Solving the Three Ancient Problems using the Graef Curves

by Edward V. Graef

Overview

The Three Ancient Problems (trisecting the angle, squaring the circle, duplicating the cube) have challenged mathematicians for over 2000 years. It's been proven that it's impossible to solve these problems using compass and straightedge alone, so mathematicians were challenged to create a curve or family of curves to solve all three. Until Edward V. Graef invented his Graef curves, no mathematician had accomplished this feat--not Archimedes, not Nicomedes, not Hippias. His solution has been verified by top mathematicians, and published in prestigious math publications.

Product Details

BN ID:	2940013418080
Publisher:	Marsha Bauer
Publication date:	09/26/2011
Sold by:	Barnes & Noble
Format:	eBook
File size:	567 KB

About the Author

Edward V. Graef

1911 - 1989

Edward V. Graef was a gifted mathematician though he had no formal training in math. He had a M.A. in Social Administration from Ohio State University and a B.S. in History and Education from City College of New York. By the time he retired, he was Executive Director of the Health Research and Services Foundation in Pittsburgh (part of the United Fund (United Way)), and was responsible for creating the Health-O-Rama, which provided free health screening.

Edward served as an Army infantry captain in the Philippines during WWII, and was awarded a Bronze Star. He first demonstrated his mathematical abilities while in the service. He devised a method of firing mortar when the sight is lost or broken, and received a Legion of Merit commendation for the effort.

While in the service, Edward married Eleanor Marshall, an American Red Cross volunteer and the daughter of a former mayor of Columbus, Ohio. They had four children: Ed, John, Howard, and Marsha. It was while helping his children with geometry homework that his interest in the Three Ancient Problems began.

Edward was also an accomplished writer. He took the daily letters he wrote to his wife during WWII, and compiled them into a memoir. Entitled "War Letters: Eleanor, My Darling--a Memoir of WWII", it is available as an eBook.

Editorial Reviews

"Graef's method [solve all three problems with one class of curves alone] passed the scrutiny of the most rigorous analysis."

Professor Henry Gould West Virginia University

An article coauthored by Edward V. Graef and Dr. Vincent C. Harris regarding duplicating the cube.

Mathematics Magazine

"Mr. Graef's discovery of the method of trisecting the angle is indeed commendable, his discovery of the method of squaring the circle is remarkable, and his finding of his method of duplicating the cube is truly almost unbelievable.... Mr. Graef accomplished the feat of solving all three problems using methods of geometry, using the same family of curves (not constructible with compass and straightedge alone) for all three."

Dr. Vincent C. Harris Professor Emeritus, San Diego State University

Dear Abby featured the Graef curves in an answer to who solved the Three Ancient Problems.

Dear Abby