Newton , Binomial Series and Power Series , and James Gregory

Question

After 2 month long search on the net, I was lucky enough to find a pdf on how Newton found the series for sine. It was a beautiful derivation mostly geometrical. But he used the Binomial Series.

Now what I look for is Newton's proof (sort of proof) of the Binomial Series . I have read that he interpolated areas under circles and hyperbolas but a succinct answer jumps in terms of what he actually did without is what I am looking for. I know he made tables then followed the pattern. But there must be a lot of geometrical work too which would have led him to this point. All sorts of interpolation.

That's what I want to understand. What did he actually do ? What was his geometrical work regarding it as well as what was his sort Interpolation of areas

Answer 1

See : Derek T. Whiteside, Newton's Discovery of the General Binomial Theorem , Math. Gaz. 45 (Oct.1961).

For an English transaltion of De analysi per aequationes numero terminorum infinitas (1669) see :

• Sir Isaac Newton's Two Treatises: Of the Quadrature of Curves and Analysis by Equations of an Infinite Number of Terms, Explained : Containing the Treatises Themselves, Translated Into English, with a Large Commentary ; in which the Demonstrations are Supplied where Wanting, the Doctrine Illustrated, and the Whole Accommodated to the Capacities of Beginners, for Whom it is Chiefly Designed (1745), page 321-on.