Sukkah 29a ~ The Causes of a Solar Eclipse - and a Special Talmudology Event

On today’s page of Talmud we read that rain on Sukkkot is a bad omen. Continuing along with this theme, the Talmud describes another bad omen: a solar eclipse:

סוכה כט, א

תָּנוּ רַבָּנַן: סִימָן רַע לְכל הָעוֹלָם כּוּלּוֹ. מָשָׁל לְמָה הַדָּבָר דּוֹמֶה? לְמֶלֶךְ בָּשָׂר וְדָם שֶׁעָשָׂה סְעוּדָה לַעֲבָדָיו וְהִנִּיחַ פָּנָס לִפְנֵיהֶם, כָּעַס עֲלֵיהֶם וְאָמַר לְעַבְדּוֹ: טוֹל פָּנָס מִפְּנֵיהֶם וְהוֹשִׁיבֵם בַּחוֹשֶׁךְ

The Sages taught: When the sun is eclipsed it is a bad omen for the entire world. The Gemara tells a parable. To what is this matter comparable? It is comparable to a king of flesh and blood who prepared a feast for his servants and placed a lantern [panas] before them to illuminate the hall. He became angry at them and said to his servant: Take the lantern from before them and seat them in darkness.

תַּנְיָא רַבִּי מֵאִיר אוֹמֵר: כל זְמַן שֶׁמְּאוֹרוֹת לוֹקִין — סִימָן רַע לְשׂוֹנְאֵיהֶם שֶׁל יִשְׂרָאֵל, מִפְּנֵי שֶׁמְּלוּמָּדִין בְּמַכּוֹתֵיהֶן. מָשָׁל לְסוֹפֵר שֶׁבָּא לְבֵית הַסֵּפֶר וּרְצוּעָה בְּיָדוֹ, מִי דּוֹאֵג — מִי שֶׁרָגִיל לִלְקוֹת בְּכל יוֹם וָיוֹם הוּא דּוֹאֵג

It is taught in a baraita that Rabbi Meir says: When the heavenly lights, i.e., the sun and the moon, are eclipsed, it is a bad omen for the enemies of the Jewish people, which is a euphemism for the Jewish people, because they are experienced in their beatings. Based on past experience, they assume that any calamity that afflicts the world is directed at them. The Gemara suggests a parable: This is similar to a teacher who comes to the school with a strap in his hand. Who worries? The child who is accustomed to be beaten each and every day is the one who worries.

The Last total Solar Eclipse over America

On Monday August 21st 2017, almost exactly three years ago, my family and I witnessed a total solar eclipse over Charleston South Carolina. It was an unforgettable event. There is another total solar eclipse coming up in a couple of years, and you won’t want to miss it. We will have more to say about that below. But first let’s focus on the science and the superstition of a solar eclipse.

What causes a solar ecplipse according to The talmud?

The classic Talmudic source on the origins of a solar eclipse is found on today’s page of Talmud, in Succah 29a:

תנו רבנן: בשביל ארבעה דברים חמה לוקה: על אב בית דין שמת ואינו נספד כהלכה, ועל נערה המאורסה שצעקה בעיר ואין מושיע לה, ועל משכב זכור, ועל שני אחין שנשפך דמן כאחד

Our Rabbis taught: A solar eclipse occurs on account of four things: Because the Av Beis Din died and was not properly eulogized, because a betrothed woman was raped in a city and none came to rescue her, because of homosexuality, and because of two brothers who were murdered together.

It is challenging to find a common thread to these four events that would satisfactorily relate them to a solar eclipse, and Rashi despaired of doing so: לא שמעתי טעם בדבר—“I do not know of an explanation for this” he write. Neither do we.

What actually causes a solar eclipse?

