It is well known that the Gemara and Mishnah uses the value of 3 for PI. However we know that PI is 3.1415926... Various answers on this discrepancy have been given, the simplest is that the Gemara/Mishnah are using 3 as an approximation to make it easy for the common man to calculate when halachah dictates a measurement that uses PI.
However, the Gemara (Eruvin 13B?) says that we learn that PI is 3 from the verse in Kings 2 Chapter 7 Verse 23:
And he made the molten sea, 10 cubits from brim to brim: round all about, and a height thereof 5 cubits: and a line of 30 cubits did compass it round about
So it seems that the prophet is telling us that PI is 3 as well!
One could say that this too is a known approximation, and is not meant to tell one the true value of PI. From an e-mail I was directed to from RationalistJudaism.com, a very interesting gematria comes out of this verse.
This gematria is attributed to the Vilna Ga'on (although there is no source to substantiate the claim that the Vilna Ga'on said it. The actual source for the insight may be credited to Matityahu ha'Kohen Munk (Frankfurt-London), who published the thought in the journals "Sinai," Tamuz 1962, and "ha'Darom," 1967).
In the verse quoted above, there is a kri/ksiv on the word "and a line" (v'kav) describing the 30 cubit line around the circle. The ksiv is spelled "v'kavah" with a hey at the end, while the kri is "v'kav" without the hey. The gematria of 'kava' is 111, while that of 'kav' is 106. With this we get something interesting:
The ratio of the kri/ksiv is: 111/106 = 1.047169811320755...
While the ratio of PI to 3 is: PI/3 = 1.047197551196598...
So the ratio of the written/actual word to the practical/read word is extremely close to the ratio of the actual value of PI to the practical/approximation of PI.
In fact, if one uses the ratio 111/106 to derive an approximation of PI, one gets:
333/106 = 3.141509433962265... which is an approximation of PI that is over 99.997% accurate!
So from this we see that it is possible that the prophet knew to very good precision the actually number PI but used a value of 3 nevertheless to show that using 3 is a good enough approximation when it comes to halachic measurements.
(And a happy PI day to everyone!)
Update: I should have posted this before, but here is additional info that is very interesting from a commenter named Ephraim:
EphraimMarch 16, 2010 at 7:10 AM
I was inclined to believe ... that the 111:106 was far from unique, so I did some programming ...
I wrote a little program to generate pairs of numbers that produce ratios close to PI:3.
Starting at 1 and continuing, I took note of the best ratios as they appeared. The first pair, is of course 1/1 which gives the ratio of 1:1 which approximates pi=3. The next pair that produces a better result is 12/11 which gives us pi=3.2727. The next is 13/12, followed by 14/13 and so forth- until 22/21 which provides the approximation 3.1428. No better pair appears until 67/64, then 89/85, then 111/106- our value here. Until 1000, there are only two more pairs that give us better results: 244/233 and 355/339. And that's the best you can get for pairs under 10,000.
It means that 111:106 is the third best ratio for numbers under 10,000. And it's the best that could be generated from a single letter difference- i.e. a kri/kesiv.