Sukkah 8a ~ Rashi’s Mathematical Errors

In today’s page of Talmud, the discussion about the minimum size of a sukkah continues, and we get into some geometry. According to Rabbi Yochanan, the minimum circumference of a round sukkah is 24 amot. The Talmud tells us that this is based on the belief that when a person sits he occupies one square amah.

Next the talmud tells us that according to Rabbi Yehuda Hanasi the minimum size of a square sukkah is 16 square amot. But according to a “mathematical rule” on today’s page of Talmud, the perimeter of a square that surrounds a circle is greater than perimeter of the circle by one one quarter of the perimeter of the square.

סוכה ח,א

כַּמָּה מְרוּבָּע יוֹתֵר עַל הָעִיגּוּל — רְבִיעַ

Now, by how much is the perimeter of a square inscribing a circle greater than the circumference of that circle? It is greater by one quarter of the perimeter of the square.

If that is true, then the circle that surrounds a 16 amot square will be 3/4 or the perimeter, or 12 amot. Why does Rabbi Yochanan require a perimeter of 24 amot, which is twice as large?

The Talmud gives the following answer:

הָנֵי מִילֵּי בְּעִיגּוּל דְּנָפֵיק מִגּוֹ רִיבּוּעָא, אֲבָל רִיבּוּעָא דְּנָפֵיק מִגּוֹ עִגּוּלָא — בָּעֵינַן טְפֵי, מִשּׁוּם מוּרְשָׁא דְקַרְנָתָא

This statement with regard to the ratio of the perimeter of a square to the circumference of a circle applies to a circle inscribed in a square, but in the case of a square circumscribed by a circle, the circle requires a greater circumference due to the projection of the corners of the square. In order to ensure that a square whose sides are four cubits each fits neatly into a circle, the circumference of the circle must be greater than sixteen cubits.

Rashi explains that a circle of diameter 16 can inscribe a square whose perimeter is 12:

בעיגולא דנפיק מיגו ריבוע - אם היקפת בחוט של ט"ז אמה בקרקע בריבועא תמצא בתוכו ארבע מרובעות ואם היית צריך לעגלו מבפנים ולהוציא קרנות ריבועו אתה עוגלו בחוט י"ב ונמצא חיצון יתר על הפנימי רביע אף כאן אם היתה מרובעת סוכה זו דיה להיות כדי שישבו בהיקיפה ששה עשר בני אדם

But Rashi here is not precise, as you can see below:

Tosafot on our page of Talmud notes this inaccuracy, and points out another one. According to this page of Talmud,

כל אַמְּתָא בְּרִיבּוּעָא אַמְּתָא וּתְרֵי חוּמְשֵׁי בַּאֲלַכְסוֹנָא

Any square of side 1 will have a hypotenuse of 1.4

But as Tosafot points out, this is not correct (אין החשבון מכוון ולא דק דאיכא טפי פורתא). In fact the hypotenuse will be the square root of 2, which is an irrational number beginning with 1.141213….

Another Rashi with wrong math

In tractate Eruvin (78a) we read the following:

עירובין עח, א

אמר רב יהודה אמר שמואל כותל עשרה צריך סולם ארבעה עשר להתירו רב יוסף אמר אפילו שלשה עשר ומשהו

Rav Yehuda said that Shmuel said: If a wall is ten handbreadths high, it requires a ladder fourteen handbreadths high, so that one can place the ladder at a diagonal against the wall. The ladder then functions as a passageway and thereby renders the use of the wall permitted. Rav Yosef said: Even a ladder with a height of thirteen handbreadths and a bit is enough, [as it is sufficient if the ladder reaches within one handbreadth of the top of the wall].

Here is Rashi’s explanation:

עירובין עח, א רשי

סולם ארבעה עשר - שצריך למשוך רגלי הסולם ארבעה מן הכותל לפי שאין סולם זקוף נוח לעלות

A ladder that is 14 handbreadths high: The feet of the ladder need to be placed four handbreadths from the wall

Here is a diagram of the setup, and the problem with Rashi:

This error is noted by Tosafot (loc. cit.)

