tag:blogger.com,1999:blog-6906205856510467947.post3798561502081227699..comments2019-06-19T19:50:02.374+03:00Comments on Rationalist Judaism: Happy Pi DayJoshhttp://www.blogger.com/profile/02405210530279411496noreply@blogger.comBlogger53125tag:blogger.com,1999:blog-6906205856510467947.post-19367414696188010682016-05-29T16:51:32.851+03:002016-05-29T16:51:32.851+03:00Certainly not - but they did have fractions.Certainly not - but they did have fractions.dlzhttps://www.blogger.com/profile/08269900740743979382noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-26614271156487206582014-09-02T09:57:43.231+03:002014-09-02T09:57:43.231+03:00"A square is greater than a circle by one-fou..."A square is greater than a circle by one-fourth," This indicates that if the circle is 1, then the square is 1 plus 1/4 or a ratio of 1 is to 1.25. The Babylonians have used the same ratio for deriving their Pi value of 3.125. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-69848707345750782062014-08-28T07:40:36.146+03:002014-08-28T07:40:36.146+03:00How did ancient people derive the ratio now known ...How did ancient people derive the ratio now known as Pi ?<br /><br />Check this link :<br /><br />https://www.academia.edu/8084209/Ancient_Values_of_PiAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-42901555372024191412013-12-27T23:40:58.527+02:002013-12-27T23:40:58.527+02:00R'Slifkin. The formula I stated above is corre...R'Slifkin. The formula I stated above is correct, but it is not Euler's formula, which is<br /><br />e^(i*x) = Cos(x) + i*Sin(x).<br /><br />No explicit Pi there, but hiding everywhere just under the surface. So you can remove my comment if you wish. Samuel Dinkelshttps://www.blogger.com/profile/04902791806700848095noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-4949688231050576682013-12-27T23:24:14.695+02:002013-12-27T23:24:14.695+02:00Pi or 2Pi?
First of all, the definition of Pi as...Pi or 2Pi? <br /><br />First of all, the definition of Pi as the ratio of circumference to diameter of a circle is only one of many, many possible definitions of Pi. Pi is everywhere in pure and applied mathematics, as is the constant e, the natural base of logarithms. <br /><br />Euler's formula, called by Richard Feynman the jewel of mathematics, has Pi not 2Pi:<br /><br />e^(i*Pi) = -1<br /><br />It has e, Pi, the imaginary number i = sqrt(-1), and -1. That's a lot of important numbers in mathematics!Samuel Dinkelshttps://www.blogger.com/profile/04902791806700848095noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-72414328224572772992013-08-26T23:27:09.466+03:002013-08-26T23:27:09.466+03:00Moshe Zeldman said:
R' Natan- where do you se...Moshe Zeldman said:<br /><br />R' Natan- where do you see Greek sources alluding to the idea that pi is an irrational number?Moshehttps://www.blogger.com/profile/06827631361659305332noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-7377245952837619632013-03-25T19:07:42.756+02:002013-03-25T19:07:42.756+02:00Avraham, I have read your paper and complimented t...Avraham, I have read your paper and complimented the effort. However, I disagree with your interpretation (or the Meiri) about R' Yochanan's statements on circular measurements being about area rather than circumference. As you admit, the language he used in Eruvin (hekeifa) alludes to circumference. Moreover, there was no direct way to measure circular areas in talmudic times. Finally, any error is attributable to R' Yochanan - not the sages of Caesaria since the latter were, indeed, referring to circular vs square areas in giving their rules for the ratios. The latter point is made by the Tosafot in Eruvin 76b(s.v. Rabbe Yochanan)Y. Aharonnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-9988186125020918752013-03-25T16:06:57.427+02:002013-03-25T16:06:57.427+02:00I'll leave it as an exercise for the reader wh...I'll leave it as an exercise for the reader whether my Diameter-vs.-Radius confusion was accidental or subversively demonstrative of the problem with Pi...<br /><br />Really, the mathematically and physically meaningful circle constant is the ratio of Circumference to Radius, Tau, or 2*pi. The number 2*pi, 6.28... shows up frequently in "real life" such as the reduced Planck constant and the Fourier transform.