Famous Problems of Geometry and How to Solve Them (Dover Books on Mathematics) by Benjamin Bold (PDF)

Description

It took two millennia to prove the impossible; that is, to prove it is not possible to solve some famous Greek problems in the Greek way (using only straight edge and compasses). In the process of trying to square the circle, trisect the angle and duplicate the cube, other mathematical discoveries were made; for these seemingly trivial diversions occupied some of history’s great mathematical minds. Why did Archimedes, Euclid, Newton, Fermat, Gauss, Descartes among so many devote themselves to these conundrums? This book brings readers actively into historical and modern procedures for working the problems, and into the new mathematics that had to be invented before they could be “solved.”The quest for the circle in the square, the trisected angle, duplicated cube and other straight-edge-compass constructions may be conveniently divided into three periods: from the Greeks, to seventeenth-century calculus and analytic geometry, to nineteenth-century sophistication in irrational and transcendental numbers. Mathematics teacher Benjamin Bold devotes a chapter to each problem, with additional chapters on complex numbers and analytic criteria for constructability. The author guides amateur straight-edge puzzlists into these fascinating complexities with commentary and sets of problems after each chapter. Some knowledge of calculus will enable readers to follow the problems; full solutions are given at the end of the book.Students of mathematics and geometry, anyone who would like to challenge the Greeks at their own game and simultaneously delve into the development of modern mathematics, will appreciate this book. Find out how Gauss decided to make mathematics his life work upon waking one morning with a vision of a 17-sided polygon in his head; discover the crucial significance of eπi = -1, “one of the most amazing formulas in all of mathematics.” These famous problems, clearly explicated and diagrammed, will amaze and edify curious students and math connoisseurs.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Good book for testing my knowledge of geometry.

⭐It’s a classic. Enough said.

⭐These little Dover books are great. Good knowledge and excellent price.

⭐wasn’t helpful

⭐This book arrived undamaged and quickly. Thanks.

⭐Good !

⭐Bold has a gem of a book here. It’s only a little bit over a hundred pages, but it’s packed full of the great geometry problems that occupied the minds of the world’s greatest thinkers for the past 2000 years.The title describes the book perfectly. These really are “Famous Problems from Geometry” and he does indeed explain how to solve them.The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) – which ones can be constructed and why. He also discusses which ones cannot be constructed and why.The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again – this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader’s level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be “obvious.”If you’re expecting simple proofs to these problems, you’re not going to find them. If they were simple, they wouldn’t have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry.

⭐Good problems.

⭐Such a wealth of information in a very small book – and with pedagogic exercises pawing the way. An amazing presentation of what can and what can’t be made with traditional and more elaborate methods. Overwhelming but also demanding; the book is dense and some intermediate equations are omitted – not easy reading. But very satisfying and complete. I definitely recommend this small book!!

⭐If you like maths, as I do, you will like this book.The puzzles are intensely interesting, and the explanations will save you a lot of frustration. Makes you wonder what was in the heads of the people who pioneered this area of maths.

⭐Ok book … presentation could be improved.

⭐excellent technical book

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