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**About this book :- **

**Geometry of Lengths, Areas, and Volumes Two-Dimensional Spaces ** written by
** James W. Cannon **.

This volume is suitable for undergraduates who understand calculus and linear algebra and who want to understand a number of those beautiful results usually quoted to the undergraduate without proof. It explains an entire string of results that teased me as an undergraduate because they were stated without proof. I sorely wanted to understand why they were true. This book is written for the “me” who was a young college student. A number of individual sections might be appropriately used as projects for an advanced undergraduate. An occasional more difficult section or exercise is included for extra challenge, and may be skipped.

The main focus in this textbook is to explore classical attempts to measure distances and areas in the plane and gives natural applications of those attempts to classical problems in geometry and to algebra, number theory, and measure theory. This volume explains all Pythagorean triples such as 32 + 42 = 52 geometrically and generalizes them to Pythagorean n-tuples. It describes Archimedes’ discovery of the volume of the sphere using weights and balance arms. The method of his discovery eventually led to the methods of calculus. This volume also calculates the volume of spheres and balls in all dimensions and explains Wallis’s product formula for p. It explains the geometry behind Farey sequences and continued fractions. It proves the unsolvability of squaring the circle, duplicating the cube, and trisecting the angle. It explains Hilbert’s proof that p and e are transcendental. The volume ends with a proof of the wonderful Hausdorff-Banach-Tarski Paradox, which shows that the 2-dimensional sphere can be broken into finitely many pieces that are too complicated and fuzzy to be assigned a well-defined area since they can be rigidly reassembled to form two copies of the original sphere.

James W. Cannon

**Book Detail :- **
** Title: ** Geometry of Lengths, Areas, and Volumes Two-Dimensional Spaces
** Edition: **
** Author(s): ** James W. Cannon
** Publisher: ** American Mathematical Society
** Series: **
** Year: ** 2017
** Pages: ** 133
** Type: ** PDF
** Language: ** English
** ISBN: ** 9781470437145,1470437147,9781470443030,1470443031
** Country: ** US

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**About Author :- **

The author **James W. Cannon ** (1943-2012) was born, in Bellefonte, Pennsylvania. James Cannon received his Ph.D. in Mathematics from the University of Utah in 1969, under the direction of C. Edmund Burgess.

He was a Professor at the University of Wisconsin, Madison from 1977 to 1985. In 1986 Cannon was appointed an Orson Pratt Professor of Mathematics at Brigham Young University. He held this position until his retirement in September 2012.

James Cannon gave an AMS Invited address at the meeting of the American Mathematical Society in Seattle in August 1977, an invited address at the International Congress of Mathematicians in Helsinki 1978, and delivered the 1982 Mathematical Association of America Hedrick Lectures in Toronto, Canada.

James Cannon was elected to the American Mathematical Society Council in 2003 with the term of service February 1, 2004, to January 31, 2007. In 2012 he became a fellow of the American Mathematical Society.

**All Famous Books of this Author :- **

Here is list all books, text books, editions, versions, solution manuals or solved notes avaliable of this author, We recomended you to download all.

** • Download PDF Two-Dimensional Spaces: Volume 1, Geometry of Lengths, Areas, and Volumes by James W. Cannon **

** • Download PDF Two-Dimensional Spaces: Volume 2, Topology as Fluid Geometry by James W. Cannon **

** • Download PDF Two-dimensional Spaces: Volume 3, Non-Euclidean Geometry & Curvature by James W. Cannon **

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**Book Contents :- **
**Geometry of Lengths, Areas, and Volumes Two-Dimensional Spaces ** written by
** James W. Cannon **
cover the following topics.

1. Lengths—The Pythagorean Theorem

2. Consequences of the Pythagorean Theorem

3. Areas by Slicing and Scaling

4. Areas by Cut and Paste

5. Areas by Counting

6. Unsolvable Problems in Euclidean Geometry

7. Does Every Set Have a Size?

Bibliography

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- Abstract Algebra
- Calculus
- Differential Equations
- Engineering Mathematics
- Linear Algebra
- Math Magic
- Real Analysis