Talmudology Redux ~ Could the Vilna Gaon Have Stayed Awake Over Sukkot?

The Vilna Gaon, (lit. The Genius of Vilna) lived from 1720-1797, and today, the 19th of Tishrei, is the Hebrew anniversary of his death. He was one of the leading talmudic and halakhic authorities of his time. His works are widely studied to this day, and his legal opinions are often cited. Today’s question is: did the Vilna Gaon stay awake for several consecutive days one Sukkot? Here is the background:

The Vilna Gaon.

In January 1788 the Vilna Gaon was involved in the kidnapping of a young Jewish man who had converted to Christianity. For his role in the kidnapping Rabbi Eliyahu (the given name of the Gaon) was arrested and held for over a month.  The case was later tried, and on September 15, 1789 (sic) the Gaon of Vilna, together with others involved in the kidnapping, was sentenced to twelve weeks in jail.  Although it is unclear how long he was imprisoned, he was there over Sukkot, and the Lithuanian authorities were hardly in the practice of providing imprisoned Jews with a sukkah.  But since one is not permitted to sleep outside of the sukkah, what was the Gaon to do?  Simple.  He’d stay awake, and by doing so he would not transgress the prohibition of sleeping outside the sukkah.  Here’s how the episode is described in the work Tosefet Ma’aseh Rav published in 1892. 

Our Leader, teacher and Rabbi may he rest in peace, when he was imprisoned on Sukkot, tried with all his strength, and walked from one place to another, and held his eyelids open, and made an extraordinary effort not to sleep outside the sukkah – not even a brief nap – until the authorities released him to a sukkah.

We don’t know for how many nights the sixty-nine year-old Rabbi Eliyahu stayed awake, but is such a claim even plausible? As it turns out, it is.

The World Record for Staying Awake

The world record for staying awake is an amazing eleven days. Eleven days – that’s 264 hours (and 24 minutes to be precise) without sleep. It was set in 1965 by Randy Gardner, who was then seventeen years old. Gardner seems to have suffered little if any harm by his marathon period of sleep deprivation. But don’t try and beat the record. The Guinness Book of Records no longer has an entry for staying awake – because it is considered too dangerous an ordeal to undertake. (You can hear a review of sleep deprivation stunts in general and a wonderful interview with Gardner himself here.)

The Health Risks of Sleep Deprivation

What are the health risks of prolonged sleep deprivation? A 1970 study of four volunteers who stayed awake for 205 hours (that’s eight and a half days!) noted some differences in how the subjects slept once they were allowed to do so, but follow-up testing of the group conducted 6-9 months after the sleep deprivation showed that their sleep patterns were similar to the pre-deprivation recordings.

Although Randy Gardner and these four volunteers seem to have suffered no long-term health consequences of staying awake for over a week, scientists have long noted that sleep deprivation is rather bad for the body. Or to be more precise, the bodies of unfortunate laboratory rats who are not allowed to sleep. In these animals, prolonged sleep deprivation causes the immune system to malfunction. This results in infection and eventual lethal septicemia. The physiologist who kept these rats awake noted that there are “far-reaching physical implications resulting from alterations in immune status [which] may explain why sleep deprivation effects are risk factors for disease and yet are not well defined or specifically localized.” In other words, sleep deprivation makes rats really sick, but we don’t know how, or why…

One possible explanation was suggested in 2013 by a group from the University of Rochester Medical Center. They demonstrated that during sleep, the space between the cells of the brain (the interstitial space) increased by up to 60%, allowing toxic metabolites to be cleared. This raises the question of whether the brain sleeps in order to expel these toxic chemicals, or rather it is the chemicals themselves that drive the brain to switch into a sleep state.

The extracellular (interstitial) space in the cortex of the mouse brain, through which cerebral spinal fluid moves, increases from 14% in the awake animal to 23% in the sleeping animal, an increase that allows the faster clearance of metabolic waste products and toxins. From Suzana Herculano-Houzel. Sleep it out. Science 2013: 342; 316

Not having any sleep is bad for your health - but so too is going without enough sleep. Chronic restriction of sleep to six hours or less per night can produce cognitive performance deficits equivalent to up to two nights of total sleep deprivation. So be sure to get a full night's rest.

