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A brief guide to the history of the integral. (Malý průvodce historiíintegrálu.) (Czech) Zbl 1076.01002

Dějiny Matematiky / History of Mathematics 6. Prague: Prometheus (ISBN 80-7196-038-1). 95 p., open access (1996).
The booklet contains a serious, careful exposition beginning with ancient methods in calculation of areas and volumes (Egypt, Mesopotamia, Greece, Eudoxus’ Method of Exhaustion, Archimedes). It further treats the rise of the infinitesimal calculus (Kepler, Cavalieri, Fermat, Newton and Leibniz). The controversial priority disputes of Newton and Leibniz are briefly mentioned. The section ends with a list of the principal results of the epoch marked by these two great men.
The section about the integration in the 18th century contains many interesting notes on d’Alembert, Lagrange, Jacob and Johann Bernoulli, l’Hospital and Laplace. The remarks on Euler give an impressive array of his results and the authors show Euler’s method how to find derivatives of functions using their differential and the corresponding series expansions.
An elementary survey of the well known types of integrals and their history are presented in a very appropriate form including Cauchy’s work, the Riemann integral, the Darboux integral, the measure of Jordan–Peano and the related Borel’s ideas.
Other topics are dedicated to the Lebesgue and Perron integral, to Mařík’s approach and to the Kurzweil integral. The text has a large and useful bibliography.
Reviewer: L. Beran (Praha)

MSC:

01A05 General histories, source books
26-03 History of real functions
28-03 History of measure and integration
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