4 ms·
Not convinced that that's what it is. It's certainly a ~3700-year-old statement that the diagonal has length sqrt(2), but there's no evidence that it is, or was
by gjm11 18y ago
Not convinced that that's what it is. It's certainly a ~3700-year-old statement that the diagonal has length sqrt(2), but there's no evidence that it is, or was intended to be, or demonstrates that anyone had, a proof.
The author of those pages says "It amounts to a dissection of the square on the hypotenuse of an isosceles right triangle into pieces which can be reassembled to make up the two squares on the sides, and I can't see why the figure is exactly what it is if it weren't understood to demonstrate this." but it seems to me that even if all you want to do is write down that the diagonal is sqrt(2) then you'll need at least the square and one diagonal, and adding the other could as well be motivated by love of symmetry as by having noticed that with it there you can dissect-and-reassemble into a 2x1 rectangle or whatever.
Remarkable, none the less.
- deleted 18y ago[deleted]
- chancho 18y agoIf you know that 1/2 * base * height = the area, then the second diagonal can be seen as the height of the triangle cut by the first diagonal. Since b = 2 h, 1/2 b h = 1/2 ==> b h = 1 ==> 1/2 b b = 1 ==> b b = 2 ==> b = sqrt(2). I can't read cuneiform so I don't know if it makes this argument.
- aneesh 18y agoYou'd first have to prove that the diagonals bisect each other (so that you can say b = 2h). I'm not sure whether that result was known at the time, but it's not too hard to prove.
- chancho 18y agoYou're right. That's pushing it for those handful of scratches on the tablet. Who knows, maybe they had some lemma tablets lying around.
- sethg 18y agoFrom my own limited experience, ancient documents tended to be very elliptical; when everything had to be copied out by hand the reader was expected to do a little more work to understand what documents meant. So this tablet may have been the equivalent of notes accompanying a lecture: "As you can see from the triangles on this diagram..."
- gjm11 18y agoThe tablet just gives the lengths of the sides and diagonals. There is no sign of any explicit reasoning on it. The author of the pages linked to from here thinks that the maker's decision to show both diagonals of the square indicates that it's meant to illustrate a proof. I am not convinced.