source file: mills2.txt Date: Wed, 5 Mar 1997 16:54:31 -0800 Subject: Re: Greeks and Pi ? From: Jonathan Wild Charles Lucy asked: >If irrationals were treated differently to rationals, how >did the ancient Greeks handle Pi? The first real proof of Pi's irrationality was not until the 18th century - I think you'll find it was either by Euler or Lambert. The proof pi is transcendental didn't appear till even later. But the way the Greeks attacked the problem of finding the ratio between a circle's diameter and area suggests they at least had doubts about pi's rationality. This is perhaps best exemplified in the way Archimedes, among others, made ever-finer polygonal approximations (of known area) to a circle, from both inside and outside, converging towards the "true" ratio which could never be completely attained. I believe he even says something to the effect that while it is false to believe that in this way you can "exhaust" the whole area of the circle, you can in fact come arbitrarily close. Persistent failure to properly "square the circle" -- definitely one of antiquity's most popular problems -- could also have sown seeds of doubt among the Greeks as to pi's rationality. One would imagine that this would have created some consternation for early Pythagoreans, who had rather hoped that the cosmos and everything in it worked according to numbers -- in fact in some sense that everything *was* number. But if the ratio of a circle's radius to its circumference was irrational, then the lengths of the celestial bodies' circular paths around the "Central Fire" would be incommensurable with their distances from it, i.e. have no common number. (This particular cosmological model, in favour with early Pythagoreans, had a Central Fire around which were placed the earth, moon, and Sun; the five planets known to the ancients; a fixed sphere of stars and a "counter-earth", perpetually invisible to us since it remained diametrically opposed to ours on its orbit around the central fire, and apparently incorporated to fudge the number of "things" out there to add up to ten, the so-called perfect number...) Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 6 Mar 1997 02:15 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07241; Thu, 6 Mar 1997 02:15:28 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA07256 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id RAA21282; Wed, 5 Mar 1997 17:13:50 -0800 Date: Wed, 5 Mar 1997 17:13:50 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu