While the commentary throughout exhibits some fantastic insights, the curious reader may find themselves less interested in the simplicity put forth by Euclid and more focused on dissecting the extensive tangents of Mr. Heath. Regardless, the general content is still very much in tact and solely focusing on it will no doubt reveal much to any reader of any mathematical background. ( )

It is difficult to argue with the fact that Euclid stands as one of the founding figures of mathematics. The ability of the ancient Greeks to perform complex mathematical calculations using only logic, a compass and a straight edge is profoundly humbling. Euclid's 13 books cover an enormous swath of math, from planar geometry to trignometry to irrational numbers and root finding to 3D geometry. At one point you feel he is on the cusp of discovering the Calculus. Considering these pages were written more than two thousand years ago I stand in awe.

That said, I have some serious problems with the way Euclid's materials are presented in this Dover Mathematics book. The book itself (a three volume set actually) is a reproduction of Sir Thomas Heath's famous Elements of 1908. This is the second Dover edition and it is unabridged. Usually I'm not a fan of abridgements but this book could certainly use it. At the very least some modernization of the notes and introductory essays would seem to be in order. Of course, if you approach this book as a mathematician, you will likely skip over the first hundred or so pages and be spared some pain. If you are a student of philosophy you aren't so lucky. Heath's notes are dense, tangential, and require the mastery of at least four languages, two of which are now dead. Latin and Greek quotes of considerable length are left untranslated as an exercise for the reader, and French and German receive similar treatment. At times the footnotes threaten to overwhelm the text and for every page of Euclid there must be at least 3 pages of commentary. References to obscure mathematical theory and little known Greek manuscripts abound. I understand that this is Victorian Age scholarly writing at its height but it makes it a tough read - and I say this as someone with a background in Latin, Greek and French as well as considerable mathematical (never got much past partial differential equations) background. Heath was a polymath of the highest order.

If you are brave enough to tackle this book you may want to grab just the volume that interests you. The first volume contains introductory remarks by Heath and most of the well known postulates related to geometry. Book I, postulate 5 (I.5) is the well know triangle inequality while I.47 is the geometric proof of the Pythagorean theorem - a thing of rare beauty. In the second volume, Books III and IV deal with circles and arcs while Book V deals with ratios. I found the proofs with respect to ratios difficult to follow owing partially to the language in which they are couched. Book VI applies the theory of ratios to geometric figures while books VII and VIII deal with factorization, multiples and primes. Book IX deals with prime numbers, perfect numbers and odd and even numbers. The third volume begins with Book X which deals at length with rational and irrational numbers. It is here that the Greek methods seem to be a little weak, requiring rather clumsy proofs which would be much simpler in modern notation. Still, it is amazing to see the math they did with what they had. Books XI and XII deal with solids - spheres, prisms, parallelpipeds and pyramids - while Book XIII deals with the platonic solids. It is here that Euclid approaches calculus with his method of proof by exhaustion. The persistent reader will, by this point, also be quite exhausted but, as a bonus, Heath throws in the sometimes attributed Books XIV and XV, both of which are brief and neither of which are by Euclid.

If you are planning on buying this book I would recommend you consider the reason carefully. If you are looking for a math text there must surely be something more modern with a more concise commentary available. If you are a student of Greek philosophy you may find the first volume useful for its introductory notes but the last two volumes are likely unhelpful. If you are fluent in Latin, Greek, French, German and English, have a background in ancient greek literature, Renaissance and 19th century mathematical theory, and love geometric proofs then this is the book for you ( )

A collection of all 13 Euclid's books of The Elements, with an introduction and commentary throughout the book.