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1. Here's Looking at Euclid: A Surprising
2. Euclid's Elements
3. Euclid's Window : The Story of
4. The Mathematicall Praeface to
5. The Thirteen Books of the Elements,
6. The Thirteen Books of the Elements,
7. The Thirteen Books of the Elements,
9. The Thirteen Books of Euclid's
10. A History of Greek Mathematics:
11. Euclid in the Rainforest: Discovering
12. Euclid - The Creation of Mathematics
13. A Long Way From Euclid
14. The Babylonian Theorem: The Mathematical
15. Euclids Elements of Geometry
16. The Mathematicall Praeface to
17. Euclid and His Modern Rivals
18. Six Books of Euclid
19. Geometry: Euclid and Beyond (Undergraduate
20. The Elements: Books I - XIII -

1. Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math
by Alex Bellos
Hardcover: 336 Pages (2010-06-15)
list price: US$25.00 -- used & new: US$15.14
(price subject to change: see help)
Asin: 1416588256
Average Customer Review: 4.5 out of 5 stars
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Editorial Review

Product Description
Too often math gets a bad rap, characterized as dry and difficult. But, Alex Bellos says, "math can be inspiring and brilliantly creative. Mathematical thought is one of the great achievements of the human race, and arguably the foundation of all human progress. The world of mathematics is a remarkable place."Bellos has traveled all around the globe and has plunged into history to uncover fascinating stories of mathematical achievement, from the breakthroughs of Euclid, the greatest mathematician of all time, to the creations of the Zen master of origami, one of the hottest areas of mathematical work today. Taking us into the wilds of the Amazon, he tells the story of a tribe there who can count only to five and reports on the latest findings about the math instinct—including the revelation that ants can actually count how many steps they’ve taken. Journeying to the Bay of Bengal, he interviews a Hindu sage about the brilliant mathematical insights of the Buddha, while in Japan he visits the godfather of Sudoku and introduces the brainteasing delights of mathematical games.Exploring the mysteries of randomness, he explains why it is impossible for our iPods to truly randomly select songs. In probing the many intrigues of that most beloved of numbers, pi, he visits with two brothers so obsessed with the elusive number that they built a supercomputer in their Manhattan apartment to study it. Throughout, the journey is enhanced with a wealth of intriguing illustrations, such as of the clever puzzles known as tangrams and the crochet creation of an American math professor who suddenly realized one day that she could knit a representation of higher dimensional space that no one had been able to visualize.

Whether writing about how algebra solved Swedish traffic problems, visiting the Mental Calculation World Cup to disclose the secrets of lightning calculation, or exploring the links between pineapples and beautiful teeth, Bellos is a wonderfully engaging guide who never fails to delight even as he edifies. Here’s Looking at Euclid is a rare gem that brings the beauty of math to life. ... Read more

Customer Reviews (16)

4-0 out of 5 stars Great book - awful formatting!
I highly recommend this book for anyone looking for a fun foray into recreational mathematics.However, I do have to agree with other comments that the formatting really needs to be fixed. Superscript and Subscript just don't render, which results in formulas being listed like:


Because the formulas are never very complex, you can follow along provided you know what math formulas are supposed to look like, but I found that this really distracted from the narrative in some areas.

Graphics such as curves, waveforms, etc are rendered truthfully, so it is only a problem with text.

But I still recommend this book, even if you have to muddle through the poor formatting a few times.

5-0 out of 5 stars wonderfully entertaining
This book should be in every library and school, and available to everyone - for those who love math, it will entertain them with stories of how clever math is; for those who aren't so sure about math, it will show them a side of the subject they probably never learned or appreciated before.

2-0 out of 5 stars Great book, but layout is horrible on Kindle
This is a very interesting and well-written book, but there are a lot of layout problems in the Kindle version (I have a six-inch Kindle 3). Fractions and exponents are particularly bad, with the figures often breaking across lines instead of staying on one line. For example, 1/6 is rendered:


I wish the editors would have paid more attention to these details in a book about math.

4-0 out of 5 stars Interesting
The author provides a cursory look at a wide range of interesting mathematical topics without using too much math.The excursions were very enlightening but I felt that they did go quite deep enough.For example, he mentioned that the Greeks were unable to find the cube root of a number using geometry but an Italian women in the early 1900's was able to do it folding paper.I found this fascinating, but the author doesn't show how she did it.As I read the book I was constantly looking for deeper explanations of the topics he discussed.The author did a great job of making these mathematical nuggets interesting but didn't have the space left over in the book to completely explain them.I found the book interesting and inspirational, and I feel that because it made me want to learn more about the topics it was a good book.

Among the topics I found particularly interesting were:

Flash Anzan: using the mental image of an abacus for great speed

Base 12: why base 12 might be better than base 10

Vedic Math: a different and faster way to multiply numbers

Origami used to solve geometric problems

Pi and the historical quest for more digits

The Pythagorean Theorem and it's many proofs

When it's OK to try to buy one of every lottery ticket

2-0 out of 5 stars Much better books are available
"Here's Looking at Euclid" seems to be targeted at laymen--people who know little mathematics and also little science or engineering.For them, there are much better books available: as informative and much more fun.

Consider the following alternatives (the ordering of this list is not meaningful):

Paul J. Nahin, "Dr. Euler's Fabulous Formula"
Willian Dnunam, "The Mathematical Universe"
Paul Hoffman, "The Man Who Loved Only Numbers"
Robert & Ellen Kaplan, "The Art of the Infinite"
Rueben Hersh, "What is Mathematics, Really?"
E.T. Bell, "Men of Mathematics"
and, of course,
James R. Newman, "The World of Mathematics"(although portions dated, this 1956 work is a classic).
The next paragraph, quoted with trivial editing from Chapter 3, contains a logical mistake that calls into question much of what Bellos writes about numbers.

The Shankaracharya of Puri spoke for about ten minutes ...: "The present mathematical system considers zero as a nonexistent entity," he declared."We want to rectify this anomaly.Zero cannot be considered a nonexistent entity.The same entity cannot be existing in one place and non-existing somewhere else."The thrust of the Shankaracharya's argument was, I think, the following: People consider the 0 in 10 to exist, but 0 on its own not to exist.This is a contradiction--either something exists or it does not.So zero exists."In Vedic literature zero is considered as the everlasting number:' he said."Zero cannot be annihilated or destroyed.It is the indestructible base.It is the basis of everything."

The problem is that Bellos confounds the name of the entity 'zero' with instances of the entity.

Surely an Oxford education in philosophy would have included 20th-century philosophy of language, which emphasizes how such elementary errors can lead to unwarranted conclusions, such as the one in this passage!
... Read more

2. Euclid's Elements
by T.L. Heath Translation
Paperback: 527 Pages (2002-08-20)
list price: US$29.95 -- used & new: US$22.23
(price subject to change: see help)
Asin: 1888009195
Average Customer Review: 5.0 out of 5 stars
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Editorial Review

Product Description
Green Lion Press has prepared a new one-volume edition of T.L. Heath's translation of the thirteen books of Euclid's "Elements" In keeping with Green Lion's design commitment, diagrams have been placed on every spread for convenient reference while working through the proofs; running heads on every page indicate both Euclid's book number and proposition numbers for that page; and adequate space for notes is allowed between propositions and around diagrams. The all-new index has built into it a glossary of Euclid's Green terms. ... Read more

Customer Reviews (19)

3-0 out of 5 stars Nice Presentation, but not the whole stroy
This is a nice edition of Thomas L. Heath's translation of Euclid's ELEMENTS.Those who rely on this edition alone will miss the considerable insights that they could get from looking at the Dover reprint of Heath's Euclid, available in three volumes.