צריך סולם ארבעה עשר להתירו - פי' בקונטרס שצריך למשוך רגלי הסולם ארבעה מן הכותל ולא דק דכי משיך ליה י' טפחים נמי מן הכותל שהוא שיעור גובה הכותל יגיע ראש הסולם לראש הכותל דארבעה עשר הוא שיעור אלכסון של י' על י' דכל אמתא בריבועא אמתא ותרי חומשי באלכסונא

A ladder 14 handbreadths heigh is needed - Rashi explains that the feet of the ladder need to be placed 4 handbreadths from the wall. This is not accurate. For when the feet of the ladder are placed ten handbreadths from the wall, (a larger measure) the ladder will still reach the top of the wall. Because the length of the diagonal [i.e.the hypotenuse] is 14 handbreadths in a 10x10 [right angled] triangle. Because for every unit along the sides of square, the diagonal will be 1.4

What Tosafot is getting at is that in a 1x1 right angled triangle, the hypotenuse must be √2. But √2 is an irrational number, meaning the calculation never ends. However, an irrational number is not useful for our real-world measurements, and so Tosafot rounds √2 to 1.4, just as in the Talmud the value of π, another irrational number, is rounded to 3. (Shalom Kelman, a loyal Talmudology reader, sent us another explanation of the errors which you can read here.)

What to make of these Errors?

We have reviewed two mathematical errors made by Rashi, but how should we view them? As ignorance, mistaken calculations, or something else?

In his classic 1931 work Rabbinical Mathematics and Astronomy, (p58) W.M Feldman is firmly in the "Rashi was ignorant” camp.

It is however, most remarkable that although Rashi displayed great genius in mathematical calculation, he was quite ignorant of the most elementary mathematical facts. He was not aware that the sum of two sides of a triangle is greater than the third, for he says if a ladder is to be placed 4 spans from the foot of a wall 10 spans high so as to reach the top, then the ladder must be 14 spans high, (i.e. the sum of the two lengths), which of course is absurdly incorrect, the real minim length of the ladder must be only √(4x4)+(10x10)= √116=10.7 spans.

Judah Landa, in his book Torah and Science suggests that the Rabbis of the Talmud (and Rashi too, I suppose) had mistaken calculations. They did not give mathematics “the serious attention it deserved and that as a consequence their knowledge of it suffered.” Later commentators, like Tosafot and Maimonides knew that π was slightly greater than three, and Tosafot “demands to know why the Rabbis of Caesarea made statements without attempted verification by measurement and experiment.”

no error here

Rabbi Moshe Meiselman believes that the rabbis of the Talmud were incapable of making an error. Of any sort. In his book Torah, Chazal & Science he dedicates eight pages and copious footnotes to explain why, in fact, there were no errors in any of the math found on today’s page of Talmud. Among the sources he cites is a commentary on the Talmud written by the fifteenth-century Rabbi Simeon ben Tzemach Duran, better known as the Tashbetz. In his Sefer Hatashbetz, he addresses the parallel discussion in the tractate Sukkah:

ובתוס' תרצו כי התלמודיים טעו בדבריהם של רבנן דקסרי …וכל זה חיזוק וסמך שאין בכל דברי רז"ל דבר שיפול ממנו ארצה כי הם אמת ודבריהם אמת

Tosafot explains that the Talmud believed that the Rabbis (of Caesarea) were mistaken…but there is not a single error in all the words of the Sages, for they are true and their words are true…

Rabbi Meiselman concludes that not one of the commentaries on the Talmud “suggests that any of the Chachamim [Sages in the Talmud] made elementary errors in calculation or were ignorant of basic principles.”

Rabbi Meisleman aside, Rashi certainly seems to have made an error in his calculations. But why should this bother us? Of the hundreds of thousands of words written by Rashi, commenting on and explaining the entire Hebrew Bible and Babylonian Talmud, an error or two is hardly unexpected. (Just ask any author who has proof-read her manuscript a dozen times only to find a typo in the published book.) Did Rashi misunderstand, or was he ignorant of Pythagoras and his Theorem? In the end it doesn’t matter much. We all make mistakes. I only hope I make as few of them as Rashi did. Now that would be an accomplishment.