Hard, But Not Impossible

The world record set by Randy Gardner has implications for a talmudic decision. In the tractate Shavuot (26b) Rabbi Yochanan ruled that since it was impossible to stay awake for more than three days, any vow to do so is considered to have been a vow made in vain – and punishment follows swiftly.  Here is how the ArtScroll Talmud explains this ruling, (based on the explanation of the Ran).

Since it is impossible for a person to go without sleep for three days, the man has uttered a vain oath. Hence, he receives lashes for violating the prohibition (Exodus 20:7): לא תשא את שם ה׳ אלוקיך לשוה, You shall not take the Name of Hashem, your God, in vain. And since the oath -being impossible to fulfill -has no validity, he is not bound by it at all and may sleep immediately.

Maimonides codified this law and also assumed that it is impossible to stay awake for three consecutive days. 

רמב"ם הלכות שבועות פרק ה הלכה כ

נשבע שלא יישן שלשת ימים, או שלא יאכל כלום שבעת ימים וכיוצא בזה שהיא שבועת שוא, אין אומרין יעור זה עד שיצטער ויצום עד שיצטער ולא יהיה בו כח לסבול ואח"כ יאכל או יישן אלא מלקין אותו מיד משום שבועת שוא ויישן ויאכל בכל עת שירצה

If a person swore that he would not sleep for three days, or would not eat anything for seven days, or something similar, this is considered a false oath. We do not tell the person to stay awake until it is impossible to do so, or fast for as long as possible until the discomfort is too great to bear, and then allow him to eat or sleep. Rather he is punished with lashes immediately for making a false oath, and is then allowed to sleep or eat as much as he wants…

Based on Randy Gardner’s feat, Rabbi Yochanan was incorrect when he ruled that it was impossible to stay awake for three days. It's certainly not impossible, but that hardly means it's a good idea to try. 

It also means that the story of the Gaon of Vilna’s sleepless nights in that cold prison might indeed have occurred. Still it is best not to disrupt your usual sleep patterns. Perhaps that is why Rabbi Chaninah ben Chachina'i in Masechet Avot  (3:4) taught that one who stays awake at night "will forfeit with his life."  Now that's a warning to heed.

רבי חנינא בן חכינאי אומר הנעור בלילה ... הרי זה מתחייב בנפשו

HAPPY SUKKOT FROM TALMUDOLOGY

“Sleep deprivation reduces learning, impairs performance in cognitive tests, prolongs reaction time, and is a common cause of seizures. In the most extreme case, continuous sleep deprivation kills rodents and flies within a period of days to weeks. In humans, fatal familial or sporadic insomnia is a progressively worsening state of sleeplessness that leads to dementia and death within months or years. ”

— Lulu Xie et al. Sleep Drives Metabolite Clearance from the Adult Brain. Science 2013. 342.317

Sukkot Redux I ~ Gauss, Tosafot, and Counting the Sukkot Sacrifices

We are currently celebrating the seven-day festival (or eight outside Israel) of Sukkot (Tabernacles). When the Temple in Jerusalem stood, it was a time when a large number of animals were sacrificed. A very large number. For each of the seven days of the festival, in addition to two rams, fourteen lambs and one goat there were a number of bulls that were sacrificed. Thirteen on the first day, twelve on the second, eleven on the third, ten on the fourth, nine on the fifth, eight on the sixth and finally seven bulls on the last day.

