Saturday, June 4, 2016

Is It Amazing That Rambam Knew That Pi Is Irrational?

(For those who are sick of Pi, feel free to skip this post! But it does clarify some important points. The reason why I am spending so much time on this topic is in a post that I drafted last week but which I am waiting another few days to publish.)

First, I have to correct a mistake that I made. Back in 2011, on Pi Day (3.14), I wrote that Rambam was the first person in recorded history to explicitly describe pi as being an irrational number (i.e. a number that cannot be expressed by a fraction). But I now discovered that it seems that he was preceded by others.

Boaz Tsaban and David Garber, in an article that is helpfully referred to in Rabbi Meiselman's book, note that "Various ancient Greek writers, including Hero, Eutocius, and Simplicius, understand the difficulty of finding an exact value for the ratio, and seem to realize the possibility of its being irrational," although they did not say so definitively. It is thus certainly no surprise that after centuries of failed efforts to calculate number precisely, people would conclude that it is indeed irrational.

The fifth-century Indian mathematician Aryabhata wrote that “Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached.” The 15th century commentator Nilakantha Somayaji interprets the original words as saying that not that is this an approximation, but that the value is irrational.

Then, at the turn of the ninth century, the Persian mathematician Muhammad Al-Khwarizmi notes that there are several different methods for calculating Pi. A marginal note (I am not sure when it was written) observes that "It is an approximation not a proof, and no one stands on the truth of this, and no one but Allah knows the true circumference of the circle, as the line is not straight and has no beginning and no end, we merely attempt to approximate and discover the root, but even the root has no definition as no one may know its exact value but Allah, and the best of these approximations that is to multiply the diameter by three and seventh as it is faster and simpler and only Allah might know it true."

The Muslim scholar Abu al-Rayhan al-Biruni, who lived from 973-1048, was familiar with Aryabhata's works. In The Encyclopedia of the History of Arabic Science, [Roshdi Rashed ed.], vol. 2, London/New York, 1996, p. 454), Boris Rosenfeld and Youschkevitch note that al-Biruni described Pi as irrational. In their discussion of medieval Arabic science, they further note that "...Arabic mathematicians repeatedly expressed their belief that the ratio of the length of a circumference to its diameter was irrational... Subsequent European mathematicians were also sure that pi is irrational but only J. H. Lambert, a native of Alsace, in 1766 succeeded in proving this."

In light of all this, it can hardly be seen as surprising that, just as the early Greeks seem to have suspected and just as the early Indian and Muslim scholars were certain, Rambam was likewise certain that pi is irrational.

Now, here is where matters become really strange. Rabbi Meiselman, in Torah, Chazal and Science p. 155, says that Rambam "was not merely repeating an accepted piece of information, since this fact was as of yet unknown to the rest of the world." Rabbi Meiselman concludes that Rambam deduced it from the Talmud, seemingly from the Talmud's approximating Pi as three rather than using a fraction. As such, Rabbi Meiselman presents this as evidence that Rambam, and in turn Chazal, possessed wisdom that was ahead of their time, and was somehow derived from the Torah or some other such supernatural source.

But Rabbi Meiselman himself, while he does not appear to be aware about Aryabhata, acknowledges in a footnote that al-Biruni already knew it! Rabbi Meiselman claims that "there is no reason to assume that the Rambam saw this work." This statement is very strange. The article from which Rabbi Meiselman took his information about al-Biruni just says that "it is not known" if Rambam saw it. Seeing that al-Biruni was a famous scientist, it is certainly highly plausible that Rambam saw it! And furthermore, it is obviously something that was known in the medieval period! How can Rabbi Meiselman write that "this fact was as of yet unknown to the rest of the world"?

Rabbi Yaakov Menken, in his review and defense of Rabbi Meiselman's book, likewise makes these strange and counterfactual claims (aside from repeatedly slandering me by falsely claiming that I denied Rambam's knowledge of pi and his attribution of that knowledge to Chazal.) He writes that "The field of mathematics did not leap forward in the centuries between the Gemara and the Rambam. It stretches credulity to imagine that Chazal made a rough estimate for no known reason, the fact that Pi is irrational then became common knowledge, and the Rambam then projected this common knowledge back upon Chazal." Accordingly, he concludes that not only was Rambam ahead of his time, but even Chazal are shown to be way ahead of their time! To quote Rabbi Menken: "The Rambam’s statement itself is evidence that Chazal possessed knowledge of the physical world beyond what was known to other cultures.”

