The early mathematics of Leonhard Euler : [the MAA tercentenary Euler celebration

The Early Mathematics of Leonhard Euler gives an article-by-article description of Leonhard Euler's early mathematical works; the 50 or so mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These early pieces contain some of Euler's greatest work, the Konigsberg bridge problem, his solution to the Basel problem, and his first proof of the Euler-Fermat theorem. It also presents important results that we seldom realize are due to Euler; that mixed partial derivatives are (usually) equal, our $f(x)$ notation, and the integrating factor in differential equations. The books shows how contributions in diverse fields are related, how number theory relates to series, which, in turn, relate to elliptic integrals and then to differential equations. There are dozens of such strands in this beautiful web of mathematics. At the same time, we see Euler grow in power and sophistication, from a young student when at 18 he published his first work on differential equations (a paper with a serious flaw) to the most celebrated mathematician and scientist of his time. It 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. Contents Preface Interlude: 1725-1727 E-1: Constructio linearum isochronarum in medio quocunque resistente E-3: Methodus inveniendi traiectorias reciprocas algebraicas Interlude: 1728 E-5: Problematis traiectoriarum reciprocarum solutio E-10: Nova methodus innumerabiles aequationes differentialis secundi gradus reducendi ad aequationes differentialis primi gradus Interlude: 1729-1731 E-19: De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeun E-9: De linea brevissima in superficie quacunque duo quaelibet puncta iungente E-20: De summatione innumerabilium progressionum Interlude: 1732 E-25: Methodus generalis summandi progressiones E-26: Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus E-27: Problematis isoperimetrici in latissimo sensu accepti solutio generalis Interlude: 1733 E-11: Constructio aequationum quarundam differentialium quae indeterminatarum separationem non admittunt E-28: Specimen de constructione aequationum differentialium sine indeterminatarum separatione E-29: De solutione problematum Diophanteorum per numeros integros E-30: De formis radicum aequationum cuiusque ordinis conjectatio E-31: Constructio aequationis differentialis axn dx = dy + y2 dx* Interlude: 1734 E-42: De linea celerrimi descensus in medio quocunque resistente E-43: De progressionibus harmonicis observationes E-44: De infinitis curvis eiusdem generis seu methodus inveniendi aequationes pro infinitis curvis eiusdem generis E-45: Additamentum ad dissertationem de infinitis curvis eiusdem generis E-48: Investigatio binarum curvarum, quarum arcus eidem abscissae respondents summam algebraicam constituant* Interlude: 1735 E-41: De summis serierum reciprocarum E-46: Methodus universalis serierum convergentium summas quam proxime inveniendi E-47: Inventio summae cuiusque seriei ex dato termino general! E-51: De constructione aequationum ope motus tractorii aliisque ad methodum tangentium inversam pertinentibus E-52: Solutio problematum rectificationem ellipsis requirentium E-53: Solutio problematis ad geometriam situs pertinentis Interlude: 1736 E-54: Theorematum quorundam ad numeros primos spectantium demonstratio E-55: Methodus universalis series summandi ulterius promota E-56: Curvarum maximi minimive proprietate gaudentium inventio nova et facilis Interlude: 1737 E-70: De constructione aequationum E-71: De fractionibus continuis dissertatio E-72: Variae observationes circa series infinitas E-73: Solutio problematis geometrici circa lunulas a circulis formatas Interlude: 1738 E-23: De curvis rectificabilibus algebraicis atque traiectoriis reciprocis algebraicis E-74: De variis modis circuli quadraturam numeris proxime exprimendi E-95: De aequationibus differentialibus, quae certis tantum casibusintegrationem admittunt E-98: Theorematum quorundam arithmeticorum demonstrationes E-99: Solutio problematis cuiusdam a celeberrimo Daniele Bernoullio propositi Interlude: 1739 E-122: De productis ex infinitis factoribus ortis E-123 : De fractionibus continuis observationes E-125: Consideratio progressions cuiusdam ad circuli quadraturam inveniendam idoneae E-128: Methodus facilis computandi angulorum sinus ac tangentes tam naturales quam artificiales E-129: Investigatio curvarum quae evolutae sui similes producunt E-130: De seriebus quibusdam considerationes Interlude: 1740 E-36: Solutio problematis arithmetici de inveniendo numero, qui per datos numeros divisus, relinquat data residua E-157: De extractione radicum ex quantitatibus irrationalibus Interlude: 1741 E-63: Demonstration de la somme de cette suite E-158: Observationes analyticae variae de combinationibus E-790: Commentatio de matheseos sublimioris utilitate Topically Related Articles Index About the Author "The Early Mathematics of Leonhard Euler describes 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. It also presents 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 provides some of the way mathematics is actually done. For example, Euler found partial results towards the Euler-Fermat theorem well before he discovered a proof of the Fermat theorem itself, and the Euler-Fermat version came 30 years later, beyond the scope of this book. The book shows how results in diverse fields are related, how number theory relates to series, which, in turn relate to elliptic integrals and then to differential equations, There are dozens of such strands in this beautiful web of mathematics. At the same time, we see Euler grow in power and sophistication, from his first work on differential equations as an 18-year old student, a paper with a serious flaw in it, to the most celebrated mathematician and scientist of his times, when, at the age of 34, he was lured away like a superstar athlete might be traded today. 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."--Publisher's website