The first thing you notice about this text is its thickness, or should I say lack of thickness. For a text that claims to go " From Mesopotamia to Modernity" it is remarkably thin, just 281 pages, and that includes the bibliography and index. Do not let its thin size deceive you. There is still a substantial amount of material here. The first 132 pages cover Babylonian and Geek mathematics in sufficient depth to give the reader a good feel for the mathematics done by these two ancient civilizations. The coverage is probably not as thorough as that found in other texts, but the topics are well chosen. There is also a chapter on Chinese mathematics and another on Islamic mathematics. While I would hesitate to call these chapters "sparse", it is fair to say that they are "to the point".
To understand the direction that this text is taking one must read the "Introduction" carefully. Hodgkin spends thirteen pages justifying the text and his approach to the history of mathematics. His philosophy is (essentially) that other textbooks in the history of mathematics are "accumulation(s) of facts". He bases his approach on a quote from E. H. Carr who wrote
What had gone wrong was the belief in this untiring and unending accumulation of facts as the foundation of history, the belief that facts speak for themselves and that we cannot have too many facts, a belief at that time so unquestioning that few historians then thought it necessary — and some still think it unnecessary today — to ask themselves the question 'What is history?' (E. H. Carr, in What is History?)
Hodgkin's goal was to write a history of mathematics book that answers Carr's fundamental question. Much time is spent on historiography and not just history. That can be a bit of a double-edged sword. I found the philosophical discussion of history in the Introduction very interesting, but at the same time it was a bit distracting, especially some of his "political" commentary. For example, in discussing a letter written by Simone Weil on the teaching of the history of science Hodgkin made a point of mentioning her "extreme-left sympathies" which had little to do with the context. Part of me wanted to say, "Let's get on with the history of mathematics!"
Once I got past the Introduction I found the exposition to be quite readable and the coverage adequate. I must say "adequate" and not "complete" because the author admits to leaving out some topics typically found in a textbook on this subject. He omits "Egypt, the Indian contribution aside from Kerala and most of the European eighteenth and nineteenth centuries." His bibliography is relatively extensive and does include what he would call "more traditional" textbooks in which the deleted material could be found.
The chapter on the calculus presents a balanced view of both Newton and Leibniz with an excellent use of diagrams. There is also a concise discussion of the "priority dispute" between Newton and Leibniz.
Exercises are sprinkled throughout the text with many having solutions appearing at the end of the chapter. The selection is adequate, but as with other aspects of this book, a bit sparse.
In conclusion, Hodgkin makes the case in his Introduction for a different perspective on the history of mathematics. His argument is well justified and I think he succeeds in producing a text that is different from others on the market. That is the good news. The bad news is that it may not be to the liking of instructors looking for a textbook that would be successful in the classroom. Practitioners of the history of mathematics will find the approach taken by Hodgkin to be interesting and insightful. It is a much tougher read for individuals new to the subject.
Herbert Kasube is Professor of Mathematics at Bradley University in Peoria, IL.