As we now understand the phenomenon, a solar eclipse occurs when the moon gets in-between the sun and the earth. When it does, it blocks some of the sunlight and casts a shadow on the earth. A person standing in that shadow (called the umbra) will see an eclipse. The time at which the moon is directly between the sun and the earth is also the start of every Jewish month (or close to it, as we will see below). And so it is clear that a solar eclipse can only occur on (or very close to) Rosh Chodesh, the start of the new Jewish month. However, we certainly do not witness a solar eclipse on every Rosh Chodesh. The reason is that the moon’s orbit is inclined at 5 degrees from the sun-earth plane, so that each month the moon may be slightly above, or slightly below that plane. An eclipse will occur only when the three bodies line up on the same plane, which only occurs infrequently.

If we know that a solar eclipse is a regular celestial event whose timing is predictable and precise, how are we to understand this page of Talmud which suggests that it is a divine response to bad Jewish conduct? We have already noted that Rashi was unable to explain the passage, but this did not prevent others from trying to do so.

1. The Maharal’s unhelpful suggestion

The Maharal of Prague (d. 1609) has a lengthy explanation in his work Be’er Hagolah which, for the sake of clarity, we shall summarize. The Maharal acknowledged that an eclipse is a mechanical and predictable event but he further suggested that if there was no sin, there would indeed never be a solar eclipse. G-d would have designed the universe differently, and in this hypothetical sin-free universe our solar system would have been created without the possibility for a solar eclipse. The conclusion from the Maharal’s writings is that in a sin- free universe, the moon would not orbit as it does now, at a 5-degree angle to the sun-earth plane. But we now need to ask where, precisely, in a sin- free universe, would the moon be? It couldn’t be in the same plane as the sun and the earth, since then there would be a solar eclipse every month. If the moon were, say, 20° above the earth-sun plane, there would still be solar eclipses, though they would be rarer than they are today. The only way for there to be no solar eclipses in the Maharal’s imaginary sin-free universe would be for the moon to orbit the earth at 90° to the sun-earth axis. Then it would never come between the sun and the earth, and there could never be a solar eclipse. But this would lead to another problem. In such an orbit, the moon would always be visible, and so there could never be a Rosh Chodesh. The Maharal’s thought experiment seems to provide more complications than it does solutions.

2. Rabbi Yonason Eibeschutz (d.1764) and his sunspots

Another attempt to explain the Talmud was offered by R. Yonason Eibeschutz (d. 1764). In 1751, Rabbi Eibeschutz was elected as Chief Rabbi of the Three Communities (Altona, Hamburg and Wandsbek), although he was later accused of being a secret follower of the false messiah Shabtai Tzvi. In January of that year Rabbi Eibeschutz gave a sermon in Hamburg in which he addressed the very same problem that Maharal had noted: If a solar eclipse is a predictable event, how can it be in response to human conduct? His answer was quite different.

He suggested that the Talmud in Succah is not actually addressing the phenomenon that we call a solar eclipse. According to R. Eibeschutz, the phrase in Succah "בזמן שהחמה לוקה" (“when there is a solar eclipse”) actually means “when there are sunspots.” Inventive though this is, there are two problems with this suggestion. In the first place, sunspots were almost impossible to see before the invention of the telescope. The first published description of sunspots in Western literature was in 1611 by the largely overlooked Johanness Fabricius and later by a contemporary of Galileo named Christopher Scheiner (though Galileo quickly claimed that he, not Scheiner, was the first to correctly interpret what they were). Because sunspots are so difficult to see with the naked eye, it seems very unlikely (though not impossible) that this is what the rabbis in the Talmud were describing. Second, according to Eibeschutz, sunspots “have no known cause, and have no fixed period to their appearance.” However, and even by the science of his day, this claim is not correct. In fact, both Scheiner and Galileo knew—and wrote—that sunspots were permanent (at least for a while) and moved slowly across the face of the sun in a predictable way. The suggestion that these spots are a response to human activity is therefore difficult to sustain. Furthermore, while a total solar eclipse is strikingly visible to those who are in its shadow, sunspots are, as we have noted, incredibly difficult to see with the naked eye. It would therefore make little sense to declare that these invisible sunspots serve as a warning (סימן רע)to humanity. Finally, the Talmud describes the phenomenon of an eclipse (ליקוי) as being visible in only some places on the earth. While this is a perfect description of a solar eclipse, sunspot activity would be visible from any place on earth, a situation that is clearly not the one described in the Talmud.

