derive the catenary arch with minimal calculus

Today, I have a different take on the catenary arch. I first conceived of this idea during a course in Engineering Statics with Dr. Dee Puntenney at PLNU as a freshman. As far as I know, this is the first such argument for a cylinder arch maybe because of its impracticality, although I have fun and build one out of candles anyway. This can perhaps be done as a student-led project as a lesson on modes of vibration and stability.
I attempt to use only force diagrams and simple shapes--to wit, circles, and finite sums. If you know those, you should be fine. The historic portraits are for general license as is the hanging chain at the beginning.
The footage of the child exploring the hanging chain is meant to show natural learning and adheres to KZbin's Child Safety guidelines: as the child's identity is kept anonymous and they are learning in a family-safe event at JPL.
Minor error correction: At time 11:25, I wrongly concluded that as the limit of N goes to infinity, the arcsinh of 1/N goes to zero. While true, I didn't multiply by that N in front. If I had distributed the 2RN first, the limit would be 2R, not zero. Then for sufficiently large N, the first term overwhelms the second which vanishes anyway, then the derivation continues as before.
Thanks to Patty McLaughlin for suggestions on candles. Thank you to JoAnne McKenny for help with setup. Thanks to Scott Seward and Family for help with audio equipment.
Piano music was written and performed by me, Sobermath.

[@trinityblackout6895]

This is one of the best KZbin videos I've ever watched. Keep 'em coming :)

[@sobermath]

Oh wow, what an endorsement! Thank you for saying so. I will certainly continue publishing videos as time allows.

[@skyeandtiger]

WOW, very detailed and thorough explanations- love the diagrams . Great job !

[@sobermath]

Thank you! It's been a long time and lot of work. I'm happy with how it turned out. And I'm grateful for you.