Newton's evaluation of 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11

I had asked the following question on MSE here, and I was directed to this exchange. There is a nice solution in the comments there, but perhaps someone here can add some additional insight?

How might have Newton evaluated the following series?

$$sqrt{2} , frac{pi}{4} = 1 + frac{1}{3} – frac{1}{5} – frac{1}{7} + frac{1}{9} + frac{1}{11} – cdots$$

The method of the this thread applies by setting $x=pi/4$ in the Fourier series for $f(x) = pi/2 – x/2$ and then subtracting the extraneous terms (which are a multiple of the Gregory-Leibniz series for $pi/4$).

I read that this series appears in a letter from Newton to Leibniz. However, I do not have access the letter which appears in this volume.

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