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)

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.

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.

fabulous!
this is the only history book I have ever liked. great stuff if you like physics or geometry

"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.

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

Product Description

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.

Customer Reviews (11)

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.

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.

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.

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.

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

Product Description
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

Customer Reviews (1)

Horrible History
This book was the worst book I have ever read.It was extremely boring and is more like a text book.It is only good if you want to copy a report word-for-word. ... Read more

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Product Description
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)

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.

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

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

Product Description
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)

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

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This Elibron Classics edition is a facsimile reprint of a 1866 edition by Longmans, Green, and Co., London. ... Read more

Product Description
This is a reproduction of a book published before 1923.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

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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)

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.

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]

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

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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

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Euclid's Elements Of Geometry - Books I. Ii. Iii. Iv., Vi And Portions Of Books V. And Xi. - By A. E. Layng -PREFACE - In this edition of those parts of Euclids Elements most commonly read in schools and required for examinations, no fundamental changes have been made, the axioms and enunciations remaining the same as in Simsons edition. Minor changes have, however, been frequently introduced, when it appeared that, either by the insertion of an additional step or reference, the path of the beginner could be made smoother, or conciseness and clearness gained by the removal of redundant and antiquated phrases. In some cases, moreover, Euclids proofs have been displaced in favour of shorter ones based on his................ ... Read more

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For more than two millennia, the Elements of Geometry by the Greek mathematician Euclid of Alexandria (ca. 300 B.C.E. ) was held to be "the supreme example of the exercise of human reason" and "a paradigm of rational certainty" (from the preface, after Simon Blackburn). The Commentary of al-Nayrizi on Book I of Euclid’s Elements of Geometry introduces readers to the transmission of Euclid’s Elements from the Middle East to the Latin West in the medieval period and then offers the first English translation of al-Nayrizi’s (d. ca. 922) Arabic commentary on Book I. ... Read more

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This is a reproduction of a book published before 1923.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

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Anthony Lo Bello’s Gerard of Cremona’s Translation of Book I of the Commentary of al-Nayrizi on Euclid’s Elements of Geometry is the first modern translation of Gerard of Cremona’s (1114–1187) Latin version of al-Nayrizi’s famous Arabic commentary. Lo Bello gives an introductory account of the twenty-two early extant Arabic manuscripts of the Elements, an annotated English translation of Gerard’s translation of al-Nayrizi’s commentary, and finally a critical analysis of the idiosyncrasies of Gerard’s method of translation. ... Read more

Product Description
This is a reproduction of a book published before 1923.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

Product Description
The Commentary of Albertus Magnus on Book I of Euclid’s Elements of Geometry is the third in Lo Bello’s series on the Elements. Lo Bello provides the first modern translation of a key Latin text of the Elements in the Middle Ages, the commentary of the Dominican scholastic philosopher Albertus Magnus (d. 1280), the teacher of Thomas Aquinas. The volume includes a translation, notes on the translation, and a critical examination of the mathematical content of the three commentaries on Euclid’s Elements of Geometry thus far treated in this series. ... Read more

Product Description
This scarce antiquarian book is included in our special Legacy Reprint Series. In the interest of creating a more extensive selection of rare historical book reprints, we have chosen to reproduce this title even though it may possibly have occasional imperfections such as missing and blurred pages, missing text, poor pictures, markings, dark backgrounds and other reproduction issues beyond our control. Because this work is culturally important, we have made it available as a part of our commitment to protecting, preserving and promoting the world's literature. ... Read more

Product Description
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