במדבר 29:12-34

וּבַחֲמִשָּׁה עָשָׂר יוֹם לַחֹדֶשׁ הַשְּׁבִיעִי מִקְרָא־קֹדֶשׁ יִהְיֶה לָכֶם כָּל־מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ וְחַגֹּתֶם חַג לַה׳ שִׁבְעַת יָמִים׃ 

וְהִקְרַבְתֶּם עֹלָה אִשֵּׁה רֵיחַ נִיחֹחַ לַה פָּרִים בְּנֵי־בָקָר שְׁלֹשָׁה עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם יִהְיוּ…׃ 

וּבַיּוֹם הַשֵּׁנִי פָּרִים בְּנֵי־בָקָר שְׁנֵים עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם׃… 

וּבַיּוֹם הַשְּׁלִישִׁי פָּרִים עַשְׁתֵּי־עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם׃… 

and so on…

Numbers 29:34

Pablo Picasso: “Bull,” 1945

On the fifteenth day of the seventh month, you shall observe a sacred occasion: you shall not work at your occupations.—Seven days you shall observe a festival of the LORD.— 

You shall present a burnt offering, an offering by fire of pleasing odor to the LORD: Thirteen bulls of the herd, two rams, fourteen yearling lambs; they shall be without blemish. 

The meal offerings with them—of choice flour with oil mixed in—shall be: three-tenths of a measure for each of the thirteen bulls, two-tenths for each of the two rams, 

and one-tenth for each of the fourteen lambs. 

And there shall be one goat for a sin offering—in addition to the regular burnt offering, its meal offering and libation. 

Second day: Twelve bulls of the herd, two rams, fourteen yearling lambs, without blemish… 

Third day: Eleven bulls, two rams, fourteen yearling lambs, without blemish…

and so on… 

So how many bulls would the manager of the Temple inventory have to make ready for the entire festival of Sukkot? Well, you could just add them up, which is not too hard (13+12+11+10+9+8+7=70), but there is another way, which is found in the medieval commentary known as Tosafot, on page 106 of the tractate Menachot. Let’s take a look. 

Another offering, another math problem

We read there that the mincha offering is accompanied by a minimum of a one-issaron measure of flour. But a mincha can also be accompanied by a multiple of that number, up to a maximum of 60 issronot. What happens if a person vows to bring a specific number of issronot of flour to accompany a mincha offering but cannot recall how many he had in mind? What number of issronot of flour should he offer? Well it’s a bit tricky. The sages ruled that a single offering using the full sixty issronot of flour is all that needs to be brought. But the great editor of the Mishnah, Rabbi Yehudah Hanasi disagreed. In a spectacular way. Here is the discussion:

מנחות קו, א

תנו רבנן פירשתי מנחה וקבעתי בכלי אחד של עשרונים ואיני יודע מה פירשתי יביא מנחה של ששים עשרונים דברי חכמים רבי אומר יביא מנחות של עשרונים מאחד ועד ששים שהן אלף ושמונה מאות ושלשים

The Sages taught in a baraita: If someone says: I specified that I would bring a meal offering, and I declared that they must be brought in one vessel of tenths of an ephah, but I do not know what number of tenths I specified, he must bring one meal offering of sixty-tenths of an ephah. This is the statement of the Rabbis. Rabbi Yehuda HaNasi says: He must bring sixty meal offerings of tenths in sixty vessels, each containing an amount from one-tenth until sixty-tenths, which are in total 1,830 tenths of an ephah.

Since there is a doubt as to the true intentions of the vow, Rabbi Yehudah HaNasi covers all the bases and requires that every possible combination of a mincha offering be brought. So you start with one mincha offering accompanied with one issaron of flour, then you bring a second mincha offering accompanied with two issronot of flour, then you bring a third mincha together with three issronot, and so on until you reach the maximum number of issronot that can accompany the mincha - that is until you reach sixty. The total number of Rabbi Yehudah HaNasi’s mincha offerings is then calculated: 1,830.

How did the Talmud arrive at that number? We are not told, and presumably you simply add up the series of numbers 1+2+3+4….+59+60, which gives a total of 1,830. That certainly would work. But Tosafot offers a neat mathematical trick to figure out the sum of a mathematical sequence like this:

שהן אלף ושמונה מאות ושלשים. כיצד קח בידך מאחד ועד ששים וצרף תחילתן לסופן עד האמצע כגון אחד וששים הם ס"א שנים ונ"ט הם ס"א ושלש ונ"ח הם ס"א כן תמנה עד שלשים דשלשים ושלשים ואחד נמי הם ס"א ויעלה לך שלשים פעמים ס"א

How did we arrive at 1,830? Take the series from 1 to 60 and add the sum of the first to the last until you get to the middle. Like this: 1+60=61; 2+59=61; 3+58=61. Continue this sequence until you get to 30+31 which is also 61. You will have 30 sets of 61 (ie 1,830).