3. Rabbi David Pardo (1718–1790)

A different suggestion was offered by R. David Pardo (1718–1790) in his work Chasdei David, published in 1796. R. Pardo acknowledged that most solar eclipses are indeed predictable events, but suggested that there are other kinds of eclipses that cannot in fact be predicted, and it is these kinds of eclipses to which the Talmud is referring. Unfortunately, this suggestion has no factual basis. There are no such phenomena as an unpredictable lunar or solar eclipse, and R. Pardo’s suggestion is untenable.

4. The explanation of the Lubavitcher Rebbe- it’s all about the weather

Rabbi Menachem Mendel Schneerson, the Lubavitcher Rebbe, also addressed the Talmudic passage, and in a 1957 responsum he wrote that while a solar eclipse was predictable, the local weather was most certainly not. It could not be predicted whether or not a solar or lunar eclipse would be visible through the clouds, and since it was this aspect that was under Divine control, it presumably could change in response to the local actions of people.

Elegant as this might be, this suggestion, too, has considerable problems. In the first place, the weather is indeed predictable, although of course the ability to predict the weather is relatively limited. But more problematic is the fact that a total solar eclipse will be completely visible whether or not there are clouds. A cloudy day will prevent a viewer on the ground from witnessing the moment of conjunction as the moon covers the disc of the sun (which, I can tell you, is pretty cool), and also prevent him from seeing the stars. However, the other effect of a total solar eclipse— darkness as though it were night—will be just as visible.

On the Molad and Astronomical Conjunction

The last solar eclipse brought to our attention another issue. It occurred on August 21, 2017, when the moon is directly between the sun and the earth (or, more technically, when the sun and the moon have the same elliptical longitude). This started at sunrise over the Pacific Ocean northeast of Hawaii, at 4:48 p.m. UTC, or 6:48 p.m. in Jerusalem. And yet the announced time for the molad of Rosh Chodesh Elul, however, was Tuesday, August 22, at 10:44 a.m. and 15 chalakim—about 16 hours later. That is odd since the molad (lit the birth of the moon) is supposed to be the exact time at which the sun, the earth and the moon all line up, at time called the astronomical conjunction.

“...the molad we announce on the Shabbat preceding Rosh Chodesh represents a theoretical time only, and has no relationship to an astronomical phenomenon. ”

The solar eclipse is therefore a visible reminder that the time of molad we announce on the Shabbat preceding Rosh Chodesh represents a theoretical time only, and has no relationship to an astronomical phenomenon. The announced molad is calculated by using the length between one new moon and the next. This figure assumes that every lunar month is of equal length, 29 days, 12 hours 44 minutes and 3 1/3 seconds. The Jewish calendar is based on the axiom that all future times of the molad are based on the theoretical time for the first molad, which was in Tishrei of the first year of Creation. This is assumed to have occurred on a Monday night, at five hours and 204 chalakim—a time that occurred only in theory since, according to Jewish tradition, the sun and the moon had not been created at that time. To determine the time of any molad since then, we simply add 29 days, 5 hours and 204 chalakim for each month from the primordial Tishrei. But this calculated time differs from the actual length of time between one new month and the next, which is not constant. For this reason, the times announced for the molad are not astronomically accurate—and, as we have seen, this can result in a discrepancy of more than 16 hours between the astronomical conjunction and the calculated Jewish conjunction. (To read more about this problem see our post here.]

