Last time on Talmudology we tried to determine the nature of the special tube that Rabban Gamliel carried with him on a boat, and which allowed him to calculate its distance from the shore. Here is a recap:
עירובין מג, ב
פעם אחת לא נכנסו לנמל עד שחשיכה אמרו לו לרבן גמליאל מה אנו לירד? אמר להם מותרים אתם שכבר הייתי מסתכל והיינו בתוך התחום עד שלא חשיכה
Once a ship did not enter the port until after nightfall on Shabbat eve. The passengers asked Rabban Gamliel, “what is the halakha with regard to alighting from the boat at this time? [In other words, were we already within the city’s limit before Shabbat commenced?]
He said to them: You are permitted to alight, as I was watching, and I observed that we were already within the city’s limit before nightfall. [The port is therefore within the area on which we may walk on Shabbat.]
Regardless of whether it was a telescope (unlikely) or a protractor of some kind (more likely, but still not certain), the question we will address today is how Rabban Gamliel used the instrument to determine his distance from the shore.
The Jerusalem Talmud Comes to the Rescue
There is nothing in the Babylonian Talmud that would suggest an answer. But the Jerusalem Talmud goes into a little more detail, and in so doing it provides us with an explanation:
תלמוד ירושלמי עירובין כח, ב
מצודות היו לו לרבן גמליאל שהיה משער בה עיניו במישר
Rabban Gamliel knew of the heights of some towers (along the coast) which he estimated with his eyes…
You can only use the trigonometry of a right-angled triangle if you know the length of one of the sides of the triangle, and one of its angles. The Yerushalmi provides the key. Rabban Gamliel knew the height of the towers that he was observing (AB in the diagram below). Here is the explanation provided by W.M. Feldman in his classic work Rabbinical Mathematics and Astronomy, first published in London in 1931.
Did the Rabbis of the Talmud know their trigonometry?
So Rabban Gamliel could have used some high-school math to determine his distance from the shore, if he knew the height of the tower (AB) the angle (ACB), and the tan of that angle. So were sines, cosines and tangents known to the talmudic world?
I had no idea. But I asked a friend who is an Associate Professor of Writing and of Mathematics at The George Washington University in Washington DC. He pointed me to this book on the history of mathematics, and made the following observations.
Based on a quick perusal of Boyer's A History of Mathematics, the notions of trigonometric ratios were well known to Aristarchus (and hence, presumably, to Archimedes). Aristarchus was doing more complex calculations than for right-angled triangles; it seems likely that he understood the right-angle case, although I didn't see explicit mention of that. Also, the Babylonians at the same time had active astronomical investigations going on, and were known to use ratios of sides of triangles in relation to angles.
However, no one had a notion of trigonometric function for more than 1800 years after that. Trigonometric ratios were just that - individual numerical ratios. This was one of the hang-ups that Newton and Leibnitz almost, but didn't quite, work through. Trigonometric results were known, but they weren't necessarily expressed, or interpreted, in the same way as we now do. So, Rabban Gamliel might not have been calculating tan(x), per se…
However, I think it's safe to say that if R"G did a calculation, it didn't look anything like what the book shows. First, there was no "tan" function. Second, fractions hadn't been invented. Third, decimal expansions hadn't been invented. Fourth, calculation of the tangent of that angle would have required a careful and explicit approximation process, and could not have been done so handily.
Which means that while Feldman’s math is correct, it wasn't the math used by Rabban Gamliel. And so the question of how Rabban Gamliel calculated the distance to the shore on that eve of Shabbat must remain a mystery.