Tosafot continues with the math lesson, and let’s us know the total number of bulls sacrificed over the seven day festival of Sukkot:

וכן נוכל למנות פרים דחג דעולין לשבעים כיצד ז' וי"ג הם עשרים וכן ח' וי"ב הם עשרים וכן ט' וי"א הם כ' וי' הרי שבעים

This method may also be used to count the number of sacrificial bulls on Sukkot, which are a total of 70. How so? [There are thirteen offered on the first day of sukkot, and one fewer bull is subtracted each day until the last day of sukkot, on which seven bulls are offered.] 13+7=20; 12+8=20; 11+9=20… [There are a total of 3 pairs of 20+ an unpairable 10]= 70.

In mathematical terms, the Tosafot formula for the sum (S) of the consecutive numbers in Rebbi’s series, where n is the number of terms in the series and P is the largest value, is S= n(P+1)/2. Which reminds us of…

Carl Friedrich Gauss

Carl Friedrich Gauss (1777-1855) was one of the world’s greatest mathematicians. He invented a way to calculate the date of Easter (which is a lot harder than you’d think), and made major contributions to the fields of number theory and probability theory. He gave us the Gaussian distribution (which you might know as the ”bell curve”) and used his skills as a mathematician to locate the dwarf planet Ceres. The British mathematician Henry John Smith wrote about him that other than Isaac Newton, “no mathematicians of any age or country have ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute rigorousness in demonstration, which the ancient Greeks themselves might have envied.”

There is a delightful (though possibly apocryphal) story about Gauss as a bored ten-year old sitting in the class of Herr Buttner, his mathematics teacher. There are at least 111 slightly different versions of the story, but here is one, as told by Tord Hall in his biography of Gauss:

When Gauss was about ten years old and was attending the arithmetic class, Buttner asked the following twister of his pupils. “Write down all the whole numbers from 1 to 100 and add their sum…The problem is not difficult for a person familiar with arithmetic progressions, but the boys were still at the beginner’s level, and Buttner certainly thought that he would be able to take it easy for a good while. But he thought wrong. In a few seconds, Gauss laid his slate on the table, and at the same time he said in his Braunschweig dialect: “Ligget se” (there it lies). While the other pupils added until their brows began to sweat, Gauss sat calm and still, undisturbed by Buttner’s scornful or suspicious glances.

How had the child prodigy solved the puzzle so quickly? He had added the first number (1) to the last number (100), the second number (2) to the second from last number (99) and so on. Just like Tosafot suggested. The sum of each pair was 101 and there were 50 pairs. And so Gauss wrote the answer on his slate board and handed it to Herr Buttner. It is 5,050.

THE Number of bulls=70

Gauss was raised as a Lutheran in the Protestant Church, and so he did not learn of this method from reading Tosafot. But it is delightful to learn that the same mathematical method that launched Gauss into his career as a mathematician predated him by at least four-hundred years and can be found on page 106a of Menachot, where it also applies to the festival of Sukkot.

Happy Sukkot from Talmudology

Eruvin 56a ~ The Identity of the Constellation Eglah

In order to set the boundaries of a city with regards to where it may be permissible to carry, the Talmud states that one should “square” it, meaning an imaginary square is drawn to include within it the entire city.

This is a simple enough instruction, but we are not done. The sides of this imaginary square are to be aligned with the four cardinal directions, North, East, South and West. We are not told why this must be done. Instead the Talmud explains how this squaring is done. Here is one suggestion.

