Eruvin 77-78 Chazal and Science III- Ladders and more square roots

We have learnt that two courtyards sharing a common wall  may make one eruv together, only if there is a suitable window or opening in the wall- otherwise, the wall serves as a closed מחיצה (partition) between the two courtyards and the eruv does not work.

The minimum size of a halachik opening is 4 tefachim by 4 tefachim, and at least some of it needs to be below 10 tefachim in height.

We also discussed the complex issue of how to ensure that a circular window fits the minimum size and concluded that it need to be large enough for a square of 4 by 4 tefachim to be inscribed inside it.

Although Rabbi Yochanan seemed to require a circle with a circumference of 24 tefachim, Assuming π equaling 3 and Ö2 equaling 1.4, the Gemara concludes that one with a circumference of 16.8 tefachim is sufficient, and that Rabbi Yochanan was relying on the judges (or Rabbis) of Caesaria’s different formula in his ruling, who believed that in order to circumscribe a square, the perimeter of a circle must be twice the perimeter of the square.

 We saw different views as to how to understand what seems like such a large error on the part of the judges of Caesaria and Rabbi Yochanan, as well as how to understand what appears to be a lack of mathematical precision on behalf of both the Rishonim and Chazal.

Though the issue of why Chazal did not use more accurate measures for rational numbers like  π and Ö2 is also essential to our discussion, even more difficult was the far greater “error” (at least according to the way Rashi explained it) of Rabbi Yochanan and the judges he relied on that the hypotenuse of an isosceles right-angled triangle is equal to the sum of the other two sides, whereas according to the ancient theorem of Pythagoras, it equals Ö2 times the width of one of the sides, less than 75% of what they claimed.

We noted that even if Rabbi Yochanan and/or the judges of Caesaria and/or Rashi were unaware of this theorem, it is so easy to see that a factor of 2 is completely off by the simplest of measuring, and that ascribing such an error to any of them is extremely problematic, even without dealing with the question of whether and what type of “ruach hakodesh” they might have had.

I suggested that it might make more sense to explain that everyone understood that the length of the hypotenuse is less than the sum of both sides, but because measuring the precise length was difficult and involved square roots which were often irrational, they preferred to use the highest possible value of the hypotenuse of a right-angled triangle, which approaches (but never reaches) the sum of its’ width and height as the ratio between them decreases. (see chart below plotting hypotenuse on the y access and width on the x axis) assuming a constant height of 10)

We also mentioned that the Gra has another approach, which “seems” not to fit the text so well, but interprets the view of the judges of Caesaria and Rabbi Yochanan in a way that fits the maths perfectly- He claims that when the judges said that the square inside the circle is half, it does not refer to the perimeter of the square but to its area, and is not comparing it to the area of the circle it is inscribed in, but the square which superscribes the circle it is inscribed in (something Tosfos suggested was the intention of the judges of Casearia but misunderstood by Rabbi Yochanan and the Gemara!) and that Rabbi Yochanan’s requirement for 24 tefachim was also referring to the perimeter of the outer square.

On our daf, we have yet another instance of this issue, and seeing how the Rishonim handle it here, might shed some light on the earlier discussion, as well as its parallel discussion in Sukkah 8a (regarding the minimum circumference of a circular sukkah.)

On Daf 77, we have been discussing other options for linking the two courtyards, and one of them is by using a ladder that allows easy access.

On daf 78, Rav Yehuda quotes Shmuel as ruling that if a wall is 10 tefachim in height, it requires a ladder of 14 tefachim to permit it (serve as the equivalent of an opening.)

Rashi explains that seeing as the ladder cannot be placed vertically and still provide easy access, its base  needs to be at least 4 tefachim away from the wall, and that given its diagonal position (forming a hypotenuse,) it needs to be at least 14 tefachim (the sum of the two sides of the right-angled triangle formed) in order to reach the top of the wall.

Once again, we see that Rashi seems to assume that the hypotenuse of a right-angled triangle equals the sum of the other two sides, which we clearly know to be incorrect.

