E101 -- Introductio in analysin infinitorum, volume 1
(Introduction to the Analysis of the Infinite, volume 1)
Summary:
In E101, together with E102, Euler lays the foundations of modern mathematical analysis. He summarizes his numerous discoveries in infinite series, infinite products, and continued fractions, including the summation of the series
1/1^{k} + 1/2^{k} + 1/3^{k} + ... for all even values of k from 2 to 26. Perhaps more importantly, the Introductio makes the function the central concept of analysis; in particular, Euler introduces the f(x) notation for a function and uses it for implicit as well as explicit functions, and for
both continuous and discontinuous functions. In addition, he calls attention to the central role of e and e^{x} in analysis, and he gives e^{x} and ln x the independent definitions
e^{x} = lim_{n → ∞} (1 + x/n)^{n},
ln x = lim_{n → ∞} n(x^{1/n} – 1),
putting them on an equal basis for the first time. Euler also proves that every rational number can be written as a finite continued fraction and that the continued fraction of an irrational number is infinite. He also shows how infinite series correspond to infinite continued fractions; in particular, Euler derives continued fraction expansions for e and √e.
The book contains a dedication by Bousquet, along with Euler's own preface. The main body of the work is divided into 18 chapters:
- De functionibus in genere.
- De transformatione functionum.
- De transformatione functionum per substitutionem.
- De explicatione functionum per series infinitas.
- De functionibus duarum pluriumve variabilium.
- De quantitatibus exponentialibus ac logarithmis.
- De quantitatum exponentialium ac logarithmorum per series explicatione.
- De quantitatibus transcendentibus ex circulo ortis.
- De investigatione factorum trinomialium.
- De usu factorum inventorum in definiendis summis serierum infinitarum.
- De aliis arcuum atque sinuum expressionibus infinitis.
- De reali functionum fractarum evolutione.
- De seriebus recurrentibus.
- De multiplicatione ac divisione angulorum.
- De seriebus ex evolutione factorum ortis.
- De partitione numerorum.
- De usu serierum recurrentium in radicibus aequationum indagandis.
- De fractionibus continuis.
Publication:
- Originally published as a book in 1748.
- Opera Omnia: Series 1, Volume 8.
- An English translation of §§140-141 (pp. 105-107) was published by F. Masères in Scriptores
logarithmici 3, London 1796, pp. 169-182 ("Euler's method of squaring the circle") [E101a].
- John Blanton has translated both E101 and E102 in full. His translation of E101, Introduction to Analysis of the Infinite, Book I, was published by Springer-Verlag in 1988.
Documents Available:
- Original Publication: E101 (Original Latin), available via the Bibliothèque Nationale de France's Gallica digital library.
- Opera Omnia 1922 Reprinting: E101 (Latin).
- French Translation: E101 (French), available via the Bibliothèque Nationale de France's Gallica digital library.
- German Translation by H. Maser (1885), available on the Euler Archive:
- E101 was discussed in Ed Sandifer's MAA Online column, How Euler Did It, in March, June, July, and October 2005.
- The Euler Archive attempts to monitor recent scholarship for articles and books that may be of interest to Euler Scholars. Selected references that discuss or cite E101 include:
- Adiga C, Berndt BC, Bhargava S, et al., “Chapter-16 of Ramanujan 2nd notebook - theta-functions and q-series.” Memoirs of the American Mathematical Society, 53 (315), pp. 1-85 (1985).
- Alder HL., “Partition identities - from Euler to present.” American Mathematical Monthly, 76 (7), pp. 733-& (1969).
- Dutka J., “The early history of the factorial function.” Archive for History of Exact Sciences, 43 (3), pp. 225-249 (1991).
- Ernst T., “A method for q-calculus.” Journal of Nonlinear Mathematical Physics, 10 (4), pp. 487-525 (Nov 2003).
- Ferraro G., “Analytical symbols and geometrical figures in eighteenth-century calculus.” Studies in History and Philosophy of Science, 32A (3), 535-555 (Sep 2001).
- Ferraro G, Panza, M., “Developing into series and returning from series: A note on the foundations of eighteenth-century analysis.” Historia Mathematica, 30 (1), pp. 17-46 (Feb 2003).
- Fraser CG., “The calculus as algebraic analysis - some observations on mathematical-analysis in the 18th-century.” Archive for History of Exact Sciences, 39 (4), pp. 317-335 (1989).
- Gilain C., “History of the fundamental theory of algebra - theory of equations and integral calculus.” Archive for History of Exact Sciences, 42 (2), pp. 91-136 (1991).
- Ku YH., “Solution of Riccati equation by continued fractions.” Journal of the Franklin Institute-Engineering and Applied Mathematics, 293 (1), pp. 59-& (1972).
- Lehmer DH., “2 nonexistence theorems on partitions.” Bulletin of the American Mathematical Society, 52 (6), pp. 538-544 (1946).
- Maor E., e: The Story of a Number
- Muses C., “Some new considerations on the Bernoulli numbers, the factorial function, and Riemann's zeta function.” Applied Mathematics and Computation, 113 (1), pp. 1-21 (Jul 2000).
- Ruthing D., “Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N..” Mathematical Intellingencer, 6 (4), pp. 72-77 (1984).
- Todd J., “Lemniscate constants.” Communications of the ACM, 18 (1), pp. 14-19 (1975).
- Volkert K., “History of pathological functions - on the origins of mathematical methodology.” Archive for History of Exact Sciences, 37 (3), pp. 193-232 (1987).
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