HalaKhic Aspects of a Solar Eclipse

There are two categories of questions surrounding a solar eclipse. The first focuses on the technical aspects of the eclipse as a natural phenomenon, and the second on the eclipse as an omen of tragedy.

1. Publicizing the date of a forthcoming eclipse

The Mishnah Berurah rules that it is forbidden to tell another person that a rainbow is visible, because this violates the prohibition of slander (מוציא דבה), since the primordial rainbow appeared after the sins of humanity that caused Noah’s Flood.

And since a solar eclipse is, according to today’s page of Talmud, a sign of human sin, it might be suggested that it would also be forbidden to announce the time of a future solar eclipse. However, unlike a rainbow, a solar eclipse may be entirely predicted, and on the basis of this, Rabbi Avigdor Nebenzahl (b. 1935) ruled that it is permitted to publicize the dates and times of a future eclipse. (See R. Avigdor Nebenzahl, Teshuvos Avigdor HaLevi (Sifrei Kedumim: 2012), p. 249 #105.)

2. Reciting a blessing on seeing a solar eclipse

There is halachic precedent for reciting a blessing on seeing an awe-inspiring vista or event. We make a berachah on seeing the Mediterranean Sea, or a rainbow, on hearing thunder and seeing lightening, and even on seeing a person of exceptional beauty. It is perfectly understandable, therefore, for a person witnessing one of the greatest of nature’s spectacles, to wish to mark the event with a blessing. However, there appear to be no halakhic authorities who would allow a berachah to be recited. Perhaps the first to write about this was R. Menachem Mendel Schneerson. In 1957, he was asked if it was permitted to say a berachah on seeing a solar or lunar eclipse, and his reply was unequivocal:

ידוע הכלל אשר אין לחדש ברכה שלא הוזכרה בש"ס (ב"י או"ח סמ"ו). וי"ל הטעם דאין מברכין ע"ז מפני שהוא סימן לפורעניות הבאה )סוכה כט, א(. ואדרבה צריכה תפלה לבטלה וצעקה ולא ברכה

There is a well-established principle that it is forbidden to institute a blessing that is not mentioned in the Talmud. And some say that the reason that no blessing was instituted is because the eclipse is a bad omen. To the contrary, it is important to pray for the omen to be annulled, and to cry out without a berachah. (Iggerot Kodesh 15:1079.)

R. Schneerson combines a halakhic justification for not reciting a berachah with the classic Talmudic teaching from our page today that a solar eclipse occurs as a result of human sin. However, there are two problems with R. Schneerson’s ruling. First, it is normative Jewish practice to recite a berachah on hearing bad news such as the death of a person, and second, the Talmud does not describe a solar eclipse as an omen of forthcoming disaster. It is a sign of sin, not of punishment.

R. Chaim Dovid HaLevi, Av Bet Din (Chief of the Rabbinic Court) of Tel Aviv and Yaffo, also ruled that we are forbidden to create new berachos, (Aseh Lecha Rav [Tel Aviv, 5749], 150) although he understood the urge to do so:

Our Rabbis instituted blessings over acts of creation and powerful natural events, like lightning and thunder and so on. However, they did not do so for a lunar or solar eclipse. And if only today we could institute a blessing when we are aware that an eclipse is indeed an incredible natural event. But we cannot, for a person is forbidden to make up a blessing. If a person still wants to make some form of a blessing, he should recite the verses “And David blessed...blessed are you, God, the Lord of our father Israel, who performs acts of creation.”

Finally, we should note the opinion of R. David Lau, then the Chief Rabbi of Modi’in, and currently the Ashkenazi Chief Rabbi of Israel. A certain David Eisen wrote to R. Lau about his experiences of observing the (partial) solar eclipse of 2001 that could be seen in Israel. He had been left wishing to make a blessing for what was, for him, an awe-inspiring cosmic occurrence. R. Lau empathized with Eisen’s feelings, but noted that since the Rabbis of the Talmud had not prescribed a blessing over an eclipse, it was not possible to institute such a blessing today. Rabbi Lau noted that his own religious response to witnessing the eclipse had been to say Psalm 19, “The Heavens tell of G-d’s glory,” and Psalm 104, “My soul will bless G-d.”

