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Euclid, The Elements
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 Greek terms.
Heath's translation has stood the test of time, and, as one done by a renowned scholar of ancient mathematics, it can be relied upon not to have inadvertantly introduced modern concepts or nomenclature.
We have excised the voluminous historical and scholarly commentary that swells the Dover edition to three volumes and impedes classroom use of the original text. The single volume is not only more convenient, but less expensive as well.
432 pages, Hardcover
First published January 1, 291
509 people are currently reading
13075 people want to read
About the author
Euclid
1,021 books218 followersEuclid (Ancient Greek: Εὐκλείδης Eukleidēs -- "Good Glory", ca. 365-275 BC) also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Stoicheia (Elements) is a 13-volume exploration all corners of mathematics, based on the works of, inter alia, Aristotle, Eudoxus of Cnidus, Plato, Pythagoras. It is one of the most influential works in the history of mathematics, presenting the mathematical theorems and problems with great clarity, and showing their solutions concisely and logically. Thus, it came to serve as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor. He is sometimes credited with one original theory, a method of exhaustion through which the area of a circle and volume of a sphere can be calculated, but he left a much greater mark as a teacher.
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Displaying 1 - 30 of 120 reviews
July 31, 2007
I am rating this primarily to boost its 4.64 average rating. I feel ridiculous reviewing it.
This book is good, and people with cerebral cortexes know that.
For reference, one person who gave it one star gave Allan Moore's "Watchmen" "graphic novel" (read: comic book) 4 stars.
. . .
If this book is not perfection, I am not sure what perfection entails.
The Elements is one of the ten most important, if not best, books ever written. There is no better course in deductive logic, much less geometry.
This book is good, and people with cerebral cortexes know that.
For reference, one person who gave it one star gave Allan Moore's "Watchmen" "graphic novel" (read: comic book) 4 stars.
. . .
If this book is not perfection, I am not sure what perfection entails.
The Elements is one of the ten most important, if not best, books ever written. There is no better course in deductive logic, much less geometry.
November 15, 2021
"The laws of nature are but the mathematical thoughts of God.”
Euclid of Alexandria was a Greek mathematician who lived somewhere between the mid-4th and mid-3rd century BCE. Barely anything is known about Euclid's life, but through his books and works, his name became a synonym for geometry. Euclid's 13 unabridged books on geometry comprise this textbook titled Euclid's Elements which was the second most printed book, following The Bible, after the invention of the printing press.
Although Euclid is known as the founder of geometry, one could assume that he possibly learned much geometry from his predecessors, such as the Pythagoreans, and the Pythagoreans from Pythagoras, and Pythagoras from the ancient Egyptians, and the ancient Egyptians for the ancient Babylonians and so on. Perhaps even the origin and inheritance of this knowledge was not so linear as I just presented. Therefore, bestowing anyone the title of being the founder of a study that deals with the fundamental symmetrical properties of space and things and their proportions and ratios could be an overstep. Nevertheless, what Euclid achieved in this volume of books was unifying the existing knowledge of geometry, introducing his own propositions, and meticulously, with firm argumentation, and a well-grounded methodology, proved and disproved the mathematical propositions of plane and solid geometry. Even though numbers were discussed a great deal, but they were nowhere in sight, as they were all illustrated by varying lengths of lines and geometrical figures and angles that made the study of integers challenging to follow.
This reading experience of Euclid had little entertainment value as I suppose it did throughout history, but entertainment does not have to be a priority in reading. I knowingly read a tedious old book once in a while to bring me to a state of relaxation.
Euclid of Alexandria was a Greek mathematician who lived somewhere between the mid-4th and mid-3rd century BCE. Barely anything is known about Euclid's life, but through his books and works, his name became a synonym for geometry. Euclid's 13 unabridged books on geometry comprise this textbook titled Euclid's Elements which was the second most printed book, following The Bible, after the invention of the printing press.
Although Euclid is known as the founder of geometry, one could assume that he possibly learned much geometry from his predecessors, such as the Pythagoreans, and the Pythagoreans from Pythagoras, and Pythagoras from the ancient Egyptians, and the ancient Egyptians for the ancient Babylonians and so on. Perhaps even the origin and inheritance of this knowledge was not so linear as I just presented. Therefore, bestowing anyone the title of being the founder of a study that deals with the fundamental symmetrical properties of space and things and their proportions and ratios could be an overstep. Nevertheless, what Euclid achieved in this volume of books was unifying the existing knowledge of geometry, introducing his own propositions, and meticulously, with firm argumentation, and a well-grounded methodology, proved and disproved the mathematical propositions of plane and solid geometry. Even though numbers were discussed a great deal, but they were nowhere in sight, as they were all illustrated by varying lengths of lines and geometrical figures and angles that made the study of integers challenging to follow.
This reading experience of Euclid had little entertainment value as I suppose it did throughout history, but entertainment does not have to be a priority in reading. I knowingly read a tedious old book once in a while to bring me to a state of relaxation.
May 3, 2025
Elementary, my dear friends: Euclid of Alexandria makes mathematics interesting and fun.
Not much is known about Euclid. The editors of the edition of The Elements that I have before me indicate that “Euclid appears to have lived in the time of the first Ptolemy, 323-283 B.C., and to have been the founder of the Alexandrian school” (p. 253). But his importance as a mathematics scholar and educator has been recognized since his time – as demonstrated by the way in which, in Raphael’s classic painting The School of Athens (1511), with its all-star lineup of great thinkers from classical Greece, one can see Euclid teaching mathematics to some young pupils. Look at that portion of the painting, next time you’re at the Vatican Museum, and no doubt you will find it engaging – as I did – that Euclid’s students are finding all his math talk interesting, in spite of themselves.
Euclid’s work is best known to people of the modern world through his treatise The Elements, in which he sets forth principles of what has come to be known as Euclidean geometry. Math – or “maths,” if you are from Great Britain – will never be my strong suit; but I found that my time reading The Elements gave me a strong sense of what mathematics aficionados love about math (or maths).
First, a few words about organization (and that sounds very math, doesn’t it?) The Elements is divided into 12 books; but, as the editors of this J.M. Dent & Sons/Everyman’s Library edition of The Elements helpfully explain, most modern readers who come to Euclid’s work in the hope of gaining in mathematical knowledge and/or cultural literacy in a general way only read Books I-VI, Book XI, and a bit of Book XII. If you want to seek out the rest of the books, with all their pages and pages, then you’re a better mathematician than I (not heavy lifting, that).
Each book of The Elements focuses on a different area of mathematics: Book I is dedicated to triangles and parallelograms; Book II to lines, segments, and rectangles; Book III to circles, and so on. Within each book, one will typically find definitions, postulates, axioms, theorems, and problems. In case you weren’t sure:
• A postulate is an assumption of the truth of something, as a basis for discussion.
