I could not figure out what she says.

Would you mind to help me about her sayings between 6:29 and 6:37 .
http://www.youtube.com/watch?v=12Qx6I0l7to
…astounding new number.
???? ????(along?) the diagonal…
…is a really very good approximation…
…to describe the root of 2.
I need it for its subtitle, thanks.
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Answer #1
Here’s what I get
… and reveals just what the Babylonians did know about the right angle triangles. It uses the principle of Pythagoras’s Theorem to find the value…. an astounding new number. Drawing along the diagonal is a really very good approximation to square root of two.
Answer #2
Here's what I get
... and reveals just what the Babylonians did know about the right angle triangles. It uses the principle of Pythagoras's Theorem to find the value.... an astounding new number. Drawing along the diagonal is a really very good approximation to square root of two.
thanks..
and another one:
http://www.youtube.com/watch?v=12Qx6I0l7to
2:29 – 2:32
So the guardsmen must have played ????
when they were bored.
Answer #3
Again this is what I get
… the board games scratched on the entrances of palaces so the guardsmen must have played at times they were bored.
Answer #4
Again this is what I get
... the board games scratched on the entrances of palaces so the guardsmen must have played at times they were bored.
Thank you so much.