In two previous papers (vol. li, p. 77 and p. 211) an account has been given, of the published work of Newton on the subject of Interpolation by means of formulas of Finite Differences, and references have been made to the important letter which Newton sent on 24 October 1676, to Oldenburg, the Secretary of the Royal Society, with the intention that it should be communicated to Leibnitz. The second paper closed with a quotation from this letter, in which Newton mentions “a method [for the “construction of tables by interpolation] which I had almost “decided to describe here for the use of computers.” It has been my good fortune to discover that Newton had prepared an account of his method for inclusion in the letter and that the draft is preserved in the University Library at Cambridge among the Portsmouth Collection of Books and Papers written by or belonging to Sir Isaac Newton. By the freely granted permission of the Syndicate of the Library, I am able to give a reproduction and transcription of the manuscript, and I have added a translation.