Benjamin Banneker and the square root of 3

At the Teaching Contemporary Mathematics Conference held Jan 21-22, 2011, at the North Carolina School of Science and Mathematics, teacher John Mahoney from the Benjamin Banneker Academic High School in Washington, DC, presented the plenary session talk Saturday morning.  The talk was about the math of Benjamin Banneker, who helped survey the District of Columbia, and John's work with Banneker's journals (but I was also very inspired by his description of his experience in moving from the Sidwell Friends school to the public school where he now teaches.)

John has studied and electronically restored copies of Banneker's journal entries (an example is at the top of this blog post), and he has been intrigued by one approximation that he found in the journals, that the square root of 3 is approximately 5/9 of pi.  Even back in 2003, John had asked about the approximation, in a post in the math forum of Drexel University.  In that post, he mentioned another Banneker approximation, which happens to be the same as approximating the square root of 3 by 26/15--which is actually better than 5/9 of pi.  A good question would be why use one approximation when another is better, and, even simpler, but that still leaves the question of where the approximation came from. I think I have found a possibility.

If you take the Fourier series expansion of arcsine of x divided by the square root of one minus x squared (you can plug "arcsin x / sqrt(1-x^2)" into WolframAlpha to find it), you get a series that starts with these first three terms: x+(2 x^3)/3+(8 x^5)/15.  If you let x be 1/2 in this series, using those first three terms, you get Banneker's formula!