3. Marriage and fasting on the day of a solar eclipse

The Chasidic leader R. Zvi Elimelech Shapira of Dinov (b. 1785), wrote in his classic work Bnei Yissaschar that a man should not marry when the moon is waning, “and particularly not during a lunar eclipse, G- d forbid.”(Bnei Yissaschar, Ma’amarei Rosh Chodesh, #2.) He does not mention whether this would apply to a solar eclipse. The Mishnah Berurah also notes the opinion of the Sefer Chasidim that one should fast on the day of a lunar eclipse, although he does not rule on the matter further (Mishnah Berurah #580:2). The matter was more recently addressed by R. Menachem Lang, who notes that it might be forbidden to marry on the day of any kind of eclipse, but ultimately ruled that there is no such prohibition. When a solar eclipse occurs on the same day as Rosh Chodesh, any fast would be forbidden under the general prohibition of fasting on Rosh Chodesh (Mishnah Berurah #580:1).

[Most of this post comes from an essay published in Hakirah in 2017. You can read the entire essay here.]

A SPECIAL ECLIPSE ANNOUNCEMENT FROM TALMUDOLOGY

Here is the great news: there is another total solar eclipse coming soon to North America!

On Monday April 8, 2024 another total solar eclipse will be visible over North America. If the weather cooperates, it will be seen along a narrow path that starts from Mexico's Pacific coast, passes through several American states, and ends on the Atlantic coast of Canada. The rest of mainland United States and Canada, and parts of the Caribbean, Central America, and Europe will see a partial solar eclipse, which is nowhere nearly as spectacular and is often not even noticeable. The next time you will get to see a total solar eclipse in the US after that will not be until August 2045. So plan now!

Image from here

Join Talmudology for the Great American Solar Eclipse II

The Talmudology Excursions Division is in the early stages of planning a special weekend of events in preparation of this amazing experience. As you can see from the map above, Cleveland Ohio will be in the center of the path of the totality, and that is likely where the Talmudology Eclipse Gathering will be.

Sign up below to receive updates about this unique event. We hope to see you in April 2024. (It will be here sooner than you think.)

Sukkah 26b ~ How do Horses Sleep?

In a discussion about various aspects of sleep, the Talmud today discusses equine sleeping patterns:

סוכה כו, ב

אָמַר רַב: אָסוּר לָאָדָם לִישַׁן בַּיּוֹם יוֹתֵר מִשֵּׁינַת הַסּוּס. וְכַמָּה שֵׁינַת הַסּוּס — שִׁיתִּין נִשְׁמֵי

Rav said: It is prohibited for a person to sleep during the day longer than the duration of the sleep of a horse. [One who sleeps for longer is derelict in the study of Torah.] And how long is the duration of the sleep of a horse? It is sixty breaths long.

Elsewhere, Rabbi Zeira extols the virtues of sleeping like a horse at night. It is what King David did.

ברכות ג, ב

רַבִּי זֵירָא אָמַר: עַד חֲצוֹת לַיְלָה הָיָה מִתְנַמְנֵם כְּסוּס, מִכָּאן וְאֵילָךְ הָיָה מִתְגַּבֵּר כַּאֲרִי. רַב אָשֵׁי אָמַר: עַד חֲצוֹת לַיְלָה הָיָה עוֹסֵק בְּדִבְרֵי תוֹרָה, מִכָּאן וְאֵילָךְ בְּשִׁירוֹת וְתִשְׁבָּחוֹת

Rabbi Zeira said: Until midnight, David would doze like a horse, [as a horse dozes, but never sleeps deeply]. From midnight on, he would gain the strength of a lion. Rav Ashi said: Until midnight, he would study Torah, as it is written: “I rose with the neshef and cried, I hoped for Your word,” and from midnight on, he would engage in songs and praise, as it is written: “At midnight I rise to give thanks.”