<br /><br />Yes, you can still do the arithmetic with Pi, but as some Tau-ists put it, that's the equivalent of using the symbol "H" for One-Half, using 2*H instead of using the number One as the Muliplicative Identity, and celebrating how the special number H shows up in the Amazing Formula (any number) * (2*H) = (any number)<br /><br />This is Rationalist-Jewishly very important. Realizing that Tau, 2*pi, is what "counts," we rightly attribute our sages' discussion of Pi rather than Tau to their adoption of the best math and science of their surrounding societies.<br /><br />To find "special" non-rationalist meaning in Pi is to, chas vashalom, accuse the Creator of misleading His people by sending "ruach-hakodesh" hints backing the wrong mathematical horse!<br /><br />See: www.tauday.com<br /><br />RLRob Levenenoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-21045241238145460392013-03-24T23:12:59.573+02:002013-03-24T23:12:59.573+02:00Y.Aharon
“… your effort to rationalize the view of...Y.Aharon<br><br /><i>“… your effort to rationalize the view of R' Yochanan … So, either R' Yochanan is wrong or the gemara is wrong (the Tosafot conclude that both were mistaken).”</i><br /><br />I can’t take the credit for the rationalization the Me'iri preceded me by a few centuries.<br /><br />If you read my paper (linked above) you will see I am carful to not assign right or wrong, I merely go through the geometry with the gemara’s and modern math assumptions. I agree with you that the language brought by the gemara is circumference but then R’ Yochanan and the Rabbis of Caesarea are very wrong. If we assume R’ Yochanan and the Rabbis of Caesarea are talking about area, with the normal assumption of pi = 3, they are exactly correct in both Succa and Eruvin. That appears to leave the gemara with a bad quote from R’ Yochanan and then, based on the text they had, they then try very hard to get close to the reality (but don’t not quite make it, see appendix B of my paper).<br /><br />AvrahamAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-88202904448017594092013-03-22T03:41:47.308+02:002013-03-22T03:41:47.308+02:00Avraham, yasher koach for your review of the comme...Avraham, yasher koach for your review of the commentators on the issue of a minimal round opening or succah and your effort to rationalize the view of R' Yochanan. However, the expression used by R' Yochanan in Eruvin is "chalon agul tzarich sheyehei behekeifo esrim varba'a tefachim [a round opening (that is just sufficient to negate the effect of a wall separating 2 yards) must be 24 tefachim). Hekeifo means circumference - not area, as can be seen in the gemara's question immediately following the above statement, "michdei, kol sheyeish behekeifo shlosho tefachim yesh bo berochbo tefach" (let's see, if something has a circumference of 3 tefachim, it has a diameter of 1). The latter expression is used often in shas. While shetach may be a later expression for area, makom is quite ancient. R' Yochanan does not state "chalon agul tzarich sheyehei b'mkomo esrim ve'arba'ah" (a round opening must have a minimal area of 24 sq. tefachim). <br /><br />Moreover, despite my prior concession that R' Yochanan in Succa may have been referring to area, his language indicates otherwise. He states, a succah made like an oven (i.e., a cylindrical succah) must be able to accomodate 24 people to be kosher. If he meant in the interior of the succah, it would hold 16 people each taking up 1x1 ama, i.e. in the enclosed 4x4 ama square (the round caps representing the areas outside the enclosed square have a maximum depth of just over 0.8 ama). The gemara concludes that he meant 24 people sitting around the perimeter, i.e., a circumference of 24. Your sincere attempt to justify R' Yochanan also leaves the gemara in Eruvin and Succah with a mistaken understanding of both R' Yochanan and the sages of Caesaria. So, either R' Yochanan is wrong or the gemara is wrong (the Tosafot conclude that both were mistaken).<br />Incidentally, I was mistaken when I stated previously that the area outside the square is 1/2 (taking pi as 3). It is actually S^2/2, where S is the side of the square, i.e., 8, for S=4. The bottom 'cap' then has an area of 2 sq. tefachim - as you stated.<br /> Y. Aharonnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-48966170752521229832013-03-21T04:24:21.057+02:002013-03-21T04:24:21.057+02:00Y.Aharon