Yet every single statement here is wrong. The field of mathematics most certainly did leap forward in the centuries between the Gemara and the Rambam. While it is certainly possible that Chazal knew pi to be irrational - after all, even the early Greeks suspected it - it does not at all stretch credulity to posit that Chazal did not know this, and gave the rough estimate of three for any number of reasons (such as those posited by Rishonim other than Rambam). It likewise does not at all stretch credulity to imagine that the irrationality of Pi became common knowledge in Rambam's time - even Rabbi Meiselman himself acknowledges that it was already stated by al-Biruni. Nor does it remotely stretch credulity to posit that Rambam projected this common knowledge back upon Chazal. People do this all the time - just look at how many recent figures have claimed that Chazal actually believed that the sun shines on the other side of the world at night. Rambam in particular is acknowledged by everyone (including the Vilna Gaon!) to have projected Greco-Islamic philosophy back into many statements in Chazal and Tanach (does anyone actually think that Maaseh Merkavah is really about Greek metaphysics?!).

And what is Rabbi Menken positing instead? That Rambam derived from Chazal that pi is irrational? Where exactly did he derive this from? Rabbi Menken does not seem to want to say, as Rabbi Meiselman does, that the source is the Gemara approximating Pi as three. But Rabbi Menken refuses to say where else the Rambam got it from. Is it some lost Gemara that nobody has ever heard of?!

It is clear that it is extremely plausible, even overwhelmingly likely, that that Chazal made a rough estimate of what they thought was a difficult (though not necessarily irrational) number for the reasons given by the other Rishonim. It is close to a fact that the irrationality of Pi later became common knowledge. And it is extremely normal and in character for Rambam to project such knowledge back upon Chazal.

Would anyone claim that the fact of the 15th century commentator Nilakantha Somayaji deriving the irrationality of pi from the fifth-century Indian mathematician Aryabhata is evidence that Aryabhata was in possession of a supernatural source of knowledge?! What stretches credulity is that Rabbi Meiselman and Rabbi Menken see Rambam's statement as "evidence that Chazal possessed knowledge of the physical world beyond what was known to other cultures"!


  1. Well said, as usual.

    Fast fact - the term 'algorithm' comes from the name of the mathematician Al-Khwarizmi, and the term 'algebra' comes from a method he describes called 'al-jabr'.

  2. "The field of mathematics did not leap forward in the centuries between the Gemara and the Rambam."

    An astonishing statement. Did he not have the time for the less than five minutes of internet searches that proves that statement false? I would be embarrassed to write something like that.

  3. Pretend these are 14 matches.
    Re-place one match to solve the problem
    xx11/v111 = 11

    1. So what's the solution?

    2. Take the final 1 before the = sign, and place it horizontally above the 11 on the right of the = sign.

      This gives xx11 / v11 = pi i.e. 22/7 = pi

      (Although technically the = sign should be curved to denote 'approximately equals')


    3. Which of the non-numbers is a match? I'm assuming the division - usually the equal sign is also matches.

    4. Haha - just got it. XXII/VII = What the whole article is about.

    5. Sorry I didn't pick up on the puzzle before, thinking that it was an 'off the walls' remark. Bad assumption. The answer is:
      Take the last '1' from V111 and place it over the 11 (as a lintel) on the right side of the equation. That then reads XX11/V11 (22/7)= pi, which is the classical approximation for pi.

      Y. Aharon

    6. I thought somebody would have got it.
      What's the subject of these many posts?
      What's the symbol for Pi?
      Now you've got it.
      Pretty clever don't you think?
      My high school teacher in London taught me that over 50 years ago.
      P.S. I haven't used Pi for anything since then.
      Hurry up R.Slifkin and reveal your Perry Mason moment!

    7. Well now we know another solution
      If 22/7=pi
      And pi =3
      So just move the 1 across from the 23 and put it after the 11

    8. (Although technically the = sign should be curved to denote 'approximately equals')

      That would risk damaging people's monitors by bending them out of shape.

  4. "And it is extremely normal and in character for Rambam to project such knowledge back upon Chazal."

    Please clarify what you mean. Do you mean that:

    a. The Rambam projected such knowledge upon Chazal incorrectly?
    b. The Rambam projected such knowledge upon Chazal correctly?

    If you mean "a", this seems quite disrespectful to the Rambam. Do you really believe he was incapable of evaluating Chazal's knowledge- less capable than yourself? Or perhaps even the Rambam was, in your lofty view, intellectually dishonest?

    If you mean "b", it is indeed remarkable that the Rambam attributed knowledge of Pi's irrational nature to Chazal.

    1. The Rambam's opinion was that a thorough understanding of (meta-)physics is required to properly understand TaNaCH and Jewish theology. His opinion was, naturally, that CHaZaL had such an understanding.

      The Rambam believed CHaZaL had such an understanding, and whether he was factually correct does not change whether the Rambam was being intellectually honest.

    2. To Avi:

      So we both agree that the Rambam was intellectually honest and truly believed that Chazal knew that Pi was irrational. Great start!

      Now: Since this was not generally known in Chazal's time, that means that the Rambam truly believed that they knew something that others didn't. Apparently, he did not think their knowledge merely reflected the general knowledge of their era - he considered it perfectly reasonable to assume that they knew more.

      For anyone who truly respects the Rambam, this should be of great interest. And it should prompt one to consider why the Rambam would have attributed superior knowledge/understanding to Chazal.

    3. Unknown: Do you believe that Rambam's explanation of Ezekiel's vision - that it refers to metaphysics - is correct?

    4. Unknown: you are making a fallacy of logic. Lets go through the stages.

      A: Rambam believed חז"ל knew pi is irrational
      B: it was not common knowledge that pi is irrational in חז"ל's time
      C: Rambam believed חז"ל had access to knowledge not commonly known in their time.

      Obviously this derivation is incorrect. To validly conclude C we need D instead of British.
      D:Rambam believed it was not common knowledge in חז"ל's time that pi is irrational.

      Otherwise the Rambam may have believed the following three statements...
      1: חז"ל knew pi was irrational
      2:חז"ל had no access to uncommon knowledge
      3: it was common knowledge in חז"ל's time that pi is irrational.

      None of which are mutually exclusive. Of course we know 3 to be false (maybe), but that does not mean the Rambam did.

    5. Yavoy said: "Lets go through the stages..."

      There's really no need for such an arichus. Obviously my argument is based on the premise that the Rambam would have been aware that Pi's irrational nature was not known in the time of Chazal. Personally, I think that this is a reasonable premise; as an extremely knowledgeable scholar, he would have been aware of writings from various eras, and the general progression of thought.

    6. Again, the Rambam explicitly states that only the foolish think that Pi is irrational. He explicitly disclaims that "knowledge" of the irrationality of Pi is anything special. All the rest is noise.

    7. He may have believed that most of the world - including the educated world - was foolish regarding this matter. But is difficult to say that Pi's irrational nature was widely known (there is extremely rare mention of this concept in writings from the Rambam's era, and none from Chazal's era); it is equally difficult to say that it was not widely known but the Rambam somehow thought it was.

      Thus, if he believed Chazal's knowledge was culled from that of general (educated) society, he would not have assumed they knew that Pi is irrational.

  5. The whole question of "why Chazal used the approximation of 3" is, IMHO, ridiculous - the gemara says WHY they used that approximation - because of the verse in Melachim.

    1. First of all, it is the Rambam who offers an alternate explanation. I hope you would not cavalierly describe his opinion as "ridiculous."

      Secondly, it is quite common throughout the Gemara that there is a derivation from a verse in addition to some logical explanation of the same idea. Generally this means either that the verse reveals the concept, or that Chazal find a "hint" (asmachta) of the idea in the verse.

    2. The Rambam is justifying the inaccuracy of the value - not explaining the original motivation for using it.
      The Mishnah Berurah implies that the verse is needed to justify using the value of 3 even when it leads to a leniency in a Torah mitzvah, BTW.

    3. While the Rambam's suggestion that 3 is used as the approximate value for pi since the latter is an irrational number that can't be exactly represented is not 'ridiculous', it is purely conjectural. Moreover, the Rambam appears to disregard the Gemara's discussion on the Mishneh's statement that the circumference of a circle is 3 times its diameter. The Gemara in Eruvin 14a appears to take the verse in Kings I about Shlomo's great basin as giving an exact ratio of 30 to 10 amot. That's why it is bothered by the consideration that the stated 1 tefach thickness (1/6 ama) of the basin will produce a circumference greater than 30 (31)- even taking pi as 3 (it is 32.5 using the more accurate 22/7 for pi). It is forced to conclude that both the diameter and circumference of the basin are interior measurements. The Tosafot there discuss this point and also conclude that the given approximation for pi is not the one used by secular learned people. Other Rishonim also take the Gemara's derivation seriously - as reflected in the cited Mishna Berura.

      The Rambam was fully aware of the mathematics and natural philosophy current in learned Arabic circles. He, then, had to rationalize the use of 3 for pi instead of the accepted and much better approximation of 22/7. It appears that he took the verse in Kings I as referring to approximate values - as opposed to the Gemara. Even if we don't accept his rationale, the fact remains that the sages of the talmud consistently used 3 for pi, and that becomes the halachic value regardless of its inaccuracy. As has been pointed out, the fact that there is a supporting verse gives it an authoritative character.

      Y. Aharon

    4. Slightly off-topic, but I just found this on the website Quora (where people ask all sorts of questions, and others offer all sorts of answers):

      Question: What is "pi"?
      Mathematician: Pi is the number expressing the relationship between the circumference of a circle and its diameter.
      Physicist: Pi is 3.1415927 plus or minus 0.000000005
      Engineer: Pi is about 3.

      It's just a joke, but we see that it's not so outlandish to employ 3 for pi in practical situations.

  6. I'm getting a little bored of this discussion about pi. Everybody's just talking over each others' heads.

    An issue which nobody has bothered to address and which strikes me as much more interesting is why chazal would have the knowledge that pi is irrational but not the knowledge to prove it. From a practical perspective, whether pi is irrational or not makes practically no difference to anyone. On the other hand, knowing how to prove that pi is irrational requires a solid basis in calculus which can literally be described as the starting point for all the scientific and technological discoveries we benefit from. These people who go around smugly pointing out that chazal knew pi was irrational but don't even know what it means to differentiate a curve really irritate me. Get your priorities straight for God's sake.

    1. "An issue which nobody has bothered to address and which strikes me as much more interesting is why chazal would have the knowledge that pi is irrational but not the knowledge to prove it."

      The whole debate is whether they might have able to derive such a fact from the Torah. Neither side is claiming that they developed or knew modern math.

      "Get your priorities straight..."

      Chazal's priority would presumably have been the correct application of the Torah. The Rambam's contention is that they would not have allowed a rounded-off Pi vi-a-vis Sukkah if an exact number were possible.

    2. Baron's point is that whatever one may dig up as evidence that Chazal might have had some sort of knowledge of some specific datum of science, the general impression one gets is so overwhelmingly the opposite that it really doesn't matter, and it just becomes ridiculous to posit that Chazal had any special knowledge of the scientific workings of the world in any relevant or useful sense. They didn't seem to know about germs, which would have been useful in the application of the Mitzvah of "V'nishmartem", and they had the wrong astronomy, even though astronomy is extremely relevant to the application of the Torah. When Biron speaks of "priorities", he means in terms of what would constitute remarkable scientific knowledge. Newton's first law of motion itself, as a method, is far far more important than the knowledge of how fast any specific object will move in specific circumstances. Galileo's insight that motion is just as much of a natural state as resting and only change in velocity needs a cause is far more important than knowledge about what it would take to make any specific object move faster, and on and on. There is not one of this type of important insights in Chazal. Rabbi Meiselman is defending a position that just seems hopeless prima facie, and doesn't get much better after all of his long arguments. Our insistence that our sages knew science better than modern western civilization is reminiscent of Islamic cultures insistence that it is much more glorious and knowledgeable than the west. If Chazal knew science so well, where were the planes and spaceships? And if you say that they could have invented these, but simply weren't interested in such frivolities, where were the M-16s and Napalm to repel the Romans? And if you say that they had decided that it was a Gzeira Mishamayim and they weren't to interfere, where was the medicine to drastically lower the infant mortality rate? Wouldn't you say that allowing babies to die while you have the knowledge to develop medicine to save them comes under the Issur of "Lo Ta'amod al Dam Re'echa"?

