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To find - How did Newton derive the general binomial theorem.

I know he approximated the area under functions.

1) But how did he approximate the area under CURVES like ( 1/((1-x^2)^(1/2))) or ( 1/((1-x^2)^(n/2))) , where n is odd.

I know he moved forward from here deriving sine inverse series and then sine series. But how did he approximate the area under the above function.

Edit

Kindly look to it. I have changed the title and the question a bit and now I don't think it's duplicate

Shashaank
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  • Regarding 2) (which is not "duplicate"; better to ask one question at a time! also, better rename this one if you want it reopened), see the Gregory references linked at this MO question. (That question was about $\tan^{-1}$, but what is said applies to $\tan$ just as much, if not more.) – Francois Ziegler Apr 01 '17 at 00:53
  • @FrancoisZiegler I have changed the title but I don't know how to ask the community to reopen it. – Shashaank Apr 01 '17 at 06:59
  • @FrancoisZiegler Also I did go through the link that you provided. After an year long search , I have only found a few pages showing the probable way in which Newton derived the sine series geometrically and lebniz derived the atan series. Most of the links in your answer had books not in English ( probably Latin). I tried to find de Analysi in English but couldn't find it. I know Newton's divided difference method proves Taylor expansion. But again I wasn't able to get anything in English on the net. It will really helpful if you know any links on these topics – Shashaank Apr 01 '17 at 07:08
  • @FrancoisZiegler & particularly on the English version of Yuktibhisha ( the wiki links are not working) – Shashaank Apr 01 '17 at 07:09

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