I recently “rediscovered” the calculations of Al-Khashi, from the 1400’s by using the same method, an approximation of pi by the circumference of a polygon. I found that this devolved onto the half-angle formula for sine, or equivalently for cosine, applied recursively. This is a trigonometric formula, but easily obtained by direct geometric construction.
I did all this on my own and I was amazed to see how closely I had followed Al-Kashi’s method. Of course it was very easy for me to carry the recursions much further, using the UNIX based extended precision calculator, bc .
Al-Kashi obtained 17 decimal digits of pi, and it is quite amazing to me how modern his thinking was concerning this recursive computation.
Yes, amazing that that precision was achieved so long ago.
Most of my math these days is converting GEO satellite orbits with inclined orbits from polar coordinates and predicted drift to cartesian coordinates for satellite terminal tracking changes. That and link budget data rate calculations.
When I was in college at University of Florida, College of Electrical Engineering I spent time trying to understand Einsteins Theory of General Relativity, but got lost at the Taylor Series Expansions.
At the time I thought that Maxwells Equations and the derivations were obvious and I thought I could have derived them if Maxwell had not. Now I have forgotten much of that and know satellite dynamics these days.
I feel dizzy all of a sudden. I'm going back to bed..........