• An axiom or common notion is a statement that is accepted as true, without needing to be proven. In other words, it is self-evident.
• A theorem is a statement that is not self-evident, but that can be proven to be true. Each proof of a theorem is followed by the initials Q.E.D. (Latin, quod erat demonstrandum, meaning “that which was to be demonstrated”).
• A problem is an opportunity to work out, on one’s own, the ideas that have been set forth and proven in a theorem. Each solution of a problem is followed by the initials Q.E.F. (Latin, quod erat faciendum, meaning “that which was to be done”).
Ordinarily, when reviewing a book, I will try to pick out some key passages from said book, so that you, friend reader, will have a sense of the content of the book as well as its style and tone. But I’m not sure that doing so will work so well in the case of Euclid and The Elements. Consider, for example, Book I, Proposition 47 (a theorem): “In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle” (p. 50).
Are you with me so far?
How about Book III, Proposition 28 (another theorem): “In equal circles, equal straight lines cut off equal arcs, the greater equal to the greater, and the less equal to the less” (p. 100).
Hey, did somebody say, “Keep on rockin’?”
Or we could try Book V, Proposition 9 (yet another theorem): “Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another” (p. 150).
I got blisters on my fingers!
Finally, consider Book VI, Proposition 20 (one final theorem): “Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides” (p. 200).
Am I buggin’ ya? I don’t mean to bug ya.
You get the idea. Quoting a theorem in isolation fails to convey what is so helpful and enlightening about reading The Elements (and, by the way, extra points to all who got all of the classic-rock allusions that I jotted down in response to each of those theorems).
The way that Euclid proceeds from definitions, axioms, and/or postulates to theorems and problems guides the reader through the process with admirable clarity. And this Dent/Everyman’s Library edition provides illustrations on every page, for every theorem or problem. I found myself drawing lines on scratch paper of my own, working out what Euclid had done to demonstrate a theorem or solve a problem. I found that Euclid makes math make sense.
And there’s one more reason why Euclid and The Elements should be important to you, even if you really really really really don’t like math. In both Great Britain and the United States of America, Euclid’s work was once a staple of the classical education, and Euclidean ideas underlay much of how the U.S.A. came together as a nation.
There’s a great scene in the mini-series John Adams (2008) when John Adams (played by Paul Giamatti), Benjamin Franklin (played by Tom Wilkinson), and Thomas Jefferson (played by Stephen Dillane) are working together on Jefferson’s first draft of the Declaration of Independence. Franklin looks at the first sentence of the draft, where Jefferson has written, “We hold these truths to be sacred and undeniable…”, and then suggests that the last phrase “smacks too much of the pulpit.”
Franklin, whose exhaustive reading of course included Euclid, approaches the drafting of this portion of the Declaration in terms of the great Alexandrian’s setting-forth of axioms or common notions that set forth self-evident mathematical truths, and suggests a fateful revision: “We hold these truths to be self-evident…” Jefferson and Adams (themselves both readers and admirers of Euclid’s work) both agree. There is Euclid! – at the very beginnings of the American republic, helping to define terms for a new nation whose existence Euclid never could have imagined.
Similarly, there’s a comparably resonant scene from the Steven Spielberg film Lincoln (2012), where Euclid and his ideas are back at work in American life and history. The film is set at the time in 1865 when President Abraham Lincoln is simultaneously leading the Union cause toward victory in the American Civil War, and guiding the Thirteenth Amendment and the abolition of slavery through the United States Congress. In a thoughtful, understated scene set at the War Department telegraph office, President Lincoln (played by Daniel Day-Lewis) explains to Union Army telegraph operator Samuel Beckwith (played by Adam Driver) how Euclid’s axioms about mathematical equality can be applied to the ideals of human equality for which the Union is fighting:
Euclid's first common notion is this: Things which are equal to the same things are equal to each other. That's a rule of mathematical reasoning and it’s true because it works – has done, and always will do. In his book, Euclid says this is self-evident. You see, there it is – even in that 2,000-year-old book of mechanical law, it is the self-evident truth that things which are equal to the same things are equal to each other.
You don’t have to like math to appreciate Euclid’s Elements – a true classic of world literature. It sets forth eloquently the truth – the self-evident truth! – that numbers don’t lie. It reminds us that, when we solve math problems, we exercise those properties of our minds that will later help us solve problems in life. And it reminds us that the ideas of mathematics have importance and relevance far beyond the math classroom.
I can’t help but think that Mr. Stroot – my 9th-grade geometry teacher at the Catholic prep school I attended in Washington, D.C. – would be pleased to know that I took up an edition of The Elements and read it, from the first axiom to the final solution of the last problem. Q.E.D., Q.E.F.
Not much is known about Euclid. The editors of the edition of The Elements that I have before me indicate that “Euclid appears to have lived in the time of the first Ptolemy, 323-283 B.C., and to have been the founder of the Alexandrian school” (p. 253). But his importance as a mathematics scholar and educator has been recognized since his time – as demonstrated by the way in which, in Raphael’s classic painting The School of Athens (1511), with its all-star lineup of great thinkers from classical Greece, one can see Euclid teaching mathematics to some young pupils. Look at that portion of the painting, next time you’re at the Vatican Museum, and no doubt you will find it engaging – as I did – that Euclid’s students are finding all his math talk interesting, in spite of themselves.
Euclid’s work is best known to people of the modern world through his treatise The Elements, in which he sets forth principles of what has come to be known as Euclidean geometry. Math – or “maths,” if you are from Great Britain – will never be my strong suit; but I found that my time reading The Elements gave me a strong sense of what mathematics aficionados love about math (or maths).
First, a few words about organization (and that sounds very math, doesn’t it?) The Elements is divided into 12 books; but, as the editors of this J.M. Dent & Sons/Everyman’s Library edition of The Elements helpfully explain, most modern readers who come to Euclid’s work in the hope of gaining in mathematical knowledge and/or cultural literacy in a general way only read Books I-VI, Book XI, and a bit of Book XII. If you want to seek out the rest of the books, with all their pages and pages, then you’re a better mathematician than I (not heavy lifting, that).
Each book of The Elements focuses on a different area of mathematics: Book I is dedicated to triangles and parallelograms; Book II to lines, segments, and rectangles; Book III to circles, and so on. Within each book, one will typically find definitions, postulates, axioms, theorems, and problems. In case you weren’t sure:
• A postulate is an assumption of the truth of something, as a basis for discussion.
• An axiom or common notion is a statement that is accepted as true, without needing to be proven. In other words, it is self-evident.
