No, really - what math did Archimedes use to calculate pi?

After reading MathWorld’s page on Archimedes’ Recurrence Formula I’m now more confused than when I started. :confused:

Basically, Big A estimated the value of pi by measuring the perimeter of a polygon which had been circumscribed around a circle with a radius of 1, then measuring the permiter of a similar polygon inscribed within that circle, then averaging the result. If you give your polygons more sides, your estimate becomes more accurate.

The problem I’m having is that the math described at MathWorld uses trigonometry. If Archimedes knew how to use trigonometric functions, then he must have known the value of pi. I was hoping that this could be a rare instance where Wikipedia trumps MathWorld, but their relevant article is just as useless, merely describing how the problem is easier to solve using integral calculus.

Was Archimedes’ exercise largely academic, or did he use non-trigonometric math? If the former, then what’s the big deal? If the latter, then exactly what was his method?

This doesn’t follow. Surely, he knew some things which would be lumped under the modern classification of trigonometry, but that doesn’t mean he knew all of it. He must, for instance, have had some system of measuring angles, but he probably didn’t use radians. Likewise, there are some angles whose sine and cosine can be determined by means other than the power series expansions or other radian-dependent methods.

This site ESCAPE.COM - PROVIDING ONLINE SERVICES SINCE 1987 explains it, although I couldn’t follow the logic because it explains it in terms of trigonometry and then “works back” to how Archimedes did it without trig. I gather that it says that he could solve the equation for two inscribed hexagons geometrically and then by extension work out the formula for doubling the sides/ halving the angles successively for 12, 24, 48 and 96 sides.

Hi SP. No, Archimedes did not use the concepts of trigonometry as we understand them today (chords, sines, cosines), although he was thoroughly familiar with the geometry of right triangles.

To see what he actually did to get an estimate of the ratio of the circumference of a circle to its diameter in his treatise Measurement of a Circle, gird up your mathematical loins and go read the treatise itself in Thomas Heath’s English translation. (It is somewhat of a paraphrase/interpretation of the original Greek rather than an exact literal translation, but it should give you quite a good idea of what Archimedes was really up to.)

You can find part of the translation reproduced online here. It only takes you up through the proof of Proposition 1, though, at which point you can switch over to the Cliffs Notes summary provided here, which somewhat simplifies Archimedes’ actual procedure but is more faithful than a trig-based explanation.

Thanks, everyone!

The last paragraph in Lumpy’s link explained it perfectly for me. I had no idea that the half-angle formula was known back then.

It wasn’t, because modern trigonometric formulas weren’t known back then. What Lumpy’s link showed was a modern trigonometric interpretation of the triangle geometry that Archimedes was actually using. What Archimedes invoked to get his “half-angle” results was not a “half-angle formula” per se but rather Proposition 3 of Book 6 of Euclid’s Elements: