Help finding a Gemara source

imasoftov

Tue, Jan 17 2017, 6:52 am

RDR wrote:

http://daf-yomi.com/DafYomi_Page.aspx?vt=1&id=3462

It's in tosefos there
And it's geometry

It seems to me that it's Rashbam there (the commentary located where Rashi usually is on the page is Rashbam here, and the relevant part continues on the other side of the page) who says not that 4x6=25 but that the diagonal of a 5x5 right triangle is the same as (or perhaps close enough to) a 4x6. Tosafot disagrees. Neither of them uses the Pythagorean Theorem

I'm also confused by what both of them are actually trying to say, Rashbam says that the diagonal of the 4x6 is 8 and of the 5x5 is 2 but they're the same so there must be a scribal error in addition to the imprecise math.

Tosafot quotes him as both having a diagonal of 2 and says this must be wrong because a 4x4 has a diagonal of 4-8/5 (not a typo, eight fifths, but I think I know where that came from, he's using the approximation of the square root of 2 as 1-2/5 and multiplying by four) so a 4x6 will have a diagonal longer by less than 2. But Tosafot also says that a 5x5 has a longer diagonal than a 4x6 because the area of the former is 25 and of the latter 24 (this is wrong twice, both about which has a longer diagonal and about the reasoning, but the arithmetic is correct).

actual lengths of diagonals, to two decimal places
1x1 1.41 (square root of 2)
4x4 5.66
5x5 7.07
4x6 7.21
6x6 8.49 (appears in Rashbam as 8-2/5)

I found someone who understands that what is meant by the diagonal being 2 is 2 longer than the longest side of the triangle (but doesn't say why) and also notices the problem with the reasoning at the end of Tosafot.

http://dafyomi.co.il/bbasra/to.....2.htm

JoyInTheMorning

Tue, Jan 17 2017, 7:29 am

Right, imasoftov, Tosafot seems not to be aware of / taking into account the fact that the greater the distance between the width and length of a rectangle (for rectangles of approximately the same area), the longer the diagonal. [The subject under discussion in the sugya is the magnitude of a distance that must be searched.] A square has, relatively speaking, the shortest diagonal.

Note also that the Babylonians had accurately approximated to six decimal places irrationals such as square root of 2 by 1600 BCE. The approximation that Tosafot gives is a lot less precise.