• A theorem is a statement that is not self-evident, but that can be proven to be true. Each proof of a theorem is followed by the initials Q.E.D. (Latin, quod erat demonstrandum, meaning “that which was to be demonstrated”).
• A problem is an opportunity to work out, on one’s own, the ideas that have been set forth and proven in a theorem. Each solution of a problem is followed by the initials Q.E.F. (Latin, quod erat faciendum, meaning “that which was to be done”).
Ordinarily, when reviewing a book, I will try to pick out some key passages from said book, so that you, friend reader, will have a sense of the content of the book as well as its style and tone. But I’m not sure that doing so will work so well in the case of Euclid and The Elements. Consider, for example, Book I, Proposition 47 (a theorem): “In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle” (p. 50).
Are you with me so far?
How about Book III, Proposition 28 (another theorem): “In equal circles, equal straight lines cut off equal arcs, the greater equal to the greater, and the less equal to the less” (p. 100).
Hey, did somebody say, “Keep on rockin’?”
Or we could try Book V, Proposition 9 (yet another theorem): “Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another” (p. 150).
I got blisters on my fingers!
Finally, consider Book VI, Proposition 20 (one final theorem): “Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides” (p. 200).
Am I buggin’ ya? I don’t mean to bug ya.
You get the idea. Quoting a theorem in isolation fails to convey what is so helpful and enlightening about reading The Elements (and, by the way, extra points to all who got all of the classic-rock allusions that I jotted down in response to each of those theorems).
The way that Euclid proceeds from definitions, axioms, and/or postulates to theorems and problems guides the reader through the process with admirable clarity. And this Dent/Everyman’s Library edition provides illustrations on every page, for every theorem or problem. I found myself drawing lines on scratch paper of my own, working out what Euclid had done to demonstrate a theorem or solve a problem. I found that Euclid makes math make sense.
And there’s one more reason why Euclid and The Elements should be important to you, even if you really really really really don’t like math. In both Great Britain and the United States of America, Euclid’s work was once a staple of the classical education, and Euclidean ideas underlay much of how the U.S.A. came together as a nation.
There’s a great scene in the mini-series John Adams (2008) when John Adams (played by Paul Giamatti), Benjamin Franklin (played by Tom Wilkinson), and Thomas Jefferson (played by Stephen Dillane) are working together on Jefferson’s first draft of the Declaration of Independence. Franklin looks at the first sentence of the draft, where Jefferson has written, “We hold these truths to be sacred and undeniable…”, and then suggests that the last phrase “smacks too much of the pulpit.”
Franklin, whose exhaustive reading of course included Euclid, approaches the drafting of this portion of the Declaration in terms of the great Alexandrian’s setting-forth of axioms or common notions that set forth self-evident mathematical truths, and suggests a fateful revision: “We hold these truths to be self-evident…” Jefferson and Adams (themselves both readers and admirers of Euclid’s work) both agree. There is Euclid! – at the very beginnings of the American republic, helping to define terms for a new nation whose existence Euclid never could have imagined.
Similarly, there’s a comparably resonant scene from the Steven Spielberg film Lincoln (2012), where Euclid and his ideas are back at work in American life and history. The film is set at the time in 1865 when President Abraham Lincoln is simultaneously leading the Union cause toward victory in the American Civil War, and guiding the Thirteenth Amendment and the abolition of slavery through the United States Congress. In a thoughtful, understated scene set at the War Department telegraph office, President Lincoln (played by Daniel Day-Lewis) explains to Union Army telegraph operator Samuel Beckwith (played by Adam Driver) how Euclid’s axioms about mathematical equality can be applied to the ideals of human equality for which the Union is fighting:
Euclid's first common notion is this: Things which are equal to the same things are equal to each other. That's a rule of mathematical reasoning and it’s true because it works – has done, and always will do. In his book, Euclid says this is self-evident. You see, there it is – even in that 2,000-year-old book of mechanical law, it is the self-evident truth that things which are equal to the same things are equal to each other.
You don’t have to like math to appreciate Euclid’s Elements – a true classic of world literature. It sets forth eloquently the truth – the self-evident truth! – that numbers don’t lie. It reminds us that, when we solve math problems, we exercise those properties of our minds that will later help us solve problems in life. And it reminds us that the ideas of mathematics have importance and relevance far beyond the math classroom.
I can’t help but think that Mr. Stroot – my 9th-grade geometry teacher at the Catholic prep school I attended in Washington, D.C. – would be pleased to know that I took up an edition of The Elements and read it, from the first axiom to the final solution of the last problem. Q.E.D., Q.E.F.
May 4, 2012
"Euclid alone has looked on beauty bare." --Edna St. Vincent Millay
This is a statement I believe more strongly as I experience more of Euclid's propositions for myself. Before encountering Euclid, I had never considered mathematics to be something beautiful. Now, however, the sheer logical clarity with which Euclid attempts to grapple with the principles of the world around him actually brings tears to my eyes. From how he uses the idea of equality as a foundation for everything else to how he presents the circle as a perfect figure capable of encompassing all else, Euclid truly has captured something captivating and universal in his mathematics. He even overcomes what he is able to himself explain by using ratios to comprehend incommensurability and, in fact, to speak what is unspeakable.
This is a statement I believe more strongly as I experience more of Euclid's propositions for myself. Before encountering Euclid, I had never considered mathematics to be something beautiful. Now, however, the sheer logical clarity with which Euclid attempts to grapple with the principles of the world around him actually brings tears to my eyes. From how he uses the idea of equality as a foundation for everything else to how he presents the circle as a perfect figure capable of encompassing all else, Euclid truly has captured something captivating and universal in his mathematics. He even overcomes what he is able to himself explain by using ratios to comprehend incommensurability and, in fact, to speak what is unspeakable.
May 1, 2016
Euclid’s Elements is one of the oldest surviving works of mathematics, and the very oldest that uses an axiomatic framework. As such, it is a landmark in the history of Western thought, and has proven so enduring that the Elements has been used nearly continuously since being written, only recently falling out of favor.
Not much is known about Euclid, other than that he must have been an unparalleled genius. Nothing about his personality can be gleaned from this book either, other than that he was very persistent.
I must admit, the primary reason I endeavored to read this book were the glowing reviews on Amazon. The very first one proclaims it the ‘best book ever written by a human’. That’s a bold statement, and especially after hearing of its supreme importance in the history of Western thought, it seemed like a good idea to give it a whirl.
I’m not sure what I was expecting when I opened the book. Math, to be sure, but 500 pages of geometric proofs? I couldn’t believe it. I will never forget the feeling of reading it for the first time. I wanted to get through 30 pages. It took me three hours. By the time I was done, I was completely exhausted and had a throbbing headache.
