Euler's numbere corresponds to shaded area equal to 1, introduced in chapter VII
Introductio in analysin infinitorum (Latin:[1]Introduction to the Analysis of the Infinite) is a two-volume work byLeonhard Euler which lays the foundations ofmathematical analysis. Written in Latin and published in 1748, theIntroductio contains 18 chapters in the first part and 22 chapters in the second. It hasEneström numbers E101 and E102.[2][3] It is considered the firstprecalculus book.
TheIntroduction 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 introducingexponentiationax for arbitrary constanta in thepositive real numbers. He noted that mappingx this way isnot analgebraic function, but rather atranscendental function. Fora > 1 these functions are monotonic increasing and formbijections of the real line with positive real numbers. Then each basea corresponds to aninverse function called the logarithm to basea, in chapter 6. In chapter 7, Euler introduces e as the number whose hyperbolic logarithm is 1. The reference here is toGregoire de Saint-Vincent who performed aquadrature of the hyperbolay = 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 theunit circle and presentsEuler's formula. Chapter 9 considers trinomial factors inpolynomials. Chapter 16 is concerned withpartitions, a topic innumber 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.
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 ofIntroductio has been assembled byV. Frederick Rickey.[8]
