## The Early Mathematics of Leonhard EulerDescribes Euler's early mathematical works - the 50 mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These works contain some of Euler's greatest mathematics: the Konigsburg bridge problem, his solution to the Basel problem, his first proof of the Euler-Fermat theorem. Also presented are important results that we seldom realize are due to Euler: that mixed partial derivatives are equal, our f(x) notation, and the integrating factor in differential equations. The book is a portrait of the world's most exciting mathematics between 1725 and 1741, rich in technical detail, woven with connections within Euler's work and with the work of other mathematicians in other times and places, laced with historical context. |

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### Contents

17251727 I | 1 |

1728 | 13 |

Nova methodus innumerabiles aequationes differentialis secundi | 22 |

17291731 | 31 |

1732 | 65 |

Observationes de theoremate quodam Fermatiano aliisque | 74 |

1733 | 89 |

De solutione problematum Diophanteorum per numeros integros | 102 |

Curvarum maximi minimive proprietate gaudentium inventio | 212 |

1737 | 227 |

Variae observationes circa series infinitas | 249 |

1738 | 269 |

De aequationibus differentialibus quae certis tantum casibus | 279 |

1739 | 299 |

De fractionibus continuis observationes | 306 |

Consideratio progressionis cuiusdam ad circuli quadraturam | 317 |

Constructio aequationis differentialis ax dx dy+y2dx | 114 |

1734 | 123 |

De progressionibus harmonicis observationes | 133 |

Additamentum ad dissertationem de infinitis curvis | 148 |

1735 | 155 |

Methodus universalis serierum convergentium summas quam | 166 |

De constructione aequationum ope motus tractorii aliisque | 176 |

Solutio problematum rectificationem ellipsis requirentium | 188 |

Solutio problematis ad geometriam situs pertinentis | 195 |

1736 | 201 |

Methodus facilis computandi angulorum sinus ac tangentes | 323 |

De seriebus quibusdam considerationes | 342 |

1740 | 349 |

De extractione radicum ex quantitatibus irrationalibus | 357 |

1741 | 365 |

Observationes analyticae variae de combinationibus | 371 |

Commentatio de matheseos sublimioris utilitate | 381 |

Topically Related Articles | 386 |

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### Common terms and phrases

Acad Acta Eruditorum algebraic analysis angle arc length axis Basel Problem binomial biquadratic calculations calculus of variations circle coefficients Commentarii constant of integration continued fractions convergence Corollary curve Daniel Bernoulli decimal places denominators denote derivatives describes differential equation divided ellipse equal Euler begins Euler finds Euler gives Euler notes Euler says Euler takes Euler tells Euler turns Euler-Fermat Theorem Euler-Mascheroni constant evolute example expression factors Fermat Figure formula function geometric series given Goldbach harmonic series infinite product Johann Bernoulli logarithm mathematics method multiplies notation number theory odd numbers Opera Omnia paper Petersburg Petropol polynomial prime numbers progression proof quantities radius of curvature reader reciprocal trajectories relatively prime Riccati equation roots sequence sine solution solve square substitutes tangent Theorem values variables writes wrote zero