7 ms·

A 3700-year-old proof that the diagonal of a unit square has length √2

54 points by sethg 18y ago

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 ago
  If 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 ago
    You'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 ago
      You'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 ago
        From 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 ago
    The 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.
smanek 18y ago
In the 6th century BC Pythagoras' followers knew that the square root of two was irrational (couldn't be expressed as the ratio of two integers). The story goes that this messed with a lot of their theories so it was ordered to be kept a secret, and anyone who dared reveal the fact was killed. It seems kind of funny to me that they actually had a cult/religion based on math. Instead of just accepting the existence of irrationals and attempting to update their theories - they tried to suppress the new evidence that contradicted their teachings.
- tome 18y ago
  But what would "accepting the existence of the irrationals" mean? As far as the Greeks knew, a "number" was a/b where a and b were integers. "Accepting" that the length of a triangle was not a "number" could have all sorts of consequences, including rejection of their belief in validity of mathematical reasoning, so I'm not surprised they were scared of it.
  - smanek 18y ago
    The Greeks did know about irrational numbers. They could draw a right triangle whose legs were 1 and 1. They knew that the length of the hypotenuse (a number that we call the square root of two) couldn't be represented as the ratio between any two integers, yet it had a length.
  - bdr 18y ago
    Actually, length was primary since geometry was the foundation of their math.
- antiform 18y ago
  In the book The Heart of Mathematics by Edward Burger and Michael Starbird, there is a quotation of a translated passage from Proclus, a 5th-century AD philosopher: "It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves."
daveying99 18y ago
Where is the proof?
- aneesh 18y ago
  I don't think there is one. At least from what I can gather, this isn't quite a proof that the diagonal of a unit square has length √2. The tablet tells us that the Babylonians knew the length was √2, but it doesn't tell us how they knew, or how they could prove they were right.
- deleted 18y ago
  [deleted]
- Avshalom 18y ago
  There is no proof specifically this is really annoying about most of the Babylonian Texts, they never go into method very much. However from the numbers they got (30, 1;24,51,10, and 30*1;24,51,10=42;25,35) it looks less like they used a proof as they did use an iterative approximation method as follows: let a be some number let a1 be an approximation of sqrt(a) such that a1 > sqrt(a) then B1 = a/a1 is also an approximation of sqrt(a) but deviates in the other direction s B1 < sqrt(a). we have now bracketed sqrt(a) so a new better bracketing can now be made by a2 = (a1 +B1)/2 and B2 = a/a2 and then a3 = (a2 + B2)/2, B3 = a/a3 etc. the answer for sqrt(2) of 1;24,51,10 happens to equal a3 in this case. note that the above is only speculation, but the method DOES produce the Babylonian numbers so and it uses only arithmetic that we know they had, so it seems pretty likely (All this information and more can be found in Mathematical Cuneiform Texts by Otto Neugebauer)
deleted 18y ago
[deleted]
Tichy 18y ago
Could also be a failed attempt at drawing the "nikolaus hut" (not sure what it is called in english: http://www.kreudenstein-online.de/AppletNikol/AppletNikol.html http://www.kreudenstein-online.de/AppletNikol/AppletNikol.ht...). That, or their alphabet is REALLY expressive.

54 points by sethg 18y ago

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 ago
If 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 ago
  You'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 ago
    You'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 ago
      From 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 ago
  The 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.

smanek 18y ago

In the 6th century BC Pythagoras' followers knew that the square root of two was irrational (couldn't be expressed as the ratio of two integers). The story goes that this messed with a lot of their theories so it was ordered to be kept a secret, and anyone who dared reveal the fact was killed. It seems kind of funny to me that they actually had a cult/religion based on math. Instead of just accepting the existence of irrationals and attempting to update their theories - they tried to suppress the new evidence that contradicted their teachings.

tome 18y ago
But what would "accepting the existence of the irrationals" mean? As far as the Greeks knew, a "number" was a/b where a and b were integers. "Accepting" that the length of a triangle was not a "number" could have all sorts of consequences, including rejection of their belief in validity of mathematical reasoning, so I'm not surprised they were scared of it.
- smanek 18y ago
  The Greeks did know about irrational numbers. They could draw a right triangle whose legs were 1 and 1. They knew that the length of the hypotenuse (a number that we call the square root of two) couldn't be represented as the ratio between any two integers, yet it had a length.
- bdr 18y ago
  Actually, length was primary since geometry was the foundation of their math.
antiform 18y ago
In the book The Heart of Mathematics by Edward Burger and Michael Starbird, there is a quotation of a translated passage from Proclus, a 5th-century AD philosopher: "It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves."

daveying99 18y ago

Where is the proof?

aneesh 18y ago
I don't think there is one. At least from what I can gather, this isn't quite a proof that the diagonal of a unit square has length √2. The tablet tells us that the Babylonians knew the length was √2, but it doesn't tell us how they knew, or how they could prove they were right.
deleted 18y ago
[deleted]
Avshalom 18y ago
There is no proof specifically this is really annoying about most of the Babylonian Texts, they never go into method very much. However from the numbers they got (30, 1;24,51,10, and 30*1;24,51,10=42;25,35) it looks less like they used a proof as they did use an iterative approximation method as follows: let a be some number let a1 be an approximation of sqrt(a) such that a1 > sqrt(a) then B1 = a/a1 is also an approximation of sqrt(a) but deviates in the other direction s B1 < sqrt(a). we have now bracketed sqrt(a) so a new better bracketing can now be made by a2 = (a1 +B1)/2 and B2 = a/a2 and then a3 = (a2 + B2)/2, B3 = a/a3 etc. the answer for sqrt(2) of 1;24,51,10 happens to equal a3 in this case. note that the above is only speculation, but the method DOES produce the Babylonian numbers so and it uses only arithmetic that we know they had, so it seems pretty likely (All this information and more can be found in Mathematical Cuneiform Texts by Otto Neugebauer)

deleted 18y ago

[deleted]

Tichy 18y ago

Could also be a failed attempt at drawing the "nikolaus hut" (not sure what it is called in english: http://www.kreudenstein-online.de/AppletNikol/AppletNikol.html http://www.kreudenstein-online.de/AppletNikol/AppletNikol.ht...). That, or their alphabet is REALLY expressive.