From then on, it became gradually easier, but it was certainly not a walk in the park. I would like to think that I somehow built some ‘mental muscle’ by slogging my way through the book. It felt like it. No book has ever taken as much effort or as much time. War and Peace took three weeks. This took six. Admittedly, I read Tolstoy all of my waking hours, while for this I found it wise not to exceed three hours per day. Any further and I could almost feel my brain seeping out of my eyes and ears.
So, is this the best way to teach geometry? Probably not. One can only spend so much time with mathematical proofs. At a certain point, you need to use some numbers and see the theorems in action. But this book should be required reading for math majors, if only to see how their discipline evolved. It also is an excellent way to practice logical arguments.
Of course, I didn’t absorb much of the book, perhaps most of it. Maybe it’s because I didn’t spend enough time on it, or maybe it’s because, in the end, it just didn’t send me. But, on those rare occasioned when I glimpsed, however dimly, what Euclid was getting at, it was a thrill.
Q.E.D.
[Note: Looking back, I'm really glad I read this. References to Euclid show up everywhere in Western classics, particularly in philosophy (the format is even copied in Spinoza's Ethics!). It seems that reading the Elements has a profound effect on a small subset of people, most of whom are destined to become mathematicians. For example, in his History of Western Philosophy, Bertrand Russell declares that this book is the perfect summation of the Grecian intellect, unequivocally better than their philosophy or literature. In G.H. Hardy's A Mathematician's Apology, he uses Euclid's proof of the infinite number of primes as an example of a beautiful piece of math. A reference to Euclid showed up in the Steven Spielberg's Lincoln, and Euclid is invoked in our Declaration of Independence ("We hold these truths to be self-evident..."). In short, the references alone make the book worth reading. And, who knows? Maybe you're destined to become a great mathematician yourself.]
Not much is known about Euclid, other than that he must have been an unparalleled genius. Nothing about his personality can be gleaned from this book either, other than that he was very persistent.
I must admit, the primary reason I endeavored to read this book were the glowing reviews on Amazon. The very first one proclaims it the ‘best book ever written by a human’. That’s a bold statement, and especially after hearing of its supreme importance in the history of Western thought, it seemed like a good idea to give it a whirl.
I’m not sure what I was expecting when I opened the book. Math, to be sure, but 500 pages of geometric proofs? I couldn’t believe it. I will never forget the feeling of reading it for the first time. I wanted to get through 30 pages. It took me three hours. By the time I was done, I was completely exhausted and had a throbbing headache.
From then on, it became gradually easier, but it was certainly not a walk in the park. I would like to think that I somehow built some ‘mental muscle’ by slogging my way through the book. It felt like it. No book has ever taken as much effort or as much time. War and Peace took three weeks. This took six. Admittedly, I read Tolstoy all of my waking hours, while for this I found it wise not to exceed three hours per day. Any further and I could almost feel my brain seeping out of my eyes and ears.
So, is this the best way to teach geometry? Probably not. One can only spend so much time with mathematical proofs. At a certain point, you need to use some numbers and see the theorems in action. But this book should be required reading for math majors, if only to see how their discipline evolved. It also is an excellent way to practice logical arguments.
Of course, I didn’t absorb much of the book, perhaps most of it. Maybe it’s because I didn’t spend enough time on it, or maybe it’s because, in the end, it just didn’t send me. But, on those rare occasioned when I glimpsed, however dimly, what Euclid was getting at, it was a thrill.
Q.E.D.
[Note: Looking back, I'm really glad I read this. References to Euclid show up everywhere in Western classics, particularly in philosophy (the format is even copied in Spinoza's Ethics!). It seems that reading the Elements has a profound effect on a small subset of people, most of whom are destined to become mathematicians. For example, in his History of Western Philosophy, Bertrand Russell declares that this book is the perfect summation of the Grecian intellect, unequivocally better than their philosophy or literature. In G.H. Hardy's A Mathematician's Apology, he uses Euclid's proof of the infinite number of primes as an example of a beautiful piece of math. A reference to Euclid showed up in the Steven Spielberg's Lincoln, and Euclid is invoked in our Declaration of Independence ("We hold these truths to be self-evident..."). In short, the references alone make the book worth reading. And, who knows? Maybe you're destined to become a great mathematician yourself.]
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March 4, 2015مبدئيا الرياضيات علم ممتع جدا .. و الهندسة بالذات بيني و بينها حكاية عشق قديم :) و حقيقي مستمتعة جدا و انا باقرأ الكتاب دا
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بسم الله :)... (عناصر إقليدس) أو (أصول إقليدس) أو (الأصول الهندسية) أو (اصول الهندسة) :) ! .. لـ إقليدس اللي كان معلم العلوم التعليمية في مدرسة الإسكندرية ، ترجمة كرنيليوس فان ديك المستشرق الأمريكي
.....
بيقولك :)
الهندسة علم موضوعه قياس المقادير ، و المقدار هو كل ما له واحد من ثلاثة أشياء وهي ... طول و عرض و عمق
رابط التحميل و القراءة المباشرة :
http://download-engineering-pdf-ebook...
*****
بسم الله :)... (عناصر إقليدس) أو (أصول إقليدس) أو (الأصول الهندسية) أو (اصول الهندسة) :) ! .. لـ إقليدس اللي كان معلم العلوم التعليمية في مدرسة الإسكندرية ، ترجمة كرنيليوس فان ديك المستشرق الأمريكي
.....
بيقولك :)
الهندسة علم موضوعه قياس المقادير ، و المقدار هو كل ما له واحد من ثلاثة أشياء وهي ... طول و عرض و عمق
رابط التحميل و القراءة المباشرة :
http://download-engineering-pdf-ebook...
November 19, 2009
Geometry is not a given, it is a mystery.
Euclid invites the modern reader to rethink what a book is, and how we might have related to math differently as a child in school if The Elements had been our textbooks. What are the elements that the title refers to? What are the complexes that the elements make up?
You cannot just read this book, I would suggest demonstrating the proofs one by one on a board in front of peers, then discussing each one.
I have read a lot of books, this book is the most instructive one I have ever experienced.
book ten is pretty boring. the fireworks are in books 1, 2, 5, 7 and 13
Euclid invites the modern reader to rethink what a book is, and how we might have related to math differently as a child in school if The Elements had been our textbooks. What are the elements that the title refers to? What are the complexes that the elements make up?
You cannot just read this book, I would suggest demonstrating the proofs one by one on a board in front of peers, then discussing each one.
I have read a lot of books, this book is the most instructive one I have ever experienced.
book ten is pretty boring. the fireworks are in books 1, 2, 5, 7 and 13
February 25, 2014
Update: Again, I find myself starting with the disclaimer that I did not actually read this in its entirety. And I never meant to. My goal was to work through the first four books and that I have done. Can I honestly say that I enjoyed it and am glad I did it. Yes, I really can. Not sure that it makes me a better or wiser person but it makes me SOUND like a better and wiser person when I get to say things like, "Oh, yes. It is like that time I worked through Book 3 of Euclid's 'Elements'..." That right there is so going to be worth the cost of admission.
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Just finished Book I. I worked through the 48 proofs by taking about 4 - 7 per morning. Yesterday morning I worked through 5 proofs and had just about enough time for two more which would have allowed me to finish the book. But I was pretty fried already. But, I love a finish line so I dug deep. Then I looked at the figure for 47 and said, "Oh, hell no." Packed it in and headed to the gym.
This morning I lit a candle, poured a coffee and sat down to look at 47 and laughed. The figure was a damn mess but even my tiny brain recognized the Pythagorean theorem in the enunciation. Okay, I could have done that yesterday. But, man, algebra certainly tidied that bit up didn't it?
So, I'm not sure how much further I'm going to take this. Part of me wants to see it through the first 4 books. More of me wants to move on to Aristotle. Dunno what to do.
Anyway, how the hell can a non-mathematician like me "rate" this book? What gives me any authority at all to have an opinion on it? Nothing. So, let's say the rating goes to the high quality of this particular edition. Also, the cover art is perfection.
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Just finished Book I. I worked through the 48 proofs by taking about 4 - 7 per morning. Yesterday morning I worked through 5 proofs and had just about enough time for two more which would have allowed me to finish the book. But I was pretty fried already. But, I love a finish line so I dug deep. Then I looked at the figure for 47 and said, "Oh, hell no." Packed it in and headed to the gym.
This morning I lit a candle, poured a coffee and sat down to look at 47 and laughed. The figure was a damn mess but even my tiny brain recognized the Pythagorean theorem in the enunciation. Okay, I could have done that yesterday. But, man, algebra certainly tidied that bit up didn't it?
So, I'm not sure how much further I'm going to take this. Part of me wants to see it through the first 4 books. More of me wants to move on to Aristotle. Dunno what to do.
Anyway, how the hell can a non-mathematician like me "rate" this book? What gives me any authority at all to have an opinion on it? Nothing. So, let's say the rating goes to the high quality of this particular edition. Also, the cover art is perfection.
September 14, 2008
I never really began to understand mathematics until I encountered Euclid. If I had had this book as a child, I feel like my eyes would have been opened to a lot more than they were.
Euclid's Elements is the foundation of geometry and number theory. There is no long-winded explanation; instead, from a set of 23 definitions, 5 postulates, and 5 common notions, Euclid lays out 13 books of geometrical proofs. In each proof, he asserts a mathematical truth or asserts that a geometrical construction is possible, and then goes step by step through it.
Most of the propositions require a bit of work to understand, but the moment of insight in which one goes from "What on Earth is this guy talking about?" to "Ohhhhhh...." is pure pleasure.
Euclid's Elements is the foundation of geometry and number theory. There is no long-winded explanation; instead, from a set of 23 definitions, 5 postulates, and 5 common notions, Euclid lays out 13 books of geometrical proofs. In each proof, he asserts a mathematical truth or asserts that a geometrical construction is possible, and then goes step by step through it.
Most of the propositions require a bit of work to understand, but the moment of insight in which one goes from "What on Earth is this guy talking about?" to "Ohhhhhh...." is pure pleasure.
August 3, 2021
The Original Maths Textbook
1 Aug 2021
So, I was sitting in a pub up in the Victorian town of Beechworth, reading this book when somebody leaned over to me to asked me what I was reading. When I told him, he laughed and made a comment that it seemed to be an appropriate book to read before going into lockdown – our fifth of the pandemic. Okay, I suspect that a lot of you out there will laugh at the fact that we are complaining about a lockdown considering that our case numbers, and our fatalities, are ridiculously low, but I guess we Australians really don’t know, or even appreciate, how good we have things.
Mind you, it does raise the question as to why anybody would want to read Euclid considering it is basically no different to reading a mathematical textbook, and more so voluntarily as opposed to being instructed to by your school/university teacher. Mind you, I don’t particularly know all that many people who would read a textbook even if instructed to. Sure, I did, to an extent, but then again I’m something of an oddity, and I’m always looking for another book to add to my Goodreads list of books read, and to write a review on it.
Anyway, if you can handle something like the following:
Then maybe Euclid’s Elements is a book for you, otherwise I suspect that your eyes will end up glazing over with boredom, as is prone to happen when you basically end up reading a mathematics textbook. However, there are quite a few interesting things relating to those works beyond it being little more than an ancient Mathematical textbook. For instance, apparently it was the second book to ever be printed by Gutenberg, with the Bible being the first. Also, it was the only maths textbook right up until the 19th Century, though I do sort of suspect that this might be an exaggeration considering that there was much more to maths than just Euclidean Geometry (and number theory – he goes into Number Theory in this book as well, and spends quite a lot of time exploring the concept of Prime Numbers).
Mind you, it isn’t as if Euclid came up with all of this stuff. In fact, it is generally accepted that Pythagoras’ theorem was floating around for quite a bit of time before somebody decided to attribute it to Pythagoras (who apparently wasn’t a mathematician). Like, the Egyptians used it (how else do you think they were able to come up with Pyramids), and the Babylonian’s used it. There is also the Golden Ratio, which was used to build the Parthenon, which was constructed around 50 years prior to Euclid’s birth. As such, much of what is contained in this book was already known, it is just that Euclid did what pretty much many of the other writers of the period did, and that codifies all of the knowledge into one work. Also, considering that he was born in Alexandria, the place with the Library, it is probably not all that surprising that he ended up codifying mathematical knowledge of the time.
The other interesting thing is that Euclid seems to use a primitive form of algebra, which is something like 1300 years before Mr Al Jebra (al-Khwarizmi) condified the concept in his book ‘The Science of Restoring and Balancing. However, Euclid’s version is basically discussing numbers based on lines, and how the lines happen to relate to each other. Even in the section on Number Theory, all of his concepts revolve around the use of lines, which shouldn’t be too much of a surprise considering most maths in those days generally involved building stuff (or lobbing rocks, but mostly building stuff), though I’m sure the merchants also used numbers to calculate their profit, and how much stock they happened to have on hand (yeah, we do have quite a lot of accounting work from the ancient time).
Mind you, like a lot of textbooks, particularly one’s dedicated to maths, this is probably not the type of book that you would read cover to cover. Okay, I read it cover to cover, but that is probably because I wanted to read one of the classics, and one that has had a huge influence upon our society, and in fact the world. Sure, it certainly isn’t an easy read, but it is really rather interesting to see the source from which the mathematical foundations were developed.
