From the Talmudology Archives: Rashi and Why Women Light the Menorah

Don't Share This With Your Young Children

The festival of Chanukah begins this evening. Ask a well-educated Jewish child about the origins of Chanukah, and she will likely tell you about the wicked Greeks who defiled the Temple, about the brave Maccabees who fought them, and about the miracle of the oil.  But in Rashi's commentary to the Talmud there is another part of the story. Here it is:

דאמר רבי יהושע בן לוי: נשים חייבות בנר חנוכה, שאף הן היו באותו הנס

 רש"י שם:  שגזרו יוונים על כל בתולות הנשואות להיבעל לטפסר תחלה  

Rabbi Yehoshua ben Levi said: Women are obligated to take part in the lighting, for they were included in that miracle...

Rashi: For the Greeks made an edict that all virgins who were about to marry must first have intercourse with the Prefect...

The great French exegete Rashi (d.1105) is referencing the Law of the First Night - Jus Primae Noctis, also known more graphically as The Right to the Thigh - Droit du Cuissage. We first encountered this when studying Ketuvot 3a. So let's go back to that daf.  

MAZAL TOV; WHEN'S THE WEDDING?

Today, when a bride and groom wish to secure a wedding day, it will depend on their budget and the availability of the caterer. My, how things have changed. In the times of the Mishnah, the wedding day was decided by the availability of the local rabbinic court, the Bet Din. Then, a wedding (of a virgin) could only take place on the night before the Bet Din convened.  This would ensure that if, after their magical first night, the groom suspected that his bride had not been a virgin, he could take his claim to court the very next day.  

“מפני מה אמרו בתולה נשאת ליום הרביעי שאם היה לו טענת בתולים היה משכים לב”ד

Why did they teach that a virgin must only marry on a Wednesday? So that if the groom questioned her virginity, he could hurry to the Bet Din...
”

— Ketuvot 3a

The Talmud (Ketuvot 3a) explains that this happy custom changed during a period of persecution. Rabbah, a forth century Babylonian sage, explained what this is all about: "[The authorities] said, "a virgin who gets married on Wednesday will first have intercourse with the governor" (הגמון). In order to avoid this awful legal rape, the wedding was moved a day early, to fly, so to speak, under the radar of the local governor. 

JUS PRIMAE NOCTIS IN THE TALMUD & MIDRASH

The law that Rabbah referenced is the same one that Rashi claims was imposed on Jewish brides by the Greeks. Its origins are further explained in the Talmud Yerushalmi, which dates it to the time of the Bar Kochba revolution:

 תלמוד ירושלמי כתובות פרק א הלכה ה  

בראשונה גזרו שמד ביהודה שכן מסורת להם מאבותם שיהודה הרג את עשו...  והיו הולכין ומשעבדין בהן ואונסין את בנותיהן וגזרו שיהא איסטרטיוס בועל תחילה התקינו שיהא בעלה בא עליה עודה בבית אביה 

In the beginning, they [the Romans] decreed destruction in Judea (for they had a tradition that Yehuda killed Esau) ... and they enslaved them and raped their daughters, and decreed that a soldier would have intercourse [with a bride] first. It was then enacted that her husband would cohabit with her while she was still in her father's house. 

A reference to Primae Noctis also appears in the Midrash Rabbah, a collection of rabbinic homilies edited sometime in the forth or fifth century. As told in Genesis 6, “the sons of God saw that the daughters of man were beautiful (tovot), and they took wives from whoever they chose.” The Midrash focuses on that word beautiful, and explains:

בראשית רבה (וילנא) פרשת בראשית פרשה כו 

אמר רבי יודן טבת כתיב, משהיו מטיבין אשה לבעלה היה גדול נכנס ובועלה תחלה, הדא הוא דכתיב כי טבת הנה, אלו הבתולות ויקחו להם נשים מכל אשר בחרו, אלו נשי אנשים, 

“Rabbi Judan said the word tovot (טבת) – beautiful – is written in the singular, [but read as a plural]. Meaning that the bride was made beautiful for her husband, but the lord of the nobles had intercourse with her first...”

