[math-fun] The value of PI
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 <--
On 30 Apr 2008 at 18:53, Bernie Cosell wrote:
Pi shows up a lot in geometry. I was wondering when "we" (ahem) realized that it was just one underlying constant.
Much thanks for the replies. I knew that Archimedes knew a lot about pi, but I didn't realize just how far he had gotten. So it *is* likely that he _did_ know that there was this single, strange and magical constant that underlay lots of things in geometry, which is what I was curious about. Of course, it'd be nearly two millennia before we actually proved that Pi was irrational (and another hundred years to prove it was transcendental). But that's a different matter.... :o) /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
=Bernie Cosell So it *is* likely that [Archimedes] _did_ know that there was this single, strange and magical constant that underlay lots of things in geometry,
Well... I'd advise being very careful about jumping to modern-colored conclusions about how earlier folks thought about things. For example you might first want to establish that Archimedes even *had* an idea of "a constant" in anything close to the modern sense. Those guys tended to talk (and presumably think) mainly in terms of relations *between* geometrical constructs, integers and the like. (Heck, so did *Newton* et al) For ancients I suspect "PI" might seem inescapably to just be an abbreviation for "the proportion between a circle's circumference and its diameter" and the idea of trying to separate it out so that "PI" somehow had an existence on its own independently from a relationship might have seemed weird. For example there is no explicit "PI" apparent in the classical statements of the relation between cylinders and spheres, just as there was no explicit "sqrt(2)" apparent in the proportion of between the edges and diagonals of squares. Moreover, they had no reason to assume that there might not be some as-yet-undiscovered tombstone-worthy nice relationship between, say, diagonals and circumferences (that is, in modern terms, between "PI" and "sqrt(2)"). Moreover, I've even heard that *one* wasn't considered "a number" in our sense, because they thought of numbers in terms of "a number of things" and a single thing was just that thing, and not a group of things that could be numbered. That may be hard for us moderns to get our heads around, but apparently there are proofs in eg Euclid that special-case out N=1 from N>1 for this reason. Given that context, I can imagine even the most radical Platonist having problems with "zero" (Dude, you say it's "the number of no things", yet itself a thing?!) let alone apprehend some "strange and magical" PI...
On 1 May 2008 at 10:41, Marc LeBrun wrote:
=Bernie Cosell So it *is* likely that [Archimedes] _did_ know that there was this single, strange and magical constant that underlay lots of things in geometry,
Well... I'd advise being very careful about jumping to modern-colored conclusions about how earlier folks thought about things.
For example you might first want to establish that Archimedes even *had* an idea of "a constant" in anything close to the modern sense.
I would guess that he did. I would have thought that he would think that *ALL* spheres and cylinders had volumes in the same ratio. That ALL circles had the same ratio of radius to circumference.
For ancients I suspect "PI" might seem inescapably to just be an abbreviation for "the proportion between a circle's circumference and its diameter" and the idea of trying to separate it out so that "PI" somehow had an existence on its own independently from a relationship might have seemed weird.
Right, but I think they would have noticed that this OTHER thing happened to have the "same proportion". They may not have had a bent to use the abstraction-machinery we do [to instantly give it a greek letter..:o) And hey: If Archimedes WERE inclined to do that, what alphabet would he have used for his constants...:o)] but I think it is pretty clear that they'd recogize "similar proportions". (e.g., that the ratio of the circumference to the radius of a circle is just twice the ratio of the area of the circle to the square of it radius). *THEN* you get the millennium-and-a-half hiatus until the proportions are abstracted and anlyzed on their own...
...Moreover, they had no reason to assume that there might not be some as-yet-undiscovered tombstone-worthy nice relationship between, say, diagonals and circumferences (that is, in modern terms, between "PI" and "sqrt(2)").
Just so!! That'll take that millennium-and-a-half before THAT kind of thing gets resolved..:o). But I still think, having now seen what Archimedes was able to do and figure out, that he would have recognized that the *same* proportionality showed up in several places. I don't need to truly nail it down but I was just trying to get a feel for where things stood until we see an actual proof that Pi is irrational... /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
I don't find it too amazing that PI shows up in some other geometric contexts. I'm quite surprised that it shows up in all sorts of other places that are not obviously geometric. Then of course, there is Euler's identity. To paraphrase Douglas Adams; surprise isn't enough, I have to resort to incredulity.
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 <--
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Bernie Cosell -
Dave Dyer -
James Buddenhagen -
Marc LeBrun