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Review: How to Read Historical Mathematics

Princeton University Press sent me a review copy of Benjamin Wardhaugh's "How to Read Historical Mathematics." I was excited to receive this book because I don't know of any other books that provide a basic introduction to the subject.

From Wardhaugh's web site:

Benjamin Wardhaugh is a historian; he does research and teaches at the University of Oxford, where since October 2007 he has been a Post-Doctoral Research Fellow at All Souls College. He is a graduate of Cambridge, Oxford, and the Guildhall School of Music and Drama in London, and holds degrees in mathematics, music, and history. ... He teaches the history of mathematics in various periods, in both the Mathematical Institute and the History Faculty at Oxford. A selection of the many things he has learned from his students will appear in his forthcoming textbook, How to Read Historical Mathematics.

How to Read Historical Mathematics is a quick read at 116 pages. Will you become an expert at reading historical Math after you read the book? Of course not. That will take years. What Wardhaugh does exceptionally well is to break the ice for readers interested in the subject. He does this largely by training readers to ask insightful questions when they read a historical text. Here's a set of questions from the book:

How they thought: What notation does the text use? What words? What concepts? How are these different from what you would use in the same situation? Does it use words or concepts you don't recognize? Can you work out what they mean, or find out what they mean from the author's definitions? Does it use familiar words or notation, but with different meanings from what you would expect?

And, here's another example:

Where did the author live? When? And where and when was the text you're looking at written? What were the time and place like for this particular person? What was going on in politics or society? What would have been the "current affairs" of the time? Who were this person's colleagues? Who were the other people--friends, colleagues, students, editors--who were involved in making this particular piece of mathematical writing happen and getting it published (if it was)? And, what about any enemies? Why was this person working on this particular piece of mathematics? What else was going on in mathematics at the time? Was this piece of writing meant to help with teaching, to further a career, to satisfy the writer's curiosity or someone else's?

What intrigues me about the many questions we're taught to ask when in front of a historical text is that reading such books is akin to solving Math problems. When we solve problems we are looking for patterns, we are looking for relationships that are not obvious on the surface, we are playing detective, and we are asking lots of questions.

Historical Math books are available, often very inexpensively, on Ebay. I bought a book published in 1863 on Ebay for $6.50, and that included shipping. Now, this was not a scholarly work. Those are much more expensive. But, you could dip your toes in the water with old textbooks at a low price. Or, you can find digitized historical monographs and books at the Cornell University Library Historical Monographs Collection and other places on the Web.

Beyond questions of the context of a particular work or the circumstances of the author, one needs to be able to deal with the nuts-and-bolts of what the mathematics actually says especially when the "notation" is very different from what we're familiar with. For example, what does this say?

When the cube and the things together
Are equal to some discrete number,
Find two other numbers differing in this one.
Then . . . their product should always be equal
Exactly to the cube of a third of the things.
The remainder then as a general rule
Of their cube roots subtracted
Will be equal to your principal thing.

— From Niccolò Tartaglia’s account of the solutions
to the cubic equation (1539) in Fauvel and Gray,
The History of Mathematics: A Reader, pp. 255–56.

Wardhaugh picks apart this example which turns out to not be particularly easy to translate as there are some ambiguities for us modern readers.

You can get a flavor of the book from a few pages of the book at Amazon or from Google Books, which has more pages from the book.

I like this little book but I don't want to set any expectation that readers will be able to sail through historical texts after reading it. Having said that, I thoroughly enjoyed the exposure "How to Read Historical Mathematics" gave me to this fascinating and huge subject.