To which the medieval French commentator Rashi add this observation

רשי שם

מתנמנם כסוס – עוסק בתורה כשהוא מתנמנם. כסוס הזה שאינו נרדם לעולם אלא מתנמנם ונעור תמיד

The horse never sleeps. Instead it naps and then always wakes

Most of us don’t keep horses, and so cannot judge what exactly is going on. So today, for all you non-horse owning folk, Talmudology asks: How do horses sleep?

An Introduction to Equine Sleep

The thing is, for a long time the way horses sleep has been a bit of a mystery. Then we developed all kinds of clever recording devices like EEGs that allowed us a harmless peek inside the brain of the sleeping horse. In his review of sleep patterns in the horse published in the September 1990 edition of Equine Practice, the veterinarian Theodore Belling wrote that “usually a horse falls asleep while standing and is in a drowsing state. The lids are partially open and the head hangs down at a medium level.” As the horse slips into a deep sleep known by its electric patterns as Slow Wave Sleep its head gradually gets lower and lower until the horse decides all is good, and while remaining asleep, actually lies down. Like this:

Once in this position, known technically as lateral recumbency, “the horse sleeps in short cycles or episodes, rather than one continuous interval, as it does in humans.” There is a short period of slow wave or deep sleep, usually lasting about five minutes, followed by a five minute paradoxical sleep, a stage also known as the rapid eye movement (REM) period. (When we enter the REM period, that’s when we dream.) The horse then enters another five minute slow wave cycle, and then wakes up. It slept for a total of about 15 minutes and will stay awake for another 45 minutes or so, before doing the whole thing again. The total sleep time of the horse is somewhere between 2.5-5 hours per night. Oh, and if you drive past a field with a horse sleeping in the day, there is nothing wrong. Some horses like to do that.

“Horses are neither diurnal nor nocturnal but have intermittent periods of rest and sleep during the day with most of their sleep happening at night, particularly when confined in a stall. Certain relaxing situations, such as grooming may even lead to episodes of sleep in normal horses. This is, perhaps, the equine equivalent of sleeping in class.”

— Aleman M. et al. Sleep and Sleep Disorders in Horses. AAEP Proceedings 2008. 54: 180-185.

EEG recordings from a pony. From From Andre Dellaire. Rest Behavior. Veterinary Clinics of North America Equine Practice 1986:2 595.

This basic outline has been confirmed in a number of other studies. A 2008 paper in Proceedings of the American Association of Equine Practitioners noted that “horses require far less sleep than most humans, averaging a total of only 3–5 h/day” though foals, being baby horses, sleep more per day than do the adults.

And so it turns out that the horse sleeps a lot less than might be expected. Here for example are the times spend awake or asleep in a few different species. The horse is the most wakeful of them all.

Time Spent (Hours) in Wakefulness and Sleep States per Day in Domesticated Species
Animal	Awake	Drowsy	Deep Sleep	Dreaming	Total Sleep Time
Cat	10	2	8.5	3.5	12
Dog	11	4	6.5	2.5	9
Swine	11	4	6.5	2.5	8
Sheep	16	4	3	<1	4
Horse	18	2	3	<1	4
From Andre Dellaire. Rest Behavior. Veterinary Clinics of North America Equine Practice 1986:2 592.

The Rav and Rabbi Zeira clearly knew about horses. The horse does indeed only take short naps, and these rabbis extol us to do the same. These are examples of a careful observation the natural world, from where many talmudic statements are derived. This might seem obvious, but it is not. As we have noted in the past, the rabbis didn’t always demonstrate an accurate understanding of the animal kingdom. Abayye thought that wild geese have a scrotum. But they don’t. And Rabbi Shimon ben Chalfta thought that ant colonies have kings. But they don’t. At the same time the rabbis knew much about the world that we had to rediscover. Like sound traveling further at night, or the periodicity of ‘Halley’s’ Comet.

