That’s What He Meant, More Or Less

Update: As ER explains in a comment, I made a grave cognitive error in formulating this post; the תפארת יעקב clearly meant exactly what the text says, although my point about the Tosafos in Eruvin stands. We shall reconsider this topic at greater length in a subsequent post, בג”ה.

In a recent comment on this blog, my friend ER makes the startling claim that the תפארת יעקב to חולין

resolves a difficulty by invoking the fact that pi is not precisely 3, rather a bit LESS!

I was flabbergasted to read this – after all, the work is a classic, and its author, Rav Ya’akov Gesundheit, was no קטלי קני באגמא, but a leading nineteenth century Polish Rav!

My perplexity deepened upon checking the citation; while the text does indeed read as ER reports, Rav Gesundheit attributes this idea to the Tosafos, and I was pretty sure that they at least knew perfectly well that Pi was greater, rather than less, than three!

ומשמע בהדיא בדברי התוספות בעירובין דכל שיש ברחבו טפח אין בהיקפו ג’ טפחים רק פחות מעט1

The resolution, of course, is that the text is corrupt, and should obviously read רק יותר מעט; this is evident both from the referenced Tosafos, as well as from the very context of his own remarks:

עירובין

כל שיש בהיקיפו ג’ טפחים יש בו רחב טפח
מנא הני מילי א”ר יוחנן אמר קרא ויעש את הים מוצק עשר באמה משפתו עד שפתו עגול סביב וחמש באמה קומתו וקו שלשים באמה יסוב אותו סביב
והא איכא שפתו אמר רב פפא שפתו שפת פרח שושן כתיב ביה דכתיב ועביו טפח ושפתו כמעשה כוס פרח שושן אלפים בת יכיל
והאיכא משהו כי קא חשיב מגואי קא חשיב2

WHATSOEVER HAS A CIRCUMFERENCE OF THREE HANDBREADTHS IS ONE HANDBREADTH IN DIAMETER.
Whence are these calculations deduced?3 — R. Johanan replied: Scripture stated: And he made the ‘molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line of thirty cubits did compass it round about. [I Kings VII, 23. As the molten sea which had a diameter of ten cubits was approximately thirty cubits in circumference, the ratio of a diameter to a circumference must consequently be 10:30 = 1:3 approx.]
But surely there was [the thickness of] its brim? [Which increased the diameter to more than ten cubits: so that the ratio between diameter and circumference was greater than 1:3.] — R. Papa replied: Of its brim, it is written in Scripture [that it was as thin as] the flower of a lily; [Its thickness, therefore, amounted to very little and might be disregarded.] for it is written: And it was a handbreadth thick, and the brim thereof was wrought like the brim of a cup, like the flower of a lily; it held two thousand baths.
But there was [still] a fraction at least? — When [the measurement of the circumference] was computed it was that of the inner circumference.4

תוספות

והאיכא משהו. משמע שהחשבון מצומצם וכן בפ”ק דב”ב (ד’ יד:) גבי שני טפחים שנשתיירו בארון ששם ספר תורה מונח שהיא בהיקפה ששה טפחים ופריך כיון דלאמצעיתו נגלל נפיש ליה משני טפחים וכן בתר הכי דמשני בספר דעזרה לתחלתו נגלל ופריך אכתי תרי בתרי היכי יתיב משמע דמצומצם לגמרי וקשיא דאין החשבון מדוקדק לפי חכמי המדות:5

So it is clear from both the Tosafos’s juxtaposition of the other passages, as well as their reference to the “scholars of measurements”, who certainly had a good idea of the true value of Pi, that they are debating whether its value is greater than, or exactly equal to, three, but certainly not less than that.

From the context of Rav Gesundheit’s own remarks it is ever more clear that he is suggesting that Pi is greater, not less, than three:

חולין

אמר גניבא אמר רבי אסי נקדרה כסלע [פירש”י: כמין ארובה עגולה ורוחב הקדירה כסלע מטבע רחב הוא] טרפה שאם תמתח תעמוד על הטפח [פירש”י: אי הוה מושכו לכאן ולכאן שיכנס העוגל לאורך או לרוחב תמצא בו טפח באותה קדירה]
אמר רבי חייא בר אבא לדידי מפרשא לי מיניה דגניבא אמברא דנהרדעא כסלע כשרה יתר מכסלע טרפה
וכמה יתר מכסלע אמר רב יוסף כגון דעיילן תלת קשייתא בציפא בדוחקא בלא ציפא ברווחא:6

