Introduction to Analysis of the Infinite: Book II

by Leonhard Euler

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra, even of all the algebraic technique so far discovered, still there are topics whose con­ sideration prepares a student for a deeper understanding. However, in the ordinary treatise on the elements of algebra, these topics are either completely omitted or are treated carelessly. For this reason, I am cer­ tain that the material I have gathered in this book is quite sufficient to remedy that defect. I have striven to develop more adequately and clearly than is the usual case those things which are absolutely required for analysis. More­ over, I have also unraveled quite a few knotty problems so that the reader gradually and almost imperceptibly becomes acquainted with the idea of the infinite. There are also many questions which are answered in this work by means of ordinary algebra, although they are usually discussed with the aid of analysis. In this way the interrelationship between the two methods becomes clear.

Genres: Mathematics

516 pages, Hardcover

First published January 1, 1945

About the author

Noted Swiss mathematician Leonhard Euler worked in analysis and algebra, including complex numbers and logarithms, and he introduced much of the basic notation in mathematics.

This pioneering physicist made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern terminology, particularly for analysis, such as the notion of a function. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

People considered Euler the preeminent mathematician of the 18th century and one of the greatest who ever lived. He is also one of the most prolific mathematicians; his collected works fill sixty to eighty quarto volumes. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, Prussia.

A statement, attributed to Pierre Simon de Laplace, expresses influence of Euler on mathematics: "Read Euler, read Euler, he is the master of us all."