If you think you have no need for Heath's exposition, ask yourself what Postulate Four, "All right angles are equal," means.Perhaps it says that all 90 degree angles are equal, but that would seem to be true by definition.The nub of this assertion is the invariance of figures, as Heath points out in his notes on Euclid and in his MANUAL OF GREEK MATHEMATICS.The invariance of figures is perhaps the deepest and most important idea underlying Euclidean (and nonEuclidean) geometry.It would be a pity for you to miss it.

The Dover books are intimidating, but you are looking at a subject that has developed over thousands of years.This Green Lion edition is a step backward, not forward.There is no royal road to geometry.Wonderful as Euclid is, you will have to work to understand him.Ignoring the commentaries of the past is not the way forward.

Nice book for your shelf.

5-0 out of 5 stars Excellent Translation
Thomas L. Heath's translation is the most authoritative translation of Euclid's Elements in English.The extensive research he put into translating the work the way he did can be found in The Thirteen Books of Euclid's Elements, Books 1 and 2, which I would recommend to anyone who wants to further investigate Euclid, and his influences.

As far as this copy is concerned, it's amazing.I got the hardcover copy, and a friend got the softcover.They are both sufficient for studying - huge margins, large pages, and diagrams on every page (so you are not constantly flipping back and forth between pages).This book was not designed for portability, so don't be surprised by its size - it's about the size of a regular math textbook.

If anyone intends to thoroughly study The Elements, this is the copy I would recommend.

5-0 out of 5 stars Review from a Greek who has read Ta Stoixeia(The Elements) in Greek.
The Elements is a work that every mathematician should read. In fact, a Math PHd degree should not be granted to anyone without such a one passing a comprehensive examination on Euclid. And by this, I don't just mean high school geometry, because The Elements contains far more than any high school course in Geometry.

Kudos to Heath. He has performed the best possible job he could. However, as with all other translations, Heath's explanations are often unclear or not immediately evident. Heath appears to write as if his audience already has a solid knowledge of mathematics, therefore the elements is not a work kind to beginners.

The ancient Greeks were intelligent beyond belief. It is not only possible that Archimedes knew how to differentiate, but highly probable that he used both integration and differentiation in his publications, many of which are now lost forever.

Every student intending to read the Elements should first read my blog on "how we got numbers". By so doing, the 'correct approach' will be in place for studying the elements. My blog on Euclid and area is also a good preparation for reading and understanding Euclid's works.


5-0 out of 5 stars What more can be said?
I feel like I cannot adequately praise such an obviously beautiful and enduring classic of mathematics.The gravity of the book makes insignificant any praise I may have for it. One does not need to praise it, it is geometry; it stands true whether we praise it or not. No math is more elegant.

So, with that said, this is a great edition of Elements. It is high quality construction: thick paper, heavy binding, thick cover. You will certainly get a lot of use out of it.

5-0 out of 5 stars There aren't enough stars
Pretty much anything I could say has already been mentioned, but I'd like to heap another perfect rating onto the pile. "Elements" is one of the most important books ever written, and the teachings it covers are the underpinnings of all modern math. The Green Lion edition is nearly peerless in terms of quality and completeness.

I often have trouble sleeping, and I'll reach for this book to lull myself with math, only to look at the clock and realize an hour or more has zipped away. Anyone with an interest in math, geometry, or philosophy should spend time with this volume. ... Read more

3. Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace
by Leonard Mlodinow
Paperback: 320 Pages (2002-04-09)
list price: US$15.00 -- used & new: US$6.42
(price subject to change: see help)
Asin: 0684865246
Average Customer Review: 4.0 out of 5 stars
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Editorial Review

Product Description

Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space -- in the living room or in some other galaxy -- have been the hidden engine of the highest achievements in science and technology.

Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.Amazon.com Review
"How do you know where you are?" asks Leonard Mlodinow in his charmingmathematical history, Euclid's Window. This question and othersabout space and time grew out of simple observations of the environment bya select group of thinkers whose lives and brains Mlodinow dissects.Starting with Euclid, geometry has flowed out over the centuries,describing the universe, and, Mlodinow argues, making modern civilizationpossible.

This is not just a history of geometry--it's a timeline of reason andabstraction, with all the major players present: Euclid, Descartes, Gauss,Einstein, and Witten, each represented by a minibiography.

Lots of examples pepper the narrative to help readers achieve their own"eureka!" And it's impossible not to be staggered at the mathematical featsof these geniuses, accomplished as many of them were in the absence ofanything but observation and intense thought. Each story buildssatisfactorily on the last, until at the end of this delightful book, onehas a sense of having climbed a peak of understanding.

A working knowledge of basic geometry is helpful but not essential forenjoying Euclid's Window, and Mlodinow's chatty style lends itselfremarkably well to explaining these deep and revolutionary concepts.--Adam Fisher ... Read more

Customer Reviews (57)

4-0 out of 5 stars Entertaining Book, Though Most of Its Coverage Isn't Unique
I was given this book as a gift, and after having it sit on my shelf for a long time, I finally read it. Or at least most of it (I'll explain later).

As with most pleasure books that are ostensibly about math, this one is really a combination of math, physics, history and a touch of philosophy. Although the title implies that it's about geometry, the book actually discusses a broad range of topics from ancient Greek philosophers, astronomy, Des Cartes and his coordinate system, Maxwell's equations, Einstein and his theories of relativity, and string theory to name only a few.

One of the things I like about this book is that it actually discusses concepts in non-Euclidean and how they apply to real life (e.g. cartography). Non-Euclidean geometry is something that seems to be very seldom discussed in laymen circles. Unfortunately, this book didn't discuss it enough. And other than this one topic, the book doesn't have anything really unique to discuss. That's not to say the read was boring, but why write about topics other authors have extensively discussed already?

Some last comments: the writing style is good (you'll probably laugh at least a couple times). There are a few egregious typos.
And why did I rate this book 4 stars instead of 5? Because it spends the last few pages discussing string theory. I personally think string theory is garbage, hence I didn't bother to read that part.

4-0 out of 5 stars Biographical history of particle physics
"Euclid's Window" traces the roots of particle physics, from the initial geometric work of the ancient Greeks, to Descartes attaching algebra to geometry, to Gauss and Riemann realizing that space need not be flat, to Einstein applying these ideas in the theories of relativity, to the particle physics and string theory as we know it today. These are just a few of the mathematicians and scientists discussed. The book is not a history of geometry as the subtitle suggests, as Mlodinow only takes the parts that are relevant to the current physics-based explanation of the world (membrane theory) and the quest for a grand unified theory and how geometry fits into it. The story along the way is very engaging and entertaining, revealing both the life and times of the people that invented the various theories we use today, as well as lucidly explaining the theories themselves (even string theory). I highly recommend the book for both entertainment value and educational value, though I must qualify this statement: Mlodinow makes a few blunders along the way with dates, fills in some details with his own imagination, and interjects his opinion quite frequently. You might walk away from the book thinking that Ed Witten is the next Einstein (not to discredit Professor Witten, as he has made very important contributions). Mlodinow most noticeably leaves out contributions from the ancient Indians and Chinese, and only briefly mentions the Arabs- basically taking a very Europe- and American-centric point of view...take it or leave it, but I can't help but agree that these are the people that took us from the parallel postulate to quarks, gravitons, and so on. Historical context is cherry-picked to support the anti-Christian and anti-antisemite (basically pro-Jewish) opinions of the author, which isn't to say the points aren't valid. As you will discover in reading the book, Christianity killed (literally) the ancient Greek science, and has impeded the return of logical thought and science ever since.

So we don't have complete historic rigor here- I say who cares. Mlodinow has written a story with few geometric sketches and even fewer equations, not a textbook. If you want the usual dry history of "and on April 12, 1652, Hermann von German discovered this phenomena while rowing a boat across a lake," or page after page of equations, then I'm sure there are many other books out there to satisfy your needs. So, take the finer points with a grain of salt (if it sounds too good to be true, it probably is- except for C.F. Gauss) and enjoy the ride of learning about the people behind the math and physics. This is still a great book that I would recommend to those interested in math and/or physics.

5-0 out of 5 stars fabulous!
this is the only history book I have ever liked. great stuff if you like physics or geometry

5-0 out of 5 stars "The book of nature is written in mathematics" Galileo
"The book of nature is written in mathematics."Galileo

If writing around 1632 Galileo was right that the book of nature is written mathematics then Leonard Mlodinow's book is a kind of Cliff's Notes version.

Mlodinow is a highly experienced writer who collaborated with Oxford's Stephen Hawking when they wrote A Briefer History of Time and his understanding of the material as well as his ability to write accessibly both abound in this work which traces mathematics from the time of Euclid to its present place of prominence on the frontlines of string theory.

Along the way, Mlodinow gives biographies of some of the critical figures like Euclid himself, Descartes, Gauss, Einstein and finally Ed Witten...interestingly enough who works out of the same Institute for Advanced Study that Einstein worked out of in the years prior to his demise.

What makes the study so fascinating is that it tracks a body of study...mathematics...which endeavors to describe reality.In the beginning the story started with Euclid and his fifth postulate...the assertion that parallel lines don't meet.

While it's true that Euclid's postulate produced a self consistent mathematical system, it's also true that eventually (and by eventually I mean like over two thousand years later) it was discovered that you can create yet another self consistent mathematical system which says that parallel lines do meet.

In other words, Euclid created a ruler which is great for measuring flat spaces but later mathematicians in collaborative effort created a special bendable ruler which can measure curved spaces...like a ball.

The significance of this later discovery was made all too obvious when Albert Einstein asserted that gravity bends space...making it more curved and less flat.

So as can be seen the story here is an important one which tells us nothing less than the true emerging story of the universe in which we live and its origins.

5-0 out of 5 stars Euclid's Window - A highly enjoyable walk through the Math Timeline!
Having a lifelong interest in 2 and 3D "geometry", this walk down memory lane into the future of mathematical theory and application was most informative,enlightening and a learning experience. Being introduced to many personalities old and new such as Edward Witten was a real treat!Mlodinow's approach caused me to think and ponder and his humorous style and personal experiences kept me very interested! I cannot wait to finish "The Drunkard's Walk". ... Read more

4. The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara
by John Dee
Paperback: 72 Pages (2010-07-12)
list price: US$9.99 -- used & new: US$9.99
(price subject to change: see help)
Asin: B003VRZ8UQ
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Editorial Review

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The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara is presented here in a high quality paperback edition. This popular classic work by John Dee is in the English language. If you enjoy the works of John Dee then we highly recommend this publication for your book collection. ... Read more

5. The Thirteen Books of the Elements, Vol. 1: Books 1-2
by Thomas L. Heath, Euclid
Paperback: 464 Pages (1956-06-01)
list price: US$14.95 -- used & new: US$8.70
(price subject to change: see help)
Asin: 0486600882
Average Customer Review: 4.5 out of 5 stars
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Editorial Review

Product Description

Volume 1 of 3-volume set containing complete English text of all 13 books of the Elements plus critical apparatus analyzing each definition, postulate, and proposition in great detail. Covers textual and linguistic matters; mathematical analyses of Euclid's ideas; commentators; refutations, supports, extrapolations, reinterpretations and historical notes. Vol. 1 includes Introduction, Books 1-2: Triangles, rectangles.
... Read more

Customer Reviews (20)

5-0 out of 5 stars Extensive copy!
This copy is intended for the reader who already knows The Elements, and wants to investigate the influences of Euclid, and a vigorous account of every single step of the translation.It is important to know that this is more of a commentary on the Elements, rather than The Elements itself.For those who want just The Elements, the copy you want is Euclid's Elements.

If this is the first time you are reading The Elements, this is probably not the copy for you.However, if you are pondering about the translations, or are curious about who might have influenced a certain proposition, this edition would be perfect.

5-0 out of 5 stars Great book on the foundation for Geometry
Great study book. It is quite detailed...very, very informative
for Geometry teachers...great tool for learning about Geometry's

5-0 out of 5 stars The Fundamentals of Math AND Philosophy
For anyone looking for an introduction to math, geometry more specifically, or philosophy at large, this is THE book, and in my estimation, THE edition to get.Heath is no fly-by-night pop-philosopher.His notes are almost as worthwhile as the text itself... a truly monumental feat.

The rigors of geometry, the method of logic learned in these proofs, are one of the best ways to prepare the mind for real philosophy.Anyone can talk about the true and the good in a manner that inspires awe through vagueness and transcendent language, but that is a very different matter than the sort of philosophy that is built upon the real work of logic and hard-earned definitions.

If you have any interest in the latter, I heartily commend this book to you.

5-0 out of 5 stars A classic, but don't try to learn geometry from Euclid
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.

3-0 out of 5 stars Profoundly humbling.
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 amathematician, 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 ... Read more

6. The Thirteen Books of the Elements, Vol. 2: Books 3-9
by Thomas L. Heath, Euclid
 Paperback: 464 Pages (1956-06-01)
list price: US$16.95 -- used & new: US$7.50
(price subject to change: see help)
Asin: 0486600890
Average Customer Review: 4.5 out of 5 stars
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Editorial Review

Product Description
The definitive edition of one of the very greatest classics of all time--the full Euclid, encompassing almost 2500 years of mathematical and historical study. This unabridged republication of the original enlarged edition contains the complete English text of all 13 books of the ELEMENTS, plus analyses of each definition, postulate, and proposition. ... Read more

Customer Reviews (4)

5-0 out of 5 stars books 3-9, not 3-4
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")

5-0 out of 5 stars Euclid's Elements Volume 2
This unabridged republication of the 2nd enlarged edition originally published by Cambridge University Press contains the complete English text...together with a critical aparatus which analyzes each defintion, postulate, and proposition in great detail. It covers textual and linguistic matters; mathematical analysis of Euclid's ideas; classical, medieval, renaissance, modern commentators and their interpretations; refutation, supports, extrapolations, reinterpretations, historical notes, all given with extensive quotes.

Book III to IX of the Elements. Circles, their properties, tangents, segments, figures described around and within circles, ratios, proportions, magnitudes of lines; of triangles; polygons and their parts; prime numbers, products, ordinary numbers, plane numbers, solid numbers, series of ratios, etc.
--- excerpts from book's back cover

4-0 out of 5 stars Classic = Elegant, if not for the notation!
We, the children of this new age, are deprived of major classics and beautiful mathematics because of the tediousness of the notation. Oh, do not be optimistic, this is not the only book with forbidding notation, see Artin's Galois theory, which is an excellent book if someone just tries to update its notation.

Aside from that the book was a merry one. It contains more books than the first one. It contains the books 3 up to 9 of Euclid's 13 books of the elements.

Book 3 is a delightful one. Its sole purpose is to characterize circles. It goes with the same style of the first two books given the first volume. Books 4 continues in the same fashion and studies circumscribing and inscribing figures by others.

Book 5 is the first attempt to bring geometry near to algebra. It deals with proportions. The notation started getting more and more cumbersome. He continues giving us things that we know already. And all through the volume until book 9 we see results commonly given in simple college algebra in the most tedious fashion.

I praise this volume only for the material on circles and I see that it is worth reading if you have a strong constitution. As for me I am not going to read the third one about the out of date commensurable numbers.