”… could well be referring to area - as...Y.Aharon<br /><br /><i>”… could well be referring to area - as opposed to the gemara's interpretation of R' Yochanan's position, that is not possible in Eruvin. … he would have used a term like 'shetach' … Since the enclosed square has a height of 4, that leaves 2 tefachim on the bottom …”</i><br /><br />The principle, and math, in both <i>Sukkah</i> and <i>Eruvin</i> is the same proportionately. Thus the idea of 24, 1 by 1 units of measure works for both <i>amos</i> and <i>tefachim</i>. and the stalemates of the Rabbis of Caesarea is independent of the units.<br /><br />I am not a linguistic expert, but I believe there was no word for area in R’ Yochanan’s time. <i>Shetach</i> came later. Area was indirectly referred to. E.g. <i>bais kur</i> an area covered by a <i>kur</i> of seed. So it would be natural to describe an area as the number of square units that would fit in a circle.<br /><br />The 2 <i>tefachim</i> on the bottom are referring to the circle. Thus either the length of the line from the bottom of the circle to the bottom of the square (.8) or the area of the segment below the square which, assuming pi is 3, would be ¼ of the area outside the 4x4 square and inside the circle, exactly 2 (or using a more accurate pi, 2.28).<br /><br />I realize this is difficult to follow without a diagram. I have written a paper that has an annotated diagram and explanation of the math. It is available at <br /><a href="http://babbageassociates.com/files/R-Yochanan.pdf" rel="nofollow"> An Approach to Rabbi Yochanan and the Rabbis of Caesarea</a><br /><br />Any comments would be appreciated.<br /><br />AvrahamAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-61159334841028784442013-03-20T05:07:03.305+02:002013-03-20T05:07:03.305+02:00Why do I bother? The last poster may not have bee...Why do I bother? The last poster may not have been serious, but shows a confusion between radius (R) and diameter (D). Pi is, precisely defined as the ratio of a circle's circumference (C) to its diameter (D). 2 pi, or tau - if you wish is the ratio of C to R. One definition is no more fundamental than the other. The area of a circle is given by pi*R^2 or (pi*D^2)/4, which is simpler than the corresponding expression for tau. While it is true that there are more equations in physics that contain 2 pi, that doesn't mean that 2 pi is mathematically more significant.Y. Aharonnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-942363064724347652013-03-18T18:35:20.337+02:002013-03-18T18:35:20.337+02:00The Pi is a Lie!
See www.halftauday.com or www.ta...The Pi is a Lie!<br /><br />See www.halftauday.com or www.tauday.com.<br /><br />There's no way chazal would have been granted supernatural knowledge of Pi, because Pi is historically interesting but mathematically irrelevant!<br /><br />Pi is nothing but half the ratio of circumference to diameter, which is 2*Pi, which we Tau-ists choose to denote with the Greek letter "tau."<br /><br />Tau, 6.28...., shows up frequently in mathematics and physics. We can forgive the ancients for this pedagogical disaster, since diameter is a simpler and more obvious measurement to take.Rob Levenenoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-86333329608583455682013-03-17T22:18:51.713+02:002013-03-17T22:18:51.713+02:00Back to basics: Why have a Kri/Kesiv on Kav in the...Back to basics: Why have a Kri/Kesiv on Kav in the first place? 111/106 x 3 is an excellent approximation for Pi. As a mathematician, I am prepared to give chazal credit for this one.Yosselnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-83234430131593611822013-03-17T19:36:28.436+02:002013-03-17T19:36:28.436+02:00Avraham, while the statement of R' Yochanan in...Avraham, while the statement of R' Yochanan in Sukkah 8b that a minimum round sukkah must fit 24 people (according to the Tannah that a minimum sukkah must be 4x4 amot) could well be referring to area - as opposed to the gemara's interpretation of R' Yochanan's position, that is not possible in Eruvin. In Eruvin 76a, As you cited, R' Yochanan refers to the hekaif (circumference) of a minimum round opening in the dividing wall of adjoining yards. That hekeif