ON ASTRONOMY. 109 



suppose the velocity of the moon to he such as to carry it from M to 

 in -one minute, how hirge a force directed towards the earth would 

 be necessary to make it move in the arc MB? If we construct a par- 

 allelogram having M for one side, M A for another, and M B (con- 

 sidered as a straight line) for its diagonal, then M A must be the 

 measure of the force necessary to keep the moon in her orbit. But 

 this line M A is the versed sine of the arc M B. Hence we know, and 

 this was Newton's mode of investigation, that the force, whatever it 

 is, which retains the moon in her orbit must be directed to the earth, 

 and must be such as to cause a body to fall through the versed sine 

 of any small arc in the same time that the body is actually occupied 

 in describing that arc. Now, the distance of the moon being known, 

 as it was proximately, and the time of revolution being also known, 

 it was easy to compute the length of the arc passed over in any small 

 portion of time and also its versed sine. Hence the distance M A 

 becomes a known distance. A body at the earth's surface falls through 

 a known distance in one minute. Call this distance h and let R be the 

 radius of the earth, and D the distance of the moon, and x the distance 

 through which the earth's attraction would cause a body to fall if 

 placed at the moon, then we should have D"^ : B^ : : /i : x. To answer 

 the required purpose, x must equal M A. This was jNewton's course 

 of reasoning. 



Taking the measure of a degree on the meridian to be 60 miles, 

 as determined by Fernel, and then generally adopted, he deduced the 

 earth's radius, and then found, by making the calculation, that the 

 attraction of the earth at the distance of the moon was insufficient 

 to produce the required effect. It equally failed whether it was supposed 

 to vary inversely as the distance simply, or as the square of the 

 distance. And the difference was too great to be attributed to any 

 error in the calculation, or in the moon's motion. 



In the s|)irit of true philosophy he rejected the hypothesis which 

 was insufhcient to account for the facts. And, singuhir as it may 

 appear, he seems, from anything now known of his labors, to have 

 dropped the investigation for nearly twenty years. 



The measurement of arc of the meridian, in France, by Picard, which 

 gave a more correct measure of the earth's radius, recalled his atten- 

 tion to the subject. And repeating his calculation with this corrected 

 measure, he had the satisfaction of finding that with the inverse ratio, 

 the square, the intensity of the terrestrial gravity was almost pre- 

 cisely that which was required in order to keep the moon in her orbit. 

 This was the " experimentum crucis." The great law of nature was 

 established. 



We now know that the mean distance of the moon is very nearly 

 60 times the earth's radius. The terrestrial gravity varying according 

 to this law, it will at that distance be 3^\,o P'^i't of wuat it is at the 

 earth's surface. Hence a body at the moon will fall through the same 

 distance (16.1 feet) in one minute, that it will fall at the earth's surface 

 in one second. And this is almost precisely the versed sine of the 

 arc for one minute described by the moon's mean motion. 



A like test was applied to the planets as influenced by the attraction 

 of the sun, and the same law was found to obtain. 



