Today, let's talk about π -- and not the kind you eat. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.
Once upon a time, it was a common sport for school children to compete over who knew the most digits of the mathematical constant, Pi (or π). (That's the ratio of the circumference of a circle to its diameter.) My interest and patience usually ran out at 3.14159265.
But finding out what those digits were has been a major mathematical challenge, ever since the invention of the wheel began stirring a real interest in circles. The earliest recorded values of π were Phoenician and Egyptian. They were 3 1/8 and 3 13/81. Both values are accurate within half a percent.
Where did those numbers come from -- from measurements? Well, you just try measuring π with a piece of string. You won't come that close. The Hebrew peoples suggested a rough empirical value of π in the Bible. It was three, and you'll find it in a text that shows up in both the 1st Book of Kings and the 2nd Book of Chronicles:
And he made a molten sea, ten cubits from the one brim to the other: It was round ... and a line of thirty cubits did compass it ... about.
From time to time you hear stories about legislative bodies that've tried to enact laws, based on this text, making π exactly equal to three. Science writer Petr Beckmann was unable to verify any of these stories, but he does report a remarkable event that took place in the 1897 Indiana State Legislature.
An Indiana doctor thought he'd solved the classical problem of squaring the circle. That means specifying the size of a square and a circle that both have the same area. If you could do that, you'd also be able to get an exact value of π. This fellow tried to get his proof enacted as law, but the text of his bill was muddled. It turns out that it would have made π greater than nine.
The House had trouble finding anyone to review the bill. They finally gave it to the Committee on Swamp Lands, who said it looked okay to them. When it cleared the House, the Senate gave it to their Committee on Temperance. Temperance could no more figure it out than Swamps could, so it got preliminary approval. After that, local academics heard what Congress was up to and started questioning legislators. The Bill mysteriously disappeared, and it was not heard from again.
All this happened fifteen years after mathematicians had shown that it was impossible either to square the circle or to evaluate π exactly. That's bizarre enough, even if extremists hadn't really tried to make it a law that π was equal to three. But historians have also found out that the accurate Phoenician and Egyptian values of π hadn't come from measurements after all.
These ancient engineers actually deduced their values, and they used elegant geometrical logic to do it. It is a sobering fact that they had clear-headed answers to questions that still troubled a lot of people four thousand years later.
I'm John Lienhard, at the University of Houston, where we're interested in the way inventive minds work.
Beckmann, P., A History of Pi. St. Martin's Press, New York, 1971.
Pi = 3.14159 26535 89793 23846 26433 83279
50288 41971 69399 37510 58209 74944
59230 78164 06286 20899 86280 34824 ...
Pi can be very well approximated as = 355/113 = 3.14159292...
This is a revised version of Episode 180.