Introductio in analysin infinitorum (Latin:[1]Introduction to the Analysis of the Infinite) is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the Introductio contains 18 chapters in the first part and 22 chapters in the second. It has Eneström numbers E101 and E102.[2][3]
Euler's numbere corresponds to shaded area equal to 1, introduced in chapter VII
The Introduction is meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of the differential and integral calculus. [Euler] made of this survey a masterly exercise in introducing as much as possible of analysis without using differentiation or integration. In particular, he introduced the elementary transcendental functions, the logarithm, the exponential function, the trigonometric functions and their inverses without recourse to integral calculus — which was no mean feat, as the logarithm was traditionally linked to the quadrature of the hyperbola and the trigonometric functions to the arc-length of the circle.[4]
Euler accomplished this feat by introducing exponentiationax for arbitrary constant a in the positive real numbers. He noted that mapping x this way is not an algebraic function, but rather a transcendental function. For a > 1 these functions are monotonic increasing and form bijections of the real line with positive real numbers. Then each base a corresponds to an inverse function called the logarithm to base a, in chapter 6. In chapter 7, Euler introduces e as the number whose hyperbolic logarithm is 1. The reference here is to Gregoire de Saint-Vincent who performed a quadrature of the hyperbola y = 1/x through description of the hyperbolic logarithm. Section 122 labels the logarithm to base e the "natural or hyperbolic logarithm...since the quadrature of the hyperbola can be expressed through these logarithms". Here he also gives the exponential series:
Then in chapter 8 Euler is prepared to address the classical trigonometric functions as "transcendental quantities that arise from the circle." He uses the unit circle and presents Euler's formula. Chapter 9 considers trinomial factors in polynomials. Chapter 16 is concerned with partitions, a topic in number theory. Continued fractions are the topic of chapter 18.
The analysis of Euler comes close to the modern orthodox discipline, the study of functions by means of infinite processes, especially through infinite series.
It is doubtful that any other essentially didactic work includes as large a portion of original material that survives in the college courses today...Can be read with comparative ease by the modern student...The prototype of modern textbooks.
English translationsedit
The first translation into English was that by John D. Blanton, published in 1988.[6] The second, by Ian Bruce, is available online.[7] A list of the editions of Introductio has been assembled by V. Frederick Rickey.[8]
Early mentionsedit
Page from Introductio in analysin infinitorum, 1748
J.C. Scriba (2007) review of 1983 reprint of 1885 German edition MR715928
^In Latin, analysis was a Neo-Latin borrowing from Greek, and the word form analysin uses the Greek accusative. Calinger, Ronald (2016). Leonhard Euler: Mathematical Genius in the Enlightenment. Princeton University Press. pp. 287–288. ISBN978-0-691-11927-4.
^"E101 -- Introductio in analysin infinitorum, volume 1". The Euler Archive. Retrieved 2020-10-15.
^"E102 -- Introductio in analysin infinitorum, volume 2". The Euler Archive. Retrieved 2020-10-15.
