Squaring the Circle, part 1

Fragment of the Ahmes Papyrus
Image via Wikipedia

Have you heard that phrase before?  I have and I didn’t know it referred to anything other than trying to do something that’s impossible, and also rather silly, right?  I mean, why would anyone try to turn a circle into in a square?

Turns out there was actually a good reason for it.  Back in the days when the best tools mathematicians had were the compass and straightedge, they needed a fairly simple way of determining the area of a circle, which is devilishly hard to do.  It’s very easy to find the area of a square, so one solution they thought of was to figure out a simple way to draw a square that had the same area as a given circle. 

This endeavor went on for millennia and turned out to be futile.

In the mean time they did a pretty good job of estimating the area of a circle.  Here’s what they did, and how we know about it.

Back around 450 B.C. the Greek historian Herodotus traveled around Egypt interviewing priests and observing all the great work that went on there, from building monuments to farming along the Nile.  He said that geometry was invented in Egypt in part because the land had to be resurveyed every year after the Nile’s flood waters receded.  (Aristotle disagreed with him – he said that it was the priestly leisure class who did all the cool math in Egypt.  I don’t see why they can’t both be right – I guess it’s the age-old rivalry between engineers, who have practical problems to solve, and mathematicians, who are frequently philosophers and deal in the purely abstract realm.)

A piece of evidence on Herodotus’s side of the story, is the Ahmes Papyrus, which resides in the British Museum.  This papyrus was named for the scribe who around 1650 B.C. wrote it, using material from about 2000 to 1800 B.C.  In the papyrus, Ahmes says that the area of a circular field with a diameter of 9 units is the same as the area of square that’s 8 units on each side.

A little earlier on he had worked a problem that shows how this relationship was discovered.

First, you draw a circle with a diameter of 9 units.  Then you draw a square around it.  This square is 9 units on each side.

 

Then you cut off an isosceles triangle from each corner . . .  

. . . which gives you an octagon with an area of 63 units that has roughly the same area as the circle, though it’s not clear whether Ahmes thought it was exactly the same, or “close enough for practical purposes.”

From this they came up with the rule that “the ratio of the area of a circle to the circumference is the same as the ratio of the area of the circumscribed square to its perimeter [A History of Mathematics, Mertzbach and Boyer, page 15].”  Apparently this works out to a mathematical constant, 4(8/9)², which they used the way we use π, but I don’t really understand that part of the story.  Mertzbach and Boyer say that it’s “a commendably close approximation” of π and I’m taking their word for it.

The Ancient Babylonians also had a method, but I’ll have to tell y’all about that another time.