4-0 out of 5 stars CAUTION
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 acquireall thirteen books of Euclid's, The Elements. ... Read more

7. The Thirteen Books of the Elements, Vol. 3: Books 10-13
by Thomas L. Heath, Euclid
Paperback: 574 Pages (1956-06-01)
list price: US$14.95 -- used & new: US$3.99
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Asin: 0486600904
Average Customer Review: 5.0 out of 5 stars
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Volume 1 of 3-volume set containing complete English text of all 13 books of the Elements plus critical apparatus analyzing each definition, postulate and proposition in great detail. Covers textual and linguistic matters; mathematical analyses of Euclid's ideas; commentators; refutations, supports, extrapolations, reinterpretations and historical notes. Includes 2,500 years of critical commentary. Total in set: 995 figures.
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Customer Reviews (1)

5-0 out of 5 stars excellent
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 andunderstand it. Propositions1.47, 2.9, 2.10, 3.35, and 3.36 areincredibly elegant and simple.the translation itself seems to be accurateenough, and while all of the notes seem to drag a bit in pedantry, they areuseful and do not detract from Euclid's work ... Read more

by Euclid
Kindle Edition: Pages (2009-12-29)
list price: US$1.05
Asin: B001CY263C
Average Customer Review: 1.0 out of 5 stars
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Customer Reviews (1)

1-0 out of 5 stars From the look of the preview, this is only an outline of the work
Euclid's Elements consist of 13 books of geometric demonstrations, called "propositions," building in complexity from triangles to multi-sided solids.However, when I looked at the preview of this book, it became obvious that it only provides a list of the titles to each proposition, but never shows the propositions themselves!This means that it is only an outline of Euclid's work, not the work itself, from which the reader can gain nothing of mathematical value.Avoid like the plague. ... Read more

9. The Thirteen Books of Euclid's Elements
by Euclid
Paperback: 404 Pages (2010-01-01)
list price: US$19.99 -- used & new: US$17.98
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Asin: 1420934767
Average Customer Review: 5.0 out of 5 stars
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Often called the Father of Geometry, Euclid was a Greek mathematician living during the reign of Ptolemy I around 300 BC. Within his foundational textbook "Elements," Euclid presents the results of earlier mathematicians and includes many of his own theories in a systematic, concise book that utilized meticulous proofs and a brief set of axioms to solidify his deductions. In addition to its easily referenced geometry, "Elements" also includes number theory and other mathematical considerations. For 23 centuries, this work was the primary textbook of mathematics, containing the only possible geometry known by mathematicians until the late 19th century. Today, Euclid's "Elements" is acknowledged as one of the most influential mathematical texts in history. This volume includes all thirteen books of Euclid's "Elements" and is translated by Thomas Heath. ... Read more

Customer Reviews (1)

5-0 out of 5 stars Fine collection - all in one book
Euclid.Everything in one binding.It doesn't have the commentary on the translation like other editions do, but this is the no-frills complete collection. ... Read more

10. A History of Greek Mathematics: Volume 1. From Thales to Euclid
by Thomas Little Heath
Paperback: 468 Pages (2000-12-27)
list price: US$29.99 -- used & new: US$29.98
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Asin: 0543974480
Average Customer Review: 5.0 out of 5 stars
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This Elibron Classics book is a facsimile reprint of a 1921 edition by the Clarendon Press, Oxford. ... Read more

Customer Reviews (2)

5-0 out of 5 stars more than just history
It should be noted that this is one of a two volume set. This author also compiled and commented upon The Elements ofEuclid in three volumes [also available here].

Theseworkswere first brought to my attention by my Greeklanguage professor nearly 40 years ago as the best English language source on Greek Mathematics.

Just as the Greeks did not view `pure' mathematics or geometry as a lifes-work so to younger readers [through collage] the methods of logic may prove most useful.

For we retired `geezers' not quite ready for Oprah reruns and made for T.V. `romances' it may be the stimulation ofthe brain by the problems [which are documented and solved infull], the history andthe `awe' of how much these did `without computers';

5-0 out of 5 stars Academically great
This is not a terribly exciting book to read, but it is a superior reference for looking up Greek mathematicians.It is apparent that the author is partial to Euclid, as his section is close to a third of the book, (see the author's version of the Elements)but being a Euclid fan myself I can forgive this easily.Even the most obscure mathematicians are covered in good detail along with what they proved, as well as how they proved it.For those interested in historical mathematics, this book is invaluable. Note:This is a two volume set.I thought it was only one and I only purchased the second.Be sure to get both. ... Read more

11. Euclid in the Rainforest: Discovering Universal Truth in Logic and Math
by Joseph Mazur
Paperback: 352 Pages (2006-07-25)
list price: US$15.00 -- used & new: US$3.21
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Asin: 0452287839
Average Customer Review: 4.0 out of 5 stars
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Like Douglas Hofstadter’s Gödel, Escher, Bach, and David Berlinski’s A Tour of the Calculus, Euclid in the Rainforest combines the literary with the mathematical to explore logic—the one indispensable tool in man’s quest to understand the world. Underpinning both math and science, it is the foundation of every major advancement in knowledge since the time of the ancient Greeks. Through adventure stories and historical narratives populated with a rich and quirky cast of characters, Mazur artfully reveals the less-than-airtight nature of logic and the muddled relationship between math and the real world. Ultimately, Mazur argues, logical reasoning is not purely robotic. At its most basic level, it is a creative process guided by our intuitions and beliefs about the world. ... Read more

Customer Reviews (12)

5-0 out of 5 stars Brought New Insights to an Old Science Reader
My only complaint is that I read the whole thing in only two days, as I couldn't put it down.Since the book cost $15.00, this comes to $7.50 per day.I was expecting a book on mathematics to take me at least a week, which would have been only $2.14 per day.Other than that, it was terrific.Mazur keeps a focus on really basic questions like, "what is truth, anyway." This sounds esoteric, but it leads to some excellent takeaway revelations.To put it another way, you don't merely learn some new facts about math, you have the opportunity to get a clearer understanding of science as well as mathematics.For instance, I had always thought of "junk science" as inferior to peer-reviewed science because their analytical processes - deduction, induction, inference, observation, etc. - were different from one another.Mazur forces you to look at a process known as "plausible reasoning" that is the hallmark of the very best peer-reviewed science, which leads to better understanding of the validity of scientific theories and laws.In my letter to the author, I told him that, "I have always been a sceptic, but now I know why I am a sceptic."I highly recommend this book to anyone who is a little sceptical - and all good scientists are a litte sceptical - about some modern scientific "discoveries." Thank you, Joe Mazur.

Dr. J. G. McCully.

5-0 out of 5 stars Why Science isn't Faith Based
I found this remarkable little book in a 'bargain basket' at a bookstore in Panama City. I took it with me to Kuna Yala where I was staying in a Kuna village without electricity, running water, or much else. Thus I had a lot of very quiet time to read it. Parts of it were very difficult for me, but the overall message of the book is wonderful.

For some reason there are 'Post Modern' scholars (sic)(Stanley Fish of Florida International University is one) who want to reserect the old canard of 'science is merely faith' as a criticism of recent books on faith by scientists and others (Sam Harris, Richard Dawkins, Christopher Hitchens).

While this book does not directly address the issue of `faith vs. science', it is a explanation of the scientific method and the cleanest example of why science is not based on faith, does not require faith, and, in fact, can be demonstrated to be counter to faith based beliefs.

When I was an undergraduate in the late Pleistocene I took a philosophy course in which the instructor took the position that `Science,' `Religion,' and `Philosophy' were equally valid approaches to understanding reality. I didn't do well in the course because I kept objecting to his basic premise. While my training is in biology I have been trying to learn Particle Physics and Cosmology in recent years and am even more convinced that science is fundamentally divorced from other approaches.

Even if you are not going to a tropical island for a week, get this little book. You will probably enjoy it and will come away with a much better appreciation for how science works and how scientists think.

5-0 out of 5 stars One of the best popular mathematics books I have ever read
Properly presented, the fundamental truths of mathematics are easy to understand. By that I mean that if they are presented in the appropriate non-technical language and with simple examples, then almost anyone can understand them. Mazur does this and does it very well. Much of the mathematics in this book is also philosophical in nature. A great deal of ink is spent in describing Zeno's famous paradox, "proving" that motion is impossible. His development of the solution to the paradox can be understood by anyone possessing the most rudimentary of mathematical backgrounds.
The role of proof in mathematics is also discussed, with questions raised as to what actually constitutes a proof. Mathematicians have debated this point since the Greeks invented the concept of the mathematical proof, and this is a good recapitulation of that debate. I consider it very healthy for the math profession to admit to the laity that mathematical proof is not necessarily fixed in concrete. It is also a point of significant honesty to admit that proofs that were considered correct for centuries contained flaws that were discovered and repaired.
There are three sections to the book:

*) Logic
*) Infinity
*) Reality.

The chapter "Does Math Really Reflect the Real World?" makes a point that often astounds mathematicians and others that work in the physical sciences. Namely, that mathematics does describe the real world, not only well, but often astonishingly well. New mathematical concepts are invented and considered to be purely abstract, there being no current practical application. However, as science progresses in other areas, that "purely abstract" idea suddenly has uses in the real world. Of course, the real world does have its flaws. It is impossible to create the perfect circle, the well-balanced coin and die do not exist and there are times when we cannot measure a value to enough decimal places to get true predictability. Those situations are also covered, which is important, as it points out that even the best mathematics does not give us absolute predictability. Fortunately, nearly all of the time, the good enough is in fact really good enough.
This is one of the best popular mathematics books that I have ever read; it covers the fundamentals that need to be covered and at a level that nearly everyone can understand.

Published in Journal of Recreational Mathematics, reprinted with permission

3-0 out of 5 stars Pleasure reading
This is a good book. You just read page after page without any brain-twisting theories. Facts and stories about math are lucidly presented. A soft way to teach truths about math and logic.

2-0 out of 5 stars Dissapointing
The book does not deliver to the promise in its title.It is yet another "discover fun in mathematics" book, mixed with a poorly written travel account. ... Read more

12. Euclid - The Creation of Mathematics
by Benno Artmann
Hardcover: 368 Pages (1999-06-10)
list price: US$79.95 -- used & new: US$29.94
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Asin: 0387984232
Average Customer Review: 4.5 out of 5 stars
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The philosopher Immanuel Kant writes in the popular introduction to his philosophy: "There is no single book about metaphysics like we have in mathematics. If you want to know what mathematics is, just look at Euclid's Elements." (Prolegomena Paragraph 4) Even if the material covered by Euclid may be considered elementary for the most part, the way in which he presents essential features of mathematics in a much more general sense, has set the standards for more than 2000 years. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem. We see how abstraction works and how it enforces the strictly deductive presentation of a theory. We learn what creative definitions are and how the conceptual grasp leads to the classification of the relevant objects. For each of Euclid's thirteen Books, the author has given a general description of the contents and structure of the Book, plus one or two sample proofs. In an appendix, the reader will find items of general interest for mathematics, such as the question of parallels, squaring the circle, problem and theory, what rigour is, the history of the platonic polyhedra, irrationals, the process of generalization, and more. This is a book for all lovers of mathematics with a solid background in high school geometry, from teachers and students to university professors. It is an attempt to understand the nature of mathematics from its most important early source. ... Read more

Customer Reviews (3)

5-0 out of 5 stars Interesting survey of the Elements
The material in Euclid's Elements may be divided into four categories of very different degrees of interest for modern readers. (a) Elementary material. To keep us interested when covering tedious proofs of obvious things Artmann discusses foundational issues (as seen by Euclid and contrasted with the modern view), the principles that guide the overall structure of the books, historical topics, etc. (b) Well-known material. This category includes some basic geometry (Pythagoras's theorem, etc.), but primarily it includes all of Euclid's number theory. This is very interesting stuff but less exotic than other parts of the Elements since these pearls have been kept polished and accessible (see, for example, the historically enlightened books by Stillwell, esp. "Elements of Number Theory" and "Numbers and Geometry"). (c) Incomprehensible material. Some parts of the Elements appear mysterious to the modern reader, especially some aspects of "geometric algebra" and of course the theory of incommensurability. A truly faithful guide to the Elements would make it its mission to clarify these things, but Artmann is not that committed, often preferring instead the easy way of looking for agreement with modern mathematical-aestetical principles and commenting on those things instead (e.g. discussions of the role of generalisations and the relation between problems and theories). (d) Constructions. This is the most rewarding part. First there is the remarkable construction of the regular pentagon in book IV. Euclid's construction draws on all previous books, in accordance with his aim to hide his masterplan and unveil it in a flash of brilliance just as we though he was getting lost in a mass of technicalities. Artmann adds helpful commentary on how the principles of construction may be understood through possible earlier constructions that used marked rulers and similar triangles (not developed by Euclid until book VI). The similar triangles proof uses a neat property of the pentagon: a side and a diagonal are in "extreme and mean ratio" (i.e. "golden ratio"), so constructing this ratio is one way to construct the pentagon. Euclid brings this up in connection with the marvellous constructions of the regular polyhedra in book XIII --- the culmination of the entire Elements. "For the construction of the dodecahedron, Euclid starts with a cube and constructs what can be called a 'roof of a house' over each of its faces". The pentagonal faces of the dodecahedron are made up of a quadrilateral piece of one roof and a triangular gable from the roof on the adjacent side. To make this work we must choose the right length for our beams, i.e. we must divide the side of the cube in extreme and mean ratio. The construction topics are not only the most rewarding in themselves but also the starting points of Artmann's most enthusiastic excursions, including the modern algebraic view of constructions as developed by Gauss, the group theoretical view of symmetries and polyhedra, appearances of these figures in art and architecture, etc.

4-0 out of 5 stars Roots of mathematics in our Western Culture
This is a Renaissance book by a Renaissance man.Artmann gives a full summary of the "Elements", using considerable modern notation.It is accurate and detailed, and the various themes he traces (such as Symmetry, or Incommensurables) let him include a wide range of topics: architecture, design, sculpture, myth, history -- even philology and poetry.Some may think he limits himself too narrowly to the classical Greeks, does too little digging in the Babylonian or Egyptian parts of the story.

To Artmann's credit, his book disregards the smallscale disputes amongst superspecialists ("all modern translations of Elements are satisfactory").He overturns the fashionable idea that the "Two Cultures" cannot communicate.So, Rilke has something to say -- perhaps not to Hilbert, but to the widely cultured mathematician, or to the general reader -- about Contradiction, or Widerspruch.

About the pre-Euclidean origins of mathematics in Greece, he overmodestly disclaims specialist knowledge.An example:he traces the earliest technical work on the dodecahedron and the icosahedron via pre-Euclideans such as Theaetetus (Plato's friend), and up to the highly abstract Group Theory work on isomorphisms of the 1990s A.D. -- and does this well and surefootedly. Too bad his modesty barred him ("I leave that to the specialists") from analyzing the pre-history of Euclid's Book XII, the classical ancestor of our integral calculus.The fact is that he knows a great deal about Eudoxus (another friend of Plato's).Perhaps more detail in a Second Edition?

His work on the so-called Euclidean Algorithm (finding a greatest common factor) is another valuable contribution.Its autobiographical flavor is reminiscent of Archimedes in "Sand Reckoner".It allows him to stake out a clear and non-partisan position on the "where is the algebra?" question, on which scholarly debates often produce more heat than light.

So multi-faceted a book, one could wish an Index fuller than a mere 2 pages.Typos are too frequent for a good house like Springer, including two I found in names of authors or book titles.But the book's cultural sweep is admirable throughout, its bibliography good.

