Decimal fractions were not used until at least 1000 years after Archimedes, so he couldn't have written Pi is about 3.1416 for example. He did find good rational approximations of Pi. Some basic history of Pi is at the The MacTutor History of Mathematics archive here: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.h... And see the "General Remarks" section here: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html for something directly relevant to your question. Jim On Wed, Apr 30, 2008 at 5:53 PM, Bernie Cosell <bernie@fantasyfarm.com> wrote:
Pi shows up a lot in geometry. I was wondering when "we" (ahem) realized that it was just one underlying constant. For example, the area of a circle is 3.14... times the square of the radius and the volume of a sphere is 4.18... times the cube of the radius. It is already profound discovering that those ratios are fixed [and to be able to calculate a bunch of digits of them]; I don't know if he did or not, but I can' easily see Archimedes managing to calculate a bunch of digits of the 4.18... constant even as he did for the 3.14. one].
But when did mathematicians realize that those weren't two separate gnarly constants, but actually "reflections" [if you will] of a single underlying constant? I don't think the Greeks had enough math machinery to figure all that out, did they? [that is, that the ratio of those two constants is exactly 3:4. Or that the ratio of the radius of a circle to its curcumferenace is really just exactly twice the ratio of the square of the radius of the circle to its area; 3.14... and 6.28... will certainly *look* like 1:3 [not clear the 3:4 ratio is as obvious..:o)], but could they actually 'know' that enough to say that that was really "proven", in some sense? If not the greeks, then when did we figure that out?
/Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
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