Solving the Three Ancient Problems using the Graef Curves
Using the same family of curves
--not
constructible with compass and straightedge alone--
to solve all
three of the Three Ancient Problems:
Trisecting the Angle
Squaring the
Circle
Duplicating the Cube
by
Edward V. Graef
Copyright 1963 Edward V. Graef. Smashwords Edition; copyright 2011 Marsha Graef Bauer (daughter). Cover illustration, copyright 2011 Marsha Graef Bauer.
All right reserved. No part of this may be used or reproduced in any manner whatsoever without permission except in the case of brief quotations embodied in critical articles and reviews. No part of this publication can be reproduced or transmitted in any form or by any means, electronic or mechanical, without permission in writing from Marsha Graef Bauer.
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Table of Contents
Chapter. Validation of the Curves
Chapter 1. A family of curves (not constructible with compass and straightedge alone) is developed to multisect angles
Chapter 2. The "trisection members" of the family are plotted "backwards" to see where the curves are coming from
Chapter 3. The same procedure used on other members suggests that one member of the family might be "missing"
Chapter 4. The "missing" member is plotted
Chapter 5. The "missing" member joins the other members of the family
Chapter 6. The "missing" member helps to "square the circle"
Chapter 7. A property of the family when plotted on concentric circles with "yardsticks" equal
Chapter 8. The construction of an automatic angle divider
Chapter 9. How to use the automatic angle divider
Chapter 10. Use of the automatic angle divider to divide a circle into any number of equal parts
Chapter 11. A member of the family helps to "duplicate the cube"
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 mountain was there . . .
By August 1963, I developed a total of nine other solutions (using curves other than those in the family) before I found a way to use one of the "first curves" in the family in a solution to the problem.
These curves were developed using geometry, and should be understandable by advanced high school and college students.
Edward V. Graef
August 1963
Chapter. Validation of the Curves
Getting the curves validated proved as difficult of a task as inventing them.
Contacting Encyclopedia Britannica
On September 3, 1965, I requested the Encyclopedia Britannica to research the following:
"Was there any mathematician during the period covered by Heath in his "History of Greek Mathematics" who was able to use the same curve or the same family of curves to solve all three of the Ancient Problems?"
The response from the Encyclopedia Britannica on November 22, 1965 was:
"Members of our staff have been investigating your request for information on the "Three Ancient Problems" and have located no evidence to indicate that any mathematician, during the period specified in your letter, was able to use the same curve or the same family of curves to solve all three of the problems."
Contacting Renown Mathematicians & Getting Published
Encouraged by the Encyclopedia Britannica response, I began corresponding with respected and well-known mathematicians who were involved with the Three Ancient Problems.