עירובין נו, א

תָּנוּ רַבָּנַן: בָּא לְרַבְּעָהּ — מְרַבְּעָהּ בְּרִיבּוּעַ עוֹלָם, נוֹתֵן צְפוֹנָהּ לִצְפוֹן עוֹלָם וּדְרוֹמָהּ לִדְרוֹם עוֹלָם, וְסִימָנָיךְ: עֲגָלָה בַּצָּפוֹן, וְעַקְרָב בַּדָּרוֹם. 

With regard to the measurements of a city’s boundaries, the Sages taught the following baraita: If, in order to measure the Shabbat limit, one comes to square a city, i.e., to extend the city’s boundaries to include all of its protrusions within an imaginary square, he squares it so that the sides of the square align with the four directions of the world. He sets the northern side of the square to align with the north of the world, and its southern side to align with the south of the world. And your sign by which you can recognize the directions of the world is as follows: The constellation of “eglah” is in the north and Scorpio is in the south. The directions of the city are determined by these constellations.

The Bull vs The Bear

These two constellations should be easy to identify. Let’s start with the second one mentioned. The word עַקְרָב means a scorpion, and Scorpio is one of the twelve signs of the zodiac. Good. But what about the first constellation eglah or agalah (there is a big difference as we will see). To what constellation might this refer? Agalah - עֲגָלָה means either a “wagon” or, when the same letters are vocalized as eglah, a “calf.” And then things really get interesting.

Eglah is Taurus

TheArtScroll English Talmud indeed identifies עֲגָלָה with Taurus, (as does the ArtScroll Hebrew translation). This would depend on vocalizing the word as “eglah” meaning a calf. This would most likely identify it as the bovine constellation we know as Taurus, the “Bull.” This was also the opinion of the great medieval commentator Rashi. He doesn’t explain the word’s meaning on this page of Talmud, but he does elsewhere. In the tractate Berachot (58b) he explains the meaning of the phrase רישא דעגלא - “the head of the eglah” as the constellation Bull, or Taurus:

רשי ברכות נח,ב

רישא דעגלא – ראשו של עגל והיינו מזל שור

But we are not done. When these two constellations are mentioned in Pesachim (94b), the medieval commentary known as Tosafot remarks that eglah cannot be Taurus, (and Scorpio cannot be Scorpio). It is all to do with a description of the universe that we cannot get into now, but will do so on February 23rd next year, when we study that page in the Daf Yom cycle. Here is that Tosafot:

תוס׳ פסחים צד, ב

מעולם לא מצינו עגלה בדרום ועקרב בצפון - צ"ל דעגלה לאו היינו מזל שור כדפי' בקו' ועקרב נמי אינו עקרב די"ב מזלות דבפ"ק דראש השנה (דף יא:) קאמר די"ב מזלות לעולם ששה למטה מן הארץ וששה למעלה וכשהא' עולה שכנגדו שוקע והכא אמר שאינן זזים ממקומן ושניהם לעולם למעלה אלא אחרים הם

So to sum, Rashi believed that eglah is Taurus - and that is the ArtScroll understanding. Tosafot claimed it cannot be Taurus, though he does not offer an alternative. Now let’s consider some more contemporary translations and explanations.

Eglah is Ursa Major

The Koren (Steinsaltz) English Talmud identifies eglah as another constellation entirely, and one that is not part of the twelve signs of the zodiac. It is called Ursa Major, “The Great Bear.” Ursa Major was called Ἄρκτος μεγάλη Arktos Megale - The Great Bear - by the second century astronomer Ptolemy, and was long associated with things north. (That’s where we derive the word arctic.) So this description could certainly have been known to the rabbis of the Talmud.

The classic Soncino English Talmud translates עֲגָלָה as “The Great Rear.” And it’s not a typo in which an “R” replaced a “B.” But why the Great Rear? Well as you can see from the image below, there are seven stars within the Ursa Major that are known as the Big Dipper. And where are they located? At the very rear of the bear.

Of course that only works if you imagine the stars forming a bear in a particular way. Here for example is how H.A. Rey - the creator of the Curious George series - depicted the The Great Bear in his wonderful book The Stars: A New Way to See Them. As you can see, the Great Bear is now made up by a very different set of lines, and the Big Dipper is no longer at its “Great Rear” but is instead part of the head of the bear.