Once again, it is hard to believe that Rashi could make such a large error. It is also perplexing that what the earlier sugya and the sugya in Beitza seemed to view as an error of Rabbi Yochanan based on the judges of Caesaria, Rashi is now attributing to Shmuel, without even saying that it was an error.

This might strengthen what we suggested that at least some of Chazal preferred to use one general rule for all right-angled triangles, and err on the side of caution using the maximum the “hypotenuse” could ever be, assuming a collapsed triangle consisting of two lines at 180 degrees to each other.

Instead of having to constantly work out square roots that vary according to the ratio between the lengths of the other two sides, the simple measure of over-estimating the hypotenuse as the sum of the other two sides is then used across the board according to this view.

It would still be difficult why Rav Yehuda would be quoting Shmuel as expressing this stringent cautious view that was seemingly rejected in the earlier sugya and in Sukkah.

Further, the Gemara itself in Sukkah suggests that perhaps Rabbi Yochanan was simply לא דק  (not being precise) and rejects that suggestion based on the size of the imprecision being way too much to contemplate.

Tosfos on our daf, however, claims that Rashi was indeed “לא דק” (not being precise,) given that the ladder would need to be placed at a distance of 10 tefachim from the wall (forming an isosceles right-angled triangle) in order for it to have to be 14 tefachim in length, given the ratio of 1.4:1 of the hypotenuse to the other side ( Tosfos on Eruvin 57a  already offered his own proof that it is slightly more than 1.4, showing  independent mathematical ability but perhaps at the same time  unfamiliarity with the theorem itself.)

Though Tosfos’ understanding of Rashi might backup my suggestion that the inaccuracy in our case  was intentional and not due to ignorance, it remains difficult why Rashi would assume that Shmuel was being imprecise to a level that the sugya in Sukkah dismissed as implausible (using Pythagoras, the required length of the wall would be Ö116 or about 10.77, far closer to the height of the wall itself!)

Either way, we seem to see that according to Rashi, using the sum of the two sides as an “estimation” of the length of the hypotenuse was not limited to Rabbi Yochanan and/or the judges of Caesaria regarding the hypotenuse of an isosceles  right-angled triangle, but extends also to Shmuel’s treatment of the hypotenuse of any right-angled triangle!

The journey continues…

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha

Eruvin 76 Chazal and Science- PI, circumscribed circles, and square roots

In an earlier post (Eruvin 14,) we saw how the Mishna teaches that the ratio between the circumference and the diameter of a circle is 3.

The Gemara derived this from the ים  של שלמה  (circular water-feature) which was 10 amos wide and 30 amos in circumference.

We raised the obvious issue that the actual value of this ratio is π, an irrational number equal to slightly more than 3.14, and we saw two basic approaches amongst the Rishonim:

  1. The Tosfos brought evidence that Chazal were being precise in their measurements, pointed out that the mathematical experts hold that it is not precise, and leave it as a difficulty.
  2. The Rambam and Tosfos haRosh both understand that this is an approximation.

On our daf, we encounter this ratio once again.

Our Mishna discusses how large a “window” in the boundary wall between two neighbors’ properties needs to be for them to be able to make one eruv between the two of them.

It rules that the window needs to be at least four by four tefachim to qualify as a פתח

(opening) and that at least part of it needs to be within 10 tefachim of the ground.

Rabbi Yochanan brings up the case of a round window and how large it needs to be.

We should recall that in the context of daf 13b, the Mishna ruled that a round pole used for the beam of a מבוי does not have to be large enough to contain a tefach-wide square beam within it- it merely needs to be a tefach wide at the diameter, or 3 tefachim in circumference. (רואים כאלו היא מרובעת)

The Gemara initially seems to have thought  that the same should be the case with our window, and that so long as it is 4 tefachim at the diameter, or 12 tefachim in circumference, it counts as if it was a square window of 4 by 4 tefachim- after all, this could just be a symbolic opening in any case.

Yet There is a strong argument to be made that this case should be different seeing as the opening might actually need  to function as a פתח, and one could never squeeze through a circular hole that is only 4 tefachim wide at the diameter.

Rabbi Yochanan, citing the famous 3 to 1 ratio, rules that it needs to be 24 tefachim in circumference, and that slightly more than 2 of them need to be below the 10 tefachim line of the wall, so that if the circle is squared, part of the square will be under the line.

This rather cryptic statement of Rabbi Yochanan has pages and pages of commentary trying to explain.

After attempting to make some sense of it myself with the little high-school math I remember and some diagram, I was immediately overwhelmed by the complexity and length of the discussion.

I knew it would not be one day’s work to even scratch the surface, but decided to take my time and try get at least some idea of what is going on, and what we can take from it into the general topic of Torah and Science that we keep coming back to.

First – the “simple” flow of the Gemara,  (if there is such a thing:)

  1. The Gemara notes the usual 3 to 1 rule, and notes that in order to get a diameter of 4 tefachim, the circle should only need to be 12 tefachim in circumference.
  • The  Gemara answers that this rule replies to a circle, but for a square, more is needed.

We need to understand what the Gemara means to say, as it is clear that we are dealing with a circle and not a square.

One possibility is that although we are dealing with a circle, Rabbi Yochanan requires a circle with a circumference equal to a square of 4 by 4, in order for it to be considered an equivalent valid opening to the square.

  • The Gemara answers that a square that circumscribes a circle is only a quarter more than the circle itself , so 16 tefachim should be sufficient. (It does not say what attribute of the square is a quarter more than the circle, but the Gemara seems to assume that this is the circumference, and to be referring to a quarter of the resulting square.)
  • The Gemara answers that this is the case with a square that circumscribes a circle (is inscribed by a circle). The internal circle is 3 (PI) times the diameter in circumference, namely 12 tefachim,  whereas the square is 4 times its width, or 16 tefachim.

However, what we need here is a square of 4 times for tefachim to be able to fit inside the circle, which means the circumference of the circle needs to be even more to cover the parts of the circle outside the square.

  • The Gemara uses another apparent approximation, the length of the hypotenuse of a right-angled triangle formed by cutting a square in two by its diagonal. Although this is the square route of two (an irrational number) times by the width, the Gemara treats it as 1.4 (1 and two fifths.) This would also be the circumference of the required circle.

Using this, it works out that circumference of our circle need only be 16.8 tefachim, in order to be able to have square of 4 by 4 tefachim inscribed in it. (using precise modern measurements, this would be 4Ö2*π, rounded to 17.77)

So why does Rabbi Yochanan require such a large circle!

  • The Gemara replies that Rabbi Yochanan was following the judges of Caesarea (some versions say “The Rabbis of Caesarea)  who said that “a circle inside a square is a quarter, a square inside a circle is a half.”

This cryptic statement is  itself subject to interpretation of course.

There are multiple ways to learn the flow of the Gemara, starting from the requirement for “2 of them and a bit” to be below the 10 tefachim line, and ending with this view of the judges of Caesarea, and it would take pages and pages to go through.

Some essential reading in in the Rishonim include  Rashi, Tosfos, Rashba, Ritva, and the Meiri who has a particularly extensive treatment of the subject.

As we have already addressed the issue of PI being approximated by 3 by Chazal, we shall not focus on that right now, although it would be in place to analysis whether the flow of the sugya here indicates that Chazal were aware of this approximation and using it intentionally or not.

What we see here in addition to this is another “approximation” of Chazal (also encountered elsewhere), namely the square root of 2, but even more significantly, that it is used in combination with the approximation of PI, creating quite a large combined “rounding error.”- after all, there is a significant different between a circumference of 16.8 and one of 17.6 (see picture and formula,) and even according to the view of Rambam and Rosh that approximation is sometimes acceptable, relying on a double approximation seems to be a significantly greater novelty.

Another fascinating issue here is the question of the accuracy of the mathematical knowledge of Chazal and the Rishonim.

The simple explanation of the sugya, as understood by Rashi and Tosfos (see also the parallel sugya on Sukkah 8a,) seems to be  that Rabbi Yochanan relied on a mistaken mathematical formula used by the judges (or Rabbis) of Caesarea which according to Rashi seems to calculate the diagonal as twice the width, rather than 1.4 times the width of the Gemara or the root of 2 used by mathematicians.

Tosfos is bothered by how the judges of Caesarea  could have erred in something that is so obviously easy to ascertain.  Interestingly enough, he seems less bothered by the fact that Rabbi Yochanan followed in their error. He even goes so far to suggest that it was not the judges who erred, but Rabbi Yochanan who erred in his interpretation of what they said!

In contrast, the Rashba seems more bothered by the fact that Rabbi Yochanan could have made such an era, and the Gra notes that we should not “chas veshalom” say that they made any era, choosing to interpret their ruling differently to most of the Rishonim.

In addition, Tosfos disagrees with Rashi’s interpretation of the “2 and a bit” rule stated earlier given that his claims are not mathematically correct and proposes another explanation which using the correct formula also does not seem mathematically correct (though in fairness to both Rashi and Tosfos, they are merely commenting on the meaning of Rabbi Yochanan and the judges of Caesarea, not on mathematical reality!)

The temptation to simply say that Rishonim did not understand basic mathematics should not be taken lightly.

Rashi’s ability to make complex calculations is well known, and the Tosfos were brilliant enough to provide their own proof that the length of the  hypotenuse of an isosceles right-angled triangle is slightly more than 1.4 times the length of its other sides- It is hard for us to imagine that people with such minds did not know mathematical formula that even the ancient Greeks were aware of so long ago.

Yet despite the above,  it might not be necessary to assume that all Rishonim were well-versed in all mathematical formula and knowledge that was known to man at the time.

Medieval Europe was not the most “enlightened” part of the world by any means, communications were not what they are today, and much knowledge that the ancient Greeks had access to was inherited by the Islamic world to the South and East, rather than France and Germany.

The Rishonim might have been brilliant enough to work out mathematical theory on their own had they dedicated their time to it, but they clearly had other priorities and did not go all the way.

It is also not so outlandish to posit that whereas some Chazal were very exposed to and familiar with the scientific knowledge of the Greek and Roman worlds, others were less so, and sometimes needed to be corrected by their colleagues or even later authorities.

This does not even have to contradict the view of the Ramban, discussed in earlier posts, that ascribes a form of wisdom-induced “ruach hakodesh” to great Talmidei-Chachomim.

As we mentioned before, this does not necessarily mean that they ALWAYS experienced this “ruach hakodesh” nor that this “ruach hakodesh” has anything to do with awareness of scientific facts through supernatural means- rather it seems from the examples given in the relevant sugya (Bava Basra ) that it has more to do with intellectual “siyata dishmaya” which allows them to come up with ideas only far greater people would normally have come up with, while still basing these ideas on the information available to them at the time.

However, I am extremely hesitant at over-using such explanations- while they can possibly account for a lack of precision regarding the root of 2 or the value of PI, Chazal and the Rishonim certainly knew how to learn things from simple observation, and larger errors, such as viewing the hypotenuse of an isosceles right-angled triangle as the sum of the two other sides, essentially treating root 2 as the same as 2, simply defy rational explanation.

As such, I tend to believe that at least in such cases, the errors made were more strategic than mathematical, and that for reasons of convenience, the judges of Caesaria chose to be stringent and require the maximum possible length of the third side of ANY “triangle” (ie approaching a straight line with 180 degrees between the two sides) which is the sum of the other two sides. It is important to stress that this is not necessarily Rashi’s view, but his explanation of their view, but it could also explain other issues where Rashi takes this approach (or make it more difficult!)

The Gra, of course, has his own novel approach to the sugya, and whereas it seems somewhat forced in the text and out of line with most of the Rishonim on the subject, the Gra most certainly was a complete expert in all of Torah as well as in mathematics, making a study of his approach particularly appealing.

There are indeed SO many questions raised in this sugya and the way  that the Rishonim handle them, and its awfully frustrating to have to leave the discussion for a different forum and start catching up on the daf, but such is life- maybe we shall get to revisit this discussion sooner than we think!

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halach

Eruvin 14 Science and Torah- Matters of PI

Recent years have brought some extremely “vibrant” and often downright hostile discussions about the correct approach to apparent contradictions between the Torah, Chazal and modern scientific knowledge.

