Daat Emet - PI

The value of PI, the ratio of the circumference of a circle to its diameter, has been determined to millions of decimal places and growing. Just plug it into the formula and determine the next place. There is no observation. It is all mathematical.

The ancients saw PI as 3. Even by Ptolemaic times, it was known to be approximately 1/7 more. Even more precisely, a circle is the superset of all points equidistant from a given point, called the center. Any observer with a stick and piece of string can see that Pi is a bit more than 3. However, the mishna in Eruvin 13b states that we can assume that it is 3. The Rambam says that it is imprecise and approximate. DaatEmet characterizes this as a "trick", bringing the gemorra and a famous pasuk. His essay is here. The argument centers on the Yam HaMutzak, the Gemorra's Pircha, and the Tosfos.

As you can read yourself, the essay asserts that the measurement of the circular molten sea was exact, and that the Tosfos relates it to the case of the round scrolls fitting in the ark, and asks on why the Gemorra uses 3 when the mathematicians are more precise. And according to Tosfos, it is a good question. However, for our purposes, it is fairly weak, given one obvious fact. The entire concept of PI only applies to perfect geometric circles. A rough circle like the Yam Hamutzak might have had a precisely 3 to one ratio. Certainly the scrolls in the ark are not perfect circles since they are flexible.

Going back to the principle that the halacha must be discernable, not to the engineering professional, but to the common person who is erecting an Eruv, it makes no sense to require him to make 3 1/7x measurements. How do you do that with no tools or particular skills assumed? Answer: round it off to something easy, that is within the parameter, like 3 to 1. For rough circles, it is close enough.

Conveniently, DE leaves out the next part of the mishna, that all circular things in halacha are considered squares for the purpose of measurement. If an Eruv is round, we square it off and one may carry in the corners of the square, even though this adds area to the Eruv. In hilchos Shabbas 12:17-18, the Rambam explains that a person may carry 4 amos. This would form a perfect geometric circle by definition. However, he is patur within the diagonal of a square of 4 amos, which 4 times the square root of 2. So the area of a circle is never an issue as one can use an approximation and still be within the schar hapinot.

The issue would arise l'kulah (like the succah case that he brings) e.g. when one needs a 4 x 4 amos square and has a circumference to measure. Halachically, if the circumference were 12 amos, the bayis, succah or whatever would be 4 x 4, but in reality, it would be a little less. The truth is though, this is also approximate by nature. Eruvin are not perfect circles, nor are ceiling beams and the halacha gives us a reasonable way to approximate. Since the method is halachic assumption, we can presume that the halacha accomplishes what it wishes to accomplish. IOW, we are not required to fulfill the halachic measurement precisely, but to be under the reasonable, halachically accepted presumption that we did so. Many, many places in halacha presuppose that we do not have the means to be exact and allow us, require us to estimate.

So, Tosfos aside, the Gemorra pircha "still we must subtract" is a moot point with regard to exact measurement. So again, the fundamentalists are confounded, but halacha is not particularly in question here.

Note: DE's point about Kav and Kavah is well taken. How do we know how to apply the gematrias to make this work. To find some particular combination that gives us an answer that we like is no big trick. How do we know to divide them? How do we know to add three? And if the purpose was to show us the exact value of Pi, instead of an approximation, why wasn't it even closer?