The other interesting thing that I picked up is that this book is probably the type of book that people in the medieval ages would have considered a ‘spell book’. The reason I suggest this is not that it is used to actually cast spells, but it is a book that gives us an insight into how the universe comes together, and it gives us insights into how we can build things that not only won’t fall down, but will actually stand the test of time. The fact that there are many buildings still standing that have stood for thousands of years goes to show how much the ancients knew, and how influential the contents of this book were.
1 Aug 2021
So, I was sitting in a pub up in the Victorian town of Beechworth, reading this book when somebody leaned over to me to asked me what I was reading. When I told him, he laughed and made a comment that it seemed to be an appropriate book to read before going into lockdown – our fifth of the pandemic. Okay, I suspect that a lot of you out there will laugh at the fact that we are complaining about a lockdown considering that our case numbers, and our fatalities, are ridiculously low, but I guess we Australians really don’t know, or even appreciate, how good we have things.
Mind you, it does raise the question as to why anybody would want to read Euclid considering it is basically no different to reading a mathematical textbook, and more so voluntarily as opposed to being instructed to by your school/university teacher. Mind you, I don’t particularly know all that many people who would read a textbook even if instructed to. Sure, I did, to an extent, but then again I’m something of an oddity, and I’m always looking for another book to add to my Goodreads list of books read, and to write a review on it.
Anyway, if you can handle something like the following:
Let ABCD be the given circle.
It is required to inscribe a square in the circle ABCD.
Draw two diameters AC and BD of the circle ABCD at right angles to one another, and join AB, BC, CD, and DA.
Then, since BE equals ED, for E is the center, and EA is common and at right angles, therefore the base AB equals the base AD.
For the same reason each of the straight lines BC and CD also equals each of the straight lines AB and AD. Therefore the quadrilateral ABCD is equilateral.
I say next that it is also right-angled.
For, since the straight line BD is a diameter of the circle ABCD, therefore BAD is a semicircle, therefore the angle BAD is right.
For the same reason each of the angles ABC, BCD, and CDA is also right. Therefore the quadrilateral ABCD is right-angled.
But it was also proved equilateral, therefore it is a square, and it has been inscribed in the circle ABCD.
Therefore the square ABCD has been inscribed in the given circle.
Then maybe Euclid’s Elements is a book for you, otherwise I suspect that your eyes will end up glazing over with boredom, as is prone to happen when you basically end up reading a mathematics textbook. However, there are quite a few interesting things relating to those works beyond it being little more than an ancient Mathematical textbook. For instance, apparently it was the second book to ever be printed by Gutenberg, with the Bible being the first. Also, it was the only maths textbook right up until the 19th Century, though I do sort of suspect that this might be an exaggeration considering that there was much more to maths than just Euclidean Geometry (and number theory – he goes into Number Theory in this book as well, and spends quite a lot of time exploring the concept of Prime Numbers).
Mind you, it isn’t as if Euclid came up with all of this stuff. In fact, it is generally accepted that Pythagoras’ theorem was floating around for quite a bit of time before somebody decided to attribute it to Pythagoras (who apparently wasn’t a mathematician). Like, the Egyptians used it (how else do you think they were able to come up with Pyramids), and the Babylonian’s used it. There is also the Golden Ratio, which was used to build the Parthenon, which was constructed around 50 years prior to Euclid’s birth. As such, much of what is contained in this book was already known, it is just that Euclid did what pretty much many of the other writers of the period did, and that codifies all of the knowledge into one work. Also, considering that he was born in Alexandria, the place with the Library, it is probably not all that surprising that he ended up codifying mathematical knowledge of the time.
The other interesting thing is that Euclid seems to use a primitive form of algebra, which is something like 1300 years before Mr Al Jebra (al-Khwarizmi) condified the concept in his book ‘The Science of Restoring and Balancing. However, Euclid’s version is basically discussing numbers based on lines, and how the lines happen to relate to each other. Even in the section on Number Theory, all of his concepts revolve around the use of lines, which shouldn’t be too much of a surprise considering most maths in those days generally involved building stuff (or lobbing rocks, but mostly building stuff), though I’m sure the merchants also used numbers to calculate their profit, and how much stock they happened to have on hand (yeah, we do have quite a lot of accounting work from the ancient time).
Mind you, like a lot of textbooks, particularly one’s dedicated to maths, this is probably not the type of book that you would read cover to cover. Okay, I read it cover to cover, but that is probably because I wanted to read one of the classics, and one that has had a huge influence upon our society, and in fact the world. Sure, it certainly isn’t an easy read, but it is really rather interesting to see the source from which the mathematical foundations were developed.
The other interesting thing that I picked up is that this book is probably the type of book that people in the medieval ages would have considered a ‘spell book’. The reason I suggest this is not that it is used to actually cast spells, but it is a book that gives us an insight into how the universe comes together, and it gives us insights into how we can build things that not only won’t fall down, but will actually stand the test of time. The fact that there are many buildings still standing that have stood for thousands of years goes to show how much the ancients knew, and how influential the contents of this book were.
April 8, 2011
In order to fully enjoy Euclid, you must first completely rid yourself of preconceptions about the world of mathematics. Euclid sets up his postulates and from there he proves what he needs to prove. There is very much a sense of wonder and excitement in reading Euclid. He proves things that we would never think to prove and he does so in a completely logical and beautiful way. With a climactic end in which he proves that the five Platonic solids - the cube, octahedron, tetrahedron, icosahedron, and dodecahedron - can be constructed and then proves that these are the only five solids which are contained by equilateral and equiangular figures, Euclid is in a way very satisfying and at the same time he gives one a desire to delve further into the world of mathematics.
May 31, 2019
We are in -290, Euclid is writing down some obvious stuff, also called axioms by educated fools. Here goes my favorite. "Let it be granted that things which are equal to the same thing are equal to one another". Also by transitivity, the author of this gem certainly was an interesting fool. A fool trying to be consistent. Nowadays, we call these peculiar animals mathematicians. From there he plays with his definitions… I am writing this blah in Greece and can tell how much of nature he had to ignore to be consistent around the birth of civilization. These days we can do much better because Descartes 1596-1650 connected algebra and geometry so Gauss 1777-1855 + Riemann 1826-1866 (*) finally build the tools to complement/challenge Euclid’s ideas. Hilbert 1862-1943 followed playing with the infinite geometries out there (or inside here, Plato). Then Einstein 1879-1955 showed that the universe is non-Euclidean. Then Gödel 1906-1978 showed that Euclid figured out the only way of constructing consistent arguments. They say this book has vast influences among our best thinkers and within civilization in a broad sense. But I didn't solve all the ancient problems within because it is okay to be incomplete. Quod Erat Faciendum (Q.E.F.). *Warning: in 2019 we are still missing the tools to do algebra with mathematicians.
