THE TIDES. 231 



movable, and as obeying that amount of attraction which is due to its 

 situation) gives rise to a relative tendency in the moon to recede from 

 the earth in conjunction and opposition, and to approach it in quad- 

 ratures." 



This language gives about the clearest presentation we have of 

 the pulling-away doctrine. But there is no " tendency in the moon 

 to recede from the earth in conjunction and opposition, and to ap- 

 proach it in quadratures." On the contrary, the tendency of the 

 moon's motion is just the reverse namely, to approach in conjunc- 

 tion and opposition, and to recede in quadratures. And if so in re- 

 gard to the moon and earth, it must be still more so in regard to the 

 earth and her waters under this influence alone, as can be demon- 

 strated. 



I am sustained in my position by the best of authority. " Thus our 

 moon moves faster, and, by a radius drawn to the earth, describes an 

 area greater for the time, and has its orbit less curved, and therefore 

 approaches nearer to the earth in the syzygies than in the quadra- 

 tures. . . . The moon's distance from the earth in the syzygies is to 

 its distance in the quadratures, in round numbers, as 69 to 70." The 

 authority I quote is Newton's " Principia." 



Let us make a calculation, and apply it to the earth and her wa- 

 ters. The moon performs its revolution in 27 d 7 h 43f m , which is equal 

 to 2,360,606f seconds. The seconds of time in which the moon makes 

 one revolution around the earth is to one second of time as 1,296,000 

 seconds in a whole circle is to a fractional part of one second of a 

 circle, which we will call x. Hence x = If-f^t^f^. = .54901141 + , which 

 is the fractional part of one second of the circle of the heavens the 

 moon describes in one second of time. The semicircumference of a cir- 

 cle whose radius is one equals 3.1 41 592653589 + . Hence one second of 

 this semicircumference equals 3 ' 1 4 VV8 2 oinr MiL = .0000048481368110 +, 

 and the fractional part .54901141 + of one second of this semicircum- 

 ference is equal to .00000266168242648 +. 



Let E3IavA E 31' represent the moon's distance from the earth, 

 MM' the arc which the moon describes in one second of time, and 

 A M the sine of this arc. Let EM equal 240,000 miles, the moon's 

 distance, in round numbers, from the earth, and E C equal one 

 mile. The arc B (7, being very small, may be regarded as equal to 

 its sine. The length of this arc we have already found. From simi- 

 larity of triangles we have the following proportion: A 31' : B C :: 

 E 31' : E C, or, by substituting the figures, A M' : .00000266168242- 

 648 :: 240000 : 1. Therefore A 31' = .6388037823552 +, which is the 

 sine of the arc passed over by the moon in one second of time. The 

 cosine E A is equal to 



s/EM' 2 A M '= \/(240000) 2 (.6388037823552) 2 + = 

 239999.9999991498535 + , which, subtracted EM, gives AM .00000- 

 08501464 + , and this fractional part of a mile, reduced to inches, gives 



