The Numbers Game

Because this article explains some basics of how Torah is taught by Chazal, I think it is appropriate to the week of Shavuos.

Question #1: Pie r squared

Yanki is supposed to be watching his weight and therefore needs to figure out how many calories are in the pie he beholds. To figure out how big the pie is, he measures the diameter of the pie, and divides it in half to get the length of its radius. He then multiplies the length of the radius by itself to get “r squared,” and multiplies the result by three so that he knows the area of the pie’s surface. Is there anything wrong with his calculation?

Question #2: Puzzled by the pasuk

“How can the pesukim tell us that the relationship between the circumference of a circle to its diameter is three-to-one, when simply taking a string and measuring around it demonstrates that it is noticeably longer?”

Question #3: Performing mitzvos accurately

“How accurate a calculation must I make when determining the size of an item to be used for a mitzvah?”

Introduction

In numerous places, both Tanach and Chazal approximate certain mathematical values, such as evaluating the ratio of the circumference of a circle to its diameter as three to one. The problem is that we can demonstrate mathematically that the ratio is greater than three and is almost 3 1/7. This leads to the following questions:

(1) Why would Chazal calculate using inaccurate approximations?

(2) When making halachic calculations, may we rely on these estimates, or do we need to be mathematically more accurate?

(3) A corollary question is: when providing an estimate, one must allow for a margin of error. Does halachah require a margin of error, and, if so, how much?

The Slide Rule versus the Calculator

Let me begin our discussion with a modern analogy, if something I remember can still be considered “modern.” When I first studied sophisticated mathematical estimates, I learned to use a slide rule, which today is as valuable to an engineer as an abacus. Relative to the calculator, a slide rule does not provide accurate measurements, and someone using a slide rule must allow a fairly significant margin of error in one direction or the other, depending on the situation.

Today, complex computations are made with calculators, which provide far more accurate results that can be rounded off, as necessary, to the nearest tenth, millionth, quadrillionth or smaller. Of course, using a calculator still requires one to round upward or downward, but because it is much more precise, the margin of error is greatly reduced.

How Irrational Are You?

Numerous halachic questions require mathematical calculations, involving what we call “irrational numbers.” An irrational number means one that cannot be expressed in fractional notation. Another way of explaining an irrational number is that its value can never be calculated totally accurately, but can only be estimated.

The two most common examples of irrational numbers that show up in Chazal are:

(1) Pi

The ratio of the circumference of a circle to its diameter, which we are used to calling by the Greek letter ∏ (pronounced like the word “pie,” and spelled in English “pi”). Since the 19th century, the letter pi has been used to represent this number, because the Greek word for peripheryis peripherion, which begins with the letter ∏. Hundreds of years earlier, the Rambam (Commentary to the Mishnah, Eruvin 1:5) noted that the ratio of the circumference of a circle to its diameter is an irrational number that can only be approximated, and that the scientists of his era used an estimate of 3 and 1/7, which is actually slightly greater than the value of ∏. The Rambam explains that since there is no accurate ratio, Chazal used a round number, three, for this calculation.

(2) The diagonal of a square

The length of a diagonal of a square, which is equal to the side of the square multiplied by the square root of two (√2). Chazal calculated the length of a diagonal of a square to be 1 and 2/5 times its side, which is slightly smaller than the value of √2. (Another way of expressing this idea is that the ratio between the diagonal and the side is 7:5.) The fact that Chazal’s figuring is somewhat smaller than the mathematical reality is already proved by Tosafos (Sukkah 8a s.v. kol).

Since both pi and the square root of two are irrational numbers, they can only be estimated, but can never be calculated with absolute accuracy.

Based on the above-quoted statement of the Rambam, we can already address one of our earlier questions: “Why would Chazal calculate using inaccurate approximations?” The answer is that any computation of the correlation of the circumference of a circle to its diameter will be an estimate. The only question is how accurate must this estimate be for the purpose at hand.

Chazal or Tanach?

Although the Rambam attributes the rounding of pi to Chazal, in actuality, there are sources in Tanach that calculate the ratio of the circumference of a circle to its diameter as three-to-one. Both in Melachim (I 7:23) and again in Divrei Hayamim (II 4:2), Tanach teaches that the Yam shel Shlomoh, the large, round pool or mikveh that was built in the first Beis Hamikdash, was thirty amos in circumference and ten amos in diameter, which provides a ratio of circumference to diameter of three-to-one. Thus, we can ask a question of the Rambam: Why does he attribute this ratio to Chazal, rather than the source for Chazal’s calculation, the pesukim?

In fact, the early commentaries to these verses already ask how the verse can make a calculation that we know is not accurate. The Ralbag suggests two options: either that the numbers used are intended to be a very broad estimate, or, alternatively, that the diameter is measured from the external dimensions of the mikveh, whereas the circumference is measured from its inside, which makes the estimate closer to mathematical reality.

