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Euclid's Elements Redux Euclid’s “Elements” Redux Daniel Callahan John Casey Sir Thomas Heath Version 2016-169 “Euclid’s ’Elements’ Redux” ©2016 Daniel Callahan, licensed under the Cre- ative Commons Attribution-ShareAlike 4.0 International License. Selections from the American Heritage® Dictionary of the English Language, Fourth Edition, ©2000 by Houghton Mifflin Company. Updated in 2009. Pub- lished by Houghton Mifflin Company. All rights reserved. Selections from Dictionary.com Unabridged Based on the Random House Dic- tionary, © Random House, Inc. 2013. All rights reserved. Questions? Comments? Did you find an error? Email me at: [email protected] Make sure to include the version number (written above). Download this book for free at: http://www.starrhorse.com/euclid/ “Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?” - Paul Halmos “Pure mathematics is, in its way, the poetry of logical ideas.” - Albert Einstein “Math is like going to the gym for your brain. It sharpens your mind.” - Danica McKellar “One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beau- tiful theories.” - Philip J. Davis Contents Chapter 0. About this project 6 0.1. Contributions & Acknowledgments 9 0.2. Dedication 9 0.3. About Euclid’s “Elements” 10 0.4. Open Textbooks 16 0.5. Recommended Reading 20 Chapter 1. Angles, Parallel Lines, Parallelograms 23 1.1. Symbols, Logic, and Definitions 23 1.2. Postulates 35 1.3. Axioms 36 1.4. Propositions from Book I: 1-26 41 1.5. Propositions from Book I: 27-48 90 Chapter 2. Rectangles 134 2.1. Definitions 134 2.2. Axioms 135 2.3. Propositions from Book II 136 Chapter 3. Circles 163 3.1. Definitions 163 3.2. Propositions from Book III 169 Chapter 4. Inscription and Circumscription 244 4.1. Definitions 244 4.2. Propositions from Book IV 245 Chapter 5. Theory of Proportions 278 5.1. Definitions 278 5.2. Propositions from Book V 280 Chapter 6. Applications of Proportions 288 6.1. Definitions 288 6.2. Propositions from Book VI 291 4 CONTENTS 5 Chapter 7. Elementary Number Theory 369 7.1. Definitions 369 7.2. Propositions from Book VII 372 Chapter 8. Proportions & Geometric Sequences 399 8.1. Definitions 399 8.2. Propositions from Book VIII 402 Chapter 9. Solutions 418 9.1. Solutions for Chapter 1 418 9.2. Solutions for Chapter 2 469 9.3. Solutions for Chapter 3 474 9.4. Solutions for Chapter 4 482 9.5. Solutions for Chapter 5 486 CHAPTER 0 About this project The goal of this textbook is to provide an edition of Euclid’s “Elements” that is easy to read, inexpensive, has an open culture license, and can legally be distributed anywhere in the world. The title Euclid’s “Elements” Redux indicates that while this edition covers each of Euclid’s results, the proofs themselves have been written using modern mathematics. This is because mathematics has almost completely changed since Euclid wrote “The Elements”. In Euclid’s Hellenistic culture, a number was synonymous with a length that could be measured with a ruler, a rope, or some other such device. This gave Euclid access to all of the positive numbers but none of the negative numbers. (The Greeks also had a difficult time working with 0.) But since the develop- ment of algebra wouldn’t begin for another 1,000 years, Euclid could not have known that the result of Book I Proposition 47 could be written as a2 + b2 = c2. This means that when a modern student encounters Euclid for the first time, he or she has two problems to solve: the logic of the proof, and Euclid’s now ar- chaic concepts of what numbers are and how they related to each other. While Euclid’s original intentions are of great interest to math historians, they can confuse beginners. This edition helps students understand Euclid’s results by using modern proofs, and these results contain material that every math student should know. This is one reason Euclid’s “Elements” was the world’s most important mathematics textbook for about 2,200 years. Let that number sink in for a moment... one math textbook was used by much of the literate world for over two thousand years. Why should this be? Lack of competition? At certain times and places, yes, but in schools where Euclid’s “Elements” got a foothold, other textbooks soon followed. Therefore, lack of competition can’t be the complete answer. 