There are those who claim that the description of King Shlomo's Pool having a diameter of 10 cubits and a circumference of 30 cubits is an example of a scientific error in the Bible. Now, I am certainly not ideologically closed to the idea of the Bible being scientifically inaccurate - there are several examples of this, such as with the kidneys, dew, firmament, etc., for which we invoke the concept of Dibra Torah k'lashon bnei Adam. However, I don't believe that King Shlomo's Pool is an example of this; it's just a convenient way of describing it. Although, it is perhaps a little problematic here because instead of saying that its circumference was 30 cubits, it says that a thirty-cubit line could encircle it.
A few days ago, a reader who is convinced that Chazal had a supernatural source of knowledge about the natural world gave King Shlomo's Pool as an example. Here is the idea, as presented by Rav Mordechai Kornfeld (and follow the link for further discussion):
A fascinating insight regarding the value of pi is attributed to the Vilna Ga'on. (Actually, 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 that the Gemara cites as the source for the ratio of the circumference to the diameter (Melachim I 7:23), there is a "Kri" and a "Kesiv" -- a word that is pronounced differently than it is spelled. The word in the verse is written "v'Kaveh" (with the letter "Heh" at the end), but it is pronounced "v'Kav" (with no "Heh" at the end). The Gematriya of the word "Kav" is 106, and the Gematriya of the word "Kaveh" is 111. The ratio of the Kesiv (111) to the Kri (106), or 111/106, is 1.0471698. This value represents the ratio of the value for pi to 3 (3.1415094/3 = 1.0471698).
The question is, does this provide evidence for Chazal having supernatural sources of knowledge? I don't think so, for several reasons.
First of all, a person could argue that the kav/kaveh curio is simply a coincidence. It's not a matter of something being accurate to seven decimal places. There are two numbers, 106 and 111, which can be manipulated to give a certain value. There are doubtless plenty of two and three digit numbers which can be manipulated to give a similar value, and there are plenty of two and three digit numbers that can be derived from a verse. Some will see this as unduly skeptical, and at the moment, I am inclined to agree, since it's just too neat that it's exactly with the word describing the circumference that this gematria is found. But I don't think that I can conclusively show that it's not a coincidence.
As for the significance of the kri/ksiv, while Malbim and (of course) Maharal ascribe significance to both kri and ksiv, according to Radak they simply reflect uncertainties that arose in transmission.
Then, even if one wants to claim that Pi is encoded in kav/kaveh, does this reflect a supernatural encoder? The value of Pi was known in ancient times to several decimal places, and a human could encode it in this way. There is a Greek Pythagorean motto "God is ever a geometer" (ἀεὶ ὁ Θεὸς ὁ μέγας γεωμετρεῖ) — the number of letters in each of the six words are the first six digits of pi. A cute and deliberately constructed device, but not one that indicates that the composer of either the phrase or the language was supernatural!
Finally, even if one does feel that this strongly points to a supernatural encoder, it is not evidence of Chazal possessing a supernatural source of knowledge. The verse is assumed to have been written with Divine Inspiration, which means that God has supernatural knowledge, not man. With regard to Chazal, it does not appear that they knew the value of Pi to any decimal places. The Gemara gives the value of Pi as being 3 (Eruvin 14a), and Tosafos points out that, based on the context, the Gemara does not seem to be giving an approximation. Of course, there are various apologetics which argue otherwise, but Tosafos apparently didn't find them convincing. Thus, if someone wants to believe that the Gemara did not mean this, they can do so, but one cannot use the topic of Pi to prove that Chazal had superior knowledge of the natural world.
Furthermore, the Mishnah (Ohalos 12:6) says that "A square is greater than a circle by one-fourth," referring to the perimeter of each when the circle is drawn to the height of the square. This is true if Pi is assumed to be 3, but given a more accurate value of Pi, the perimeter of the square is actually closer to one-fifth longer than that of the circle.
Some readers will doubtless find it hard to accept that Chazal believed Pi to be 3. The question is whether there is basis for their disbelief, and an analysis of the Gemara and Rishonim reveals that there were much more basic mathematical errors committed by some (but not all) of Chazal. Tosafos (Eruvin 76a) says that Rabbi Yochanan and the Gemara in Sukkah misunderstood a statement by the judges of Caesarea to mean that the diagonal of a square is equal to twice the length of its side. Tosafos states that Rabbi Yochanan subscribed to this understanding of the judges of Caesarea, and that the Gemara in Sukkah rejected it precisely because it is mathematically inaccurate. Rashba expresses surprise at Tosafos attributing a simple mathematical error to Chazal, and he gives an alternate explanation, but he does not deny that Tosafos does indeed say this! Ran likewise expresses surprise that the judges of Caesarea erred in a simple mathematical matter, and cites an alternate explanation of Rabbi Yochanan’s misunderstanding of what the judges of Caesarea were saying, which somewhat lessens the error, but still leaves Rabbi Yochanan making genuine errors of both interpretation and mathematics. Tosafos HaRosh states similarly. Given all this, there is no reason not to take the Gemara's statement about the ratio of a circle's circumference to its diameter at face value.