    3. Exactly. This is why the whole PI discussion is absurd. This is the point R. Slifkin should be making.

    4. Oh I believe that people believe this. I live in it.

    5. Shai said: "If Chazal knew science so well, where were the planes and spaceships? And if you say that they could have invented these, but simply weren't interested in such frivolities, where were the M-16s and Napalm to repel the Romans?"

      No one (at least no party to to this debate - I can't vouch for your neighbors) claims that Chazal knew science "so well," or that they were GENERALLY more advanced than others regarding science. R' Meiselman specifically states that Chazal's mastery of Torah was not sufficient to derive all knowledge of the world from it. The argument is only that particular facts about the world were derived from the Torah, and were not limited to the general knowledge of their time.

      Regarding technology: It seems to me that for one studying the world from the inside out - i.e., through underlying meta-physical principles - practical technology would only come waaay at the end of the road.

      A mashal ONLY: Einstein might have mastered many underlying principles of physics, but he probably wouldn't have known how to build a car or an airplane. If this is true of one whose focus is on underlying physical principles, it is certainly true of one whose focus is on underlying metaphysical principles.

  7. Chinese mathematicians seem to have worked Pi out earlier, and more completely, for longer than any other geographical location/people group. Check out Liu Hui's algorithm and the work of Zu Chongzhi for example. With all due respect to the amount of research that R'Slifkin puts into this site (and I know it's immense and it shows), there is undeniably a very Western bias to the research. Several times before when the subject of secular science and the like has come up, and only Western examples (or Talmudic) only were given when in fact China had already knowledge and use of the said topic--I remained quiet. Now, I think it is appropriate to encourage us to expand our research criteria to a global perspective and remember that China made tremendous advances in the sciences and lead the world in many areas for millennia. Joseph Needham's "Science and Civilisation in China" and Li Shi Zhen's "Ben Cao Gang Mu" are great resources. Of special interest to R'Slifkin may be the the "Guideways through Mountains and Seas" Classic which is a compendium of Chinese animals both mythological and real and their location(s). This would be useful as a comparison to the work already done in that field here.

    1. While I am certainly in no position to dispute any of your information, the reason why the discussions are "Western-centered" is not due to a bias of some sort. Rather, the crux of the issue is whether Chazal obtained their natural-world knowledge from the scholars of the surrounding culture, or whether they derived it from the Torah in some manner. Therefore, the relevant point is what they knew relative to those around them. While historically interesting, what the Chinese knew or didn't know does not really enter into this discussion one way or another.

    2. East-west contacts, though sporadic, surely existed at Chazal times. For example silk was found on Egyptian mummy dated c. 1070 BCE.

    3. Yeshoshua the very fact that you don't see the connection is the bias. :-)

      1.) Of course it matters if the Chinese had a more coherent understanding of Pi before the West. Revelation is about timing. One of the points of discussion is whether or not Pi was derived from light of historical evidence it seems more likely that Chazal learned it from their immediate culture since it showed up in their writings around the same time as Western development, as opposed to a much earlier time than the surrounding secular scholars. That latter scenario would possibly inform some sort of "higher knowledge" that R'Meiselman and mainstream Haredi thought espouse of Chazal vs. Chazal just being very aware of the science/mathematics of their day. It is about putting the development of the idea in its entire context and timeline.

      2.) My point was to be applied in ALL discussions that I've seen on this page--there are other articles I've seen where R'Slifkin says to his knowledge "X" is only found in the Talmud--and it wasn't (I apologize for not being able to remember the most recent example before this). Expanding our research base globally would have revealed that info. It's a glaring hole in data gathering that can be easily remedied by including knowledge from the East--this is my suggestion.

      3.) Constantine is correct--hence the importance of a timeline and global discussion. Is it realistic for that Chinese knowledge of Pi to come down the Silk Road through Persia and into Arab lands and then to the West? We already know Persia owes a lot of its medical philosophy to China as well as ideas from the West so it's not so far fetched--but I was attempting to look at dates and cultural exchange timelines, etc.

    4. A couple of relevant quotes:
      "Chinese Scholars abandoned the idea of a Supreme Being with personal and creative properties. No rational Author of Nature existed in their universe; consequently the objects they meticulously described did not follow universal principles... In the absence of a compelling need for the notion of general laws - thoughts in the mind of God, so to speak - little or no search was made for them." - Edward O Wilson "It appears that the idea of a single Supreme Deity was foreign to the early Chinese, and as a consequence the fate of natural science in that culture was a curious stillbirth." - JD Barrow

    5. Liu Hui was 4rd century CE while Archimedes was 3rd Century BCE. Not to take anything away from non-Western mathematics, but the Greeks came first on this one.

    6. It's kinda weird that we're so quick to disagree as opposed to really seeking something out to see the full context of someone's point isn't it?

      @Wagner: Good job! You found guys who didn't do serious research on the topic and who disagree with Joseph Needham (the expert on this subject) and the majority of researchers (insert playful sarcasm with a smile). Read Needham's work (cited above) volume 1 and/or the "Tao of Physics" and let me know what you think. As someone who currently uses "Chinese science" daily I must say that it is greatly superior to "Western science" in many ways, and in many ways not. They are both useful, complimentary and necessary ways of looking at the world in my opinion.

      @David Ohsie: Actually the Babylonians and Egyptians are credited with the first understanding of pi before anyone else (2,000 B.C.E.), so that beats Archimedes by quite a bit (insert smile and friendly tone). However, my point was that the Chinese worked out pi at a higher level much earlier than the West did and for a lot longer--so by R' Meiselman's estimation that means that they had "special knowledge" (as a humorous aside), but in a serious way I'm curious if it traveled down the Silk Road. Also, so much of the Jewish religion comes from Babylon--I wouldn't be surprised if we picked it up whilst there...just thinking out loud.

      For a reference to exact point regarding pi, David I recommend "The Genius of China: 3,000 yrs of Science, Discovery, and Invention" by Robert Temple and the following: Liu Hui was able to obtain both his upper limit 3.142704 and lower limit 3.141024 with only an inscribed 96-gon. Furthermore, both the Liu Hui limits were tighter than Archimedes's: 3.140845 < 3.141024 < π < 3.142704 < 3.142857. With his method Zu Chongzhi obtained the result: 3.1415926 < π < 3.1415927, which held the world record for the most accurate value of π for 1200 years, even by 1600 in Europe, mathematician Adriaen Anthoniszoom and his son obtained π value of 3.1415929, accurate only to 7 digit, still 3 digits short of Zu's result.

  8. There was a frum rov, not so just;
    Who ate Slifkin Pi fit to bust;
    Although he sure knew it;
    That we all could see through it;
    What finished him off was lost trust.

  9. Is Pi Irrational, or is it this discussion?

    3.14159…, or round 5 off to 6
    A cumbersome number that fosters some tricks
    Whether Pie in the sky, or Pie in his eye
    Obsession seems part of this mix.
    Eeny meeny Rav Meiselman mo
    I think it is really time to go
    You saw such great sights
    These past days and past nights
    Even Rambam might say
    In his most learned way
    “Off the stage, and get on with the show”

    (I know, I know, just skip the pi and wait for a better dessert -- and your amazing explanation/revelation -- or rebel-ation)

  10. Aryabhata calculated pi to 3.1416 which is to the nearest 10,000th in the 5th century. That is quite unbelievable!

  11. "Does anyone actually think that Maaseh Merkavah is really about Greek metaphysics?!"

    So would you say it's the standard kabbalistic explanation, then? :)

    1. If you want to know, the information is publicly available.

      "Qabala" was essentially an attempt to rewrite earlier forms of mysticism in such a way that they became formally compatible with the formulations of orthodoxy that the Rambam had been successful in propagating (i.e. re-conceptualizing angels being prayed to as hypostases of G-d חס ושלום). It's a bit more complicated than that, of course, but the important point to understand is that Qabala is not the mysticism talked about in Hazalic sources and if, for whatever reason, you feel that we should incorporate Ma'asei Merkava into our contemporary religious practice you first have to dump Qabala.

    2. The Rambam writes in the introduction to Part III of Moreh Nevuchim that his interpretation of Ma'aseh Merkavah is not based on any tradition, but purely on his own insights. From the Friedlander translation:

      "To give a full explanation of the mystic passages of the Bible is
      contrary to the Law and to reason; besides, my knowledge of them
      is based on reasoning, not on divine inspiration [and is therefore
      not infallible]. I have not received my belief in this respect from
      any teacher, but it has been formed by what 1 learnt from Scripture
      and the utterances of our Sages, and by the philosophical
      principles which I have adopted. It is therefore possible that my
      view is wrong, and that I misunderstood the passages referred to.
      Correct thought and divine help have suggested to me the proper
      method, viz., to explain the words of the prophet Ezekiel in such a
      manner that those who will read my interpretation will believe that
      I have not added anything to the contents of the text, but only, as it were, translated from one language into another, or given a short exposition of plain things. Those, however, for whom this treatise has been composed, will, on reflecting on it and thoroughly examining each chapter, obtain a perfect and clear insight into all that has been clear and intelligible to me. This is the utmost that can be done in treating this subject so as to be useful to all without fully explaining it."

  12. I have no stake in whether Rambam was the first recorded as saying pi was irrational, don't care either way, but the fact that these couple of Arab scholars gave approximations for it or saying we don't know it exactly is NOT the same thing as stating it is an irrational number.

    It sounds like based on what you wrote AL biruni was the first to say it IF we can trust Rosenfeld and youschevitch, of which I'm not so sure. Either your paraphrase or their original work is overly muslimophile because they make a point (anti western point) of stressing that a European did not PROVE pi was irrational until the much later 1766, but how is that relevant? The Muslims also did not prove it, and as they admitted European scholars believed it was irrational before they proved it so, so they were on no different footing than the Muslim scholars (if we take their claim about AL biruni at face value that he indeed stated it was irrational - the muslimophilic framing of the statement would cause me to double check their claim for accuracy that it wasn't just some need of theirs to stress the Muslim scholars were "better" sooner than europeans).

    Likewise wih the Indian sources, the fifteenth century "interpretation" of 5th century statement is colored by 15th century knowledge. Its clear that he gave an approximation. The 15th century commentator, since it was known by then that pi is irrational comments to point this out but clearly the 5h century comment did not point it out.

    Of course we should factor in China as Boaz suggested.

  13. I have no stake in whether Rambam was the first recorded as saying pi was irrational, don't care either way, but the fact that these couple of Arab scholars gave approximations for it or saying we don't know it exactly is NOT the same thing as stating it is an irrational number.

    The problem is that stating it is definitely irrational is actually less accurate as long as you don't have a proof; if so, then you conjecture it is irrational, but you don't know. In addition, Rambam stated emphatically that spontaneous generation occurs. So Rambam stating something as fact doesn't give any indication that he actually knew it to be so.

    If the Rambam was the first to state, without proof, that Pi is irrational, then that is something interesting and notable, but not evidence of exceptional knowledge.

    Either your paraphrase or their original work is overly muslimophile because they make a point (anti western point) of stressing that a European did not PROVE pi was irrational until the much later 1766, but how is that relevant?

    It's relevant to the interesting point that various people may have correctly conjectured, in writing, the irrationality of pi before it was proven. The reference to 1766 is just to show how far away people were from actually having a proof at the time of the conjecture. No one with any understanding of math is claiming that conjecturing pi irrational is anywhere near as significant as proving it it irrational.

    1. Right without proof it could be a lucky guess

    2. Not even that. It is more like seeing that the weather forecast calls for a 90% chance of rain, stating that it will definitely rain, and then claiming advanced knowledge when it ends up raining.

    3. "If the Rambam was the first to state, without proof, that Pi is irrational, then that is something interesting and notable, but not evidence of exceptional knowledge."

      Again, I don't care and it wasn't the point of my post. I don't need the Rambam to have had "exceptional knowledge." I believe he had an exceptional mind and was an exceptional scholar, but I don't care if he had scientific knowledge "beyond his time" (doubt it!)

      "It's relevant to the interesting point that various people may have correctly conjectured, in writing, the irrationality of pi before it was proven. "

      But that includes BOTH the European scholars, as well as, *possibly Muslim scholars.
      And apparently, maybe the Chinese too.

    4. I've got the comment below awaiting moderation on Menken's post. It behooves us to keep it in mind.
      While both sides accept that our sages knew about pi's nature (at least according to Rambam), it seems to me they're using the word "knowledge" differently.

      The rationalists are using it in the sense of "conviction based on inductive reasoning." They're content that Chazal learned it from the mathematicians of their day, even without rigorous proof, and for Rambam, that will suffice.

      Conversely, the non-rationalists are using it in the sense of "proven information." For them, Chazal's source is Divine; otherwise Rambam could not say what he did. The rest of the world had to wait for Lambert before they too would acquire "knowledge."

      It may help to keep this distinction in mind while reading this page.

  14. There is evidence that the Rambam had both studied and mastered Arabic mathematics and science. He even uses spherical trigonometry in elucidating the calculation of zemanim based on star/constellation observations in the more esoteric parts of hilchot Kiddush Hachodesh. His statement about the irrationality of pi can therefore be viewed as reflecting the conclusion of the Arabic geometers and not his own innovation - much less something derived from Talmudic tradition. What is of greater significance (to me at any rate) is the apparently independent demonstration by the Tosafists in Eruvin and Succah of the area of a circle in terms of the radius, and that the diagonal of a square is a bit more than the Talmudic 7/5 of the side. They use a variation on the methodology in integral calculus to get that area. This involves conceptually filling a circle with helically wound string, cutting the string from the edge to the center, and spreading out the cut string. The latter is seen to form an isosceles triangle. They then cut the triangle symmetrically and rearrange the triangular pieces to form a rectangle. The sides of that rectangle is seen to be pi*r and r, giving an area of pi*r*r, i.e., the area of the circle. While they don't use algebraic notation (r), their method works with the above algebraic generalization.
    Their more accurate calculation of the diagonal of a square is based on dividing a square into 4 quadrant squares - each of which forms a 5x5 collection of unit squares. The total number of unit squares, or area, in then 100. Taking the diagonal lines joining the midsections of the large square, we see logically that the diamond shaped inner square is half of area of the large square - or 50 unit squares. However if we use the rule of the diagonal being 7/5 of the side, then the diagonal length is 7 (since the side is 5). Then the area should be 7x7 or 49, whereas it is really 50. Hence the length of the diagonal is a bit more than 7/5 of the side (it is closer to 1.414 than to 1.4). This doesn't negate the Talmudic approximation which is only off by 1%.
    What is disappointing, given their brilliance, is an apparent ignorance of the Pythagorean theorem equating the square of the diagonal of a rectangle to the sum of the squares of the 2 sides. This theorem can be proven using a geometric construction. Take a square and mark off unequal lengths, a and b in a consistent fashion, i.e., the sides are a,b;a,b;a,b,;a,b in order. Now, connect the points on adjacent sides by lines. These lines form an inner square with sides, c. The area of the outer square is (a+b)*(a+b). This is algebraically equal to a*a + b*b + 2a*b. Now this area is seen to be the sum of the areas of the inner square (c*c) and the 4 surrounding triangles each of which has an area of (1/2)*a*b. The total area is then c*c + 2a*b. Subtracting out the common factor 2a*b gives a*a + b*b = c*c. Knowledge of this geometric relationship would have helped in some sugyot, but the sages of the Talmud weren't aware of it either.

    Y. Aharon


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