So the next time you see horses lying in a field, or even dozing while standing, don’t disturb them. They are doing what comes naturally. Instead, do what the rabbis did. Observe, and ask: what can I learn from this moment?

NEXT TIME ON TALMUDOLOGY:

SOLAR ECLIPSES

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.

Sukkah 7b ~The value of Pi in the Talmud

We have discussed the value of pi in the Talmud on several previous occasions. But today’s page of Talmud contains one of the important sources on this topic. So let’s take another look at this most magical of numbers, pi, written as the Greek letter π.

The Talmud is discussing the minimum legal size of a sukkah, the temporary booth that Jews must live in for seven days each fall. The Talmud then makes this decalaration:

סוכה ז, ב

כֹּל שֶׁיֵּשׁ בְּהֶקֵּיפוֹ שְׁלֹשָׁה טְפָחִים יֵשׁ בּוֹ רוֹחַב טֶפַח

For every three handbreadths circumference in a circle, there is a diameter of one handbreadth.

This means that the value of value of π is 3. Elsewhere the Talmud determines this value, and uses a verse in the Book of Kings:

מלכים א פרק ז פסוק כג 

ויעש את הים מוצק עשר באמה משפתו עד שפתו עגל סביב וחמש באמה קומתו וקוה שלשים באמה יסב אתו סביב 

And he made a molten sea, ten amot from one brim to the other: it was round, and its height was five amot, and a circumference of thirty amot circled it.

One of the vessels in the Temple of Solomon was ten amot in diameter and 30 amot in circumference. Since π is the ratio of the circumference of a circle to its diameter (π=c/d), π in the Book of Kings is 30/10=3. Three - no more and no less. From this verse another page of Talmud teaches a general rule:

תלמוד בבלי מסכת עירובין דף יד עמוד א 

כל שיש בהיקפו שלשה טפחים יש בו רחב טפח. מנא הני מילי? - אמר רבי יוחנן, אמר קרא : ויעש את הים מוצק עשר באמה משפתו עד שפתו עגל סביב וחמש באמה קומתו וקו שלשים באמה יסב אתו סביב 

Whatever circle has a circumference of three tefachim must have a diameter of one tefach.

But PI is more than three

However, this value of π =3 is not accurate. It deviates from the true value of π (3.1415...) by about 5%. In another tractate, the medieval gloss on the Talmud known as Tosafot is bothered by this too.

תוספות, עירובין יד א

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

Tosafot opens the objection with these words: “But [pi] is a little more [than 3]. Which means that the value [of pi] is rounded down” Tosafot can't find a good answer to this obvious problem, and concludes "this is difficult, because the result [that pi=3] is not precise, as demonstrated by those who understand geometry." 

PI IN THE RAMBAM

In his commentary on the Mishnah on which today’s discussion is based, (Eruvin 1:5) Maimonides makes the following observation:

פירוש המשנה לרמב"ם מסכת עירובין פרק א משנה ה 

צריך אתה לדעת שיחס קוטר העיגול להקפו בלתי ידוע, ואי אפשר לדבר עליו לעולם בדיוק, ואין זה חסרון ידיעה מצדנו כמו שחושבים הסכלים, אלא שדבר זה מצד טבעו בלתי נודע ואין במציאותו שיודע. אבל אפשר לשערו בקירוב, וכבר עשו מומחי המהנדסים בזה חבורים, כלומר לידיעת יחס הקוטר להקיפו בקירוב ואופני ההוכחה עליו. והקירוב שמשתמשים בו אנשי המדע הוא יחס אחד לשלשה ושביעית, שכל עיגול שקוטרו אמה אחת הרי יש בהקיפו שלש אמות ושביעית אמה בקירוב. וכיון שזה לא יושג לגמרי אלא בקירוב תפשו הם בחשבון גדול ואמרו כל שיש בהקיפו שלשה טפחים יש בו רוחב טפח, והסתפקו בזה בכל המדידות שהוצרכו להן בכל התורה