TL Heath's 1933 report about the Cambridge undergraduate, so struck by Euclid ("a book to be read in bed or on a holiday") may have been exaggerated, making him over into a Young Werther.But Artmann's charming and learned book really is hard to put down, on or off holiday.

[note: this is a lightly revised version of a review I submitted a few days ago.-Malcolm Brown]

4-0 out of 5 stars Roots of mathematics in our Western Culture
This is a Renaissance book by a Renaissance man.Artmann gives a full summary of the "Elements", using considerable modern notation.It is accurate and detailed, and the various themes he traces (such as Symmetry, or Incommensurables) let him include a wide range of topics: architecture, design, sculpture, myth, history -- even philology and poetry.

He largely disregards smallscale battles amongst the superspecialists ("all modern translations of Elements are satisfactory").He overturns the fashionable idea that the "Two Cultures" cannot communicate.(Rilke has things to say, perhaps not to Hilbert, but to the widely cultured mathematician, about Widerspruch!)

About the pre-Euclidean origins of mathematics, he overmodestly disclaims specialist knowledge.An example:his tracing of the earliest technical work on dodecahedrons and icosahedrons via pre-Euclideans such as Theaetetus (Plato's friend), and on up to the Group Theory work on isomorphisms of the 1990s A.D. is done well and surefootedly. Too bad his modesty barred him ("I leave that to the specialists") from analyzing the pre-history of Euclid's Book XII, the classical ancestor of our integral calculus.The fact is that he knows a great deal about Eudoxus (another friend of Plato's).Perhaps more detail in a Second Edition?

His work on the so-called Euclidean Algorithm (finding a greatest common factor) also contributes importantly.Its autobiographical flavor is reminiscent of that of Archimedes' in "Sand Reckoner".It allows him to stake out a clear and non-partisan position on the question "where is the algebra?" question, on which scholarly debates often produce more heat than light.

So multi-faceted a book, one could wish a fuller Index.But the cultural sweep is admirable throughout.TL Heath's 1933 report about the Cambridge undergraduate, so struck by Euclid ("a book to be read in bed or on a holiday") may have exaggerated, making him over into a Young Werther.But Artmann's charming and learned book really is hard to put down, even at vacationtime. ... Read more

13. A Long Way From Euclid
by Constance Reid
Paperback: 132 Pages (2010-01-10)
list price: US$22.16 -- used & new: US$22.16
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Asin: 1153010585
Average Customer Review: 3.5 out of 5 stars
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Publisher: Thomas Y. Crowell CompanyPublication date: 1834Notes: This is an OCR reprint. There may be numerous typos or missing text. There are no illustrations or indexes.When you buy the General Books edition of this book you get free trial access to Million-Books.com where you can select from more than a million books for free. You can also preview the book there. ... Read more

Customer Reviews (2)

2-0 out of 5 stars Good intentions, bad execution
Constance Reid selected a fascinating series of mathematical topics to present to lay readers, tied together with the unifying thread of what Euclid pioneered. Her writing is lively, much of it is informative, and I would guess that this book, originally published in 1963 and now reprinted by Dover, has been widely read.

It being understood that a popular book must necessarily slough over technicalities in order to convey general ideas, I am nevertheless shocked that Reid, with all her mathematical contacts, including her famous sister and brother-in-law (alive when she wrote this), did not have professional mathematicians check the correctness of what she wrote.

Here are a few inaccuracies I found:

p.60. "Their [Greek] geometry was based on an axiom which stated in essence that parallel lines never meet ..." (By definition, parallel lines are lines in the same plane that do not meet. No axiom is needed to guarantee that!)

p.152. "... the fifth postulate, which makes a statement very roughly equivalent to our common statement that parallel lines never meet."

p.136 Reid states falsely that Gauss was the first person in the history of mathematics to question the age-old assumption that the four classical constuction problems could be solved using straightedge and compass alone. Descartes, for one, preceded him.

p.157 "The true surface of hyperbolic geometry ... is what is called the pseudosphere, a world of two unending trumpets."

p.255 She states incorrectly that the domain of arithmetic presented by Hilbert in his Grundlagen is that of the constructible numbers. It is only a subfield of that.

p. 278 She states that elementary algebra is a decidable theory according to Tarski, without specifying (as she did for "elementary geometry") what that is. Ditto her statement that elementary arithmetic is decidable.

I was also disappointed that Reid hardly gave any references and has no bibliography. Surely many readers became interested in topics she presented and would wish to read more about them.

Reid still can fix these defects in a subsequent edition.

5-0 out of 5 stars Constance Reid: the best non-fiction writer ever!
It was very hard to get a copy of this book, and it cost me a ton of money.I do not regret this purchase in the least.If you have any interest in math (or if you don't you should read it and maybe you'llbecome interested) this book is incredible.I have reread it at least 10times.It tries for nothing less than a story of the major advances ingeometry from the greeks to the present.It is now a little out of date,(it says that Fermat's theorem is unproven) Constance Reid is such a goodwriter, that it does not matter.You should without a doubt try to get acopy of this book.Chapters 7, 9, 12 are exceptional. ... Read more

14. The Babylonian Theorem: The Mathematical Journey to Pythagoras and Euclid
by Peter S. Rudman
Hardcover: 248 Pages (2010-01-26)
list price: US$26.00 -- used & new: US$13.85
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Asin: 159102773X
Average Customer Review: 3.5 out of 5 stars
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In this sequel to his award-winning "How Mathematics Happened", physicist Peter S Rudman explores the history of mathematics among the Babylonians and Egyptians, showing how their scribes in the era from 2000 to 1600 BCE used visualisations of how plane geometric figures could be partitioned into squares, rectangles, and right triangles to invent geometric algebra, even solving problems that we now do by quadratic algebra. Using illustrations adapted from both Babylonian cuneiform tablets and Egyptian hieroglyphic texts, Rudman traces the evolution of mathematics from the metric geometric algebra of Babylon and Egypt - which used numeric quantities on diagrams as a means to work out problems - to the non-metric geometric algebra of Euclid (ca. 300 BCE). Thus, Rudman traces the evolution of calculations of square roots from Egypt and Babylon to India, and then to Pythagoras, Archimedes, and Ptolemy. Surprisingly, the best calculation was by a Babylonian scribe who calculated the square root of two to seven decimal-digit precision. Rudman provocatively asks, and then interestingly conjectures, why such a precise calculation was made in a mud-brick culture.From his analysis of Babylonian geometric algebra, Rudman formulates a "Babylonian Theorem", which he shows was used to derive the Pythagorean Theorem, about a millennium before its purported discovery by Pythagoras. He also concludes that what enabled the Greek mathematicians to surpass their predecessors was the insertion of alphabetic notation onto geometric figures. Such symbolic notation was natural for users of an alphabetic language, but was impossible for the Babylonians and Egyptians, whose writing systems (cuneiform and hieroglyphics, respectively) were not alphabetic. Rudman intersperses his discussions of early math conundrums and solutions with "Fun Questions" for those who enjoy recreational math and wish to test their understanding. This is a masterful, fascinating, and entertaining book, which will interest both math enthusiasts and students of history. ... Read more

Customer Reviews (3)

5-0 out of 5 stars College-level math and science collections especially will find this an intriguing math analysis
The Babylonian Theorem: The Mathematical Journey to Pythagoras and Euclid offers a fine sequel to HOW MATHEMATICS HAPPENED and comes from a physicist who explores the early history of math and how it was used to solve amazing problems. Illustrations from early Egyptian texts shows how math evolved and presents a 'Babylonian Theorem' which he shows was used to drive the Pythagorean Theorem. College-level math and science collections especially will find this an intriguing math analysis.

