Civilizational Collapse (Part 5)

A tale of civilizational decline and rebirth.

Around 250 BC Archimedes found a general algorithm for computing pi to arbitrary accuracy, and used it to prove that 223/71 < π < 22/7. This seems to be when people started using 22/7 as an approximation to pi.

By the Middle Ages, math had backslid so much in Western Europe that scholars believed pi was actually equal to 22/7.

Around 1020, a mathematician named Franco of Liège got interested in the ancient Greek problem of squaring the circle. But since he believed that pi is 22/7, he started studying the square root of 22/7.

There’s a big difference between being misinformed and being stupid. Liège was misinformed but not stupid. He went ahead to prove that the square root of 22/7 is irrational!

His proof resembles the old Greek proof that the square root of 2 is irrational. I don’t know if Liège was aware of that. I also don’t know if he noticed that if pi were 22/7, it would be possible to square the circle with straightedge and compass. I also don’t know if he wondered why pi was 22/7. He may have just taken it on authority.

But still: math was coming back.

Liège was a student of a student of the famous scholar Gerbert of Aurillac (~950–1003), who studied in the Islamic schools of Sevilla and Córdoba, and thus got some benefits of a culture whose mathematics was light years ahead of Western Europe. Gerbert wrote something interesting: he said that the benefit of mathematics lie in the “sharpening of the mind”.

I got most of this interesting tale from this book:

• Thomas Sonar, trans. Morton Patricia and Keith William Morton, 3000 Years of Analysis: Mathematics in History and Culture, Birkhäuser, 2020. Preface and table of contents free here.

It’s over 700 pages long, but it’s fun to read, and you can start anywhere! The translation is weak and occasionally funny, but tolerable. If its length is intimating, you may enjoy the detailed review here:

• Anthony Weston, 3000 years of analysis, Notices of the American Mathematical Society 70 1 (January 2023), 115–121.