A great deal has been written about accomplished mathematician Karl Menger’s attempt to reform traditional calculus, since the first publication of his text in the mid 20th century. Much of the focus has been on his development of notation designed to eliminate confusion caused by relaxed (and often incorrect) approaches used by others. The author details many common flaws and ambiguities, and attacks some for making it “hard to distinguish… minor results from tremendous discoveries.” While this “Mengarian” approach is both admirable and original, it can become burdensome to the reader, who must be on careful lookout for distinctions provided by Roman versus italic print and other equally subtle typographical variations. Menger apparently predicted the difficulty many would have with his different notational approach, and provided tables throughout the text to translate between his and more standard notation. Suffice it to say, the book is a great read — for some, because of notational matters; for others, in spite of the same.
The book is organized in an interesting fashion, with Menger postponing any mention of the limit until the fifth chapter. Even then he foregoes the usual delta-epsilon rigor, instead using phrases such as “sufficiently close.” Prior to this, Menger provides a geometrical development of calculus, working with linear and then piecewise linear functions to illustrate the notions of differentiation and integration. Almost immediately, he mentions the relationship between the two concepts, and continues to emphasize this idea throughout the book in various forms. By the time Menger presents the fundamental theorems of calculus (or the reciprocity laws, as he calls them), even the beginning reader is well acquainted with the concepts. This is a wonderful characteristic of the text — conceptual understanding and intuitive development come prior to any formal presentation of theorems, rules, or laws. In this manner, Menger really does provide an alternative to the standard reversed method.
Along with the theoretical development, the author places emphasis on the role of calculus in science. In addition to a full chapter on the topic, Menger frequently mentions topics of high importance in applied mathematics: approximation, error analysis, stability. Also noteworthy is that he devotes an entire chapter to the mean value theorem, highlighting the fact that determining actual mean values is often difficult and unnecessary. It is the existential inference of such values and the resulting consequences of the theorem that are far reaching. Again, this is a welcome improvement over many traditional calculus texts whose presentations trivialize the subject.
This is a valuable text for beginners and non-beginners. For the former, Menger stresses that certain aspects of calculus (determining anti-derivatives, for example) require trial, error, and creativity. This is frequently difficult — the student should not be terminally discouraged by initial failures in this regard. For the latter, the text offers a precise, logical, thought provoking, and appealing new approach to an “old” subject.
Trent Kull is an assistant professor of mathematics at Winthrop University in South Carolina.