H.A Rey. The Stars: A New Way to See Them. Houghton Mifflin 1980. 35.

So not everyone looks at a constellation and draws the same images. Here is another example, from Goldshmidt’s German translation of the Talmud which reads the word not as eglah but as as agalah: “der Wagen.” Actually, Ursa Major or more precisely seven of its stars that are called the Big Dipper was once called Charles’ Wain, a name that came from the

…Middle English Charlewayn, from Old English carles wǣn, apparently from a common Proto-Germanic *karlas wagnaz (cognate with forms in other Germanic languages). It seems that this common Germanic name originally meant the ‘peasant's wagon’ (the churls' wagon) in contrast to the ‘woman's wagon’ (Ursa Minor). Later it was interpreted as ‘Charles's wagon’ and associated with Charlemagne.

So in another culture the seven stars of the Big Dipper were seen as a wagon. Which is precisely how you could vocalize the Hebrew word in question: agalah. If you take a look at the stars it is easy to see why. But in Holland the stars are popularly known as the "Saucepan" (Steelpannetje). Which you can also make out. It’s all in the eye of the beholder.

Here is a summary of what we found:

The Meaning and Pronunciation of the Constellation  "עגלה"
	Eglah = Calf	Agalah = Wagon
Rashi	Taurus	X
Tosafot	Not Taurus	X
Goldscmidt (German)		The Wagon = The Big Dipper
Soncino (English)	X	The Big Rear = The Big Dipper
ArtScroll (English & Hebrew)	Taurus	X
Koren (English)	X	Ursa Major

Which of these possibilities, Taurus, Ursa Major, or the Big Dipper, is the most likely? To find out let’s do some astronomy.

Taurus is a large constellation that is best seen (in the Northern Hemisphere) from November to February. In late November and December it can be seen the entire night. However by late March it appears for only a short time before sunrise and then almost completely disappears in the summer months. Although Taurus is always found in the northern sky moving from northeast to northwest, because it is sometimes only barely visible for an hour or so right before sunrise it could not always be used to find North.

Taurus. Image from the excellent app StarWalk 2.

Ursa Major “The Great Bear” is the third largest constellation in the sky, and is visible for the entire year. This constellation is circumpolar, meaning it never sets below the horizon. And because it is always near the north celestial pole, it is always in the northern part of the sky. So it could reliably be used year round to identify north.

As part of Ursa Major, the Big Dipper is also circumpolar. In fact it can be used to identify Polaris, the Pole star, around which the stars seem to revolve each night. And the Pole Star is also known as the North Star, because it is always in the north.

Ursa Major. Image from StarWalk2.

So Which is the Most likely?

While Taurus, Ursa Major, and the Big Dipper are all found in the northern sky, the most reliable of them for finding which direction is north are the last two, and particularly the Big Dipper. Here is how H.A. Rey draws its relationship to the North (Pole) star:

The relationship of the Big Dipper (aka the Wagon aka the Saucepan) and two of its “pointer” stars to the Pole star and hence to the North.

So if you are ever lost in the wilderness without GPS or a compass, remember this page of Talmud, look for the agalah (and not the eglah), and find your way back home.

Talmudology Redux - Readers Suggest How Rabban Gamliel Did It…

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The Story So Far…

Last time on Talmudology we discussed the mysterious tube that Rabban Gamliel used to estimate the distance of his ship from the shore, described on Eruvin 43b. The tube was likely some kind of protractor which measured the angles between two objects. But how this enabled the calculation of distance was not explained, and trigonometry had not yet been invented. We concluded that it was a mystery.

“And so the question of how Rabban Gamliel calculated the distance to the shore on that eve of Shabbat must remain a mystery.”

Talmudology Readers to the Rescue

But Talmudology readers don’t like to leave things a mystery. Several wrote in with various solutions, and with their permission we are sharing them with you.

Jeffrey Lubin suggested that the answer might be something to do with making some markings on the glass at the end of the tube (assuming of course that it did indeed contain glass).

