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Conceived by the author as an introduction to "why the calculus works" (otherwise known as "analysis"), this volume represents a critical reexamination of the infinite processes encountered in elementary mathematics. Part I presents a broad description of the coming parts, and Part II offers a detailed examination of the infinite processes arising in the realm of number--rational and irrational numbers and their representation as infinite decimals. Most of the text is devoted to analysis of specific examples. Part III explores the extent to which the familiar geometric notions of length, area, and volume depend on infinite processes. Part IV defines the evolution of the concept of functions by examining the most familiar examples--polynomial, rational, exponential, and trigonometric functions. Exercises form an integral part of the text, and the author has provided numerous opportunities for students to reinforce their newly acquired skills. Unabridged republication of Infinite Processes as published by Springer-Verlag, New York, 1982. Preface. Advice to the Reader. Index.