June 28, 2012
The original bible of geometry! Plato had an inscription above the entrance to his Academy: "Let no man enter here who is ignorant of geometry." From Plato's time to the 20th century, Euclid's "Elements" was the gold-standard for learning this most basic of the mathematical disciplines. When you read it, you will understand why. Every proof and every construction is worked out meticulously, step-by-step, such that there is zero doubt about the final result. Required reading for all truly educated people!
August 23, 2016
Most exquisite.
February 7, 2022
"The Elements" is a series of books on mathematics written by Euclid. It is an ancient textbook, and for centuries it was the second most published work in the world behind the Bible.
My copy takes the text from a 1908 translation. Each proposition and theorem gets an explanation. The book is lengthy and exhaustive; it makes the book dull. Finally, due to its size, it is hard to carry.
On the other hand, "The Elements" is groundbreaking. As a textbook, it works to some extent. It wasn't questioned until the 19th century, giving it plenty of staying power. I suppose the main reason to read it now is for historical reasons. There has to be some reason for "The Elements" being so influential. My copy contains the original Greek text and a translation of what it says.
We know little of Euclid, but he might have been a contemporary of Plato. We do know that Euclid did not originate these results. Instead, he is a great expositor. Euclid placed the results into a single framework, making it easy to reference. Furthermore, he builds on each axiom, logically leading from first principles to advanced ideas.
Thanks for reading my review, and see you next time.
My copy takes the text from a 1908 translation. Each proposition and theorem gets an explanation. The book is lengthy and exhaustive; it makes the book dull. Finally, due to its size, it is hard to carry.
On the other hand, "The Elements" is groundbreaking. As a textbook, it works to some extent. It wasn't questioned until the 19th century, giving it plenty of staying power. I suppose the main reason to read it now is for historical reasons. There has to be some reason for "The Elements" being so influential. My copy contains the original Greek text and a translation of what it says.
We know little of Euclid, but he might have been a contemporary of Plato. We do know that Euclid did not originate these results. Instead, he is a great expositor. Euclid placed the results into a single framework, making it easy to reference. Furthermore, he builds on each axiom, logically leading from first principles to advanced ideas.
Thanks for reading my review, and see you next time.
March 21, 2024
This book genuinely changed the way I think about math and geometry. So amazing. I love shapes. Holy shit.
December 9, 2012
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.
July 6, 2021
Didn't read the entire thing (not insane enough to). 5 stars primarily for the historical significance and Euclid's overwhelming genius. Despite the anachronistic grammatical structure of the translation, it's actually quite an enjoyable read. The construction proofs are fun to figure out on your own and seeing the propositions of each book fit together is like watching a time-reversed video of a shattering cup. I recommend looking through at least the first few books just to get an idea for the rigor involved in laying the foundations of two millennia of math.
August 10, 2010
There are three crucial science books in the history of mankind. This is one of them. I have no idea how this guy could devise these ideas so soon. Enormously influential. Forget sudokus, the problems here are more interesting.
February 24, 2023
Euclid's Elements is a timeless masterpiece of mathematical reasoning and logical deduction that has stood the test of time. Despite being written over two thousand years ago, its insights and methods still inspire and challenge contemporary mathematicians and scientists.
Thomas Little Heath's translation is a remarkable rendition of the original text that maintains its elegance and precision. The clarity of the explanations and the inclusion of helpful diagrams make the book accessible to readers with varying levels of mathematical expertise. Additionally, the commentary and notes provide an informative and illuminating analysis of the historical and cultural context of Euclid's work, enriching the reader's understanding of the material.
Euclid's Elements contains the principle that "the whole is greater than the part," which serves as the basis for many crucial theorems in geometry and arithmetic. Euclid rigorously proves this statement by assuming the opposite and deriving a contradiction, demonstrating the importance of axioms and the power of logical deduction in mathematics. Heath's translation fully captures the elegance and beauty of Euclid's reasoning, making it a valuable resource for anyone interested in mathematics. Students, teachers, and enthusiasts alike are highly recommended to explore this masterpiece of mathematical reasoning and logical deduction.
Thomas Little Heath's translation is a remarkable rendition of the original text that maintains its elegance and precision. The clarity of the explanations and the inclusion of helpful diagrams make the book accessible to readers with varying levels of mathematical expertise. Additionally, the commentary and notes provide an informative and illuminating analysis of the historical and cultural context of Euclid's work, enriching the reader's understanding of the material.
Euclid's Elements contains the principle that "the whole is greater than the part," which serves as the basis for many crucial theorems in geometry and arithmetic. Euclid rigorously proves this statement by assuming the opposite and deriving a contradiction, demonstrating the importance of axioms and the power of logical deduction in mathematics. Heath's translation fully captures the elegance and beauty of Euclid's reasoning, making it a valuable resource for anyone interested in mathematics. Students, teachers, and enthusiasts alike are highly recommended to explore this masterpiece of mathematical reasoning and logical deduction.
December 9, 2012
This edition of Euclid’s Elements presents the definitive Greek text—i.e., that edited by J.L. Heiberg (1883–1885)—accompanied by a modern English translation, as well as a Greek-English lexicon. Neither the spurious books 14 and 15, nor the extensive scholia which have been added to the Elements...
more...
August 1, 2021
"Let no one ignorant of geometry enter here." It took me almost 2 years to trudge through this thing so I'm REALLY hoping Plato will let me into his Academy now. Glad to have Euclid knocked off my bucket list!
Want to read
December 31, 2008Wikipedia: "Euclid's Elements is the most successful and influential textbook ever written." Sounds promising :]
November 22, 2024
Lecture académique — Automne 2024
3.
Je n’ai lu que le premier Livre de 13 dans le cadre de mon cours, et la lecture me laisse assez indifférent. Les démonstrations ont une élégance non négligeable, et il peut être plaisant de suivre les conséquences des axiomes avec Euclide, mais n’empêche qu’il s’agit davantage d’un manuel que d’un essai. Aussi n’y a-t-il pas vraiment de passages dignes de mention, car même si les stratégies d’Euclide peuvent changer, son ton demeure froid et scientifique, mathématique: géométrique.
3.
