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Articles 811 - 831 of 831
Full-Text Articles in Mathematics
De Formis Radicum Aequationum Cuiusque Ordinis Coniectatio, Leonhard Euler
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.
Von Der Gestalt Der Erden, Leonhard Euler
Von Der Gestalt Der Erden, Leonhard Euler
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No abstract provided.
Methodus Generalis Summandi Progressiones, Leonhard Euler
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.
Observationes De Theoremate Quodam Fermatiano Aliisque Ad Numeros Primos Spectantibus, Leonhard Euler
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.
Specimen De Constructione Aequationum Differentialium Sine Indeterminatarum Separatione, Leonhard Euler
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.
Mechanica, Volume 2, Leonhard Euler
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, …
Mechanica, Volume 1, Leonhard Euler
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 …
Curva Tautochrona In Fluido Resistentiam Faciente Secundum Quadrata Celeritatum, Leonhard Euler
Curva Tautochrona In Fluido Resistentiam Faciente Secundum Quadrata Celeritatum, Leonhard Euler
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No abstract provided.
De Innumerabilibus Curvis Tautochronis In Vacuo, Leonhard Euler
De Innumerabilibus Curvis Tautochronis In Vacuo, Leonhard Euler
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No abstract provided.
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.
Constructio Aequationum Quarundam Differentialium, Quae Indeterminatarum Separationem Non Admittunt, Leonhard Euler
Constructio Aequationum Quarundam Differentialium, Quae Indeterminatarum Separationem Non Admittunt, Leonhard Euler
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No abstract provided.
Solutio Problematis De Invenienda Curva, Quam Format Lamina Utcunque Elastica In Singulis Punctis A Potentiis Quibuscunque Sollicitata, Leonhard Euler
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.
Nova Methodus Innumerabiles Aequationes Differentiales Secundi Gradus Reducendi Ad Aequationes Differentiales Primi Gradus, Leonhard Euler
Nova Methodus Innumerabiles Aequationes Differentiales Secundi Gradus Reducendi Ad Aequationes Differentiales Primi Gradus, Leonhard Euler
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No abstract provided.
De Linea Brevissima In Superficie Quacunque Duo Quaelibet Puncta Iungente, Leonhard Euler
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.
Problematis Traiectoriarum Reciprocarum Solutio, Leonhard Euler
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.
Dissertatio De Novo Quodam Curvarum Tautochronarum Genere, Leonhard Euler
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."
Tentamen Explicationis Phaenomenorum Aeris, Leonhard Euler
Tentamen Explicationis Phaenomenorum Aeris, Leonhard Euler
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No abstract provided.
Meditationes Super Problemate Nautico, Quod Illustrissima Regia Parisiensis Academia Scientiarum Proposuit, Leonhard Euler
Meditationes Super Problemate Nautico, Quod Illustrissima Regia Parisiensis Academia Scientiarum Proposuit, Leonhard Euler
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No abstract provided.
Methodus Inveniendi Traiectorias Reciprocas Algebraicas, Leonhard Euler
Methodus Inveniendi Traiectorias Reciprocas Algebraicas, Leonhard Euler
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No abstract provided.
Dissertatio Physica De Sono, Leonhard Euler
Dissertatio Physica De Sono, Leonhard Euler
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Euler first explains his theory of what makes up the atmosphere; the basis of his theory lies in the theory of elasticity of the teacher he had before Johann Bernoulli. He also states without proof a formula for the speed of propagation and derives from it numerical values of the correct order of magnitude for air.
Constructio Linearum Isochronarum In Medio Quocunque Resistente, Leonhard Euler
Constructio Linearum Isochronarum In Medio Quocunque Resistente, Leonhard Euler
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No abstract provided.