ter. The second point Newton did not himself attack, but in 1682, when attending a meeting of the Royal Society in London, he heard of Picard's recent measurements of the earth's degree. On returning to Cambridge he inserted the new value in his old calculations of 1665, measuring the distance from earth center to moon center. Finding as he advanced that the result was manifestly going to produce the long-wished-for answer, he found it impossible—so the story goes—to proceed. With the aid of a friend the calculation was completed, and Newton had reached another milestone on the way towards his cherished goal. The figures tally exactly—has he not solved the problem and may he not proclaim the answer to a waiting world? None but Newton knew the distance yet to be traveled before complete success should be his. Nothing short of this could satisfy his truth-loving mind, and the world must wait. The third point may be stated thus: "Given a central force varying as the inverse square of the distance, show that the orbit is an ellipse with the force-center at one focus." This Newton did before the year 1684, for in August of this year, when Halley, disgusted with Hooke's bombast, came to Cambridge, he asked Newton without delay the following question: "What path will a body describe if it be attracted by a center with a force varying as the inverse square of the distance?" To this Newton at once replied, "An ellipse with the center of force at one focus." "How on earth do you know?" exclaimed Halley in amazement and delight. "Why, I have calculated it," and Newton rummaged for the paper. Failing to find it, he promised to forward it to Halley by post. This promise Newton fulfilled in November. It is not known how much ground was covered in this paper, but, of course, the desired demonstration of the third point above noted was given. Newton must now have realized that he must solve the fourth point and thus complete the work so nearly finished. Something of this may have been expressed in his letters to Halley, for in December, 1684, Halley again visited him and urged him to continue his investigations. Thus far he had shown that Kepler's laws called for the inverse square law of gravity—that Picard's value of the earth's radius fitted exactly into the theory. It but remained to prove that he is correct in taking the distance from center to center of the earth and moon. For weeks and months he works over this proof and finally, some time in 1685, it yields to his unremitting toil. The approximate date of the achievement we know from a letter of Newton's to Halley dated June 26, 1686, in which he says, "I never extended the duplicate proportion lower than to the superfices of the earth, and before a certain demonstration I found the last year, have suspected it not to reach accurately enough down so low." The answer, mathematically proved in Prop. LXXIV. of Book I. of the "Principia,"
- "Principia," Book II., Prop. XXIV., Theorem XIX.