Eruvin 76b ~ Mathematical Error

The Talmud is discussing the necessary dimensions of a window that would allow two private domains separated by a wall to merge as one. Rabbi Yochanan rules that a round window must have a circumference of at least twenty-four tefachim. Then we learn of a mathematical rule stated by the “Rabbis of Caesarea.”

ערובין עו,ב

עיגולא מגו ריבועא ריבעא ריבועא מגו עיגולא פלגא

A circle that comes out of a square - one fourth; a square that comes out of a circle- one half

Rashi explains the second part of this cryptic statement: One half of the square’s perimeter is what you lose by going from a circle’s perimeter to that of a square that fits inside of it. Consider a circle, says Rashi, with a circumference (i.e. a perimeter) of 24 (inches, meters, amot, whatever). Inside of this circle is a square. The perimeter of the square will be 16. Half of 16 is 8, and the circumference of the circle is 8 more than the perimeter of the square. Another way of thinking about this is that in this particular geometric setup, the circumference of the circle will be always be 1.5 times larger than the perimeter of the square.

But this is not correct. In the circle below, the value of the diameter d is also the same as the value of the hypotenuse of an isosceles triangle of length 4. And as you can see, following the simple math, it isn’t. According to Rashi, d=8, but in fact d=5.65.

The same discussion appears in the tractate Sukkah (8a) and Tosafot in both places notes that the Rabbis of Caesarea, together with Rabbi Yochanan who cites the rule in their name, were wrong. There is also an error in Rashi, whose explanation we have just seen..

תוס עירובין עו, ב

דהא קא חזינא דלאו הכי הוא שכל האורך והרוחב לא הוי אלא תרי אמה…וקשה היאך טעו דייני דקיסרי הא קא חזינן דלאו הכי הוא

It is not true that an [isosceles] triangle with sides [one unit long] has a hypotenuse of two…how could the Rabbis of Caesarea have made such an error, for we can measure and see that this is not correct…

“A square inside a circle inside a square”

To save Rabbi Yochanan, the Rabbis of Caesarea and Rashi from this embarrassing error, Tosafot suggests that the Rabbis of Caesarea were never discussing the circumference; instead, they were discussing the area. And they were discussing a square that is inside a circle that is inside a square. Like this:

Tosafot suggests that the phrase “A circle that comes out of a square - one fourth” (עיגולא מגו ריבועא ריבעא) means that the area of the inner circle is one fourth less than the outer square.

Tosafot also suggests that the phrase “a square that comes out of a circle- one half” (ריבועא מגו עיגולא פלגא) means that the area of an inside square is one half that of the outside square. Here is the demonstration, assuming that π=3.

Another Rashi with wrong math

In a couple of pages, (Eruvin 78a) we read

עירובין עח, א

אמר רב יהודה אמר שמואל כותל עשרה צריך סולם ארבעה עשר להתירו רב יוסף אמר אפילו שלשה עשר ומשהו

Rav Yehuda said that Shmuel said: If a wall is ten handbreadths high, it requires a ladder fourteen handbreadths high, so that one can place the ladder at a diagonal against the wall. The ladder then functions as a passageway and thereby renders the use of the wall permitted. Rav Yosef said: Even a ladder with a height of thirteen handbreadths and a bit is enough, [as it is sufficient if the ladder reaches within one handbreadth of the top of the wall].

Here is Rashi’s explanation:

עירובין עח, א רשי

סולם ארבעה עשר - שצריך למשוך רגלי הסולם ארבעה מן הכותל לפי שאין סולם זקוף נוח לעלות

A ladder that is 14 handbreadths high: The feet of the ladder need to be placed four handbreadths from the wall

Here is a diagram of the setup, and the problem with Rashi:

This error is noted by Tosafot (loc. cit.)