This applies across the board from astronomy, medicine, geography, physics, mathematics, and through archeology, even to history.

There are those like my good friend, the famous “Zoo Rabbi,” Rabbi Nathan Slifkin, who after his books touching on the subject were banned by various Chareidi authorities, has made it a major part of his life’s work to restore the popularity of “Rationalist Judaism.”

This, loosely speaking, encompasses the approach of many of the Geonim, the Rambam, and many other great early authorities that statements made by Chazal which appear to conflict with nature and science are not to be taken literally, and that when Chazal express their views on scientific matters, they are basing themselves on the accepted science of the time, and not on neo-prophetic revelation.

In contrast, Rabbi Moshe Meiselman, a renowned Chareidi Rosh Yeshiva who also has an academic background, has written his own work “Torah, Chazal, and Science” with the primary intention of both condemning and refuting this view, as well as attempting to prove that even the oft-quote protagonists amongst the Rishonim have been misunderstood.

In all humility, in a series of shiurim of my own on Agada, and a Hebrew analysis that I am still working on, I have brought numerous sources, including the introduction to the Talmud attributed to Rabbeinu Shmuel haNagid (printed at the back of most traditional versions of Masechta brachos,) which explicitly say that Agada is not comparable to halacha in its divine source and authority, but rather consists of Chazal’s own interpretations of the pesukim.

I also brought the words of the Ran (drasha 5) in his Derashos that seem to say the exact opposite and define anyone who does not believe that every word of Chazal’s Agadot are sourced at Sinai as a heretic.

As complex and sometimes aggressive this debate tends to be, it is exponentially more problematic when Chazal’s apparently out of date scientific knowledge forms the basis for practical halachik rulings.

In such cases, agreeing to disagree is not even an option, as huge areas of halacha are affected and a practical decision must made. What in one view invalidates a Sukkah or an Eruv, for example, can be essential to making it valid in the other. What renders a fish permitted according to one view, might render it forbidden according to another.

The above applies both ways, but although some authorities do indeed take into account discrepancies between modern scientific knowledge and that which Chazal were presumed to have, it is virtually always when it results in a stringency and not in a leniency.

It is not my intention to take sides in this longstanding and critical debate, but rather, as is my way in general, to examine the relevant sugyas on their own merit, together with the way the Rishonim interpret them, and see what we can learn from them

In our mishna, we are told that the minimum width of the pole used to “close” the open side of a מבוי is 1 handbreadth.

If the pole is round, we are to view it as if it is square and go by the width of its diameter.

As directly measuring the diameter of a solid cylinder is tricky, the Mishna advises us to measure its circumference, something far more practical and rely on a universal ratio between the circumference of a circle and its diameter to calculate the diameter.

The ratio given by the Mishna is the number three, according to the formula:

“כל שיש בהקיפו שלשה טפחים יש בו רוחב טפח”

“any (circle) whose circumference is 3 handbreadths has a diameter of one handbreadth.

As such, it follows that so long as the circumference of the round pole is at least 3 tefachim, we can assume that the diameter meets the minimum width of 1 tefach (handbreadth).

The same principle is employed (Sukkah 7b) to measure the diameter of a circular Sukkah to ensure it meets the minimum width of 4 amos. In the same sugya, the square-root of 2 is also assumed to be 1.4.

Every student of basic mathematics is immediately faced with the fact that the Mishna’s ration of 3 to 1 appears extremely inaccurate.

The universal ration between the circumference and diameter of any circle is of course the constant pi, a little more than 3.14, which has been known for some time already to be an irrational number.

Tosfos on our daf is so bothered by this apparent contradiction that after pointing out that it seems that our Gemara understood our Mishna’s ratio of 3 to be precise, based on the continuation of the sugya and another sugya in Bava Basra, he notes that this requires further investigation, since the mathematical experts hold that 3 is certainly not the precise ratio.

One should note that Tosfos leaves this question unanswered- he does not suggest explicitly that either Chazal or the contemporary mathematicians were wrong!