Finally, we have Rambam on record as being the first person in recorded history to explicitly describe Pi as being an irrational number. I don't know whether it is amusing or sad that some people co-opt the Rambam for anti-rationalist purposes. Jonathan Rosenblum declared that Rambam's statement about Pi is evidence that Torah scholars have supernatural sources of knowledge about the natural world. But first of all, while Rambam was the first to write this explicitly, it had already been hinted at by earlier Greek writers. Secondly, the idea that Rambam knew this via kabbalah or some other such source is ludicrous and a distortion of Rambam's fundamental ideology. Rambam himself wrote that even Chazal had no such supernatural sources of knowledge; he certainly did not consider himself to be privy to kabbalistic secrets!
Have a happy Pi day, and let's not undermine the credibility of Torah and Judaism by making extreme claims that do not stand up to scrutiny. There's enough to be proud of in our religion without having to resort to such shtick!
26 comments:
Can you post a source for Rambam's statement about pi? Thanks! Nice article!
Actually wouldn't you say it's very clever: the very word(s) to describe the circumference "kav...yasov", the way it is said represents the index of the way PI is always "said" in Chazal - 3, whereas the written but unspoken, or "underlying" Gematria represents the index of PI "below the surface"? KAVA would be gramtically incorrect, and in Divrey Hayamim 2: 4:2 it says KAV without the alternative reading so they obviously knew the correct reading. Come on, there's something going on here...
It's very clever, but maybe a person noticed the inaccuracy of the possuk, and introduce the variant reading to make up for it!
In divray hayamim it says the SAME passuk with just KAV so "they" should have just corrected it, and it's not really an accidental inaccuracy but rather a purposeful one, because a KAV/KAVA KERI/KETIV is also done in Zecharia 1:16 in a different context, and it is referring to a measuring tool (the mefarshim don't agree on what it's measuring exactly - but you could make the same allusion to the common measuring methods and the accurate ones). So what you're saying, Chaim, does not seem to be plausible.
I've always thought that this was a fascinating calculation and - even though I'm heretical in my beliefs - still don't think that the k'ri k'siv ratio is coincidence. But, as you say, it isn't proof of divine knowledge.
Did chazal believe that pi=3? A very simple measurement of a circle with a string will show that it isn't. It is much more likely that a whole number was used because it is immensely more practical (unless, of course, one needs a more precise value for - say - architecture, something that chazal was not concerned with.)
But forget the Greeks; this is historically too late re Shlomo's pool. A quick wiki on the history of pi says that "the earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value."
For the interested reader, here are two papers which analyse the "π = 3" topic in detail.
Do Scripture and Mathematics Agree on the Number π?
http://u.cs.biu.ac.il/~tsaban/Pdf/ElishakoffPines.pdf
On the Rabbinic approximation of π
http://u.cs.biu.ac.il/~tsaban/Pdf/latexpi.pdf
From rationalist "it's just an approximation" to mystical "in the temple, pi was actually 3", it's all discussed there.
"Rambam himself wrote that even Chazal had no such supernatural sources of knowledge"
Where did he write that?
Well, although I like the drash and I hate to diminish it, the math is wrong:
Pi is 3.141592653589793.....
NOT 3.1415094 like the e-mail says
So we have:
111/106 = 1.047169811320755...
pi/3 = 1.047197551196598...
So it is accurate to 5 sig-figs not 8.
Even so, it could be that 5 sig-figs is enough for almost any purpose, i.e. if we take the 111/106 ratio to find an approximate value to pi we get:
333/106 = 3.141509433962265...
which is a 99.997% accurate approximation of PI which is pretty darn close!
It amazes me that on PI-day nobody noticed the e-mail was using the wrong value for pi! :)
Are there any Jewish commentators who try to resolve the dimensions of King Shlomo's pool in the same way some Christians do, by referring to the inner and outer diameters?
http://www.bibleprobe.com/pi.htm
(Heh, note how they, ahem, borrowed the kri/ksiv idea!)
