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The mishna on thursday’s daf says: A korah needs to be wide enough to hold an ariach, which is 1.5 tefachs, so therefore it needs to be 1 tefach wide (bec with mud on each side it will hold an ariach). If it is round, then you know it is a tefach wide if it is 3 tefachim around. (roughly pi)
The gemara says, how do we know this measurement system? Because the passuk says shlomo made a pool that was 10 across and 30 around.
The gemara asks: but what about the thickness of the walls? (the gemara apparently thinks the 10 is inside measurement but the 30 is outside).
Gemara answers that the walls were thin.
Gemara asks: but what about the ????
Gemara answers: no, both are inside measurements (prob could have said both are outside measurements.
Tosfos asks on gemara’s question of the ????, that the ???? ????? say that the amount is really a bit more than 3 anyway.
So what can we do with this gemara? Did the gemara know that pi is really some amount more than 3, or did they think it was precisely 3?
Presumably, everyone will agree they knew as much as the mathematicians of the time, who did know that the circumference of a circle is more than 3. Moreover, anyone who has ever measured it knows it is something more than 3. So it is quite dochek to think they thought it was precisely 3.
So here’s what I’m thinking.
??? ????, why does the mishna say you can do this calculation? Is it a halacha? Or is the mishna just telling you how to measure stuff?
And why does the gemara ask for a source from a mishna, instead of just saying that everyone knows you can measure circles and find the diameter?
I think the real chiddush here is that the measurement of the korah, and of other measurements, does not have to be precise. And that is the real halacha of the mishna. And that is what the gemara is asking for a source. And proves from the passuk that you are allowed to use standard measuring techniques which assume a diameter is about 1/3 of a circumference.
so God was being intentionally inexact when he wrote the passuk, to show us that we can be inexact too? trippin. wish he put sum spelling mistakes up in there too.
yo I thought navi was wit dat divine inspiration. also dat kav/kavei trick is supatrippy.
PBA: Artscroll quotes the Tosfos HaRosh who basically explains what Brony says. We know that Halachic pi is 3 but that real pi is 3.14. The Gemara asked for a source (which it doesn’t need because it’s easily measurable) that Halachic pi is 3, not for a source for what real pi is.
PBA: I disagree. The Gemara has a concept that we can approximate sometimes (Lo Dak; see around Sukkah Daf 8), but always only L’Chumra. We need a Passuk to tell us that we can always assume that pi is 3, even though it will have Nafka Minos that are L’kula as well.
In the ???? about the ?? ???? ????, there’s a ??? ???? on the word ??, which has an extra ?. The ratio between the ??? and the ???? is almost the same as the ratio between halachic pi and the scientific pi; using the pi yielded by this formula would be acceptable in any architectural application.
Heard ??? ???”? ???????
Here are the numbers:
The actual number for scientific pi is 3.14159.
I think popa is right. The Tosafos HaRosh appears to be saying that the pasuk is a raya for Chazal to approximate, because the pasuk does so too.
I don’t know what “Halachic pi” means.
PI is not exactly a number, it is a formula. You can never finish the function and get a result.
I’ve seen Popa b. Abba’s explanation before.
By the way, what bothered Tosafos is the fact that we worried about a tiny width of the lid, since the whole thing is an approximation in the first place.
There is no Pasuk for the hypotenuse, is there?
Decimals were invented relatively recently ( I believe by the Indians, and introduced to the western countries by the Arabs). Mathematics in the time of the gemarra, regardless of whether you were Jewish, Greek, Roman or Mesopotamian, had to “fudge” a bit as a result. They also didn’t have a “zero” until fairly recently.
There is a concept in science and engineering called significant digits. For example, if you are weighing something with a scale that can have an error of one pound or greater, the reading on the scale of tenths of pounds is really meaningless.
Whenever I encounter PI in my engineering work I will try control the related phenomenon with parts that have a possible error of +/-15%. Hence, the only significant digit of PI is the “3”, not the digits to the right of the decimal point, which affect the accuracy by less than 5%.
Josh31: Significant digits is irrelevant until you invent decimals (and invent zero) – which was after the time of the gemara. If it would make a difference, I doubt anyone today would allow using 3.0 rather than a better estimate of PI now that we can be more accurate.
HaLeivi: I’m pretty sure the Pythagorean theorem is mentioned somewhere in Sukkah. (Or Pe’ah?) My brother showed it to me once.
Thanks for a really interesting topic, popa.
The Pethagorean theorem is not mentioned. The Gemara uses 1 and 2/5 which only works for a square, as Tosafos points out, and is not exact.
Without decimals Chazal could easily have said to measure 3 and a seventh, or Shlosha Ushviis, or at least Shalosh Umashehu as we do by the Techum. They could also have mentioned Ve’od.
yo u must be british or sumthing cuz here in are country theyre called significant figures. gotta science right my man.
I don’t know what “Halachic pi” means.
I meant it as the number three, which the Torah recognizes as a valid representation of pi (the right to approximate).
DaasYochid – Ha gufa. That’s what popa was saying, and I agreed. Sam was using the term as an alternative pshat. That’s what my comment was directed at.
YW,
I didn’t think anyone was disagreeing here.
As mentioned above, the gemara is establishing a basis of measure independent of the physical fact that the circumference is more than 3. Once established, this will be the unit of measure to be used when questions of this nature need resolution. The Torah is the source of nature itself and supersedes facts as we know them.
It is interesting to note that in the following few blatt much discussion revolves as to whether 2.9 or 3 tefachim is lavud. This is the basis of a whole machlokes.
Assuming an amah is say 22 inches, 1 tefach will be 1/6 of that or 3.67 inches. 1/10 of that is .37 of an inch and this forms the basis of many blatt of machlokes and no one questions as to whether this difference in actual fact will prevent a goat from slipping under a gap.
The Torah tells us when to make these minute distinctions and when not to.
Do not panic, I will not embark on a whole discussion on the Brisker shiurim by the seder for tonight.
One shiur is important to remember.
There is no limit on how much to love, honour, and respect your spouse and children.
Chag kosher and someach to all of Klall Yisroel and may we be zocheh to see the end of this long and terrible golus so that Eliyahu Hanavi will finally come and take the Yidden out of golus and the golus out of the Yidden.
DaasYochid – Ah, I see. You’re right.
Popa, with your permission, if I can add to this thread another question that has been bothering me on this subject.
However, what’s bothering me is that as the Gemara concludes, the pool consisted of 3 amos of a square tub, followed by 2 amos of a round tub. If my calculations are correct, the ratio of height-to-width is larger in the case of a round tub than a square tub. So the premise Rashi sets forth on the height/width ratio, is this true for round tubs or for square tubs? And in this case we have a combination of the two.
Presumably, everyone will agree they knew as much as the mathematicians of the time, who did know that the circumference of a circle is more than 3.
I always wondered how the Gemorah would have been written if the works of Archimedes had been published earlier.
Correction to my previous post. The gemara is referring to water vs dry matter, instead of water vs heavier liquids. Can’t fathom why I mis-learned the gemara like that. In any event, my kashya is still valid, and I did find over Pesach that the Ritva asks the kashya and he is madchik a teretz that indeed the calculations of 2:3 of liquid:dry is applicable in this unique case where the first two Amos are round and the remaining 3 are square.
Wondering if anyone had the opportunity to see a good teretz to this kashya. Thank you.
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