If he was familiar with the height of the towers (per the Yerushalmi); Is it possible that he was able to place upper and lower markings (with grease or etches or something else) on the glass at the end of his tube, and if the entire towers fell within those markings he was still too far out and once the towers reached the full distance between the markings he knew he was within 2000 amot?

This same device could be useful if he was on land or on sea and since he knew the heights of the towers, he had likely been to the area before. Perhaps he traveled there regularly. It’s easier than measuring angles and doing complex mathematical calculations in the moments before shabbat came in. (He may even have had multiple Glass attachments for different ports - with the markings in different places on the glass)

Another suggestion was made by Ori Pomorantz:

He [Rabban Gamliel] didn't need a general solution to measure distances. He just needed to know whether the distance is above or below 2000 cubits. On a weekday he could have gone 2000 cubits from the tower and measured the angle the tower appeared to be. Then, Friday afternoon, he could have checked against that particular angle. If the tower appears as a bigger angle, they're within Tchum Shabbat. If it appears smaller, they aren't. 

Shalom Kelman suggested that the key lies with similar triangles.

The ancients were certainly familiar with this area of mathematics. They might not have had tangent tables but using proportions would have been sufficient for the task at hand. See the Hebrew Art Scroll for a worked-out example. So if Rabban Gamliel could measure an angle from his sextant to a tower of known height, the calculation should be straight forward. 

Here is the original note to which Shalom referred in the ArtScroll Hebrew Talmud, together with a free translation. It too relies on a prior measurement of 2,000 amot and securing the tube and the angle subtended when viewing that precise distance.

We are discussing a tube attached to the top of a pillar, which is pointing downwards and aligned so that a person looking through it will see a distance of 2,000 amot. This distance is [previously] determined by placing the pillar on flat terrain, measuring out 2,000 amot, and placing an object at that distance. Then the observer points the tube down towards the object until it can be seen through the tube.

With the tube at that angle, a triangle is formed between the pillar, the ground and the hypotenuse, which is the line of sight from the tube to the ground. Then the distance from the base of the pillar to where the line of sight meets the ground must be 2,000 amot. When a person looks through the tube his line of sight is along the hypotenuse and where it meets the ground must be a distance of 2,000 amot from where he is standing, which is the limit of where he may walk on Shabbat.

Marvin Littman, another Talmudology reader, seems to have a very good memory. He sent us a paper published in the American Journal of Ophthalmology, which he chanced upon when he was in a university library “some time in the 1970s.” The paper, A Telemeter without Refractive Optics came from the Vision Research Laboratory at Hadassah University Hospital. It is based on “nonconcentric overlapping monocular fields”, and the author claims that Rabban Gamliel’s tube used this method to measure distances, making it “the earliest telemeter in the world.” Here is the brief paper, for your delight and consideration:

And finally, Avi Grossman wrote to inform us that “even without the fancy calculations and knowledge of tangents, anyone can calculate the distance using some drawn-to-scale sketches and a given angle and length.”

Avi referred us to the American cartoonist Carl Barks (1901-2000), who created and drew the first Donald Duck stories. In one of these stories, Donald Duck, Huey, Dewey and Louie Duck, known as the Junior Woodchucks, face off against the all-female Chickadee Patrol. You can read the full story here, but the bit that suggests an answer to how Rabban Gamliel calculated his distance from the shore is reproduced below:

In the end, the Woodchucks use a different method to build their bridge, and (spoiler alert) both teams tie. Might Rabban Gamliel have used a similar method to calculate his distance from the shore? And are there any other examples of a passage of Talmud being elucidated by a Donald Duck cartoon?

There are, to be sure problems with each method; for example, there is no suggestion in the Talmud that Rabban Gamliel had done any prior calculations, or even that he knew the geography of the port into which he sailed that Friday evening (although the Yerushalmi’s note the he knew the height of the towers might suggest a familiarity with the area). But the fact we received these different solutions is a testament to the Talmudology readership. And a reminder that the work of explaining the Talmud does not end with Rashi and the other famous commentaries. It is an ongoing process, and sometimes even involves cartoon ducks.