Solving the Three Ancient Problems using the Graef Curves
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About this ebook
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.
"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 College (now University).
"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
The Graef curves were proven in analytic geometry in an article coauthored by Edward V. Graef and Dr. Vincent C. Harris: "On the Solutions of Three Ancient Problems," published in Mathematics Magazine. This published Mathematics Magazine article was subsequently favorably reviewed in the German journal "Zentralblatt."
Another paper was coauthored by Edward V. Graef and Dr. Vincent C. Harris: "A Method of Duplicating the Cube," which was published in Mathematics Magazine.
Dear Abby featured the Graef curves in an answer to who solved the Three Ancient Problems.
Edward V. Graef
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.
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Solving the Three Ancient Problems using the Graef Curves - Edward V. Graef
Chapter. Introduction
The Graef curves (which resemble the capital letter G) came into existence when I became interested in the children's efforts to invent a curve that could trisect an angle, and helped them to get the job done.
After inventing a curve to trisect an angle, I became interested in using the same family of curves to solve all three of the Three Ancient Problems:
--Trisecting the angle
--Squaring the circle
--Duplicating the cube
These Three Ancient Problems have been challenging mathematicians for over 2000 years.
Using the same family of curves was something it was believed the ancient Greek mathematicians could never do, though there is mention in the literature of Greek mathematicians who had used the same curve, or the same family of curves (also not constructible with compass and straightedge alone) to solve two out of the three.
In the problem of squaring the circle,
all my efforts to use any one of the new curves proved fruitless until an analysis of the bases upon which the various curves had been constructed pointed up the possibility that one member of the family might be missing.
The realization that the missing member
(if it did exist) would require a yardstick
equal to the circumference of the base circle was the clue that led ultimately to the solution of the problem.
No missing member
came to the rescue, however, when I attempted to use any member of the family in a solution to the problem of duplicating the cube.
Trying to fit
a curve that had been invented for the solution of a specific problem--the trisection of any angle
--into the solution of another and unrelated problem seemed, at times, an impossible and pointless task. But the