It could be argued that through a philosophic lens, one way of telling the story of analysis is to tell the story of humankind’s quest to understand the continuum. In our contemporary parlance, the continuum consists of the notions of and distinctions between the continuous and the discrete. The atomic and the divisible. Indeed one could argue that the study of this dichotomy eventually led to the invention of calculus. As noted in Bressoud's

*Calculus Reordered* a great many people over long periods of time, and vast geographic expanses, contributed to the knowledge edifice that emerged as calculus. At the dawn of calculus, some examples of this dichotomy then in vogue were Cavalieri’s notion of an

*indivisible*, highlighting the atomistic perspective, while Newton’s and Leibnitz’s continuous notion of an

*infinitesimal* ultimately proved more satisfactory. (Of course, there were other reasons as well. Cavalieri’s methods were heavily tethered to geometry while the calculus of Newton and Leibnitz was more algebraic and exploited the recent discovery of analytic geometry.) And if you really want to understand the story of the continuum, you have to go back to its origins in Ancient Greek philosophy and follow its historical development.

The book *3000 Years of Analysis* by Tomas Sonar is a history of mathematics that focuses on and emphasizes the history of analysis, broadly construed, embedded within the historical and cultural contexts in which these ideas emerged. As a history of mathematics, it contains more philosophy and general history than such books usually do. It is an English translation of the second German edition. The book’s title refers to the time period going back to about 1000 BCE. The editors point out in the preface that the book is not merely a translation but contains new material as well as a reworking of some of the sections in the German edition.

Professor Sonar states in the first few pages that the 3000 years in the book’s title were a somewhat arbitrary choice due to the fact that we do not really know when ideas we would term analytical first emerged in the mists of human history. To most mathematicians, the term analysis conjures up the spectre of the calculus of Newton and Leibnitz along with its subsequent development into the various analytical branches of mathematics which have their roots in the processes of differentiation, integration, and other infinite processes. Thomas Sonar’s book makes use of a somewhat more general conception of analysis as the human endeavor to understand infinity.

A common thread that runs through the book is the idea of the continuum and its contradistinction to *atomism*. These ideas surely precede the classical Greek philosophers, but their written history seems to have begun with Aristotle in his Physics, Aristotle’s investigation of motion, change, time, space, matter, and the infinite, and his refutation of the paradoxes of Zeno of Elea such as Achilles and the Tortoise. The ancient Greek words for this dichotomy are συνέχεια, synécheia, which means continuous or continuity and ἄτομος , átomos, which means indivisible. Other words that express the distinction between the continuum and the atom include infinitesimal and the indivisible or the continuous and the discrete.

In preparing this review I found it useful to make comparisons with several other books on my bookshelf besides the book by Bressoud mentioned in the first paragraph, namely

*An Introduction to the History of Mathematics* by Howard Eves,

*A History of Mathematics: An Introduction* by Victor Katz, and

*Mathematical Thought from Ancient to Modern Times* by Morris Kline. I would describe the book as an account of the history of mathematics with an emphasis on the ideas of analysis that is aimed at a mathematically literate but general audience. Like other textbooks, histories of mathematics often cover the same topics with some variations. These books will typically cover the history of mathematical ideas and the biographies of individual mathematicians. This is true of Thomas Sonar’s book. But in addition to these topics, this book contains more socio-political-cultural history than is found in the other books just referenced.

Some of the strengths of Sonar’s book:

- The book is well referenced. Assertions made are typically supported by references to their source in the literature. Many of these sources are books or articles in German less familiar to native English speakers but valuable to know about.
- The diagrams and other visuals are attractive and inviting. Individual personalities or figures in the history of mathematics are discussed in greater depth and color than that in other histories of mathematics.
- A survey of some ideas of nonstandard analysis brings the saga of Newton and Leibnitz’s “ghosts of departed quantities” full circle.

A few shortcomings:

- The book may be seen by some as a bit Eurocentric. It reflects the tradition that deductive mathematical proof was invented by the Greek philosopher Thales of Miletus. There is a good deal of coverage of the roots of analysis in Europe and the fertile crescent, but not much on the ideas of analysis in India, China, or Sub-Saharan Africa.
- Some statements are presented as facts but are actually at best speculative and often disputed. These include references to the House of Wisdom (which some scholars argue did not exist), the poisoning of Descartes with arsenic (an intriguing and plausible but ultimately speculative proposition), and the psychosis of Newton (the nature of Newton’s well documented eccentric or erratic behavior remains an open question). Of course all historiography must be tempered with an understanding that historical accounts involve gray areas of argument and interpretation. Readers must in the end weigh the evidence for and against propounded interpretations to draw their own conclusions.
- Some typographical errors and some minor translational solecisms appear.
- The book has no exercises.

To conclude, I very much enjoyed Sonar’s book and can see possible uses for its material in undergraduate history of mathematics courses. Anyone interested in the history of mathematics will find some unique things in this book.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D. from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.