4-0 out of 5 stars It all adds up through history
Get this book only if you love math, algebra, and geometry. //The Babylonian Theorem// is a mind-boggling examination of ancient Babylonian and Egyptian mathematical prowess. Until reading this book, this reviewer mistakenly thought advanced math was mostly a modern invention. Rudman shows us how Old Babylonian and Egyptian scribes avoided nonterminating fractions, though today the electronic calculator makes them less maddening. Rudman includes a generous portion of square root calculations courtesy of Archimedes, the most famous mathematician in ancient history. The chapter on pyramid volume will intrigue anyone interested in how the ancient Egyptians might have attempted to determine the needed material to build pyramids. Rudman delves into Pythagoras' founding of a religious cult with the motto "All is number," or as Rudman interprets it, "All is rational numbers." We learn that Pythagoras and his followers studied patterns that are formed by numbers, which led to number theory. Rudman concludes with a chapter on Euclid, the ancient Greek mathematician who wrote history's most popular textbook. Throughout the book, Rudman posts "Fun Questions" which are mathematical problems for the reader to solve. This is a mathematics book that includes a little history of ancient people and cultures. It's a good read for number lovers.

Reviewed by Grady Jones

2-0 out of 5 stars I don't get it
This is a very good idea for a book and perhaps some day someone will do it right.Combining history and mathematics is a wonderful way of teaching students and I really wish that this book had been done right.

I have to say that it got off to a fairly good start, with a good description of Egyptian and Babylonian number systems and an explanation for how they might have evolved.Although some of the related equations are not difficult to derive, I think that a quick derivation would have been helpful.I also would not have been able to figure out what a greedy algorithm was from the explanation given if I did not already know it, but these are relatively minor points.

The problem comes when the author starts talking about what he calls the Babylonian Theorem mentioned in the title.He claims that the Babylonians knew how to prove the Pythagorean Theorem and he gives as justification a geometric diagram.Now the diagram does geometrically show that (a-b)^2 + 4ab = (a+b)^2, but I have hard time seeing how the Pythagorean Theorem follows, because the diagram contains no right triangles.There is a related diagram that can be used to prove the Pythagorean Theorem, but the author makes no reference to it, and I am not convinced that the Babylonians could have made use of it, because there is some algebraic manipulation required that they might not have been able to handle.

Okay, so at the very least the author showed how the Babylonians came up with a way of solving a particular type of quadratic equation.The author then claims to show how this was used to solve problems.He gives the following problem from a Babylonian text: A number subtracted from its inverse is equal to 7.I was guessing that in modern terms this would be: x - 1/x = 7, though neither this or any other interpretation is presented. My interpretation must be incorrect because it is stated that the equation has an integer solution and you can tell by inspection that this will not be true for my equation.There is then shown how the Babylonian student solved the problem and I have no idea how the manipulations relate to the original problem.

Later on, it is stated that Euclid proved the Babylonian Theorem using the Pythagorean Theorem.What is shown is a simple way of constructing a right triangle have a hypotenuse of (a+b) and a side of (a-b).Since there is a simple general method of constructing right triangles using straightedge and compass, I am not sure what this particular construction proves.

I would strongly suggest that the author do some serious editing of the book, providing explanations.It may yet prove to be useful, but in its present form it is one big mess.
... Read more

15. Euclids Elements of Geometry
by Euclid
Paperback: 234 Pages (2010-03-27)
list price: US$10.12 -- used & new: US$9.10
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Asin: 1154865770
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The book has no illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from the publisher's website (GeneralBooksClub.com). You can also preview excerpts of the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Original Publisher: Cambridge [Cambridgeshire] : at the University Press; Publication date: 1893; Subjects: Geometry, Non-Euclidean; ... Read more

16. The Mathematicall Praeface to the Elements of Geometrie of Euclid of Megara
by Dr. John Dee
Hardcover: 62 Pages (2010-05-23)
list price: US$31.95 -- used & new: US$23.13
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Asin: 1161359834
Average Customer Review: 4.0 out of 5 stars
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1570. Dee's occult philosophy of mathematics. ... Read more

Customer Reviews (1)

4-0 out of 5 stars reproduction of an interesting text
This is a photopgraphic reproduction of the preface from a 16th century English edition of Euclid. Dee's English isn't itself difficult to read, but the text here can be had to make out in places. I can't say whether this is due to poor reproduction, or to the poor quality of the printing of the original. It's probably a combination of both. This text is of historical interest because it anticipates to some degree the central role mathematics was to take among the sciences in Europe in the context of the 17th century scientific revolution. Dee was writing about 50 years prior to Galilieo's famous publications, and was praising the almost universal practical applicability of mathematical, specifically geometrical, knowledge. ... Read more

17. Euclid and His Modern Rivals
by Lewis Carroll
Paperback: 316 Pages (2010-02-16)
list price: US$29.75 -- used & new: US$17.57
(price subject to change: see help)
Asin: 1144550947
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Editorial Review

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

18. Six Books of Euclid
by Werner Oechslin
Hardcover: 464 Pages (2010-05-25)
list price: US$59.99 -- used & new: US$37.78
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Asin: 3836517752
Average Customer Review: 5.0 out of 5 stars
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Editorial Review

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A rare and beautiful geometry primer from the 19th century

Red, yellow, blue – and of course black – are the colours that Oliver Byrne employs for the figures and diagrams in his most unusual 1847 edition of Euclid, published by William Pickering and printed by Chiswick Press, and which prompt the surprised reader to think of Mondrian. The author makes it clear in his subtitle that this is a didactic measure intended to distinguish his edition from all others: "The Elements of Euclid in which coloured diagrams and symbols are used instead of letters for the greater ease of learners." Byrne is not content to trust solely in the supposed intuitive “logical” structure of Euclid’s axioms and theorems – who doesn’t know the first famous sentences of Euclid’s Elements: “I. A point is that which has no parts. II. A line is length without breadth”? –, but translates them into colourful diagrams and symbols. He thereby thinks in terms of the school classroom: he compares his colours to the dyed chalks in which figures are drawn on the blackboard.

Oliver Byrne (c. 1810–c. 1880) was an Irish author and civil engineer. Little is known about his life, though he wrote a considerable number of books. As Surveyor of Her Majesty's Settlements in the Falkland Islands, Byrne had already published mathematical and engineering works, but never anything like his edition on Euclid. This remarkable example of Victorian printing has been described as one of the oddest and most beautiful books of the 19th century.

Each proposition is set in Caslon italic, with a four-line initial, while the rest of the page is a unique riot of red, yellow and blue. On some pages, letters and numbers only are printed in colour, sprinkled over the pages like tiny wild flowers and demanding the most meticulous alignment of the different colour plates for printing. Elsewhere, solid squares, triangles and circles are printed in bright colours, expressing a verve not seen again on the pages of a book until the era of Dufy, Matisse and Derain.

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Customer Reviews (2)

5-0 out of 5 stars The most beautiful math book of the 1800's
The Byrne book is absolutely gorgeous.Over 130 years ago, Byrne had the idea of removing almost all words from Euclid, and replacing the words with visually obvious color pictures.

His demands on the publishers were tremendous.No color printing of this magnitude had been done before (in the 1800's), and they feared the high cost of the book and the high printing cost would bankrupt them.

They were right.This book destroyed them.It remained an obscure collectors item.Most mathematicians know nothing about this book, and are stunned when they see the pages.

The pages aren't viewable here, but are available at the publisher's site.Well worth a look.

Oh... and Euclid's Elements.Most famous math book in the past 2500 years.This is arguably the best presentation of this material ever made.