Reuven Meir: Note the superfluous hei in v'kav. Perhaps it somehow alludes to taking the ratio of v’kav and v'chameish - the value of hei. If you do so, it just so happens that the ratio of the numerical values of v'chameish (355) and v'kav (113) [including one for the word itself] equals 3.14159292...which is precisely six decimal places of accuracy – one part in a million of pi's exact value, 0.00000026 more than pi, and 99.9999915...% similar to pi. It turns out that there is no better approximation for pi as a ratio of the numerical values of any two words, or of any other whole numbers less than ten-thousand. This calculation is even internally consistent due to the fact that it is veiled within a pasuk containing the ratio of circumference to diameter.
Simple question, Did Chazal have the tools to describe pi (i.e. decimal points)?
Yossi
Anyway I'm looking at the Gemara, it seems to me that Chazal IS questioning the circumference.
And yes, they do suggest the inner/outer diameters.
It seems to me that the p'shat in the Gemara is that for all practical purposes (i.e. t'chum shabbat), we CAN use 3 to calculate the circumference, and we learn this law from the Book Kings.
Doesn't the shape of the bowl, and the place where you measure the bowl make a difference here? We aren't talking about theoretical circles and theoretical shapes.
>Simple question, Did Chazal have the tools to describe pi (i.e. decimal points)?
No, but neither did the Greeks or anyone else. They described pi as a ratio
Oh, that wikipedia...
[edit] Biblical value
It is often claimed that the Bible states that π is exactly 3, based on a passage in 1 Kings 7:23 (ca. 971–852 BCE) and 2 Chronicles 4:2 giving measurements for the round basin located in front of the Temple in Jerusalem as having a diameter of 10 cubits and a circumference of 30 cubits. Rabbi Nehemiah explained this in his Mishnat ha-Middot (the earliest known Hebrew text on geometry, ca. 150 CE) by saying that the diameter was measured from the outside rim while the circumference was measured along the inner rim. This interpretation implies a brim 0.22535 cubit (or, assuming an 18-inch “cubit”, some 4 inches) thick, or roughly one “handbreadth” (cf. 1 Kings 7:24 and 2 Chronicles 4:3).
The interpretation of the biblical passage is still disputed[13][14], however, and other explanations have been offered, including that the measurements are given in round numbers (as the Hebrews tended to round off measurements to whole numbers[citation needed]), or that cubits were not exact units, or that the basin may not have been exactly circular, or that the brim was wider than the bowl itself. Many reconstructions of the basin show a wider brim (or flared lip) extending outward from the bowl itself by several inches.[15]
The issue is discussed in the Talmud and in Rabbinic literature.[16] Among the many explanations and comments are these:
In 1 Kings 7:23 the word translated 'measuring line' appears in the Hebrew text spelled QWH, but elsewhere the word is most usually spelled QW. The ratio of the numerical values of these Hebrew spellings is 111⁄106. If the putative value of 3 is multiplied by this ratio, one obtains 333⁄106 = 3.141509433... – within 1/10,000th of the true value of π, a convergent for π which is more accurate than 22⁄7, although not nearly as good as the next one 355⁄113.
Maimonides states (ca. 1168 CE) that π can only be known approximately, so the value 3 was given as accurate enough for religious purposes. This is taken by some[17] as the earliest assertion that π is irrational.
Surely they can divide a string into halves, quarters, eighths, sixteenths, etc!
There's an additional problem with the kri/ksiv argument. Taken to its logical conclusion it would imply that the author thought that that slightly more accurate number was the exact value of Pi which is of course wrong. So if one really believed that argument, it would potentially come out to an argument against the divine inspiration of the text.
In contrast, the verse using a simple approximation (or using a rope that stretched slightly or both) are much easier.
Incidentally, the claim about the Greek phrase is particularly interesting- the decimal system is very much a modern innovation which is also arbitrary. To a culture using any other base, that phrase would have zero significance.
(Somewhat related, I was under the impression that there's a Gemarrah that uses the 22/7 value of Pi, but I don't know where it is and you don't mention it here. Am I misinformed?)
Yossi, to answer your question, no Chazal did not have decimal points. That wasn't invented to around 500 years ago. But rational numbers have been well-understood for a very long time.
Explicit use of continued fractions (http://en.wikipedia.org/wiki/Continued_fraction ) occurred around 500 c.e. or so although when it came to the West isn't clear. But you don't need continued fractions to do rational approximation. It just helps it make much more sense (the theory of continued fractions explains why 22/7 is a surprisingly good approximation for such a small denominator. 22/7 = 3.14285... as opposed to Pi= 3.14159... so the difference is about 1/790 which is much smaller than 1/7 (which is about where you would very naively expect the approximation to be).