צריך סולם ארבעה עשר להתירו - פי' בקונטרס שצריך למשוך רגלי הסולם ארבעה מן הכותל ולא דק דכי משיך ליה י' טפחים נמי מן הכותל שהוא שיעור גובה הכותל יגיע ראש הסולם לראש הכותל דארבעה עשר הוא שיעור אלכסון של י' על י' דכל אמתא בריבועא אמתא ותרי חומשי באלכסונא

A ladder 14 handbreadths heigh is needed - Rashi explains that the feet of the ladder need to be placed 4 handbreadths from the wall. This is not accurate. For when the feet of the ladder are placed ten handbreadths from the wall, (a larger measure) the ladder will still reach the top of the wall. Because the length of the diagonal [i.e.the hypotenuse] is 14 handbreadths in a 10x10 [right angled] triangle. Because for every unit along the sides of square, the diagonal will be 1.4

What Tosafot is getting at is that in a 1x1 right angled triangle, the hypotenuse must be √2. But √2 is an irrational number, meaning the calculation never ends. However, an irrational number is not useful for our real-world measurements, and so Tosafot rounds √2 to 1.4, just as in the Talmud the value of π, another irrational number, is rounded to 3. (Shalom Kelman, a loyal Talmudology reader, sent us another explanation of the errors which you can read here.)

What to make of these Errors?

We have reviewed two mathematical errors made by the Rabbis of Caesarea and Rashi, but how should we view them? As ignorance, mistaken calculations, or something else?

In his classic 1931 work Rabbinical Mathematics and Astronomy, (p58) W.M Feldman is firmly in the "Rashi was ignorant” camp.

It is however, most remarkable that although Rashi displayed great genius in mathematical calculation, he was quite ignorant of the most elementary mathematical facts. He was not aware that the sum of two sides of a triangle is greater than the third, for he says if a ladder is to be placed 4 spans from the foot of a wall 10 spans high so as to reach the top, then the ladder must be 14 spans high, (i.e. the sum of the two lengths), which of course is absurdly incorrect, the real minim length of the ladder must be only √(4x4)+(10x10)= √116=10.7 spans.

Judah Landa, in his book Torah and Science suggests that the Rabbis of Caesarea (and Rashi too, I suppose) had mistaken calculations. They did not give mathematics “the serious attention it deserved and that as a consequence their knowledge of it suffered.” Later commentators, like Tosafot and Maimonides knew that π was slightly greater than three, and Tosafot “demands to know why the Rabbis of Caesarea made statements without attempted verification by measurement and experiment.”

no error here

Rabbi Moshe Meiselman believes that the rabbis of the Talmud were incapable of making an error. Of any sort. In his book Torah, Chazal & Science he dedicates eight pages and copious footnotes to explain why, in fact, there were no errors in any of the math found on today’s page of Talmud. Among the sources he cites is a commentary on the Talmud written by the fifteenth-century Rabbi Simeon ben Tzemach Duran, better known as the Tashbetz. In his Sefer Hatashbetz, he addresses the parallel discussion in the tractate Sukkah:

ובתוס' תרצו כי התלמודיים טעו בדבריהם של רבנן דקסרי …וכל זה חיזוק וסמך שאין בכל דברי רז"ל דבר שיפול ממנו ארצה כי הם אמת ודבריהם אמת

Tosafot explains that the Talmud believed that the Rabbis of Caesarea were mistaken…but there is not a single error in all the words of the Sages, for they are true and their words are true…

Rabbi Meiselman concludes that not one of the commentaries on the Talmud “suggests that any of the Chachamim [Sages in the Talmud] made elementary errors in calculation or were ignorant of basic principles.”

Rabbi Meisleman aside, Rashi certainly seems to have made an error in his calculations. But why should this bother us? Of the hundreds of thousands of words written by Rashi, commenting on and explaining the entire Hebrew Bible and Babylonian Talmud, an error or two is hardly unexpected. (Just ask any author who has proof-read her manuscript a dozen times only to find a typo in the published book.) Did Rashi misunderstand, or was he ignorant of Pythagoras and his Theorem? In the end it doesn’t matter much. We all make mistakes. I only hope I make as few of them as Rashi did. Now that would be an accomplishment.