JUS PRIMAE NOCTIS...IN THE MOVIES

There are numerous references to Primae Noctis in ancient and modern literature, from the Epic of Gilgamesh to The Marriage of Figaro. One recent example can be seen in the movie Braveheart, when the evil King Edward gallops into a village, to interrupt a wedding celebration. “I’ve come to claim the right of Primae Noctis. As lord of these lands, I will bless this marriage by taking the bride into my bed on the first night of her union.”  And as the groom is restrained by Edward's henchmen, Edward reminds the peasants “it is my noble right.”  

Jus Primae Noctis. Is there a more fearsome example of feudal barbarism? Of what one scholar called “a male power display…coercive sexual dominance…and male desire for sexual variety”?  But the legend, despite its appearance in many guises, is, fortunately, likely to be nothing more than just that: a legend.  

JUS PRIMAE NOCTIS...IS A LEGEND

Perhaps the most comprehensive investigation of the legend of Primae Noctis is The Lord's First Night: the Myth of the Droit de Cuissage, by the French social scientist Alain Boureau. His careful analysis is particularly important since, as we have seen, Rashi, our favorite French commentator, cites this legend twice. After a meticulous two-hundred page review of every alleged appearance of the legend, Boureau is clear:

“[T]he droit de cuissage never existed in medieval France. Not one of the arguments, none of the events insinuated, alleged or brandished, holds up under analysis.
”

— Alain Boureau, The Lord's First Night.

Others scholars agree with Boureau. In 1881, the German historian Karl Schmidt concluded that the right never existed.  In 1973, the historian J.Q.C. Mackrell noted that there is "no reliable evidence" that it existed. And Prof. Tal Ilan, now at the Free University of Berlin, addressed the myth of Primae Noctis in a magnificently titled 1993 paper: Premarital Cohabitation in Ancient Judea. Prof. Ilan noted that that “all medieval literature that evokes the custom of Jus Primae Noctis has been proven to be folkloristic and has no historical basis.” But what about the evidence from the Talmuds, and the Midrashim? Don’t they provide evidence that Primae Noctis was indeed practiced in the time of the Talmud? Not so, claims the professor:

If a motif of this sort could have appeared in a sixteenth-century document and upset the entire history of medieval Europe for the next two centuries, the same motif likewise could have cropped up in the fourth -or fifth-century Palestinian Talmud, falsely describing events of the second century.

Instead, Prof Ilan suggests that the Talmud used the myth of Primae Noctis to excuse the behavior of some prospective couples, who would engage in sexual relations before they married.  “the jus primae noctis was conveniently drawn in order to explain and justify a custom that seemed to the rabbis to undermine their view of proper conduct in Jewish society.”

“Some events do take place but are not true; others are—although they never occurred.”

— Elie Wiesel, Legends of Our Time

There is some further support to the claim that primae noctis never existed, and it is not one I have seen suggested before.  It is a claim from silence.  I've checked over 100,000 responsa, and there is not one on this topic. Not a single one.  If primae noctis really was a law of the Greek and Roman empires, and a feudal right across medieval Europe, then why were its implications for the Jewish community never discussed in the responsa literature?  This silence supports the conclusions of work done by Boureau, Ilan and others: it never existed. In fact Boureau wonders what muddled thinking would lead anyone to believe it existed in the first place: 

It has been clear from the start that no matter what social restrictions were put on conduct and the management of wealth, and no matter how violent mores became, the principle of free choice of an unfettered matrimonial life was the most sacred area of individual liberty in medieval Europe. The Church, European society's principal normative center, very early removed all restrictions on the marriage of dependents, and it imposed consent as a sacramental value. No juridicial form, no custom, could attack that principal...sanctified in the twelfth century by the establishment of the sacrament of matrimony.

HISTORY AND HERITAGE

The historian David Lowenthal has explained the differences between history and heritage. While history "seeks to convince by truth," heritage "passes on exclusive myths of origin and endurance, endowing us alone with prestige and purpose." Heritage, continues Lowenthal, commonly alters the past: sometimes it selectively forgets past evils, and sometimes it updates the past to fit in with our modern sensibilities. Sometimes it upgrades the past, making it better than it was, and sometimes it downgrades the past, to attract sympathy.  And so, how we read the Talmud will depend on whether we see it as a work of history or as a book of our heritage.  