...The ratio of the diameter to the circumference of a circle is not known and will never be known precisely. This is not due to a lack on our part (as some fools think), but this number [pi] cannot be known because of its nature, and it is not in our ability to ever know it precisely. But it may be approximated ...to three and one-seventh. So any circle with a diameter of one has a circumference of approximately three and one-seventh. But because this ratio is not precise and is only an approximation, they [the rabbis of the Mishnah and Talmud] used a more general value and said that any circle with a circumference of three has a diameter of one, and they used this value in all their Torah calculations.

Is the Value of Pi hidden in the Bible?

There are lots of papers on the value of pi in the the Bible. Many of them mention an observation that seems to have been incorrectly attributed to the Vilna Gaon.  The verse we cited from מלאכים א׳ spells the word for line as קוה, but it is pronounced as though it were written קו.  (In דברי הימים ב׳ (II Chronicles 4:2) the identical verse spells the word for line as קו.)  The ratio of the numerical value (gematria) of the written word (כתיב) to the pronounced word (קרי) is 111/106.  Let's have the French mathematician Shlomo Belga pick up the story - in his paper (first published in the 1991 Proceedings of the 17th Canadian Congress of History and Philosophy of Mathematics, and recently updated), he gets rather excited about the whole gematria thing:

A mathematician called Andrew Simoson also addresses this large tub that is described in מלאכים א׳ and is often called Solomon's Sea. He doesn't buy the gematria, and wrote about it in The College Mathematics Journal.

A natural question with respect to this method is, why add, divide, and multiply the letters of the words? Perhaps an even more basic question is, why all the mystery in the first place? Furthermore, H. W. Guggenheimer, in his Mathematical Reviews...seriously doubts that the use of letters as numerals predates Alexandrian times; or if such is the case, the chronicler did not know the key. Moreover, even if this remarkable approximation to pi is more than coincidence, this explanation does not resolve the obvious measurement discrepancy - the 30-cubit circumference and the 10-cubit diameter. Finally, Deakin points out that if the deity truly is at work in this phenomenon of scripture revealing an accurate approximation of pi... God would most surely have selected 355/113...as representative of pi...

Still, what stuck Simoson was that "...the chroniclers somehow decided that the diameter and girth measurements of Solomon's Sea were sufficiently striking to include in their narrative." (If you'd like another paper to read on this subject,  try this one, published in B'Or Ha'Torah - the journal of "Science, Art & Modern Life in the Light of the Torah." You're welcome.)

Did the rabbis of the Talmud get π wrong?

So what are we to make of all this? Did the rabbis of the Talmud get π wrong, or were they just approximating π for ease of use?  After considering evidence from elsewhere in the Mishnah (Ohalot 12:6 - I'll spare you the details), Judah Landa, in his book Torah and Science, has this to say:

We can only conclude that the rabbis of the Mishnah and Talmud, who lived about 2,000 years ago, believed that the value of pi was truly three. They did not use three merely for simplicity’s sake, nor did they think of three as an approximation for pi. On the other hand, rabbis who lived much later, such as the Rambam and Tosafot (who lived about 900 years ago), seem to be acutely aware of the gross inaccuracies that results from using three for pi. Mathematicians have known that pi is greater than three for thousands of years. Archimedes, who lived about 2,200 years ago, narrowed the value of pi down to between 3 10/70 and 3 10/71 ! (Judah Landa. Torah and Science. Ktav Publishing House 1991. p.23.)

Still, don't be too hard on the rabbis of the Talmud. The rule that the circumference of an object is three times its diameter is pretty close to being correct, and is usually a good enough approximation. But it is not accurate, and never will be.