EUCLID. The Thirteen Books of The Elements: Volume 2: Books 3 to 9 [Paperback]

This is a word of warning to potential purchasers of this volume; it is only one of three volumes. This is not made clear in the presentation so just be aware. Look for the other two volumes if your intent is to acquire all thirteen books of Euclid's, The Elements.

This is a review of the Dover edition with Heath's commentary. The review combines Vol. 1 (Books I-II) and Vol. 2 (Books III-IX) because I don't think many people besides possibly historians will only buy Vol.I.

I think this is the definitive edition, the best choice for scholars, the mathematically mature, and beginners alike. The main competition seems to be the Green Lion edition, which is also Heath's translation but without the commentary. It's been a while since I've looked at the GL edition so if anything I write is incorrect please let me know and I'll edit appropriately. I believe it has some commentary in the difficult Book V, but nothing as detailed as Dover. Most people will need a good amount of help in Book IX as well, which the Dover edition provides. I'll attempt a short comparison between the editions. You'll notice that GL gets short shrift. This isn't because I prefer the Dover edition, but rather the other way around. I prefer the Dover edition because there are so many good things to say about it.

IN FAVOR OF GREEN LION:

**Cheaper. The three Dover books will cost about 13 dollars more. If you don't plan to read past, say, Book VI, like many people, then vol.1 and 2 of the Dover editions are only about 3 more dollars. Although if you really want the best price, the entire Elements is free online (Google D.E Joyce).

**Figures are redrawn on the next page in case a proof carries over. This seems like a minor thing, but I think it's a really nice touch.

**Slightly easier to find a proposition. Ok, I'll give this to them, but their preface exaggerates quite a bit when it says that in the Dover edition it's "difficult to find propositions." How one encounters this difficulty when at the top of each left hand page is the book number and on the right the proposition number is unclear to me. But like I said, it is slightly easier in GL so they win this one.

IN FAVOR OF DOVER:

**If your goal is to learn even a tiny bit about Greek mathematics or math history in general, then read no further. This is the edition for you.

**This is the best edition for beginners. This directly contradicts some previous reviewers, so some explanation is in order. The main argument against the use of this edition for beginners more or less asserts that Heath's commentary is only for the scholar- just dense, esoteric filler material to be skipped by the "real" mathematicians. Maybe that's a little harsh to those other reviewers and slightly exaggerated, but I put myself in the shoes of someone who has never seen this edition and that's essentially the feeling I got. This isn't the full truth. The first two thirds or so of volume 1, before he even gets to Euclid's text, might fit this description, as do some of the in-depth comments following Euclid's definitions, but then again a lot of those comments are necessary to know what Euclid is trying to say. This is the case with quite a few theorems as well. Now, it could be the case that I can't understand them because I'm stupid; that's still a conjecture at this point, although most people believe it's true. But to appeal to authority, De Morgan called some of the propositions in Book V "unintelligible to modern ears", and this description is definitely not limited to that book. Sure, some beginners will be ok with no help, but just like that one friend we all have who always claims to win at the casino, most people who claim to be ok on their own are probably lying. Heath also, especially in Book IX, goes through difficultly-worded proofs in the same style as Euclid but with modern notation, which is a tremendous help. Another thing to keep in mind is that one can understand something at only a superficial level. If not for Heath's commentary, I wouldn't have really understood Proposition VI.27. When I saw "The importance of VI.27-29 from a historical point of view cannot be overrated", I knew I wasn't getting it. I would never have noticed the connection to quadratic equations without the commentary. As far as GL is concerned, you're on your own with all this stuff. Good luck with that.

Still keeping the beginner in mind, Euclid's habit of only proving one case and leaving any others to the reader could be confusing. Take I.24, for instance. Readers with some mathematical maturity will immediately wonder why F is drawn outside of triangle DGE, realize that this is just one case, and prove the others. A beginner might either not notice it at all, which is a bad habit to get into, or become frustrated at not understanding why (they think) F has to be outside the triangle. Heath always explicitly mentions and proves all but the most obvious missing cases. On a related note, Heath explains why lines that Euclid claims will meet, actually do meet, usually invoking the parallel postulate. I think this is important for the beginner because while they "obviously" meet from the diagrams, it gets them into the habit of thinking axiomatically for when diagrams can be misleading or are simply impossible to draw.

**In addition, there's plenty of MATH in the commentary, which not many people have mentioned. There are alternate proofs of several theorems, some of which are arguably superior to Euclid's. There are also extensions that will be new to many readers. In particular, there's a nice extension to the Pythagorean Theorem credited to Pappus, and in Book VI Heath proves, surprisingly easily, that the locus of a point such that its distances from two given points are in a given ratio (not equal to 1) is a circle. I guess this wouldn't be extremely difficult to do algebraically, but it would probably be tedious and nowhere near as pretty as a geometric proof. These are just two I recall at the moment. There are many more.

I think that's reason enough to get the Dover edition.

Contrary to the reviewer who says it is not clear that this is only one volume of 3, I thought there was 6 or 7 volumes altogether. The title clearly says "The Thirteen Books of the Elements, Vol.2: Books 3-4", so obviously this is volume 2 of a collection. Knowing that the original Elements had 13 books, my assumption was there must be several more volumes in this new edition. Actually, Amazon's title of the book is wrong! If you look at the front cover, or the index of this book, you can see that it says "Vol.2 (Books III-IX)". And if you are interested to know, volume 1 covers books I and II, and volume 3 covers books X through XIII so altogether they cover the original Elements entirely.

Update (10/14/2010):

At the time I wrote the previous review, the book's title was "The Thirteen Books of the Elements, Vol.2: Books 3-4". Obviously Amazon must have realized the mistake and corrected the title to "...: Books 3-9")