6 0. ABOUT THIS PROJECT 7 Elegant proofs? Most of Euclid’s proofs can and were rewritten as new math was developed. So while the originals remain a model for how proofs can be written, they aren’t irreplaceable. Content? Until the development of algebra and calculus, Euclid’s “Elements” covered everything a novice mathematician needed to know. And even after calculus was developed, “The Elements” still had its place. Isaac Newton’s assistant claimed that in five years of service, he heard Newton laugh just once: a student asked if Euclid were still worth studying, and Newton laughed at his ignorance. But most students only studied the first six Books (i.e., the first six of the thirteen scrolls that made up the original edition of “The Elements”). Therefore not everyone studied “The Elements” exclusively for its content. So, whatever is valuable about “The Elements” must be present in the first half of the book; but each Book begins with a secure intellectual foundation (definitions and axioms) and builds upwards one proposition at a time. And that, I think, is the key to answering this question: not only does “The Elements” help a student learn geometry, but it also trains the student’s mind to use logic properly. Learning algebra helps a student to learn calculus and statistics, but learning geometric proofs helps a student to think clearly about politics, art, music, design, coding, law... any subject that requires rational thought can be better understood after working through “The Elements”. The reason “The Elements” was needed in the past, and why it’s needed today, is that it helps students learn to think clearly. But if it’s such a good textbook, why was it abandoned after the 19th century? The last quarter of that decade saw an explosion of new mathematics, includ- ing the construction of formal logic and a new rigorous approach to proof writ- ing. Suddenly, Euclid began to look dated; and as calculus became the gate- way to engineering and the sciences, algebra became the course every student seemed to need. Geometric proofs became a luxury rather than a necessity. This was a mistake which has hindered generations from learning to think logically. While the mathematical developments of the late 19th century led to a mathe- matical Golden Age (one which we are still in), one of the best pedagogical tools ever written was discarded. Perhaps if “The Elements” were rewritten to con- form to current textbook standards, its importance in geometry, proof writing, and as a case-study in logic might once again be recognized by the worldwide educational community. This edition of Euclid can be used in a number of ways: 0. ABOUT THIS PROJECT 8 • As a means of introducing students to proofs and to writing proofs. • As a supplement to any geometry or analytical geometry textbook. • As a textbook for self-study and homeschooling. This edition also contains homework problems and a partial answer key. Euclid’s “Elements” Redux has been released under the Creative Commons Attribution-ShareAlike 4.0 International License. This means that you are free to copy and distribute it commercially or otherwise without penalty. If you wish to add to it, your work must be licensed under the same Creative Commons license. This textbook requires its student to work slowly and carefully through each section. The student should check every result stated in the book and not take anything on faith. While this process may seem tedious, it is exactly this attention to detail which separates those who understand mathematics from those who do not. The prerequisites for this textbook include a desire to solve problems and to learn mathematical logic. Some background in proportions, algebra, trigonom- etry, and possibly linear algebra may be helpful. However, there is no mistak- ing the fact that the homework in this textbook is unlike that of most other elementary math textbooks. Rather than posing questions which require cal- culations to answer, these problems require a written proof. Students who have no knowledge of proof writing should only be asked to solve the simplest of problems at first; students or parents of students who have no idea how to proceed are encouraged to read Richard Hammack’s Book of Proof, 2nd ed. or a similar proof-writing textbook (see section 1.5 for details). But it is vitally important to understand that to mathematicians proofs are mathematics. Most mathematicians at the Ph.D. level leave the calculation of difficult prob- lems to physicists and engineers; the job of a mathematician is to prove that an answer to the problem exists before the calculations begin. So, despite the fact that most students take classes described as “math” or “maths”, most peo- ple (including those who claim to hate math) do not know what mathematics actually is; in essence, math is proofs. The figures in this textbook were created in GeoGebra. They can be found in the images folder in the source files for this textbook.
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