Introduction to the Analysis of the Infinite
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''Introductio in analysin infinitorum'' (
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
: ''Introduction to the Analysis of the Infinite'') is a two-volume work by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
which lays the foundations of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
. 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.
Carl Boyer Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics. Novelist David Foster Wallace called him the " Gibbon of math history". It has been written that he was one of few hist ...
's lectures at the 1950 International Congress of Mathematicians compared the influence of Euler's ''Introductio'' to that of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's '' Elements'', calling the ''Elements'' the foremost textbook of ancient times, and the ''Introductio'' "the foremost textbook of modern times". 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. The first translation into English was that by John D. Blanton, published in 1988. The second, by Ian Bruce, is available online. A list of the editions of ''Introductio'' has been assembled by V. Frederick Rickey. Chapter 1 is on the concepts of
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
s and
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s. Chapter 4 introduces
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
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. ulermade 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. 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 Duckworth may refer to: * Duckworth (surname), people with the surname ''Duckworth'' * Duckworth (''DuckTales''), fictional butler from the television series ''DuckTales'' * Duckworth Books, a British publishing house * , a frigate * Duckworth, W ...
Euler accomplished this feat by introducing
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
''a''''x'' for arbitrary constant ''a'' in the
positive real numbers In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
. He noted that mapping ''x'' this way is ''not'' an
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
, but rather a
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed alge ...
. 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: : \exp(z) = \sum_^ = 1 + z + + + + \cdots 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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
and presents
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
. Chapter 9 considers trinomial factors in
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s. Chapter 16 is concerned with
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
s, a topic in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
.
Continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s are the topic of chapter 18.


Early mentions

* J.C. Scriba (2007) review of 1983 reprint of 1885 German edition


Reviews of Blanton translation 1988

* Doru Stefanescu * Marco Panza (2007) * Ricardo Quintero Zazueta (1999) * Ernst Hairer & Gerhard Wanner (1996) ''Analysis by its History'', chapter 1, pp 1 to 79, Undergraduate Texts in Mathematics #70,


References

{{Authority control 1748 books Mathematics textbooks Mathematical analysis Leonhard Euler 18th-century Latin books