According to the second approach of the Ralbag, no Biblical source uses an estimate of three-to-one as a substitute for pi. This will explain why the Rambam attributed the estimation of pi as three to Chazal, rather than to the Tanach. The Rambam was fully aware that one could interpret the verses according to the second approach of the Ralbag, in which case, there is no proof from the verse. He, therefore, attributed this estimate to Chazal.

Gemara Eruvin

The Ralbag’s approach reflects an earlier passage of Gemara. The Mishnah in Eruvin (13b) states that if the circumference of a pole is three tefachim, its diameter is one tefach, which means that the Mishnah assumes a ratio of three-to-one. The Gemara questions how the Mishnah knows that the ratio is three-to-one, and then draws proof from the above-quoted verse that the Yam shel Shlomoh was thirty amos around and ten amos across. The Gemara then debates whether the calculations of the Yam shel Shlomoh indeed result in a ratio of three-to-one, because one must also include the thickness of the poolitself, which offsets the computation. The Gemara eventually concludes that the verse was calculating from the inside of the pool, not its outside, and therefore the thickness of the pool’s containing wall is not included in the calculation (Eruvin 14a).

Nevertheless, this Gemara’s discussion leaves the mathematician dissatisfied, a question already noted by Tosafos. If the internal diameter of the Yam shel Shlomoh was ten amos, its circumference must have been greater than thirty amos, and if its circumference was thirty amos, then its internal diameter must have been less than ten amos.

A Different Question

The Rosh, in his responsa, is bothered by a different question, based on Talmudic logic rather than on mathematical calculation. He finds the Gemara’s question requesting proof for the ratio between a circle’s circumference and its diameter to be odd. The ratio between a circle’s circumference and its diameter is a value that one should calculate. By its nature, this is not a question that requires a Biblical proof or source.

In the literature that we have received from the Rosh, he asks this question in two different places. In his responsa (Shu’t HaRosh 2:19), we find a letter that he wrote to the Rashba, in which he asked the Rashba a series of questions that the Rosh notes bother him tremendously, and to whom he has no one else to turn for an answer. One of the questions the Rosh asks is: “Why does the Gemara ask for a Biblical source for a mathematical calculation?”

It is curious to note that a later commentary mentions that, in all the considerable literature that we have received from the Rashba, we have no recorded answer of the Rashba to this question of the Rosh (Cheishek Shlomoh to Eruvin 14a).

Another Comment of the Rosh

The Tosafos HaRosh commentary to Eruvin, which was published for the first time relatively recently, is the second place where the Rosh asks why the Gemara wanted a Biblical source for a mathematical calculation.  There, the Rosh provides an answer to this question:  Since the calculation of three-to-one is not accurate, the Gemara wanted a biblical source as proof that we are permitted to rely on this estimate.

(The Cheishek Shlomoh, whom I quoted above, provides the same answer to this question as does the Rosh in his Tosafos. The Cheishek Shlomoh never saw the Tosafos HaRosh, which had not yet been printed in his day.)

Curiosity about the Tosafos HaRosh

There is an interesting historical point that can presumably be derived from the fact that, in the Tosafos HaRosh, the Rosh answers the question that he raised and accredits this answer to himself. This should be able to prove which work the Rosh had written earlier, and also whether he ever received an answer to his question from the Rashba. This analysis is based on the following question: Why did the Rosh cite an answer in his Tosafos¸but not in his responsum, which was addressed as a question to the Rashba. There are three obvious possibilities:

(1) Although the Rosh wrote this answer in his Tosafos, he was dissatisfied with it, and therefore wrote a question to the Rashba. I would reject this answer because, if it is true, then, in his correspondence to the Rashba, the Rosh would have mentioned this answer and his reason for rejecting it.

(2) The Rosh indeed received an answer, either this one or a different answer, from the Rashba. I reject this approach also, because, were it true, the Rosh would have quoted the Rashba’s answer in his Tosafos and, if need be, discussed it.

(3) Therefore, I conclude that the Rosh, indeed, never received an answer to the question he asked of the Rashba and subsequently reached his own conclusion as to how to answer the question, which he then recorded in the Tosafos HaRosh. This would lead us to conclude that the Tosafos HaRosh was written later in his life than his responsa, or, at least, this responsum.

Mathematical Accuracy

At this point, we can address one of earlier questions.When making halachic calculations, may we rely on these estimates, or do we need to be mathematically more accurate? We might be able to prove this point by noting something in the Mishnah in Eruvin quoted above. The Mishnah there ruled that, under certain circumstances, an area that is fully enclosed on three of its sides and has a beam a tefach wide above the fourth side is considered halachically fully enclosed, and one may carry inside it. The Mishnah then proceeds to explain that if the beam is round and has a circumference of three tefachim, one may carry inside the area because, based on the calculation that the relationship of its circumference to its diameter is three-to-one, the beam is considered to be a tefach-wide. However, as the Rambam notes, a beam that has a circumference of three tefachim is actually less than a tefach in diameter, and therefore one should not be permitted to carry in this area!