^H. J. M. Bos (1980) "Newton, Leibnitz and the Leibnizian tradition", chapter 2, pages 49–93, quote page 76, in From the Calculus to Set Theory, 1630 – 1910: An Introductory History, edited by Ivor Grattan-Guinness, DuckworthISBN0-7156-1295-6
^V. Frederick Rickey A Reader’s Guide to Euler’s Introductio
December 18, 2023
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Introductio in analysin infinitorum Latin 1 Introduction to the Analysis of the Infinite is a two volume work by Leonhard Euler which lays the foundations of mathematical analysis Written in Latin and published in 1748 the Introductio contains 18 chapters in the first part and 22 chapters in the second It has Enestrom numbers E101 and E102 2 3 Euler s number e corresponds to shaded area equal to 1 introduced in chapter VII Contents 1 Contents 2 Impact 3 English translations 4 Early mentions 5 Reviews of Blanton translation 1988 6 ReferencesContents editChapter 1 is on the concepts of variables and functions Chapter 4 introduces infinite series through rational functions According to Henk Bos The Introduction is meant as a survey of concepts and methods in analysis and analytic geometry preliminary to the study of the differential and integral calculus Euler made of this survey a masterly exercise in introducing as much as possible of analysis without using differentiation or integration In particular he introduced the elementary transcendental functions the logarithm the exponential function the trigonometric functions and their inverses without recourse to integral calculus which was no mean feat as the logarithm was traditionally linked to the quadrature of the hyperbola and the trigonometric functions to the arc length of the circle 4 Euler accomplished this feat by introducing exponentiation ax for arbitrary constant a in the positive real numbers He noted that mapping x this way is not an algebraic function but rather a transcendental function For a gt 1 these functions are monotonic increasing and form bijections of the real line with positive real numbers Then each base a corresponds to an inverse function called the logarithm to base a in chapter 6 In chapter 7 Euler introduces e as the number whose hyperbolic logarithm is 1 The reference here is to Gregoire de Saint Vincent who performed a quadrature of the hyperbola y 1 x through description of the hyperbolic logarithm Section 122 labels the logarithm to base e the natural or hyperbolic logarithm since the quadrature of the hyperbola can be expressed through these logarithms Here he also gives the exponential series exp z k 0 z k k 1 z z 2 2 z 3 6 z 4 24 displaystyle exp z sum k 0 infty z k over k 1 z z 2 over 2 z 3 over 6 z 4 over 24 cdots nbsp Then in chapter 8 Euler is prepared to address the classical trigonometric functions as transcendental quantities that arise from the circle He uses the unit circle and presents Euler s formula Chapter 9 considers trinomial factors in polynomials Chapter 16 is concerned with partitions a topic in number theory Continued fractions are the topic of chapter 18 Impact editCarl Benjamin Boyer s lectures at the 1950 International Congress of Mathematicians compared the influence of Euler s Introductio to that of Euclid s Elements calling the Elements the foremost textbook of ancient times and the Introductio the foremost textbook of modern times 5 Boyer also wrote The analysis of Euler comes close to the modern orthodox discipline the study of functions by means of infinite processes especially through infinite series It is doubtful that any other essentially didactic work includes as large a portion of original material that survives in the college courses today Can be read with comparative ease by the modern student The prototype of modern textbooks English translations editThe first translation into English was that by John D Blanton published in 1988 6 The second by Ian Bruce is available online 7 A list of the editions of Introductio has been assembled by V Frederick Rickey 8 Early mentions edit nbsp Page from Introductio in analysin infinitorum 1748J C Scriba 2007 review of 1983 reprint of 1885 German edition MR715928Reviews of Blanton translation 1988 editDoru Stefanescu MR1025504 Marco Panza 2007 MR2384380 Ricardo Quintero Zazueta 1999 MR1823258 Ernst Hairer amp Gerhard Wanner 1996 Analysis by its History chapter 1 pp 1 to 79 Undergraduate Texts in Mathematics 70 ISBN 978 0 387 77036 9 MR1410751References edit In Latin analysis was a Neo Latin borrowing from Greek and the word form analysin uses the Greek accusative Calinger Ronald 2016 Leonhard Euler Mathematical Genius in the Enlightenment Princeton University Press pp 287 288 ISBN 978 0 691 11927 4 E101 Introductio in analysin infinitorum volume 1 The Euler Archive Retrieved 2020 10 15 E102 Introductio in analysin infinitorum volume 2 The Euler Archive Retrieved 2020 10 15 H J M Bos 1980 Newton Leibnitz and the Leibnizian tradition chapter 2 pages 49 93 quote page 76 in From the Calculus to Set Theory 1630 1910 An Introductory History edited by Ivor Grattan Guinness Duckworth ISBN 0 7156 1295 6 Carl Boyer April 1951 The Foremost Textbook of Modern Times American Mathematical Monthly Mathematical Association of America 58 4 223 226 doi 10 2307 2306956 JSTOR 2306956 Leonhard Euler J D Blanton transl 1988 Introduction to analysis of the infinite Book 1 Springer ISBN 978 0 387 96824 7 Introductio in analysin infinitorum V Frederick Rickey A Reader s Guide to Euler s Introductio Retrieved from https en wikipedia org w index php title Introductio in analysin infinitorum amp oldid 1176125974, wikipedia, wiki, book, books, library,