In contrast, both the Rambam and Tosfos haRosh on this Mishna are adamant that the Mishna is simply giving an estimation, and each have their own approaches as to why and how this is acceptable.

Whereas this approach certainly seems more logical, we obviously need to learn the sugya and its parallel sugyas properly to see if this fits into the flow of the Gemara.

Please join me on this exciting journey:

The Gemara opens its analysis on this part of the Mishna towards the bottom of the first side of today’s daf.

It asks the simple question: מנא הני מילי – from where are these words?

The very fact that the Gemara is looking for a verse to prove a mathematical reality that should be known to all is itself indicative of something deeper at work.

The Gemara answers that we derive this from the description of the circular ים (lit “sea” but probably referring to a water feature)) that Shlomo haMelech made, which had a diameter of 10 amos and a circumference of 30.

By describing the precise measurements of this circular feature, the passuk seems to be telling us that the ration of a circle’s circumference to its diameter is 3.

Once again, the fact that a verse is brought to teach us a simple mathematical fact seems very strange.

This question is strengthened by the fact that the ancient Greeks were very familiar with the concept of PI, and although they could not measure it precisely (though they might have suspected it was an irrational number,) it seems from my research that they certainly knew that it was more than 3, and could approximate it to at least 2 decimal points as around 3.14 .

It is hardly likely that Chazal, who took their mathematics very seriously, were unaware of this common knowledge of their time.

The Gemara then asks how we account for the width of the rim itself, which needs to be added to the actual diameter before working out the ratio with the circumference.

The Gemara responds that the possuk also tells us that the rim was extremely narrow (and thus negligible in the calculations.)

Seemingly unsatisfied by the assumption that the passuk was even nominally inaccurate in its workings, the Gemara points out that however narrow the rim was, it still would widen the exterior diameter slightly and effectively change the ratio accordingly.

The Gemara concludes that the circumference of 3 tefachim mentioned in the passuk was also measured from the inside, excluding the rim.

By now, it seems blatantly obvious that Chazal seem to take this measurement extremely precisely, and Tosfos’ observations to this affect are more than understandable.

It is harder to understand the Rambam’s approach, where he claims that any fraction that cannot be accurately measured is rounded off by Chazal.

If this is true, why were Chazal so bothered by the fact that the passuk could be doing exactly the same thing?

The Tosfos haRosh goes further and interprets the flow of the Gemara entirely differently in a way that he feels backs up his claim that we are dealing with approximations.

He understands that the Gemara’s original question, “from where are these words” is not referring to the precise value of PI but rather to the very rule that it is permissible to rely on approximations.

He understands that this leniency is sourced from the very passuk that described the properties of the circle in a way that is clearly an approximation, and quite a large one at that.

He does not say how large an approximation is needed, not under which circumstances it becomes acceptable- it could be that he agrees with Rambam who limits this to an irrational number, but is also possible that he would hold the same for other improper fractions that are hard to work with.

What remains is to understand how both the Rosh and the Rambam would explain the continuation of the sugya which certainly seems to be require precision rather than approximation.

Furthermore, even if we are able to reinterpret the rest of the flow of the sugya in a way that fits with this, or to distinguish between the approximations that are permitted and the one’s the Gemara adamantly seems to reject, we are faced with a very strong difficulty from another related sugya (Bava Basra 14b)

There, the Gemara describes how in addition to the tablets, a sefer Torah was also placed in the ark that rested in the holy of holies.

Based on the view that the circumference of a Torah needs to be 6 handbreadths, the Gemara uses our ratio to show how the 2 tefach wide Torah could fit into the 2 empty tefachim that remained in the Ark after the tablets where placed therein.

The Gemara then notes that an item with a precise width of 2 cannot fit into a precise space of 2 (presumably due to friction.)

It therefore concludes that the Torah was rolled in a way that was not precisely round (the last part was folded on top of the “cylinder”), and the width therefore was less than a third of its circumference.

It seems clear once again that the Gemara is assuming the value of PI to be precisely 3- after all, if it were more than three, a circumference of 6 would produce a width of less than 2 which would easily fit in the remaining space.

Hopefully to be continued

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha.