More math!!!
Yesterday we had the law of relativity. Today – trigonometry! My math mind is blown. We read:
Rabban Gamliel had a special tube through which he would look and see a distance of two thousand cubits on land, and also determine a corresponding distance of two thousand cubits at sea. In general, one who wishes to know the depth of a valley can bring such a tube and look through it, and he will know the depth of the valley. The Gemara cites another statement with regard to measurements: One who wishes to know the height of a palm tree, but does not want to actually climb the tree to measure it, can measure his own height, and the length of his own shadow, and the length of the shadow of the height of the palm tree, and calculate the proportions, and he will know the height of the palm tree.
My engineering heart is full. Our rabbis knew trigonometry. And had technology that they used to determine distance using those math concepts. When I first read this, I assumed that Rabban Gamliel had a telescope (and he might of). However, the telescope was not patented for another 1500 years . . . Which makes me want to claim (like the father in My Big Fat Greek Wedding) that the Jews invented everything and here is the proof. But, alas, I think that what follows this statement is more likely to tell us what is really happening – and that’s where trigonometry comes in.
How can we look at a shadow and determine height? The same way we can look at a building, calculate it’s relative size, and determine distance. According to the Jerusalem Talmud: Rabban Gamliel knew of the heights of some towers (along the coast) which he estimated with his eyes…
This tidbit of information is what we need to calculate distance. 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 teaches that 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 (From http://www.talmudology.com/ a blog about math in the Talmud that I found when I got so excited about seeing trigonometry in the Talmud and discovered I was not alone)


Did Rabban Gamliel know trigonometry? Well,probably not in the same was our 11th graders do. However, what the Talmud describes, as comparing your shadows length to your height in order to determine the height of a tree based on its shadow – that’s a skill of ratio and comparison that lead to the discovery of the trig law you see above.
So, I guess the moral is that observation of the world can bring us closer to God. And that, perhaps, those of us who love math and love Talmud are not so very rare after all.