Moreover, mathematicians in Greece and Babylon were working with approximations to Pi very early on without any use of decimals. And the same goes for the square root of 2 (which is implicitly what is being used in the issue of the diagonal of a square mentioned by Rabbi Slifkin in the original post). There's demonstrably no issue of a lack of techniques at the time since other people were able to work with them fine.
"There are doubtless plenty of two and three digit numbers which can be manipulated to give a similar value...But I don't think that I can conclusively show that it's not a coincidence."
I was inclined to believe the same thing- that the 111:106 was far from unique, so I did some programming to show that "There are doubtless plenty of two and three digit numbers..."
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. (The latter number was alluded to in Simcha's comment.) 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.
Coincidence? Perhaps, but it far less coincidental than the skeptic would have it.
I wrote: "Are there any Jewish commentators who try to resolve the dimensions of King Shlomo's pool in the same way some Christians do, by referring to the outer and inner diameters?"
Shimon writes: "Rabbi Nehemiah explained this in his Mishnat ha-Middot (the earliest known Hebrew text on geometry, ca. 150 CE) by saying that the diameter was measured from the outside rim while the circumference was measured along the inner rim."
Thanks!!!
'Tis a favourite hobby of mine
a new value to pi to assign
I'd fix it at three
For it's simpler - you see
then three point one four one five nine
Could you please post the source for RADAK regarding kri/ktiv. thank you
lawrence kaplan
Moshe Steinberg: See Guide 3:14, the end of the chapter.
The Ra'avad in his hakdomoh to sefer yetzirah believes that pi is 3.2.
"The Ra'avad in his hakdomoh to sefer yetzirah believes that pi is 3.2."
Does he 'believe' it to be that value exactly? I find it hard to believe that someone who would know what pi is, would not know that 3.2 is just an approximation. More likely, he gave the value to the nearest fifth. Just as we measure many things to the closest quarter inch....
If this discussion of a "sea", or large bowl, had been referring to what is called an "ideal" bowl (a mathematical object, not existing in a physical sense, and having no thickness that could be felt or handled), then the text would indeed be claiming that the value of pi is three. But the text is referring to a real-world physical object, having the thick sidewalls necessary to support its own weight.
Now that you know how to measure cubits, can you see that it would be rather difficult to measure the curved surface of a bowl in cubits? Instead, a straightened rope would be used to measure the length. The rope would then have been moved to outline a circle with the desired circumference. Also, Hiram would not have just tossed some brass in the furnace and waited to see what came out. He would have designed the piece and would have given his workmen instructions.
To make a "sea" like this would likely have required a mold. The outer mold would have one dimension, and the inner mold would have another. Hiram would have given his workmen instructions regarding these measurements.
Now that you have some background information, let's look at the numbers:
The Calculations
Here again is the quote being referred to:
"And he [Hiram] made a molten sea, ten cubits from the one rim to the other it was round all about, and...a line of thirty cubits did compass it round about....And it was an hand breadth thick...." — First Kings, chapter 7, verses 23 and 26
The bowl is said to have had a circumference of thirty cubits and a diameter of ten cubits. The diameter is said to be "from one rim to the other", so this would be the outer diameter; that is, the diameter of the outer mold used to make the bowl.
The circumference is not specified as being the inner or outer circumference, but since using the outer circumference would give us the "ideal" bowl (with no width or thickness), let's instead use the inner circumference, which also, reasonably, would have been the circumference of the mold used to form the inside of the bowl. That is, we will use the two measurements which were necessary for the casting of the piece.
Using eighteen inches for one cubit, we have the following:
outer diameter: 10 cubits, or 180 inches
outer radius: 5 cubits, or 90 inches
inner circumference: 30 cubits, or 540 inches
To find the "Jewish" or "Bible" value for pi, we need to have the inner radius. Once we have that value, we can plug it into the formula for the circumference and compare with the given circumference value of 540 inches.
Since the thickness of the bowl is given as one handbreadth, then the inner radius must be:
90 – 4 = 86 inches
Let's do the calculations:
inner radius: 86 inches
inner circumference: 540 inches
The circumference formula is C = 2(pi)r, which gives us:
540 = 2(pi)(86)
540 = 172(pi)
Solving, we get pi = 540/172 = 135/43 = 3.1395348837..., or about 3.14.
Um... Isn't "3.14" the approximation we all use for pi? Hmm.... I guess the Torah was fairly accurate after all.
Based on an article by Stapel, Elizabeth, "The 'Jewish' or 'Bible' Value of 'pi'"
See David Garber and Boaz Tzaban in HISTORIA MATHEMATICA 25 (1998), 75–84
ARTICLE NO. HM972185 for three three explanations to the accuracy of the biblical pi.
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