“There you have it...some of it fact, and some of it fiction, but all of it true, in the true meaning of the word”

— Miles Orvel, The Real Thing: Imitiation and Authenticity in America

There are stories both wonderful and terrible from our Jewish past. Some are factual, and some are not, and a measured approach to how we might approach these stories has been suggested by Judith Baumel and Jacob J. Schacter. They explored the claim (published in The New York Times) that in 1942, ninety-three Beis Yaakov schoolgirls in Cracow committed suicide rather than face rape by their German captors. They concluded that the evidence to support the truth of the story is not conclusive one way or the other. 

Whether or not it actually happened as described is difficult to determine, but there is certainly no question that it could have happened...in response to those claiming that the incident was "unlikely" to have occurred, let us remind the reader that the period in question was one during which the most unlikely events did occur, when entire communities were wiped out without leaving a single survivor...Maybe it did happen. But maybe again it didn't. Could it have happened? Of course.

So Why Should Women Light the Menorah?

It seems very unlikely that Rashi's explanation for why women should light the Menorah has any factual basis; the legend of Primae Noctis is not likely to have been true. But some stories are true, even though they never happened. Ask yourself, from what you know about Jewish history, could it have been true? Yes. And that's what makes it all the more terrifying. Sadly, we have plenty of tragic stories from our Jewish history, and there is no need to create one that probably never happened. 

But if Rashi's reasoning was based on a myth, then why should women light the menorah on Chanukah, since they are exempt from a positive time-bound command?  For an answer that should satisfy moderns, we need look no further than the Code of Jewish Law - the Sulchan Aruch (written about 460 years after Rashi's death).   

שולחן ערוך אורח חיים הלכות חנוכה סימן תרעה סעיף ג 

 אשה מדלקת נר חנוכה, שאף היא חייבת בה

 A women should light a light on Chanunkah, for she is obligated to do so...

So there you have it. Women should light...because they should light. We need no more of a reason than that.

Happy Chanukah.

Tomorrow on Talmudology: Was There A Chanukah Miracle?

Pesachim 9b ~ Talmudic Probability Theory

Our new tractate Pesachim, deals with all things Paschal. (Well, nearly all). What happens if there were nine piles of matzah and one pile of forbidden leavened bread known as chametz, and along came a mouse and took a piece from one of the piles and carried it into a house that had already been searched for chametz. Must the house be searched a second time? To find an answer, the Talmud quotes a Baraiasa that deals with an analogous question.

פסחים ט, ב 

דִּתְנַן: תֵּשַׁע חֲנוּיוֹת, כּוּלָּן מוֹכְרִין בְּשַׂר שְׁחוּטָה, וְאַחַת מוֹכֶרֶת בְּשַׂר נְבֵלָה, וְלָקַח מֵאַחַת מֵהֶן, וְאֵינוֹ יוֹדֵעַ מֵאֵיזֶה מֵהֶן לָקַח — סְפֵיקוֹ אָסוּר.

With regard to nine stores in a city, all of which sell kosher meat from a slaughtered animal, and one other store that sells meat from unslaughtered animal carcasses, and a person took meat from one of them and he does not know from which one he took the meat, in this case of uncertainty, the meat is prohibited.

וּבַנִּמְצָא — הַלֵּךְ אַחַר הָרוֹב

And in the case of meat found outside, follow the majority.

What this boils down to is this. If most stores in the city sell kosher meat then a piece of meat that is found in the city (that is “outside”) is assumed to be kosher, since the majority of the stores sell only kosher meat. But if a person bought meat from one of the ten stores, but he cannot recall whether or not it was from a kosher store, the meat may not be eaten. In this latter case, we assume that there were simply an equal number of kosher and non-kosher stores. There is a 50-50 chance that the meat comes from a non-kosher store, and it may not be eaten.

By analogy, if the mouse took the morsel from one of the piles, the legal status of the morsel is that of an equally balanced uncertainty concerning whether it was taken from a pile of matzah or a pile of chametz. Consequently, the owner is required to go back and search the house all over again.

