Over the course of the last few years, I have, on two occasions, taught a one-semester course in the history of mathematics. In both courses, the students seemed particularly interested in the dispute between Newton and Leibniz over who should get credit for the discovery of calculus. I suppose this is hardly surprising — the idea of two titans of mathematics and their proxies engaged in charges and counter-charges of bad faith and plagiarism must seem pretty amusing to an undergraduate.

Professional mathematicians, on the other hand, don’t seem to find this matter particularly funny. I was an undergraduate at Brooklyn College, which had as a faculty member the late mathematical historian Carl Boyer, and I distinctly remember sitting in on his history course and hearing him act rather embarrassed about the whole matter. This was reflected in his written work as well: in his 1949 book *The History of the Calculus and its Conceptual Development*, for example, he simply refuses to address the issue:

Furthermore, inasmuch as we are here more concerned with ideas than with rules of procedure, we shall not discuss the shamefully bitter controversy as to the priority and independence of the inventions by Newton and Leibniz.

In his more recent *A History of Mathematics*, coauthored with Uta Merzbach, the priority dispute was addressed in a few paragraphs, and was referred to as “disgraceful”. See also Cooke’s *The History of Mathematics* (“One of the better known and less edifying incidents in the history of mathematics”) and Suzuki’s *A History of Mathematics* (“One of the more unfortunate events in the history of mathematics”; “particularly pointless”).

Descriptions like this reflect the fact that nobody in this controversy really behaved in a manner that is above reproach. It is now unquestioned that Newton developed the calculus more than a decade before Leibniz did, but did not publish his findings. It is also the consensus of the historical community that Leibniz developed his version of the calculus (which avoids mention of fluxions and fluents and uses a notation that influenced the one taught in calculus classes today) independently of Newton. Although plagiarism is not involved, however, less-than-admirable behavior on all sides certainly is. It is interesting, in fact, that Thomas Sonar (the author of the book now under review) changed his mind about the behavior of the principals as he researched this book — originally very much pro-Leibniz, he came to realize that matters were not quite as one-sided as he originally thought. He explains why, in some detail, on pages 492–494 of this book.

In addition to brief discussions in history of mathematics textbooks, and a chapter in *Great Feuds in Mathematics* by Hellman, there have also been a couple of full-length accounts of this controversy. Two that I have glanced at, but not read thoroughly, are Hill’s *Philosophers at War* and Bardi’s *Calculus Wars*. The book by Hill is generally regarded as excellent scholarship but rather dry and dense. The book by Bardi, on the other hand, has not been well-received. It is described in the preface to the book now under review, by Thomas Sonar, as “less recommendable, since lurid” and, in addition, is thoroughly eviscerated in a review by Blank that appeared in the *Notices of the AMS* and which itself provides an interesting account of the controversy. (Blank uses terms like “appalling” and “specious” to describe Bardi’s book, thereby suggesting that Sonar’s description in the preface may have been diplomatic.)

Sonar’s book — a translation, by him, his thesis advisor, Thomas Morton, and Morton’s wife Patricia, from a German edition published a year or two earlier — is not like either Hill’s or Bardi’s. It is, unlike Bardi’s text, an example of excellent scholarship, written by a mathematician who obviously understands the subject matter at a deep level. Though the book is filled with lots of quotes and footnotes and is not really recommendable as summer beach reading, Sonar has tried hard to write in a lively and engaging style, and this book generally seems more accessible and reader-friendly than the one by Hill.

It also differs from both the Bardi and Hill books in another particular: unlike both of these books, Sonar’s text actually talks about mathematics. Sonar believes — wisely, in my view — that when discussing a dispute about the history of calculus, one can’t really fully understand what the fuss is all about unless you have at least some understanding of calculus to begin with.

Accordingly, the first chapter of Sonar’s book begins with an informal introduction to calculus, explaining the basic ideas in nonrigorous terms and giving an indication of how useful the subject is. The presentation is in the spirit of Leibniz and gives a preview of his terminology and notation. Mathematical discussions, at an appropriate level, also appear throughout the book (but, Sonar tells us, can be easily ignored by those who find them impenetrable).

After this introductory chapter, we begin the history of the priority dispute. Sonar’s recounting of the story actually begins before the dispute itself and sets the groundwork for the events that follow, and also continues beyond the dispute, discussing the consequences and repercussions of this historic event.

Like any good historian, Sonar knows that historical events must be understood in the context of the time and place in which they occurred. Accordingly, chapter 2 addresses the political and cultural climate of England, France and the Netherlands in the 17th century, with special emphasis on several prominent people (Pascal, Huygens) of that time. Chapter 3 then details the early life of the two principal protagonists in the dispute, Newton and Leibniz. Among other things, we are told how, in the period 1664–1666, Newton developed his version of the calculus, based on “fluxions” and “fluents”. Sonar also points out other achievements of Newton during these “magic years”. Additionally, he also mentions disputes that Newton had with others, such as Hooke, which had an effect on Newton’s personality and his interactions with others.

Chapters 4 through 8, which comprise the bulk of Sonar’s book, discuss the history of the priority dispute during the lifetimes of the two protagonists. Highlights of these chapters include extensive discussions of: correspondence to Leibniz (including two famous letters from Newton called the *Epistola Prior* and *Epistola Posterior); *the publication of Newton’s *Principia* and its effect on the dispute; charges of plagiarism made against Leibniz by (among others) a man named Fatio de Duillier, referred to by Sonar as “Newton’s monkey”; and the resolution of the plagiarism dispute by the Royal Society (a “kangaroo court” if there ever was one, given that Newton was the President of the Society and wrote the final decision).

Leibniz died in 1716, but his death did not end the priority dispute, which contained to be waged by surrogates, including Johann Bernoulli on behalf of Leibniz and John Keill on behalf of Newton. The period from shortly before Leibniz’s death to Newton’s death in 1727 is discussed in chapter 9.

Chapter 10 addresses some of the critical response to both the calculus of Leibniz and of Newton. For the latter, there is an excellent discussion of Bishop Berkeley and his* Analyst*; also discussed is a Dutch mathematician named Bernard Nieuwentijt, who objected to Leibniz’s use of second and higher order infinitesimals (but was not bothered by the first order ones). Over the years, Berkeley’s objections (based on, primarily, the fact that quantities were treated as simultaneously being both zero and nonzero) had much greater impact, and his phrase “ghosts of departed quantities” has become famous.

Finally, in chapter 11, the aftermath of this controversy is discussed. Two principal aspects of this aftermath are highlighted. First, the author addresses the efforts of de Morgan to rehabilitate Leibniz’s reputation; then, he discusses how mathematical progress towards a development of analysis stagnated in Britain, since the British embraced Newton’s version of the calculus, which had inferior notation.

As mentioned earlier, the author’s style of writing is vivid, interesting and accessible. Other good features of the book include numerous photographs and illustrations (many in color), and an extensive ten-page, small print, list of references. The good pictures and references are features that this book has in common with *5000 Years of Geometry*, another excellent book translated from the German that, together with this one, comprise the *Mathematics in History and Culture* series published by Birkhäuser. I look forward to the publication of other books in this series.

My guess is that this book will soon come to be viewed as a (if not *the*) definitive source for information as to this important dispute in mathematics. It belongs on the shelf of anybody interested in the history of calculus, and it certainly belongs in any good university library.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.