Download The Thirteen Books of the Elements, Vol. 1: Books 1-2 ePub

Euclid hardly needs reviews after two millennia of endorsements. Until the advent of mass-produced texts, endorsements came by way of large sums of money or time, or both. Therefore, if we do not understand what Euclid is writing about, there is overwhelming evidence that this failure is ours, not Euclid's. If we decry the unfamiliarity of Euclid's way of reasoning and his manner of writing his mathematics as being less clear or efficient than our own, we are simply expressing our faith--perhaps misplaced--in our own mathematical culture. Clearly, if one's purpose is to learn geometric techniques and results, other books may serve as well or better; if one's purpose is to understand mathematics, the thirteen books of the Elements are without equal.

The Heath edition of Euclid's Elements actually consists of three volumes: volume 1 has Euclid's Books I and II; Heath's volume 2 contains Euclid's Books III - IX; and his volume 3 encompasses Euclid's remaining Books X - XIII. Books VII, VIII, and IX are about "arithmetic," not "geometry"--a feature of the Elements often left unstated. Throughout, Heath intersperses his notes and comments, so the three volumes actually consist of as much Heath as Euclid. (Just Heath's translation, alone, is reproduced in the Great Books of the Western World, published in 1952 by University of Chicago.) Up until recently, maybe as late as the nineteenth century, a typical reader of Euclid would be quite familiar with Plato and therefore know that arithmetic and geometry are the philosophical branches of mathematics; music and astronomy are the remaining branches of mathematics, although somewhat contaminated since--in the Greek understanding as expressed by Plato--music and astronomy introduce motion, which is not strictly a mathematical topic.

Niceties such as these, and there are many others, would be lost to us if Euclid were transformed by using modern symbolism. Consider proposition 47 of Book I, the so-called Pythagorean theorem: Euclid talks about constructing squares on the sides of a triangle and never even hints at the possibility of the sides being "numbers." In fact, Euclid and all of his notable contemporaries and successors up to about the 15th century would consider the term "irrational number" as utter nonesensical babble--something more dangerous than an oxymoron such as a "square circle" because "square" and "circle" are not fundamental ideas. These comments may raise more questions than they purport to answer, but they give background to reviewing Heath, rather than Euclid.

Heath's edition, taken in toto, would have been very difficult to improve. His notes and collecting together of earlier commentaries represent a remarkable achievement in scholarship. He certainly made errors, but he provided nearly the best edition of Euclid possible at the opening of the last century. Heath made several efforts to explain the contents of Euclid by appealing to contemporary ideas and notations and, at least for me, these explanations simply reinforced the view that Euclid dealt with profound unanswerable questions that remain unanswered in contemporary mathematics.

Heath translated and edited several Greek primary sources, including Archimedes and Apollonius. Comparing his earlier translations with his later (in his career) Euclid, one immediately sees that Heath tried to preserve more faithfully Euclid's manner of speaking than he did Apollonius's or Archimedes'. This historigraphic point is important: if we are to respect the ancient Greeks by trying to understand or know their culture and values on their terms, we must have access to their culture with as few filters as possible. This line of arguing suggests that we should first study ancient Greek and then read Euclid, perhaps an ideal approach. Very few readers of Euclid take this approach. Hence, for an English reader (which includes readers of many other languages), a more faithful rendering of the Greek into English has greater importance because it does not filter the implicit culture as much as a less faithful rendering.

These views are my historian views. As a mathematician, I think of mathematics as timeless and critique any mathematical work on the basis of whether it represents good (read this as "my") mathematics. Heath knew his mathematics; he frequently calls on ideas from Cantor, who at this time is in the middle of his seminal publications. I would take the same critical approach if I were a philosopher--is Euclid good philosophy in that he provides answers to philosophical questions, regardless of whether many refinements have been formulated since Euclid? (By the way, there is no explicit philosophy in Euclid, but a lot of implicit philosophy.) In terms of editing a crucial historical document, Heath's work has withstood the test of about one century, and rightly so in my judgment. His Euclid is likely found among the personal books of people with a high regard for education.

Very interesting work that takes a serious look at geometry building from the ultimate basic assumptions and building from there. I definitely love Thomas L. Heath's commentary. It is daunting to read the entire commentary as it is quite longer than Euclid's book; however, many times the commentary shows the depth of Euclid. Why did Euclid choose the proof he did and not an alternative, and what the alternatives are. I found those comments the most interesting, though he offers many more types of insights. For example he gives a history in the beginning and a discussion about axioms and proofs, with much much more.

I like the writing style. As a foreigner, Heath's English is kind of different from today's newspaper and blogs. For example, the first paragraph of the preface to second edition is just ONE sentence, but it takes nine lines. Such long sentences are here and there. Long sentence with complicated structure is elegant and an indicator that the writer is well educated. And Heath likes to use subjunctive mood, which many non-native speaker do not feel comfortable about. It is a good book to learn English writing.

Euclid's "Elements" may very well be the most influential mathematical text in all of history. This fact alone justifies purchasing this book, which is the first of three volumes of Thomas L. Heath's English translation of this classic. This volume contains a lengthy introduction, and the actual mathematics covers plane geometry. Highlights include the construction of the regular 15-gon using straightedge and compass.

The actual text of Euclid's work is not particularly long, but this book contains extensive commentary about the history of the Elements, as well as commentary on the relevance of each of the propositions, definitions, and axioms in the book. As such, this book is a good scholarly reference for English readers interested in the historical evolution of Euclidean geometry. For example, there is considerable discussion on the well-known fifth postulate about parallel lines.

All this being said, do not try to learn geometry from this book. The content is more suited for readers who already know geometry and want to learn about the historical origins of the subject of geometry. There are many modern books written for readers new to geometry (some good, some bad). It's probably true that Abraham Lincoln studied the Elements as a young lawyer, but there are easier (if not better) ways to learn geometry nowadays. The Elements will be much more enlightening if the reader has a good grasp of the actual mathematics in the book prior to reading it.

The essence of ALL fundamental reasoning distilled into three very affordable and faithful translations...I would think giving it five stars would be an understatement, but that's as much as I can do! Definitely a must for, well, everyone.