Je n’ai lu que le premier Livre de 13 dans le cadre de mon cours, et la lecture me laisse assez indifférent. Les démonstrations ont une élégance non négligeable, et il peut être plaisant de suivre les conséquences des axiomes avec Euclide, mais n’empêche qu’il s’agit davantage d’un manuel que d’un essai. Aussi n’y a-t-il pas vraiment de passages dignes de mention, car même si les stratégies d’Euclide peuvent changer, son ton demeure froid et scientifique, mathématique: géométrique.
December 4, 2021
The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Love it.
May 24, 2022
Difficult, beautiful, worth working through. I needed help to understand it. But I am glad to have worked through it.
April 15, 2024
Euclid compiled his Elements from a number of works of earlier men. Among these are Hippocrates of Chios (c. 440 bce). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle. The older elements were at once superseded by Euclid’s and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own, culminating in the construction of the five regular solids, now known as the Platonic solids.
A brief survey of the Elements belies a common belief that it concerns only geometry. This misconception may be caused by reading no further than Books I through IV, which cover elementary plane geometry. Euclid understood that building a logical and rigorous geometry (and mathematics) depends on the foundation—a foundation that Euclid began in Book I with 23 definitions, five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions. Book I then proves elementary theorems about triangles and parallelograms and ends with the Pythagorean theorem.
Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book II contains a construction of “the section,” the division of a line into two parts such that the ratio of the larger to the smaller segment is equal to the ratio of the original line to the larger segment. (This division was renamed the golden section in the Renaissance after artists and architects rediscovered its pleasing proportions.) Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines. Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon.
Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus to Eudoxus of Cnidus. While Book V can be read independently of the rest of the Elements, its solution to the problem of incommensurables (irrational numbers) is essential to later books. In addition, it formed the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century. Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means.
Books VII–IX contain elements of number theory, where number (arithmos) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences; and Book IX proves that there are an infinite number of primes.
According to Proclus, Books X and XIII incorporate the work of the Pythagorean Theaetetus (c. 417–369 bce). Book X, which comprises roughly one-fourth of the Elements, seems disproportionate to the importance of its classification of incommensurable lines and areas (although study of this book would inspire Johannes Kepler [1571–1630] in his search for a cosmological model).
Books XI–XIII examine three-dimensional figures, in Greek stereometria. Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’s method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters. Book XIII culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere.
Almost from the time of its writing, the Elements exerted a continuous and major influence on human affairs. Euclid set a standard for deductive reasoning and geometric instruction that persisted, practically unchanged, for more than 2,000 years.
A brief survey of the Elements belies a common belief that it concerns only geometry. This misconception may be caused by reading no further than Books I through IV, which cover elementary plane geometry. Euclid understood that building a logical and rigorous geometry (and mathematics) depends on the foundation—a foundation that Euclid began in Book I with 23 definitions, five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions. Book I then proves elementary theorems about triangles and parallelograms and ends with the Pythagorean theorem.
Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book II contains a construction of “the section,” the division of a line into two parts such that the ratio of the larger to the smaller segment is equal to the ratio of the original line to the larger segment. (This division was renamed the golden section in the Renaissance after artists and architects rediscovered its pleasing proportions.) Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines. Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon.
Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus to Eudoxus of Cnidus. While Book V can be read independently of the rest of the Elements, its solution to the problem of incommensurables (irrational numbers) is essential to later books. In addition, it formed the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century. Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means.
Books VII–IX contain elements of number theory, where number (arithmos) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, antanaresis (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences; and Book IX proves that there are an infinite number of primes.
According to Proclus, Books X and XIII incorporate the work of the Pythagorean Theaetetus (c. 417–369 bce). Book X, which comprises roughly one-fourth of the Elements, seems disproportionate to the importance of its classification of incommensurable lines and areas (although study of this book would inspire Johannes Kepler [1571–1630] in his search for a cosmological model).
Books XI–XIII examine three-dimensional figures, in Greek stereometria. Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’s method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters. Book XIII culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere.
Almost from the time of its writing, the Elements exerted a continuous and major influence on human affairs. Euclid set a standard for deductive reasoning and geometric instruction that persisted, practically unchanged, for more than 2,000 years.
May 18, 2013
Quite a thorough work. From reading this masterpiece and cornerstone of geometry, one can understand how impressive the original development was as a human achievement. Euclid wrote his Elements around 300 BC. He was one of Plato's younger students, but older than Archimedes. The 13 books of his Elements cover angles, line segments, triangles, rectangles, squares, the irrational numbers, parallelagrams, parallelapipeds, spheres, cones, and polygons. He gives a full treatment of area and never quite defines it. Many of his proofs are much simpler with the tools of algebra and Cartesian space. It's impressive that Euclid reached the same conclusions without the benefit of either. The hundreds of propositions are completely proven, using a handful of definitions that begin each book and the previously proven propositions. Some of these propositions are interesting because there is no strong intuitive basis. Some are uninteresting and never again referenced. Some are fundamental building blocks. And many are "how to" type proofs, ensuring that a given figure can be created under certain circumstances, which again are used in other proofs when auxiliary figures are needed. Among his propositions are the Pythagorean theorem, a number of points regarding equal areas and relationships between areas and volumes, and examples of a variety of techniques. When he uses proof by contradiction, his "absurd" and "which is impossible" are so clear and final that one can understand how the early philosophers latched onto the new mathematics. His use of "absurd" is very reminiscent of a Hobbes' style of argument, as well as, I'm sure, so many others who applied the same approach to political and religious philosophy. Other areas he concentrates on include odd and even numbers, means, squares, ratios, and an assortment of types of "irrationals." The latter are much more detailed than anything I ever studied. Much of his middle chapters seem less applicable to anything other than number theory. Two anecdotes involve his early students. In response to a kingly student's wanting to know if there was an easier way, Euclid responded "There is no royal road to geometry." Another student asked what was the value of this study, to which Euclid replied: "Give him a coin since he must needs make gain by what he learns."
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March 12, 2010Okay, so Euclid is hugely important because he more or less originated the idea of a closed, axiomatic system of thought, the sort of thing that would go on to influence people like Vico and Spinoza, not to mention pretty much every mathematician who ever lived. Reading the Elements is about as exciting as reading a cookbook. You start with these parts, mix them together in this way, and get this whole. This way of thinking is probably too heavily ingrained into our mindsets after all these millenia to seem like anything other than a rote, obvious exercise. If you really, REALLY love 2-D geometry, you'll dig it. If not, it will probably just feel like a textbook.
January 25, 2011
I tried to get through Elements in high school at the insistence of Fr. Rathmusen. Geometric construction was huge at Don Bosco Tech as were geometric proofs. I'm much more appreciative of the work today. It puts you in a very logical, regimented frame of mind; an essential for mathematics. This particular edition has notes, clear graphics comments that clarify the 19th century language nuances. This is a beautiful edition of a pillar of literature and science.
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