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

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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Elements has been the crowning achievement of geometry and it has only been in the twentieth century that his book has been replaced by other texts.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.

At the time of this writing, the sales summary points out "Vol. 1", but it does not point out that it is "Volume 1 of 3". Volume 1 provides a historical summary of work that followed _Elements_, along with a detailed translation of Book I and Book II. Heath includes bracketed references to justify each critical step of each proof. The text surrounding each Euclidean statement is detailed, but often very lengthy; at times, this detracts from the reading of the _Elements_ itself. This set is for the scholar of the history of _Elements_, and not the best source for a first-time reading of Euclid. Even with these minor quibbles, however, my copy of Volume I is a well-worn, beloved volume with frequently-annotated margins. All of the major "players" in the development of Geometry are detailed within, as well as their contributions.

I recommend it highly for any scholar that wishes to understand _Elements_ thoroughly, through a close reading of a detailed text.

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.

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

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 behind it all. It is possible to understand the Elements without his commentaries, but you will perhaps not appreciate all the subtle nuances (and there are many) that make us sometimes think difficult to accept that someone could write such monument to the human industry.

Lastly, I can only say that eternity is little to this work.

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 at least 3 good reasons to read this book:

* Even though it's 2300 years old, it's still a great way to learn geometry! That's because Euclid was one of the great pedagogues of all time.
* It's one of the classics. Literally, in that it dates from the classical era. Figuratively, too: it's one of the most read books of all time, and is responsible for a particular style of (mathematical) writing.
* It's a necessary pre-requisite to reading the work of any European mathematician from the medieval period through the 18th century. You can't appreciate Descartes or Newton, for example, if you don't know your Euclid.

Euclid organized his work into 13 chapters, which are called Books. This edition is organized into three volumes. If you want to get all of Euclid's Elements in one volume, the recent Green Lion edition is superb choice. So why buy this edition? For the wonderful introduction and notes by Thomas L. Heath. If these thorough and scholarly additions will bore you, then go for the Green Lion. But if you're interested in the history and influence of Euclid's Elements, this is the edition to read.

What we consider to be basic plane geometry is contained in Books 1, 3, 4 and 6 of Euclid's Elements. (So if you want to cover that material, you'll need to buy this volume and volume 2 of the same edition.) Along the way, Euclid covers the Theory of Proportions in Book 5 and "Geometric Algebra" in Book 2. While the material in Books 1, 3, 4 and 6 is still taught today, with slightly different notation and terminology, those other two books are out of step with modern mathematical practice.

This volume contains a very long introduction, plus Books 1 and 2. Book 1 covers triangles and parallelograms in 48 propositions. Book 2 involves 14 geometrical propositions that have approximately equivalent algebraic versions. The greatest hits are: The Pythagorean Theorem (Book 1, Proposition 47) and the construction of the geometric mean (Book 2, Proposition 14).

Euclid teaches us step-by-step how to prove the most fundamental and complex concepts of geometry in such a systematic and understandable way. By learning Euclid's propositions, we also find ourselves thinking and speaking in a more ordered fashion. I recommend these books to anyone interested in math as well as those who want to improve their debating and reasoning skills.

If you like long, tedious introductions and the need to sort through endless words to find what you're looking for, then you might want this version of Euclid's work. On the other hand, if you want to get to the point and prefer a clear resource for study, the version published by Green Lion is FAR superior to this one.

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.