must be 24 tefachim in his view. Had he been referring to the area - a measurement that could not have been easily made, he would have used a term like 'shetach'. Moreover, how do you get an area of 2 sq. tefachim below the inscribed 4x4 tefach square? The difference in areas between the circle and inscribed square is (pi-2)/2. If pi is taken as 3, the total area outside the square is 1/2. The bottom segment of the 'excess' area is then 1/8 - not 2. However, if R' Yochanan's circle has a circumference of 24 based on his assumed square diagonal of 8, then the vertical diameter of the circle is 8. Since the enclosed square has a height of 4, that leaves 2 tefachim on the bottom (and 2 on top), i.e., this understanding of R' Yochanan in Eruvin is both consistent with the language used and with his ostensible 'mathematics'. Y.Aharonnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-672465379696078222013-03-16T21:42:48.537+02:002013-03-16T21:42:48.537+02:00Even in plain geometry we find Chazal determining ...Even in plain geometry we find Chazal determining laws based on homiletics, and only afterwards trying to make the facts fit these laws. In Tractate Eiruvin 14a the Talmud says:<br /><br />"Anything which has, in its circumference, 3 tefachs, has one tefach in diameter. How do we know this? Rabbi Jochanan said, it is written in the Scripture: 'And he [Solomon] made a molten sea, ten amahs from one brim to the other. It was round all about, and its height was five amahs. And a line of thirty amahs circled it' (I Kings 7:23)."<br /><br />The Talmud rules that the ratio between a circle's circumference and its radius, known as pi, is 3. In fact, this number is irrational (impossible to represent as a finite common or decimal fraction), and taken to 10 decimal places, pi=3.1415926536.<br /><br />One might say that Chazal also knew that true pi is more than 3 and only tried to find a Halachically valid approximation of this number -- but this is impossible because of the Gemara in Bava Batra 14b:<br /><br />"And if you think about the Torah Scroll [of the Temple] which had 6 tefachs in circumference, provided that everything that has 3 tefachs in circumference has one tefach in diameter and provided that the Torah scroll was rolled to its middle [i.e. it was rolled on two wooden shafts like our Torah scrolls are], we have more than 2 tefachs between one handle and another -- so how could it enter the 2 tefachs of free space [in the Holy Ark]? Rav Acha the son of Jacob said: the Torah scroll of the Temple was rolled to its beginning [i. e. it was rolled on one wooden shaft only]. And yet, since it was 2 tefachs in diameter, how could it enter 2 tefachs of free space [in the Ark]? Rav Ashei said: they did not wind all the Torah scroll on the pivot, but left a part of it unwound, put the scroll into the Ark, and then folded the remaining part of the scroll onto it."<br /><br />They thought a Torah scroll 6 tefachs in circumference to be exactly 2 tefachs in diameter, so they considered it to be practically impossible to put such a scroll into a space of exactly 2 tefachs, unless one does not wind all the parchment of the scroll on its wooden shaft, thus leaving some free space to adjust the scroll in the Ark. Only after he puts the scroll into the Ark does he folds the remaining parchment and put it above the scroll.<br /><br />Of course, were the Sages aware of the real value of pi -- or at least of the approximation 22/7 known to ancient Greeks centuries before the Talmudic era, they would have understood that the real diameter of a scroll 6 tefachs in circumference is about 1.9 tefachs and that nobody would need any special tricks to put it into 2 tefachs of free space. It is not difficult to determine that pi is significantly more than 3. All one needs is a ruler and a measuring rope. Nonetheless Chazal preferred to determine reality from verses and law instead of basing law on reality.DYNAMIC WEIGHT LOSShttps://www.blogger.com/profile/06166140559223756523noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-77796039054328384032013-03-16T21:37:25.657+02:002013-03-16T21:37:25.657+02:00I find it amusing that there is a post on a ration...I find it amusing that there is a post on a rationalist blog about an irrational number, in honor of a "holiday" that makes no sense in the writer's country of residence, nor his country of birth.Avihttps://www.blogger.com/profile/15987913203857736139noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-73005786754484819272013-03-15T23:53:18.656+02:002013-03-15T23:53:18.656+02:00As was pointed out above:
