Deriving the Leibniz Formula for Pi/4

A while back, I picked up a fascinating book by Clifford Pickover ("The Math Book" - not exact, "a" is alpha, "B" is capital beta. I have yet to learn to put math symbols onto blogspot, and I'm feeling lazy right now) summarizing the chronological history of mathematical discoveries.

Today, I read the entry on the "Discovery of Series Formula for Pi" (page 110 - c.1500) and realized that while I had learned this, I had never appreciated its significance nor how it was derived.

So, given the sparse details of the entry, my question is:

"How did Leibniz, Gregory and (possibly) Somayaji independently derive the series formula for Pi? Was there a first-principles approach?"

Wikipedia, sadly, provides the proof using the Arctan series:

http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

My impression, having read the text, was that the series for arctan and pi/4 were independently derived, with arctan having been found by Gregory and Somayaji. Apparently, Gregory had not realized that arctan(1) was pi/4, which seemed a little strange.

Time to find out!