Talmudic Probability

As Dov Gabbay and Moshe Koppel noted in their 2011 paper, there is something odd about talmudic probability. If we find some meat in an area where there are p kosher stores and q non-kosher stores, then all other things being equal, the meat is kosher if and only if p > q.This is clear from the parallel text in Hullin (11a) where the underlying principal is described as זיל בתר רובא – follow the majority. Or as Gabbay and Koppel explain it:

Given a set of objects the majority of which have the property P and the rest of which have the property not-P, we may, under certain circumstances, regard the set itself and/or any object in the set as having property P.

In other words, what happens is that if there are more kosher stores than there are non-kosher, the meat is considered to have become kosher. It's not that the meat is most likely to be kosher and may therefore be eaten.  Rather it takes on the property of being kosher. 

We encountered another example of talmudic probability theory when we studied the tractate Ketuvot. There, a newly-wed husband claims that his wife was not a virgin on her wedding night. The Talmud argues that his claim needs to be set into a context of probabilities:

1. She was raped before her betrothal.

2. She was raped after her betrothal.

3. She had intercourse of her own free will before her betrothal.

4. She had intercourse of her own free will after her betrothal.

Since it is only the last of these that renders her forbidden to her husband (stay focussed and don't raise the question of a husband who is a Cohen), the husband's claim is not supported, based on the probabilities. Here is how Gubbay and Koppel explain the case - using formal logic:

Oh, and the reference to Bertrand's paradox? That is the paradox in which some questions about probability - even ones that seem to be entirely mathematical, have more than one correct solution; it all depends on how you think about the answer. One if its formulations goes like this: Given a circle, find the probability that a chord chosen at random will be longer than the side of an inscribed equilateral triangle. Turns out there are three correct solutions. Gubbay and Koppel claim that just like that paradox, the solution to many talmudic questions of probability will have more than one correct answer, depending on how you think about that answer.

Rabbi Nahum Eliezer Rabinovitch, who died in May of this year at the age of 92 was the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also had a PhD. in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  Rabbi Rabinovitch seems to have been the first to point out the relationship between Bertrand's paradox and talmudic probability theory in his 1970 Biometrika paper Combinations and Probability in Rabbinic Literature. There, the Rosh Yeshiva wrote that "the rabbis had some awareness of the different conceptions of probability as a measure of relative frequencies or a state of general ignorance."

James Franklin, in his book on the history of probability theory, notes that codes like the Talmud (and the Roman Digest that was developed under Justine c.533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the late mathematician and Rosh Yeshiva, Rabbi Rabinovitch.

“(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.”

— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.

Eruvin 100b ~ Spousal Rape

After a discussion about walking on grass on Shabbat, the Talmud switches to another topic entirely. Spousal rape:

עירובין ק, ב

ואמר רמי בר חמא אמר רב אסי אסור לאדם שיכוף אשתו לדבר מצוה שנאמר ואץ ברגלים חוטא

Rami bar Chamah said in the name of Rav Asi: It is prohibited for a man to force his wife in the conjugal mitzvah, [i.e., sexual relations,] as it is stated: “And he who hastens with his feet sins” (Proverbs 19:2). [The term “his feet” is understood here as a euphemism for intercourse.]

This statement of Rav Asi is codified in all of the major codes of Jewish law. And as we will see, it is cited in a decision of the Israeli Supreme Court which ruled spousal rape to be, well, rape. First, Maimonides, in his Mishnah Torah:

משנה תורה, הלכות דעות ה׳:ד׳

וְלֹא יֶאֱנֹס אוֹתָהּ וְהִיא אֵינָהּ רוֹצָה אֶלָּא בִּרְצוֹן שְׁנֵיהֶם וּבְשִׂמְחָתָם

A husband may not force himself sexually on his wife, if she does not consent. Rather [intercourse] should be with the consent and happiness of both husband and wife…

The Shulchan Aruch (Code of Jewish Law) addresses this in two separate rulings:

שולחן ערוך אבן אורח חיים ר״מ:י׳

אם הי' לו כעס עמה אסור לשמש עד שיפייסנה ויכול לספר עמה קודם תשמיש כדי לרצות

A man is forbidden to have intercourse with his wife if he is angry with her…

שולחן ערוך אבן העזר כ״ה:ב׳

ולא יבעול אלא מרצונה ואם אינה מרוצה יפייסנה עד שתתרצה

He may not have intercourse without her consent, and if she is not interested he should appease her until she is interested.