p (Pi) – is the ratio of...As was pointed out above:<br />p (Pi) – is the ratio of the circumference of a circle to its diameter. It is an irrational number and cannot be expressed as an exact fraction. p is ~= 3.14159265358979 accurate to 15 decimal digits. The closest approximation in a one-digit fraction is 1 divided into 3 = 3, which is used by the Gemara and is accurate to 1 decimal digit. The closest approximation with a two-digit fraction is 7 divided into 22 ~= 3.142857, which is accurate to 3 decimal digits and is also used by Chazal . The closest approximation with a three-digit fraction is 113 divided into 355 ~= 3.141592920 which is accurate to 7 decimal digits.<br><br />However, Gemara in Succah 8b and Eruvin 76b both quote a ma’amar from Rabbi Yochanan that a 4 by 4 square can be circumscribed by a circle that has an hakayfoh of 24. The Gemara then brings proof from the Rabbis of Caesarea who state, “a circle inscribed in a square, is one-fourth; a square which is inscribed in a circle, is one-half”. The normal meaning of hakayfoh is circumference or perimeter. This leads to a large difference between the 24 of R’ Yochanan and the 16.8 the Gemara calculates (or even the 17.77 with more accurate calculations).<br><br />Based on the Me'iri, who says so explicitly, and Tosafos and the Ritva, from whom we can derive it, Rabbi Yochanan is talking about <b>area</b> and is correct in both Succah and Eruvin. Rabbi Yochanan's statement that “the circumference of the Sukah must be large enough to seat 24 people in it” does not mean that the <b>circumference</b> must be 24 Amos, but that there must be room for 24 people occupying 24 square Amos <b>inside the circumference</b> -- in other words, the area of the circle must be 24 square Amos!<br><br />The Rabbis of Ceasarea are then brought as proof and state that the area of a circle that is drawn around a square which is 4 by 4 is calculated by subtracting ¼ of the area of the circumscribing square or adding ½ of the area of the inscribed square and is exactly equal to 24 square Amos.<br><br />This is what Rabbi Yochanan meant when he said that the circle must have within its circumference an area of 24 in both Succah and Eruvin.<br><br />Further when Rabbi Yochanan states in Eruvin that in order to get the inscribed square of 4 by 4 Tefachim below a height of 10 Tefachim, at least 2 Tefachim and a bit of the circular window must be below ten Tefachim; he is talking about the area of segment CBL which is exactly 2 square tefachim . If 2 square tefachim and a bit of the circle are below 10 tefachim, the bottom of the 4 by 4 square will therefore be below 10 tefachim and the Pesach (opening) is valid, and allows the Chatzeros to be joined in one Eruv.<br><br />Thus if we view the statements of Rabbi Yochanan and the Rabbis of Caesarea as referring to <b>area</b> and with the value of p equal to 3, they are exactly correct.<br><br />AvrahamAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-15436079470316205422013-03-15T17:02:26.806+02:002013-03-15T17:02:26.806+02:00Ashkenazim have more than just a minhag not to eat...<i>Ashkenazim have more than just a minhag not to eat locusts </i><br /><br />No, they have LESS than a minhag. They just lack a tradition of eating them; they do not have a tradition NOT to eat them. You might as well say that Ashkenazim have a minhag not to live in Eretz Yisrael.<br /><br />You seem to think that I am some sort of maverick in this regard. In fact, many Ashkenazi poskim say that it is perfectly legitimate to accept the North African mesorah on locusts.<br /><br /><i>The way you pick and choose</i><br /><br />Everybody picks and chooses. Some just do it with more analysis, consistency and self-awareness than others.<br /><br />Please direct further comments to the appropriate post.Natan Slifkinnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-45764262924934428652013-03-15T14:55:26.696+02:002013-03-15T14:55:26.696+02:00Hi just saw your piece on locusts.