The Aruch HaShulchan (Orach Chayim 363:22; Yoreh Deah 30:13) notes this problem and concludes that one may carry in this area. He contends that this is exactly what the Gemara was asking when it requested Scriptural proof for a mathematical calculation. “Upon what halachic basis may we be lenient in using this estimate of three-to-one, when this will permit carrying in an area in which the beam is less than a tefach wide? The answer is that this is a halachah that we derive from the verse.”

To clarify this concept, the Chazon Ish notes that the purpose of mitzvos is to draw us nearer to Hashem, to accept His reign, and to be meticulously careful in observing His laws. However, none of this is conflicted when the Torah teaches that we may use certain calculations, even if they are not completely mathematically accurate. In this instance, relying on these estimates is exactly what the Torah requires (Chazon Ish, Orach Chayim 138:4). As expressed by a different author, the Gemara (Eruvin 4a; Sukkah 5b) teaches that the measurements, the shiurim, required to fulfill mitzvos are all halachah lemoshe misinai, laws that Moshe Rabbeinu received as a mesorah on Har Sinai. Similarly, these estimates of irrational numbers mentioned above are all halachah lemoshe misinai that one may rely upon to fulfill mitzvos, whether or not they are mathematically accurate. The same Torah takes these calculations into consideration when instructing us which dimensions are required in order to fulfill specific mitzvos (Shu’t Tashbeitz 1:165).

In the context of a different halachah in the laws of Eruvin, the Mishnah Berurah makes a similar statement, contending that we can rely on Chazal’s estimates, even when the result is lenient. However, the Mishnah Berurah there vacillates a bit in his conclusion, ruling that one can certainly rely on this when the issue is a rabbinic concern (Shaar Hatziyun 372:18). In a responsum, Rav Moshe Feinstein questions why the Mishnah Berurah limits relying on this approach, and Rav Moshe rules unequivocally that one may rely on these estimates even when it involves leniency in de’oraysa laws (Shu’t Igros Moshe, Yoreh Deah Volume3 #120:5).

How Straight Are My Tefillin?

Personally, I find the context of Rav Moshe’s teshuvah very interesting. There is a halacha lemoshe misinai that requires that the boxes of the tefillin, the batim, must be perfectly square. Rav Moshe was asked whether there is a halachic preference to use scientific measuring equipment to determine that one’s tefillin are perfectly square. Rav Moshe rules that there is neither a reason nor a hiddur in measuring the tefillin squareness this accurately. Since Chazal have used the calculation of 1.4 or a ratio of 7:5, which we know is an estimate, to determine the correct diagonal of a square, there is no requirement to make one’s tefillin squarer than this, and it is perfectly fine simply to measure the length of each of the sides of one’s tefillin and its two diagonals to ascertain that the ratio between the diagonal and the side is 7:5.

In the above-cited responsum, Rav Moshe notes that he had heard that the Brisker Rav, Rav Yitzchak Zeev Soloveichek, had ruled that it was preferable to check one’s tefillin in the most scientific method available. Rav Moshe writes that he finds this suggestion very strange and disputes its being halachically correct (Shu’t Igros Moshe, Yoreh Deah Volume3 #120:5).

When making halachic calculations, may we rely on these estimates, or do we need to be mathematically more accurate?” The answer is that, indeed, the purpose of Chazal’s making these estimates was that observing halachah does not require that these calculations be mathematically precise, provided they meet the criteria that the halachah established.

An Alternate Approach

Although the majority of late authorities conclude that the calculations of Chazal are, indeed, part of the halachos of shiurim, this is not a universally-held position. The Tashbeitz, a rishon, wrote a lengthy responsum on the topic, in which he presents two ways to explain why Chazal used estimates that are not precisely accurate. His first approach reaches the same conclusion as we have already found in the later poskim, that these measurements are included within the halachos of shiurim that are part of the halachah lemoshe misinai.

The second approach of the Tashbeitz, however, differs with the above-mentioned halachic conclusion. In his second approach, he contends that all the above estimates were meant for pedagogic, but not halachic purposes. The rounding of pi to three and the diagonal of a square to 1.4 were provided to make the material easily comprehensible to all students, since every individual is required to know the entire Torah. Thus, when Chazal used these estimates in calculating the laws, their intent was to enable the average student to comprehend the halachic material, not to provide the most accurate interpretation. When an actual halachic calculation is made, it must be totally accurate. Any halachic authority involved would realize that he must use a highly accurate mathematical computation and then round either upward or downward as necessary for the specific application. (A similar position is held by Chiddushim Uviurim, Ohalos 5:6.)

Conclusion

Certainly, the majority of late halachic opinions conclude that the estimates of Chazal are meant to be halachically definitive and not simply pedagogic in nature. However, I leave it to the individual reader to ask his or her posek what to do when a practical question presents itself.