I 6 EXACT ASTRONOMY. 



division of the product by the periodic time of a hypothetical 

 satellite, conceived to revolve around the earth at the dis- 

 tance i, and the involution of the quotient to the fractional in- 

 dex ' for the moon's mean distance from the center of her mo- 

 tion. Conceiving resistance eliminated by compression of the 

 earth's volume, the periodic time in seconds, of such satel- 

 lite, is the product of three factors, to wit : the terrestrial 

 orbital arc x, the square of the solar day in hours, d 2 , and 

 the reciprocal of (i +m)«; ^+^ = ^^ = ^^^±^==^5. 



r \ / ' xd 2 -4-(i + m)£ xd a 5075-4532287233088 * ^ 



0996460061535(1 4- mf, involved to the index 2 , equals 

 60.029400472.2072(1 +m)l The simple ratio of the dis- 

 tances of sun and moon : 23408.45694842835-^60.029400- 

 4722072(1 4- m )s — 3 8 9-94 9 8 7 o<5349°9t > All that precedes is rigorously 



demonstrated by the absolute identity of the sesquiplicate 

 ratio of the astronomical periodic times of earth and moon, 

 and the simple ratio of their mean distances: T -^ t(ia ~ n ; y ' — 



i x xd - 



™L_ = 7700.3979709322854-^(1 +m) s . The cube root of 

 the square of same equals 389.9498/o6349o844-(i +m) 5 . 

 The difference, 0.0000000000007 x 60.0294 x 20926084. x 12 

 = 0.01 inch, is the necessary consequence of incommensura- 

 bility. With Herschel's value of the sidereal year, which is 

 0^.3 greater, the above discrepancy of 0.01 inch is increased 

 to S^.S^ miles. Consequently, no variation whatever from 

 the values employed in the computation is possible ; which 

 is admittedly the highest order of proof. I will assume sV 

 for the moon's mass, and then demonstrate that it cannot be 

 either more or less : (1 + m)» = (1.012 5)*= 1.0020725. 



Consequently, the arc in terms of radius that the 

 hypothetical satellite (under the conditions specified) would 



describe in one second : 6.2831853071794x1.0020725 =0.001240^21 1761. 



5075-4532287233088 ^ ^ ' 



An arc so small cannot be sensibly different from its tangent. 

 Hence, (1 +curv.) 2 — i 2 = tan. 2 = arc 2 = 0.000001538892788- 



352. 