Geniba said in the name of R. Assi: If a circular hole was cut out [in the rumen having a diameter] of a sela’, it is trefah, for then if you were to stretch out [the circumference thereof] it would amount to a handbreadth.
R. Hiyya b. Abba said: Geniba explained it to me on the bridge of Nehardea thus: A hole [having a diameter] of a sela’ is permitted; if it is more than a sela’ it is trefah.
What, for example, is a hole larger than a sela’? — Said R. Joseph. A hole through which three date stones with some of the fruit attached could pass with pressure, or easily without any fruit thereon.7

תפארת יעקב

שם כסלע כשרה כיתר מסלע טריפה. לכאורה משמע דהשתא לא חשבינן משום שאם תמתח תעמוד על טפח דהא בכסלע הוי טפח, אבל הרמב”ם פסק כיתר מסלע ונתן טעם שאם תמתח וכו’, ובאמת לא קשה מידי דעיקר הטעם לפי שאין החשבון מדוקדק לפי חכמי המידות ומשמע בהדיא בדברי התוספות בעירובין דכל שיש ברחבו טפח אין בהיקפו ג’ טפחים רק פחות מעט הילכך בעינן כיתר מכסלע דאז אם תמתח תעמוד על טפח:

Rav Gesundheit is resolving Rambam’s lenient ruling regarding a hole with the diameter of a סלע by proposing that since Pi is greater than three, a diameter of a סלע does not imply a circumference sufficiently large to render the animal טריפה.
Q.E.D.

  1. תפארת יעקב חולין נ: ד”ה שם כסלע כשרה כיתר מסלע טריפה – קשר []
  2. עירובין יד. – קשר []
  3. “This is the only instance where a doubt is raised in the Talmud in connection with a mathematical statement. This, as Zuckermann points out (Das Mathematische im Talmud, p. 23) proves that the Rabbis were well aware of the more exact ratio between the diameter and circumference and that the ratio of 1:3 was accepted by them simply as a workable number for religious purposes. Hence the question, ‘Whence are these calculations deduced?’ V. Feldman, Rabbinical Mathematics etc., p. 23.” []
  4. Ibid., translation and notes of the Soncino edition – PDF. []
  5. תוספות שם ד”ה והאיכא משהו []
  6. חולין נ: – קשר []
  7. Ibid., translation and notes of the Soncino edition – PDF. []
This entry was posted in mathematics and tagged , , , . Bookmark the permalink.

2 Responses to That’s What He Meant, More Or Less

  1. CS says:

    Nice post!

  2. ER says:

    I don’t see how you fixed it.

    The תפארת יעקב’s problem is not with Rambam’s ruling itself, which is consistent with the Gemara’s conclusion that only a hole greater then a sela is treifa.

    His problem is the wording of the Rambam.

    The Gemara originally stated that the stretched circumference of a precisely sela-sized hole was one tefach: “נקדרה כסלע טרפה שאם תמתח תעמוד על טפח”.

    Therefore, says R Gesundheit, at the conclusion of the Gemara that a sela is kosher and the hole must be larger than a sela to be treifa, we must no longer be using the criterion of אם תמתח תעמוד על טפח, for a hole that size is still kosher: “משמע דהשתא לא חשבינן משום שאם תמתח תעמוד על טפח דהא בכסלע הוי טפח”.

    (The measure of אם תמתח is either not used at all, or else the criterion for treifus is now אם תמתח תעמוד על יתר מטפח.)

    The problem is that the Rambam writes “יתר מסלע… טריפה… שאם ימתח קרע זה יעמוד על טפח”, giving the Gemara’s explanation of the preliminary reading that exactly a sela is treifa, and applying it to the final reading that only greater than a sela is a problem. The phrase שאם ימתח קרע זה יעמוד על טפח seems wrong according to the Gemara itself. (See ערוך השלחן YD 48 29.) The din is fine, but the reasoning is not.

    Zugt der תפארת יעקב, in fact the deciding criterion remains שאם תמתח תעמוד על טפח. The “עיקר הטעם” for the Gemara’s conclusion is, that since pi is LESS than 3, it would take a hole greater than a sela to produce the necessary circumference. In other words the Gemara originally overestimated pi and therefore arrived at a more stringent ruling. So the conclusion of the Gemara is that כסלע כשירה, because it actually produces a circumference less than שאם תמתח תעמוד על טפח, but יתר מכסלע טרפה, because it then fits the bill of שאם תמתח תעמוד על טפח.

    The Rambam is now readable.

    If you change the text to יותר מעט then the resolution doesn’t work.

    What am I missing?

Leave a Reply

Your email address will not be published. Required fields are marked *