algebraicam Constituant, Leonhard Euler

All Works by Eneström Number

Methodus Universalis Series Summandi Ulterius Promota, Leonhard Euler

All Works by Eneström Number

Inquisitio Physica In Causam Fluxus Ac Refluxus Maris, Leonhard Euler

All Works by Eneström Number

Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, Leonhard Euler

All Works by Eneström Number

Methodus Computandi Aequationem Meridiei, Leonhard Euler

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

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Additamentum Ad Dissertationem De Infinitis Curvis Eiusdem Generis, 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

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

All Works by Eneström Number

Solutio Problematum Quorundam Astronomicorum, Leonhard Euler

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

All Works by Eneström Number

De Linea Celerrimi Descensus In Medio Quocunque Resistente, Leonhard Euler

All Works by Eneström Number

De Motu Planetarum Et Orbitarum Determinatione, Leonhard Euler

All Works by Eneström Number

Orbitae Solaris Determinatio, Leonhard Euler

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De Summis Serierum Reciprocarum, Leonhard Euler

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De Progressionibus Harmonicis Observationes, Leonhard Euler

All Works by Eneström Number

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

All Works by Eneström Number

Tentamen Novae Theoriae Musicae, Leonhard Euler

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De Communicatione Motus In Collisione Corporum, Leonhard Euler

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De Curvis Rectificabilibus Algebraicis Atque Traiectoriis Reciprocis Algebraicis, Leonhard Euler

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Solutio Singularis Casus Circa Tautochronismum, Leonhard Euler

All Works by Eneström Number