Original sources are key when studying the history of anything, including mathematics. Most readers, however, have neither the time nor the knowledge nor the access to be able to read original sources, so we rely on secondary works, such as Journey through Mathematics: Creative Episodes in its History. But too many secondary works just rely on other such works, thus repeating erroneous information or leaving the reader to wonder what the original really said. Not so for this book. The author, Enrique González-Velasco, has crafted a very readable account of selected episodes in the history of math, all supported by numerous direct quotations from and photographs of original sources.
As a linguist, I was especially fascinated by the attention to detail in the mathematical terminology in many different languages and the ways in which we write the names of mathematicians from many different cultures. Not the main point of the book, of course — but still of great interest to me, and I hope to at least some other readers. English translations are all very well — essential, in fact — but they’re no substitute for access to the original languages. The snippets of Greek, Latin, French, German, and Arabic resonated deeply with me, since I try to provide similar snippets when teaching my own math classes (except that I have very little knowledge of Arabic). I was just talking to my precalculus class about the origins of the names of the six trigonometric functions, explaining them both geometrically and linguistically, so I was pleased to read this quotation from Journey through Mathematics:
“Sinus cuius libet portionis circuli est dimidium corde duplicis portionis illius” (The sine of any portion of a circle is one half the chord of double the portion)
Of course the English translation of the Latin translation of the medieval Arabic original is probably confusing to modern audiences, so you’ll have to read González-Velasco’s entire context to make sense of it.
As you can see from this one-sentence example, the foundations of trigonometry form one of the topics of Journey through Mathematics. Later chapters deal with logarithms, complex numbers, infinite series, calculus, and convergence.
And that brings up the subject of who the audience for this book might be. it’s based on a college course, designed and taught since 2000 by González-Velasco, who discusses the issues of audience and topics in his preface:
First, there are always students in this course who are or are going to be high-school teachers, so my selection should be useful and interesting to them. Through the years, my original selection has varied, but eventually I applied a second criterion: that there should be a connection, a thread running through the various topics through the semester, one thing leading to another, as it were. This would give the course a cohesiveness that to me was aesthetically necessary. Finally, there is such a thing as personal taste, and I have felt free to let my own interests help in the selection.
Bravo to all three! As I’ll discuss in a later post, cohesiveness in math courses has been a common theme of mine since I first started teaching, and the other two criteria are equally valuable. Not that I share the author’s tastes in all cases — who would expect that anyway? — but I deeply value the choice of criteria. (My own tastes lean more toward ancient Greek geometry, number theory, cryptography, trigonometry, the beginnings of algebra, complex numbers, and modern approaches to fractals and iteration. But the fact that only two of these overlap the topics in this book doesn’t diminish its value for me. In fact, it might be all the more useful for telling me about topics that I know less about.)
You might think that a history of math would have to be a dry book (unless, like Bell’s Men of Mathematics, it was full of inaccurate anecdotes). But Journey through Mathematics is not dry, and it’s scrupulously free of made-up anecdote. It is peppered with a subtle wit that would be easy to miss. For example, in writing about Leibniz’s integral formulas:
The next one is about the cycloid, a curve about which we are not as emotional today as they used to be back then.
Finally, the book ends with an extremely useful and useful bibliography. The task of reading these works and integrating them into a cohesive whole must have been daunting indeed.
Full disclosure: I may be a little biased, as the author is married to one of my colleagues, with whom I worked closely over 15 years.