336 THE POPULAR SCIENCE MONTHLY 



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, dis- 

 gusted 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. 



