Read Oughtreds which he understood thou not entirely, he having some difficulties about what the Author calls Scala secundi et tertii gradus, relating to the solution of quadratick Cubickand Cubick Equations. – Took Descartes's Geometry in hand, thou he had been told itt would be very difficult, read some few pages in it, then stop't began again, went a little farther than the first time, stopt again, went back again to the beginning, read on till by degrees he made himself master of the whole, to that degree that he understood Descartes's Geometry better than he had done Euclid.

Read Euclid again & then DesCartes's Geometry for a second time. Read next Dr Wallis's Arithmetica Infinitorum, & on the occasion of a certain interpolation for the quadrature of the circle found that admirable Theorem for raising a Binomial to a power given. But before that time a little after reading DesCartes Geometry, writt many things concerning the vortices Axes Diameters of Curves, which afterwards gave rise to that excellent tract de Curvis secundi generis. In 65 & 66 began to find the method of Fluxions, and writt several curious problems relating to that method bearing that date which weerewere seen by me above 25 year ago. In 65 being in the country in a garden, it came into his thought, that the power of gravity was not limited to a certain distance from the earth, but that this power would exert it self at any distance from it, & even would reach as far as the Moon, then he drew this conclusion that if it was true that this power reached as far as the Moon, it was very likely that this power ofor force of gravitation towards the earth was what contained the Moon in her orbit, whereupon he fell a calculating what would be the effect of that supposition, but he found himself disappointed for a while, the reason of which was that he took it for granted that a degree of the earth did contain 60 miles exactly, which made it that his calculation did not agree with the Theory, but he entertained a notion that with the force of gravity, there might be a mixture of that force which the Moon would have if it was carried along in a vortex, but when the Tract of Picard's of the measure of the earth came out, he began his calculations anew, & found it perfectly agreeable to the Theory. I beleive we may gather from the Transactions an account of a dispute that he had with Linus a Jesuit of Liege concerning his discovery of light 2r & Colours wchwhich would be very entertaining.

In 1673 Dr Hook writt to him to send him something new for the transactions, whereupon he sent him a little dissertation to confute that com̄mmon objection which is, that if it were true that the Earth moved from West to East, all falling bodies would be left to the West, & maintained on the contrary they swould fall a little Eastward, & having described a curve with his hand to represent the motion of a falling body, he drew a negligent stroke with his pen, from whence Dr Hook took occasion to imagine that he meant the Curve would be a Spiral, whereupon the DrDoctor writt to him that the Curve would be an Ellipsis & that the body would move according to Kepler's notion wchwhich gave him an occasion to examine the thing thoroughly & for the foundation of the Calculus he intended laid down this proposition that the areas described in equal times were equal, which thou assumed by Kepler was not by him demonstrated, of which demonstration the first glory is due to SrSir Isaac. In 1684 Dr Halley came to visit him at Cambridge, after they had been some time together, the DrDoctor asked him what he thought the curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from it. SrSir Isaac replied immediately that it would be an Ellipsis, the Doctor struck with 3r * joy & amazement asked him how he knew it, why saith he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, SrSir Isaac looked among his papers but could not find it, but he promised him to renew it; & then to send it him, SrSir Isaac in order to make good his promise fell to work again, but he could not come to that conclusion wchwhich he thought he had before examined with care, however he attempted a new way which thou longer than the first, brought him again to his former conclusion, then he examined carefully what might be the reason why the calculation he had undertaken before did not prove right, & he found that having drawn an Ellipsis coursely with his own hand, he had drawn the two Axes of the Curve, instead of drawing two Diameters somewhat inclined to one another; whereby behe might have fixed his imagination to any two conjugate diameters, which was requisite he should do, that being perceived, he made both his calculations agree together.

After this Dr Halley was (I think) sent down to Cambridge by the Royal Society to prevail with SrSir Isaac to print his discoveries wchwhich gave rise to the Principia —

Dr Halley has often valued himself to me for having been the Ulysses who produced this Achilles —