The Thirteen Books of the Elements, Vol. 3: Volume 3

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 f

Heath does a better job than most in his notes-almost all commentary written in modern editions of great scientific works is hilarious-usually some half brite clown trys to find a million faults in the writing of someone who is obviously one hell of a lot more intelligent. Heath just gives the likely facts surrounding Euclid's life, works, and the evolution of the math contained in The Elements. This is math that is accesible if you're willing to put in the time, because it starts with principles we're all familiar with and can agree on (such as the whole being greater than the part), and slowly and methodically works it's way to comparisons of the 5 Platonic solids. Along the way he covers number theory, plane and solid geometry, and provides an early basis for calculus and even certain branches of physics, although the terminology is obscure if you're familiar with more modern methods. Approach this work as a puzzle book, and try to solve the proofs yourself, or even try to disprove them; proceed slowly, it will take more than a year to work through all 13 books, but you will understand these things much better than the average math teacher when you're done. It's also more fun to try to understand the work of one of the greats than it is to study from one of those overpriced college calculus books-don't worry. The principles of Math and Physics don't change, this book is as valid now as ever!

There are two aspects that must be reviewed: Euclid's text itself and Heath's commentaries. I shall begin with the first.The Elements can be understood by anyone, although appears to have been written for adults. It begins with a system of definitions, postulates and axioms (if you do not know then difference between a postulate and an axiom, Heath's commentary explains it), and proceeds to a logical development of the ideas that appear in connection with our intuition of space. The first book treats lines (intersections, parallels), triangles and paralelograms and most of it is contained in the elementary school curriculum. The second book is also taught at elementary level, but with algebraic symbols. It is interesting to see how the ancients, that didn't have such a good notation as ours, treated problems in general with the methods used in this second book. The third book contains the geometry of the circle; the fourth treats polygons inscribed and circunscribed in circles; again, both are taught at school. The fifth is not taught at elementary level and contains one of the most precious gems of the Greek thought: the theory of Eudoxus, that has many analogies with Dedekind theory of irrationals. Indeed, it has served as a general inspiration for nineteenth century mathematics because of its clear presentation of the meaning of a magnitude. So it's not surprising that, in its endeavours to understand what is a number, the mathematicians looked for light in this beautiful book. The sixth contains the theory of similar polygons and has a lot of features taught at school, but not all. The seventh, eight and ninth treats arithmetic, again without our notation, but are interesting for the same reasons as the second book. The tenth book aplies the theory of the fifth book to geometry and contains the theory of the incomensurables. The last three books contains the Greek version of Spatial Geometry, called by them Stereometry (there are some things that you learn in high school that were not treated by Euclid because they were not known yet, but not very much). Summing all up, you learn a lot of Euclid in school and high school, but probably not with the precision and beauty that he endeavours to treat in this monumental work. Few scientists and mathematicians after Euclid can be said not to have used his work. The beauty of all is that the work still can be classified as one of the most precise, elegant and understandable book of mathematics, even after two thousand years. You can only understand the why reading it. No reviewer can catch in words the essence of the Elements.Heath's commentary is very important because he explains in detail things that would appear difficult for us to understand. For example, why Euclid chooses the order of topics he chooses in his treatment, what is the meaning of every proposition to the whole of the thirteen books, the deficiencies of the work (in today's point of view) and how to correct them, and the history be

I have taught high school geometry for nearly ten years now. It is a subject of which I am very fond. And yet, even though we call the subject Euclidean geometry, very few people, even those of us who teach it, have a clear idea of what exactly it was that Euclid did. We might use the compass and straightedge occasionally but not with Euclid's methodology. I think that this is too bad.Over the course of the past year or so, I have made it a quest to prove the propositions of The Elements in Euclid's style. Thus far (and at a leisurely pace), I have made it through the first two books outlined in this volume. It has been a wonderful experience that has deepened my knowledge of this subject and, hopefully, has made me a better teacher of it to my students. I am looking forward to going through the remaining eleven books of the last two volumes.Some things of which a reader should be aware: this volume only contains Euclid's first two books which, in and of themselves, are not very long; however, this volume also contains 150 pages of introduction and significant commentary on nearly every definition, postulate and proposition by Sir Thomas L. Heath. I found much of this very enlightening and was glad to have it included. Still, this material could easily be a stumbling block for weaker students and people interested in Euclid alone. Heath's notes are very detailed and assume a knowledge of certain things (such as classical languages) that are not a common part of the modern curriculum. But, remember, this commentary was written nearly 100 years ago. Don't let it stand in your way. It can be a bonus but, if you have trouble connecting with it, skip it. The notes and commentary should be considered gravy for the prime component here: Euclid's text.There has never been a writer of mathematics as successful as Euclid. For well over 2000 years the work that Euclid did in compiling The Elements has been the crowning achievement of geometry and it has only been in the twentieth century that his book has been replaced by other texts. There are good reasons for this but, on another level, it is sad that his genius is being diluted. Anyone with a decent handle on high school geometry could get a lot from Euclid himself. The propositions would be familiar and anyone truly interested in understanding how mathematics has become the powerful tool it is today would be remiss in not reading Euclid.

euclid knew how to express the true beauty inherrent in mathmatics with a simple logical progression. Definatly a good contrast for anyone too taken up in the numbers and rules of math, who need to really step back and understand it. Propositions 1.47, 2.9, 2.10, 3.35, and 3.36 are incredibly elegant and simple. the translation itself seems to be accurate enough, and while all of the notes seem to drag a bit in pedantry, they are useful and do not detract from Euclid's work