Solving the Three Ancient Problems using the Graef Curves
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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 UniversityThe 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.