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 between 2 and 26, inclusive. Perhaps more importantly, the Introductio makes the function the central concept of analysis; moreover, Euler introduces the notation f(x) 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 an Epistola dedicatoria by Bousquet and Euler’s own Praefatio.
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
French Translation: E101, French translation
German Translation:
E101 is discussed in Ed Sandifer's How Euler Did It March, June, July, and October 2005 columns published online by the MAA.
The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found 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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