A History of Spousal Rape Law

So spousal rape (also called marital rape) has been forbidden in Judaism since talmudic times. But other legal traditions did not consider it a crime until recently. Take for example, criminal law in England, as outlined in The History of the Pleas of the Crown (vol. 1, ch. 58, p. 629), written by Lord Hale and published in 1736.

But the husband cannot be guilty of a rape committed by himself upon his lawful wife, for by their mutual matrimonial consent and contract the wife hath given up herself in this kind unto her husband, which she cannot retract.

Hale had died 60 years earlier, but as Chief Justice his words reflected common law of the time. It remained the law of the land until it was formally overturned by a ruling in Britain’s House of Lords in 1992, which found that there were no grounds against “declaring that in modern times the supposed marital exemption in rape forms no part of the law of England.”

“The remaining and no less difficult question is whether, despite that view, this is an area where the court should step aside to leave the matter to the Parliamentary process. This is not the creation of a new offence, it is the removal of a common law fiction which has become anachronistic and offensive and we consider that it is our duty having reached that conclusion to act upon it.”

— R. v R [1991] UKHL 12. House of Lords. (23 October 1991)

Spousal Rape In ameRican law

In 1962 the American penal code still exempted spousal rape as a crime. Back then a man could only be guilty of rape if the victim was “not his wife”:

§ 213.1. Rape and Related Offenses. (1) Rape. A male who has sexual intercourse with a female not his wife is guilty of rape if…

This changed over the next thirty years, and today spousal rape is recognized as a crime in all fifty states, although the circumstances and reporting requirements vary, and they are sometimes different to those in which the victim is not the spouse.

In New York for example, the exemption of spousal rape was ruled unconstitutional in 1984, in a ruling from the New York Court of Appeals. A man had forced sexual intercourse with his wife while they were separated (and after the wife had obtained a protection order from the court that required him to move out and remain away from the family home). The husband raised the spousal exception as a defense for rape. Here is part of the that ruling:

We find that there is no rational basis for distinguishing between marital rape and non-marital rape. The various rationales which have been asserted in defense of the exemption are either based upon archaic notions about the consent and property rights incident to marriage or are simply unable to withstand even the slightest scrutiny. We therefore declare the marital exemption for rape in the New York statute to be unconstitutional.

Spousal Rape in Israeli Law

In 1980 Israel’s Supreme Court ruled that spousal rape was a crime as heinous as any other kind of rape. One of the three justices, David Bechor (1910-2000) cited a number of Jewish sources, including Maimonides’ Mishnah Torah and the Shulchan Aruch. He also cited Ravi Asi on this page of Talmud, as well as a number of contemporary rabbis. Here is an excerpt of the opinion of Justice Bechor, and you can read the entire decision here.

...סקירת התלמוד מלמדדת אותנו, שקיון יחסי אישות באונס אינו מותר והנו בגדר עבירה...המסקנה אליה הגעתי, עולה בקנה אחד גם עם עקרונות היסוד של כבוד האישה כאדם חופשי ולא כשפחה, הנתונה לחסדי בעלה בעניין כה רגיש ועדין, עקרונות אשר, לדאבון הלב, לא מומשו בחקיקה ובפסיקה של מדינות נאורות ומתקדמת בעולם

A review of the Talmud teaches us that forced sexual relations are not  permitted and are unlawful…the conclusion at which I arrive is also in keeping with the fundamental principles that a woman is to be respected as a human being and not as a servant, who depends on her husband’s kindnesses in this sensitive area. Sadly, these principles have not been enshrined in the laws of many enlightened and advanced nations…

There are, regrettably, areas in which Jewish marital law has not caught up with modern notions of justice and equality. But on the subject of spousal rape Jewish law set a precedent which was adopted only hundreds of years later by modern western legal systems. Rav Asi’s ruling, cited by the Israeli Supreme Court, is succinct. No means no.

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.