I'm interested in reading more about how Newton developed his method of interpolation and also the proofs he developed to this topic. I'm currently reading "Analysis by its history" which gives a good first insight and mentions "The Mathematical Papers of Isaac Newton - volume 8" as a source, this seems, however, not be available as a pdf or ebook and only as a paperback version. Can anyone recommend me some resources where I can read more about this topic which are eventually accessible for free?
After a bit of research I stumbled across this paper which might be a good starting point. As often in mathematics, Newton discovered his method of divided differences through pattern finding whereby the paper gives a slightly adjusted approach after some examination of Newton's thoughts which will give you a good idea why his method works.