A quick experiment to prove that a catenary is not a parabola

I sampled 9 points on this curve. The x coordinates have constant increments (equally spaced horizontal coordinates).

I then measured the y coordinates — that’s what the numbers at the top are (in millimeters). Then I calculated the differences. And then I calculated the differences of those differences. That’s the bottom line. The numbers in the bottom line are spread out, it varies a lot. And that means that it’s not a quadratic. So it’s not a parabola!

This was inspired by Are all u-shaped graphs quadratic? by James Tanton and How to build a Giant Dome by Numberphile, but Simon has built a quick experiment to prove the same statement. He was very excited a couple of weeks ago when he first watched the Numberphile video where Tom Crawford gave a lecture about the history of St.Paul’s Cathedral in London. Apparently, they didn’t know back then how to express a catenary (hyperbolic curve) mathematically and yet managed to create one for one of the cathedral’s domes. We dug into that story and fund out Sir Isaac Newton and other prominent mathematicians of the time chipped in and let architect Christopher Wren pick their brain.