



 
What a marvelous find, Marleen! I think they were building archways. Let's say you need to know the distance across an arch 2m below the top (we call that 2 m line the sagitta):
Babylonian Method: Let's write this in using algebra, with C being the length of the circumference, x being the length of the chord and P being the perpendicular.
That's 2P.
20 is C/3. So this is C/3  2P.
That makes (C/3  2P)^{2}.
400 is (C/3)^{2}. So we now have (C/3)^{2}  (C/3  2P)^{2}.
So they get x = sqrt( (C/3)^{2}  (C/3  2P)^{2} ). That's interesting. Let's compare that to what we get if we use modern mathematics. I'll draw another diagram: Modern Method: Now the radius is in blue and we know that the radius is the circumference divided by 2π: r = C/(2π). But the gray line's length is P, so the green vertical line (that's called the apothem) is r  P. That means its length is C/(2π)  P. But we have a right triangle here, so the Pythagorean theorem applies. The length of the red chord is x, so the length of half the red chord (the length of the top of the triangle) is x/2. Pythagorus tells us that
If we multiply by 4 on both sides, we get: x^{2} = (C/π)^{2}  (C/(π)  2P)^{2}. Take the square root of both sides and we have: x = sqrt( (C/π)^{2}  (C/π  2P)^{2 }). Comparison: Now we can take the square root of both sides and compare our answer to the Babylonians':
So their equation is identical to ours, except that they use a 3 in place of π. Not too bad: it's within 5% of the real value of π. Thanks for the fun,  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 