5-0 out of 5 stars a beautiful replica of Byrne's unique printing
Oliver Byrne's 1847 printing of Euclid's Elements, in which letters and symbols are replaced with yellow, red, blue, and black diagrams, was a tremendous technical printing feat in the mid 19th century, and remains the most unique Elements ever printed.Instead of "triangle ABC is equal to triangle ACD," the triangles are drawn out, in identifying colors, with an equals sign between.Boldly-colored shapes are queerly compared on every page, making the book as surreal as it is playful.Byrne took one of the most important works of Western thought and made a book, a physical book, as unique and memorable as its intellectual content.Edward Tufte's effusive praise for the work is well-deserved.

This reprinting, for the first time in 160 years (!), is a remarkable page for page copy of the original.The corrigenda page and matching errors in the text are preserved - as they of course should be - and the last page even has the "Chiswick: printed by C. Whittingham." type at the bottom.It's such an exact replica, I don't know how they did it.The originals tend to be foxed and browned, but Taschen either found a copy in incredible condition to emulate, or they recreated the whole thing from scratch (?!).The print and paper colors are near-perfectly replicated.The only noticeable difference between this and an 1847 original is the missing thickness of the diagrams.They look thick and painted in the original, while they are flat and printed in the Taschen edition - they're no longer tactile.All in all, though, mark me down as very impressed.

The binding and clamshell case look good on the shelf, and in the case is also an informative (though dry) booklet on Byrne's work.

If you're looking for a more typical copy of Euclid's Elements, I recommend Green Lion Press's edition, which contains all 13 books in a single volume. ... Read more

19. Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)
by Robin Hartshorne
 Paperback: 532 Pages (2010-11-02)
list price: US$64.95 -- used & new: US$52.25
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Asin: 1441931457
Average Customer Review: 4.5 out of 5 stars
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This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.

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Customer Reviews (11)

5-0 out of 5 stars Good introduction to the relationship between algebra and geometry
The book begins with a quotation from Gauss that suggests the elegance of treating Geometry in the "pure spirit of geometry" i.e. without using real numbers. And Hartshorne follows this principle by developing Euclidean geometry at first from the Elements of Euclid and then (after remarking their weaknesses) by using Hilbert's axioms. However the book is not about the foundations of geometry and much attention is given to the meaning of these axioms in the context of ruler and compass constructions and how this topic is related to analytical methods which lead directly to the theory of field extensions and Galois groups.

I think one of the main purposes of this book is to show how the abstract structure of a Field arise naturally both in Euclidean and Non-euclidean geometry and in this way prove that their typical algebraic models are categorical (that is, they are unique up to isomorphism) which is interesting for its own sake.

So this is not the usual approach to Geometry based on groups of transformations which can be found on other books, but a more "classical" one. But even if the approach is classical, the study of classical problems is always connected with modern algebraic facts, the most striking of them (for me) is the use of algebraic invariants to solve Hilbert's third problem which can be perfectly formulated (but not solved) in elementary terms.

5-0 out of 5 stars Geometry - anything else you need?
So this book answers one of the questions I always had. Call me an ignorant if you please, but I had never had a complete reference of the axiomatization of geometry in my hands before.

I had read Proffessor Hartshorne's book "Agebraic Geometry" before andI thought he was one of those algebrists that hide themselves inside the name of "Algebraic Geometers". Note that I like Algebraic Geometry myself, but I see it more as an "algebraic" branch of mathematics than a "geometric" one. Anyway, this book proved me wrong yet again. After reading it, I found out that Proffessor Hartshorne is really good explaining geometry.

Since I was told some years ago that Geometry could be Axiomatized, I had always hoped to see the structure being constructed. This book finally fulfilled my curiosity. I am indeed grateful with professor Hartshorne just for writting this book.

5-0 out of 5 stars Where was this book when I was a student?
This is a great book, a mature and lively treatment of a familiar subject made new again.If I'd had a text like this as an undergraduate I'd likely still be in math.Most of the serious advances in pre-20th century geometry get subsumed in the typically more topological, or algebraic, but in either case more abstract, treatment one finds today in a typical undergraduate course.Lost in this approach is the intuitive grounding which makes more modern approaches meaningful and not just mere formalism. This book, which would lend itself to self-study as well as to classroom use, goes a long way to restoring that lost grounding.Very highly recommended.

2-0 out of 5 stars Bring your copy of Elements!
I'm still working through this text but I should warn prospective buyers of one thing: The book's early chapters makes heavy references to Euclid's propositions in his books The Elements. I don't just mean references like "Remember that Proposition 43 from Book 2 that says...". No, would that it were so. He'll just give the number and assume you've got your copy of Elements handy.

In that way, it's not really a complete survey of geometry from the start. You'll want to order a copy of Elements with this book. Dover publishes eleven of the books in two volumes.

5-0 out of 5 stars a wonderful book by a world famous geometer
This book reveals the love professor Hartshorne has for geometry and euclid.I became excited about the subject just reading the introduction.The book assumes the student knows high school geometry. which unfortunately eliminates many college students, but I am going to try to use it at least for the second part of my college course.

This is a really well written, expert, wonderfully enthusiastic book, about a great, absolutely classic topic, by a powerful world famous authority in geometry.

The organization assumes the student is reading euclid concurrently.then prof hartshorne explains the difficullties with euclids treatment and shows how to remedy them. e.g. he observes euclids proof of SAS uses a principle of superposition without stating it, then although he adopts the Hilbert option of making this an axiom, he also presents an alternative treatment in which the principle of superposition is an axiom, and SAS is then proved exactly as euclid does.this sort of thing shows very clearly that euclids proofs become correct, merely by clarifying his implicit assumptions.

i love this and think it enhances the subject enormously.

the exercises are so ambitious and far reaching I at first dismissed them as unrealistic, but soon became infected with dr hartshornes enthusiasm for putting the students in touch with their best abilities, and challenging them to reach as deeply as they can.

This book is a remarkable work of scholarship, with far more content than one course can use.The student has here a work that will repay years of study.again the price makes it a bargain compared to far inferior works at double the price. ... Read more

20. The Elements: Books I - XIII - Complete and Unabridged
by Euclid
 Paperback: 1243 Pages (2006)
-- used & new: US$34.25
(price subject to change: see help)
Asin: 0760763127
Average Customer Review: 5.0 out of 5 stars
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5-0 out of 5 stars No experience like it...
In Plato's Symposium, Socrates recounts Diotima's words to him: "This, my dear Socrates," said the stranger of Mantineia, "is that life above all others which man should live, in the contemplation of beauty absolute," and, to me, reading Euclid is as close as one can get to experiencing what Diotima meant.Euclid starts with a few simple definitions and axioms, and then elegantly builds a whole system of mathematics.The reader is led from one proposition to another until he starts to perceive the immense shining beauty of mathematical truth, as if, after a lifetime in a cave, he were shown one beautiful, brightly lit object after another and gradually shown the sun.

It does not matter that Euclidean mathematics has been supplanted by modern algebra and that mathematicians have devised non-Euclidean geometries.It does not matter that David Hilbert had to come up with several more axioms which Euclid should have included, and it does not matter that Kurt Gödel proved that Euclid's dream (in fact, the dream of all mathematicians since antiquity) - to create a perfect system of mathematics, consisting of statements whose truth can be derived from a few postulates and definitions - is impossible.What matters is the experience: it is an awesome thing to be led by the Master (which he must be called, even if many of the theorems that he proves are not his own) to see the beautiful Truth, and it is genuinely humbling.Even the best mathematicians realize that they can never replicate his incredible feat.

Once upon a time, the Elements were a part of a good liberal education, and thinkers from Newton to Russell found this work an indispensable part of their education.Alas, this has been lost, so the curious reader has to find it for himself.Fortunately, we have this edition, and, while it may be old and somewhat difficult (it is a translation from the 1920s), it is heavily annotated and even presents some of the Greek text.It is not easy going (I had intense difficulty with Euclid's presentation of number theory in Book X), but it is worth the effort.I am glad that it is now available in one volume, which makes it much more affordable and portable. ... Read more

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