Quite what Halley expected to get from him when he made his unannounced visit in August 1684 we can only guess. But thanks to the later account of a Newton confidant, Abraham DeMoivre, we do have a record of one of science's most historic encounters:
In 1684 Dr Halley came to visit at Cambridge (and) after they had some time together the Drasked him what he thought the curve would be that would be described by the Planets supposing the force of attraction toward the Sun to be reciprocal to the square of their distance from it.
This was a reference to a piece of mathematics known as the inverse square law, which Halley was convinced lay at the heart of the explanation, though he wasn't sure exactly how.
Sr Isaac replied immediately that it would be an (ellipse). The Doctor, struck with joy and amazement, asked him how he knew it. 'Why,' saith he, 'I have calculated it,' whereupon DrHalley asked him for his calculation without farther delay, SrIsaac looked among his papers but could not find it.
This was astounding—like someone saying he had found a cure for cancer but couldn't remember where he had put the formula. Pressed by Halley, Newton agreed to redo the calculations and produce a paper. He did as promised, but then did much more. He retired for two years of intensive reflection and scribbling, and at length produced his masterwork: the Philosophiae Naturalis Principia Mathematica or Mathematical Principles of Natural Philosophy, better known as the Principia .