Geometric proof of expansions of series I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn't contain the  'normal' calculus methods that we use today. His methods were more geometrical. Also Gregory had proved the sine and the cosine series geometrically. Has anyone read their methods or knows their approach or how they got the expansion of series geometrically. Or if anyone knows some articles or books have their methods because I have searched quite a lot on the internet without much result.
 A: It seems to be published by Newton in De analysi per aequationes numero terminorum infinitas (Latin).
There is a link to an online version.

But just by seeing the diagrams and guessing the Latin , I don't think
  the page covers Newton's method to derive the sine or cosine series.

Just search for 5040, you will see several instances of the sine series. 

Content
The exponential series, i.e. tending toward infinity, was discovered
  by Newton and is contained within the Analysis. The treatise contains
  also the sine series and cosine series and arc series, the logarithmic
  series and the binomial series.

I am not sure, if an English version will make you much happier, because not only the language of science is different from today, also the terminology and notation and the kind of reasoning is often different.
So more helpful would be a version which is translated in language and with help from an expert to bridge the difference in mathematics over this time. I am not sure if such exists for this text, put I am pretty sure such exist for part of his main works. That might give you a bit more insight.
I am pretty sure a figure like this one here

showed up in his derivation of the fundamental theorem of calculus.
From the few glances I had so far, I can not tell if he used his iteration method. (I might try again)
I put the occasional sentence through Google translate to get a better clue.