I fail to under...Hi just saw your piece on locusts.<br />I fail to understand something; you see by the bris of your son you wrote the reason you did metziza (eventhough we are aware today that there are no medical benefits) is because in your words "why tamper with a minhag"<br />Yet Ashkenazim have more than just a minhag not to eat locusts yet you happily disregard it for the reasons you stated.<br />The way you pick and choose is an embarassment to mature and broad minded jews.As a Jewish Grammar educated boy one would have expected to see more consistancy from you. <br /> Vine Stnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-7415702541384262622013-03-15T08:54:07.542+02:002013-03-15T08:54:07.542+02:00Here's a very important point: Gematria is not...Here's a very important point: Gematria is not mentioned once in Tanach. Some say it's a Greek import, but regardless, the author of Melachim would probably not have known about it.<br /><br />Still, it's really cool.Nachumhttps://www.blogger.com/profile/11292162031685942549noreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-40452145642266190842013-03-15T05:21:12.384+02:002013-03-15T05:21:12.384+02:00Rabbi Slifkin, since you are updating this, you sh...Rabbi Slifkin, since you are updating this, you should probably heed Reuven Meir's comment. By using the words "being accurate to seven decimal places" you seem to be implying that 111/106 gives pi to 7 decimal places, when in reality it is only 4. Not a big deal, but you may as well fix it.<br />Adamnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-79678864554944117412013-03-15T03:54:21.101+02:002013-03-15T03:54:21.101+02:00The sages took the value of pi (the ratio of the c...The sages took the value of pi (the ratio of the circumfrence of a circle to its diameter) as 3 in many places in the talmud. Their conjectures about the brazen water reservoir of Shlomo in T.B. Eruvin 14a had to do with the understandable assumption that the 10 ama diameter of the round vessel that is mentioned in the cited verse was the inner diameter (only the i.d. is of interest in a volume determination), while the stated 30 ama circumference was the outer dimension (the only one easily measured with a measuring line). They speculate that the vessel had a fine edge like a chalice so that the o.d. and i.d. were very nearly the same. The objection is still raised that there remains a difference, and the answer given is that the stated circumference of the vessel is the inner one. The Tosafot conclude from this evident understanding of the gemara that pi was assumed to be exactly 3.<br /><br />I am not excited with the kav/kava correction ratio that converts 3 (30/10) into a better approximation for pi. It is a bit of cute numerology derived on an ad-hoc basis. As has been pointed out 355/113 gives a decidedly better approximation. If one wishes to become 'mystical' about the matter, consider an arithmetic series of pairs of odd numbers, i.e., 113355. If one takes the 1st 4 digits of the series (1133)and divides the latter pair by the former, one gets 3 (the approximation used by the sages). If one includes the next pair (55) and divides the 2nd set of 3 (355) by the 1st (113), one gets the best approximation using numbers below 1000.<br /><br />The outstanding question about R' Yochanan's halacha in T.B. Eruvin 76a is that he appears to assume that the diagonal of a square is given by the sum of the sides. Such an assumption is implicit in the idea that a circular opening in a wall must have a circumference of 24 tefachim in order to enclose a square space of 4x4 tefachim. If the diagonal is the sum of the sides of the square, i.e., 8 tefachim and pi is 3 then the circumference of the round opening is 24. This understanding also rationalizes his stipulation that the bottom of the circular opening must be within 8 tefachim of the ground in order for the bottom of the enclosed square opening to be within 10 tefachim of the ground. The problem with this understanding of R' Yochanan is that it is not only mathematically and logically wrong (a straight line distance between 2 points on a plane must be less than any broken path), but that a much better approximation for the diagonal of a square relative to its side was used by other sages, i.e., 1.4 or 7/5 - not 2. This value of 1.4 for sqrt 2 (1.414..) is only 1% off the accurate value while using 3 for pi is 4.5% off.<br /><br />Tosafot in Eruv. 76b not only allude to this error, but they also explain that the citation from the sages of Ceasaria that a circle is 1.5 times the enclosed square (assuming that pi = 3) refers to the areas - not the perimeters. Indeed, the ratio of the areas is 1.5 (for pi=3). They also beautifully demonstrate how to calculate the area of circle. Y.Aharonnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-29700882447350203012013-03-15T03:09:45.824+02:002013-03-15T03:09:45.824+02:00There is a Greek Pythagorean motto "God is ev...<i>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!</i><br /><br />Device? for what? That number of letters conceit looks at pi in a positional base 10 system. Wikipedia FWIW states that the Babylonians used a positional base 60 system, so reading pi into the the Pythagorean motto wouldn't have worked there, and while the Greeks did indeed use a decimal system, I don't think it was positional. <br /><br />Interesting, though.Yoel Bnoreply@blogger.comtag:blogger.com,1999:blog-6906205856510467947.post-48798388412655834132013-03-15T01:29:54.881+02:002013-03-15T01:29:54.881+02:00This is idiotic...
The text of Melahim speaks of ...This is idiotic...<br /><br />The text of Melahim speaks of the semi-spherical bowl, with outer diameter of 10 cubits, outer radius 5 cubits and inner circumference of 30 cubits. This results in outer circumference of the bowl 10Pi or about 31.1 cubits.<br /><br />So the exact value of Pi was most certainly known back then. <br /><br />I think a lot of people get confused because the text uses outer diameter and INNER circumference to describe the bowl, which is not conventional from the modern point of view.Aleksandr Sigalovhttps://www.blogger.com/profile/08622904545106070847noreply@blogger.com