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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.
