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Articles 288631 - 288660 of 288673

Full-Text Articles in Physical Sciences and Mathematics

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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Observationes De Theoremate Quodam Fermatiano Aliisque Ad Numeros Primos Spectantibus, Leonhard Euler Dec 1737

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

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Euler shows that the fifth Fermat number, 225 +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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Specimen De Constructione Aequationum Differentialium Sine Indeterminatarum Separatione, Leonhard Euler Dec 1737

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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Constructio Aequationis Differentialis AxN Dx = Dy + Y2 Dx, Leonhard Euler Dec 1737

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

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No abstract provided.

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Mechanica, Volume 2, Leonhard Euler Dec 1735

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 Dec 1735

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 Dec 1734

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

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No abstract provided.

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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 Dec 1734

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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No abstract provided.

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De Innumerabilibus Curvis Tautochronis In Vacuo, Leonhard Euler Dec 1734

De Innumerabilibus Curvis Tautochronis In Vacuo, Leonhard Euler

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No abstract provided.

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Constructio Aequationum Quarundam Differentialium, Quae Indeterminatarum Separationem Non Admittunt, Leonhard Euler Dec 1732

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

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No abstract provided.

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Solutio Problematis De Invenienda Curva, Quam Format Lamina Utcunque Elastica In Singulis Punctis A Potentiis Quibuscunque Sollicitata, Leonhard Euler Dec 1731

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

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No abstract provided.

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Nova Methodus Innumerabiles Aequationes Differentiales Secundi Gradus Reducendi Ad Aequationes Differentiales Primi Gradus, Leonhard Euler Dec 1731

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

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No abstract provided.

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De Linea Brevissima In Superficie Quacunque Duo Quaelibet Puncta Iungente, Leonhard Euler Dec 1731

De Linea Brevissima In Superficie Quacunque Duo Quaelibet Puncta Iungente, Leonhard Euler

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This article arose out of a homework assignment that Euler did for Johann Bernoulli. Bernoulli asked Euler to find the shortest line between two points on a surface. This work provided some of the first analytical foundations for the calculus of variations.

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Problematis Traiectoriarum Reciprocarum Solutio, Leonhard Euler Dec 1728

Problematis Traiectoriarum Reciprocarum Solutio, Leonhard Euler

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This article contains Euler's first published use of complex numbers and a many-axis geometric construction. Euler also defines even functions f(x) as those for which f(x) = f(-x), perhaps the first use of this term.

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Dissertatio De Novo Quodam Curvarum Tautochronarum Genere, Leonhard Euler Dec 1728

Dissertatio De Novo Quodam Curvarum Tautochronarum Genere, Leonhard Euler

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According to Eneström: "Intrigued by a clock invented by D. Sully, Euler busied himself with the construction of curves which cause a pendulum to swing isochronally."

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