On August 1st, 2008 my father died of stage 4 prostate cancer at the age of 69 (technically only, he turned 69 in the brief weeks he was in the hospital before he died). Two years prior to his death he sent this e-mail to Gilbert Strang, professor of mathematics at MIT:
Dear Dr. Strang: I am a 67 year old biochemist but have always been interested in Mathematics. In fact, I use your wonderful text book on Calculus when I attempt to assist younger people with their studies. Recently, a young man asked why in the world such a beautiful, graph and equation as the ellipse couldn't be solved for arc length. Of course I couldn't answer that question so I picked up my pencil and paper and went to work. While doing so, I discovered an approach that I had not seen before and I wish to ask your advice. I believe it has merit. Briefly, if one were to draw an ellipse in standard format on an ax/by axis with ax=sin(Q), and by=sin(a)cost, (a stands for alpha which is the angle a circle is sloped on an ax/by/so axis, and Q is the original angle describing the circle), one can easily develop a parallelogram with the shorter side being (1-sin(a)) and the longer side being (1+sin(a)). The larger angle is PI-2Q and the smaller angle is 2Q. Now, the diagonal of the parallelogram represents the rate of change of the arc length with respect to Q. (Try it yourself) When drawing the diagonal, note that two very important angles are created. I call these E and D representing Sum and Delta. They are the sum of Q and Qy where Qy is the angle created by the first derivative of y vs x, and Delta representing the difference between these two angles. Note that perpendicular from the apex of the parallelogram at (PI-2Q) to the diagonal represents the 'acceleration' of the above diagonal which represents the rate of change of the elliptical arc length. Also note, that (1-sin(a)) times sin(E) = (1+sin(a)) times sin (D). This creates many opportunities. It is too difficult to type out my subsequent calculations as I am on a home computer which doesn't have the software I would like to use. However, if you are interested or have a comment, please let me know. I have done many calculations in my life, and this one was like wandering through a forest in the dark with no flashlight or compass. (LOL) Finally, I apologize for bothering you but as many have told me before, I am a persistent one. Thanks for your time. Sincerely, John C. Coggeshall
The young man my father is speaking about was me. Many years ago now when I was still at Kettering University I was speaking with my father complaining about classes. Knowing my father loved math because of its purity, I asked him to justify how this could be so when something as simple to understand as an ellipse could only be approximated through calculus. That was nearly 6 years ago at the time I write this but it only took my father 4 years to answer my question.
For the remaining years of my father's life he spent countless hours working on this problem and, according to him during our conversations before his passing he was able to solve for the arc length of an ellipse. In fact, when this e-mail was written to Mr. Strang he had done it -- he spent the last 2 years of his life exploring how his discovery impacted the rest of the math he loved so dear.
This e-mail to Mr. Strang is the only evidence I have that he told anyone what he had done. I had often encouraged him to reach out to others but he was paranoid his work might be stolen as it had in the past. That said, upon his passing I found stuck within his calculus book a nearly folded stack of papers which I know is the solution mentioned in the e-mail to Mr. Strang above.
I'm not a math guy, never really have been. I looked at my father's work and had flashbacks to my Calculus III course and I realized there wasn't any way I was brushed up enough to make heads or tails of the complexities. For about a year now I've thought about what to do with them and I've decided to post them here in the hopes that one of my fellow geeks has more time and math skill to perhaps identify and transcribe my father's work into a form that can be shared with the world.
To this end, I've scanned all of the documents I took from my father's work area. Included in this is the set of papers neatly folded in the calculus book which I believe is the solution to the arc length of an ellipse and a number of other unknown equations that my father apparently had been working on before his passing. I wasn't sure what was valuable or not, so I grabbed both. Also included in this scan is the e-mail my father sent to Mr. Strang and his response -- I have only blacked out my parents phone number for privacy.
It is also worth mentioning that I have labeled each page using a simple system of letter/number. The pages with the same letter represent a "grouping" of pages (i.e. I found them as a group) where the number represents the page # for that group. So I1, I2, I3 are all the same "group" of pages number 1 2 and 3 respectively.
Some other useful pieces of information..
There isn't a particular order to the pages, other than they are grouped and order maintained.
The "I" and "H" series of pages look the most familiar to me from the numerous attempts my father attempted to explain to a distracted son, and probably a great place to start if you want to dig
I have all of the original documents as well, if something didn't scan cleanly
I of course want to believe that my father did what he said, but who am I to know for certain? That's the point of me posting all of this to the public. If it turns out my father forgot to carry a 2 somewhere and he's totally wrong please feel free to tell me, but at least be polite about it -- I did love him and he is dead now after all.
All of that said..
I would love to find out someday if there was any merit to my father's equations and to actually learn them if they were truly what he believed they were. All I ask is that my father receive the recognition for this work if it was legitimate. I don't know if you can do such things but I am going to attempt to license this document and say this PDF is provided under the General Public License version 3.
