Squeak, I still question why in say a very clear Tosfos like in Sukkah 8a where he shows that the 1.4 value is actually less than the true sqrt(2), he doesn’t bring up the Pythag Thm. I am not aware of any mention in classical sefarim of it,, although diagonals are brought up a number of places in shas. I still don’t buy that the Rishonim were aware of it, although, as before, I take back that they made an actual error, since it may be a bad girsa.
Joseph, let me amplify my earlier post to you of “singing to the choir”. Suppose you were running a kiruv seminar and advertised that the Talmudic Rabbis knew modern science. You get a large crowd, and really have their interest piqued. The people are anxious for some proof. They expect some discussion of inventions and some slides showing advanced physics and math. They are all excited. Instead, you say, “Folks you know what the proof is, Rabbi B says that Rabbi A knew science. Rabbi C says we must believe everything Rabbi B said. Rabbi D once referred to Rabbi C as the wonder of the generation who knew everything. Rabbi E says that anybody who doubts Rabbi C is a complete heretic. This concludes my presentation.”
Joseph, have you made a convincing case to these non-religious people? They are not interested in what Rabbis say about other Rabbis. They want you to show them the pudding. You need to come up with specific examples of advanced science, not long lists of quotes.
Interestingly, the Chazon Ish in Emuna uBitachon tries to do that, and shows some sources, like the fact that in Tanach, they removed the spleen of runners to make them go faster. The Egyptians also used to sterilize horses by removing the womb (or maybe cows, don’t remember) to make it harder to breed them so they could keep prices high. Also, a few other examples. He tries to say that earlier generations in general did not try to invent, just to study chochma. However, I am not at all sure I buy this argument.
Finally, let me openly state my negiah here. My point in being skeptical about this principle that the gedolim knew science from Torah, is because it is a convenient excuse for denying a generation any secular knowledge, and telling them they can know everything they need even if they don’t complete more than 7th or 8th grade, as long as they learn. The kids are told that ignorance is a virtue, and are crippled when they enter the job market. Then the rabbonim are only too happy to write them a letter to carry around saying they need support from the klal. I simply don’t believe it. However, if one can find me a single knowledgeable gadol today in science or math who did not study it from secular sources, but only from Torah, I will eat my hat. (I don’t wear a hat, but I will go out and buy one especially to eat. I will serve an entire seudah of hat to many guests.)