Pashuteh Yid
….It is clear that both of these Rishonim made a clear mathematical error. They did not know how to properly calculate the diagonal of a 4×6 rectangle. This, despite the fact that Pythagoras showed how to do it a thousand or so years before. They were unaware. Not only does it seem from there that Chazal were not ahead of their times, but were in fact behind their times. I was stunned when I saw it……..
BUT THEN YOU SAY….
….Furthermore, all you need to do is to go measure the rectangle with a ruler or string and see for yourself……
An oiber chochom. So according to your understanding, Chazal were not only behind their time, they were also too stupid CH”V to do what any 2nd grader can do, i.e. take a string and measure? Klop al chait again. If you have a hard time imagining that Chazal knew more about physics than we do now, I can understand your position. But do you really think CH”V that Chazal couldn’t even measure something properly?
Now, if they CAN measure properly, why did they make that mistake? So you have a question. Fun a kashe shtarb men nischt. Uber men ken leben mit a teretz, right? 😉
This Gemara may have been news to a poshute Yid, but it’s not to me. Nor to any mathematically inclined individual who has studied Gemara. In fact, there are many Gemaros that are “famous” to the mathematically inclined. I can name you a few other places where the Gemara and the Rishonim discuss mathematical ideas and are quite imprecise (to be polite). One example is Succah daf 8a where they discuss the relationship between the circumference and the area of a circle (pool of Shlomo Hamelech). The quantity they seek is Pi, but their approximations are not even close to those that the Greeks produced thousands of years earlier! Behind the times? No, it’s not that at all. Slow down and think.
This is how I would explain it: In my understanding, there has to be a purpose to mathematical precision. In the case of these Gemaros, there is no purpose to precision – it would only serve to confuse those who could not understand such an esoteric concept. You are right that nowadays general education is far better than it was, and that now there are more lay people who appreciate precision. But when we are discussing matters of halacha, most of which have to be implemented by the average Poshite Yid, who may not (or most likely not, I should say) be able to perform accurate mathematical calculations – requiring precision would cause errors in halacha. Therefore, in these matters of Halacha Chazal intentionally used approximations.
Don’t believe me? Well, a few months ago I posted about Bircas Hachama. This is an event that occurs once in 28 years. But does it really occur that often? No. It is only due to an approximation that Chazal used to calculate the event. What about the equinox and solstice? Do you know when we start saying V’sain Tal U’matar in CH”L? It is 60 days after the fall equinox. Is it really? No. It is really on December 4th or 5th. Why? Because we don’t use the REAL equinox in the calculation – as that would cause confusion. Rather, we use an approximate equinox so that the time is fixed. Most of the time the approximation is far off, but if the halacha was set with precision, once again the Poshite Yid would err and start counting 60 days at the wrong time.
So does that mean that Chazal could not perform accurate calculations? CH”V! Of course they could – and did. In fact, when it comes to matters of halacha that are implemented only by Beis Din the calculations are extremely precise. One example is the Molad. We have been calculating the exact Molad for Rosh Chodesh for ages. It’s on our calendars, so we don’t need to calculate it ourselves.
There are many examples, but I know if my post gets too long you will just skim and I think you need to read everything I wrote. ‘Watershed event’ indeed. Please don’t be so quick to jump to conclusions, especially when you are CH”V “judging” Chazal.
G’mar Chasima Tova.