ENCOUNTERS WITH EUCLID

As a math teacher who has taught high school geometry for nearly 30 years, I am always interested in a new book on Euclid and his Elements. I am particularly interested in books for the general reader that I can share with my students. Though I find this book to be uneven, there are parts of it that definitely fit the bill.To be clear, this is not really a book about Euclid. It is a book about how the Elements, Euclid’s book has impacts people in the 2300 or so years it’s been around. It has often been said that, after the Bible, no other single book has done so much to shape human history. Since over the past 100 years the study of geometry has gradually been moving away from examining the Elements itself, it can be difficult for a modern reader to understand how influential this single book has been for so long.Mr. Wardhaugh examines the influence of the Elements by giving us dozens of short vignettes on people throughout history who can be associated with it. This accounts, I believe, for why the book is rather uneven. Some of the vignettes are very interesting; others, less so.Some themes develop across the vignettes. One of the best is the idea that we cannot really know much about the Elements as Euclid wrote them. There is no manuscript that reaches nearly far enough back in time. Unfortunately, this also leads him into a lot of discussion about translations, which can sometimes be great and sometimes be quite dull. I also enjoyed how he often returned to the use of the Elements as a teaching tool though, again, in some cases, the anecdotes seemed somewhat tenuous in their connection to the theme.In the end, despite some weak point, I think Mr. Wardhaugh ultimately succeeds in his mission. He makes in clear why the Elements is important, and not only to mathematicians. For anyone who has sat through a high school geometry class, there is something to be gained here.

A lot of research clearly went into this book. The author covers the history of the ancient Greek mathematician, Euclid and his work on mathematical geometry, titled "Elements," from the time it was written in roughly around 300BC to the 21st century. The author starts by giving us a run down of the history of the Elements from its early recording in ancient Alexandria through to its transcribing into Arabic, Latin, Urdu, Chinese (Mandarin?) and English, and the history of the text from Alexandria to ancient Baghdad and Constantinople (during the Byzantine Empire), through to it manifestation in printed form with the development of the printing press in Europe.One interesting theme explored in the book is the difference in perspectives taken by some of those who have read Euclid's work. Though out history some people seem to have found that the Elements, with its formal propositions and proofs and its depictions of perfect, idealised or archetypal geometrical constructs, was describing something of great spiritual significance which was perhaps even a window into the mind of God. On the other hand, others have reacted to Elements by finding it to be a rather austere, sterile text with little to say about anything of relevance to experience, nature or God. The former idea about the Elements, and mathematics and geometry in general, is a philosophical notion we would now call Platonism or neo-Platonism, after the great philosopher Plato. In reading this book, I was surprised to learn that ironically, Plato did not really believe that learning much in the way of geometry or mathematics was of any great intellectual, or presumably spiritual, benefit to the mind. Later chapters of the book look at how Elements has impacted on people throughout our history and how those who have read Euclid's work have tried to either revise Euclid's ideas or been influenced to develop their own ideas from his work. For example, the book has a brief chapter on the influence of The Elements on the great physicist Sir Isaac Newton; the renaissance painter and early progenitor of perspective art, Peiro della Francesca and more recently, German painter, Max Ernst. There are quite a few other brief chapters on individuals who have been influenced by Euclid's text.Overall, I think I would have perhaps enjoyed this book more if there had been a deeper focus on either certain moments in history and/or certain individuals and their relationship to Euclid's work and ideas. We are given only a brief peek into the history of the Euclidean geometry and the lives of those influenced by it and no topic is deeply fleshed out, leaving the book feeling a little superficial. I guess the reader can always research some of the subject matter in greater detail themselves? The reader should also be aware that there is really not much discussion on the actual content of The Elements. No one reading this book will be required to understand too much about geometry or algebra, perhaps other than Euclid's Five Postulates. The famously controversial fifth postulate regarding parallel lines is explored a little bit in the book's short chapter on Lobachevskian/hyperbolic geometry.The author only gives a few mentions to another great ancient mathematician and geometer, Archimedes. Euclid may have been a mentor to Archimedes, but Archimedes apparently found The Elements to be mathematically trivial and The Elements was considered to be an inadequate work by Archimedes' own reckoning. It is generally believed that Archimedes took Euclidean geometry to another level in terms of geometrical reasoning and problem solving. Archimedes went far beyond the rudimentary Elements and went so far as to start calculating in geometry with infinitesimals, a fact that Isaac Newton was aware of and seemingly influenced Newton's own development of integral calculus. In my opinion, saying that Newton was influenced by Euclid without mentioning Archimedes leaves a huge gap in the picture as to how mathematical thought evolved from Euclid to Newton.Read this book if you have an interested in the history of mathematics and how Euclid's work spread across the globe from North Africa, to Europe, to India, the Middle East and to China.