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Articles 291601 - 291630 of 291657

Full-Text Articles in Physical Sciences and Mathematics

Inquisitio Physica In Causam Fluxus Ac Refluxus Maris, Leonhard Euler Dec 1740

Inquisitio Physica In Causam Fluxus Ac Refluxus Maris, Leonhard Euler

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No abstract provided.

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Methodus Universalis Serierum Convergentium Summas Quam Proxime Inveniendi, Leonhard Euler Dec 1740

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 Dec 1740

Methodus Computandi Aequationem Meridiei, Leonhard Euler

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No abstract provided.

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Solutio Problematum Rectivicationem Ellipsis Requirentium, Leonhard Euler Dec 1740

Solutio Problematum Rectivicationem Ellipsis Requirentium, Leonhard Euler

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No abstract provided.

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Theorematum Quorundam Ad Numeros Primos Spectantium Demonstratio, Leonhard Euler Dec 1740

Theorematum Quorundam Ad Numeros Primos Spectantium Demonstratio, Leonhard Euler

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No abstract provided.

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

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

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No abstract provided.

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Additamentum Ad Dissertationem De Infinitis Curvis Eiusdem Generis, Leonhard Euler Dec 1739

Additamentum Ad Dissertationem De Infinitis Curvis Eiusdem Generis, Leonhard Euler

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No abstract provided.

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Solutio Problematis Arithmetici De Inveniendo Numero, Qui Per Datos Numeros Divisus Relinquat Data Residua, Leonhard Euler Dec 1739

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 Dec 1739

Solutio Problematum Quorundam Astronomicorum, Leonhard Euler

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No abstract provided.

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De Infinitis Curvis Eiusdem Generis Seu Methodus Inveniendi Aequationes Pro Infinitis Curvis Eiusdem Generis, Leonhard Euler Dec 1739

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

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No abstract provided.

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De Linea Celerrimi Descensus In Medio Quocunque Resistente, Leonhard Euler Dec 1739

De Linea Celerrimi Descensus In Medio Quocunque Resistente, Leonhard Euler

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No abstract provided.

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De Motu Planetarum Et Orbitarum Determinatione, Leonhard Euler Dec 1739

De Motu Planetarum Et Orbitarum Determinatione, Leonhard Euler

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No abstract provided.

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Orbitae Solaris Determinatio, Leonhard Euler Dec 1739

Orbitae Solaris Determinatio, Leonhard Euler

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No abstract provided.

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De Minimis Oscillationibus Corporum Tam Rigidorum Quam Flexibilium. Methodus Nova Et Facilis., Leonhard Euler Dec 1739

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

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No abstract provided.

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De Summis Serierum Reciprocarum, Leonhard Euler Dec 1739

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 En. 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 Dec 1739

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 Dec 1738

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 Dec 1738

Tentamen Novae Theoriae Musicae, Leonhard Euler

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No abstract provided.

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De Communicatione Motus In Collisione Corporum, Leonhard Euler Dec 1737

De Communicatione Motus In Collisione Corporum, Leonhard Euler

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No abstract provided.

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De Curvis Rectificabilibus Algebraicis Atque Traiectoriis Reciprocis Algebraicis, Leonhard Euler Dec 1737

De Curvis Rectificabilibus Algebraicis Atque Traiectoriis Reciprocis Algebraicis, Leonhard Euler

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No abstract provided.

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Solutio Singularis Casus Circa Tautochronismum, Leonhard Euler Dec 1737

Solutio Singularis Casus Circa Tautochronismum, Leonhard Euler

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No abstract provided.

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Problematis Isoperimetrici In Latissimo Sensu Accepti Solutio Generalis, Leonhard Euler Dec 1737

Problematis Isoperimetrici In Latissimo Sensu Accepti Solutio Generalis, Leonhard Euler

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No abstract provided.

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De Solutione Problematum Diophanteorum Per Numeros Integros, Leonhard Euler Dec 1737

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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De Progressionibus Transcendentibus Seu Quarum Termini Generales Algebraice Dari Nequeunt, Leonhard Euler Dec 1737

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

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No abstract provided.

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Quomodo Data Quacunque Curva Inveniri Oporteat Aliam Quae Cum Data Quodammodo Iuncta Ad Tautochronismum Producendum Sit Idonea, Leonhard Euler Dec 1737

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

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No abstract provided.

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De Formis Radicum Aequationum Cuiusque Ordinis Coniectatio, Leonhard Euler Dec 1737

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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Von Der Gestalt Der Erden, Leonhard Euler Dec 1737

Von Der Gestalt Der Erden, Leonhard Euler

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No abstract provided.

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De Indorum Anno Solari Astronomico, Leonhard Euler Dec 1737

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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De Summatione Innumerabilium Progressionum, Leonhard Euler Dec 1737

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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Methodus Generalis Summandi Progressiones, Leonhard Euler Dec 1737

Methodus Generalis Summandi Progressiones, Leonhard Euler

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This paper leads to Bernoulli numbers from an integral of an infinite series and is called a beautiful triumph by Euler.

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