The mishna on daf 41A says that Rabban Gamliel looked to see if they were within the techum at nightfal. The Gemara in 43B says that he had an instrument which he looked through. Can you please send me a picture of this insturment and explain that mathematics of how it enabled him to locate the 2000 mark for the techum please? Thanks very much !
MEYER, Israel
The most detailed explanation I found about this is a on a website which brings three possible ways to explain the math, and also includes an explanation based on Euclidean geometry. See:
https://rmbhanes.wixsite.com/rmbh/copy-of-9
I also found a discussion in Daniel Sperber's Nautica Talmudica (p. 109) where he calls this a "viewing-tube" and describes its features.
It would seem that the instrument used is a simple tube which can be calibrated to measure distances. The tube is placed in such a way that the measured distance can be seen with the naked eye (by measuring the angle) and then the person holding it walks backwards until he reaches flat territory and measures the excess area he has walked. The two sums represent the distance to the bottom of the valley.
An alternative method involves measuring congruent triangles at the edge of the site. This is done by placing the tube in a way that it creates a triangle over the valley, then twisting it round so that it can proscribe a similar triangle over the flat area. It can then be used to measure the angle beneath and, using the theory of congruent triangles, by creating a small triangle and a larger one.
The site explains that this was also Euclid's way of measuring the depth of a well.
Yoel Domb