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 quadratures."
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 approach it in quadratures." On the contrary, the tendency of the moon's motion is just the reverse—namely, to approach in conjunction and opposition, and to recede in quadratures. And if so in regard 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 demonstrated.
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 quadratures. . . . 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 waters. The moon performs its revolution in 27d 7h 4349m, which is equal to 2,360,60623 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 129600023606062/3 .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 circle whose radius is one equals 3.141592653589 +. Hence one second of this semicircumference equals 3.141592653589648000 .0000048481368110 +, and the fractional part .54901141 of one second of this semicircumference is equal to .00000266168242648 +.
Let E M and E M’ represent the moon's distance from the earth, M M' the arc which the moon describes in one second of time, and A M’ the sine of this arc. Let E M’ equal 240,000 miles, the moon's distance, in round numbers, from the earth, and E C equal one mile. The arc B C, being very small, may be regarded as equal to its sine. The length of this arc we have already found. From similarity of triangles we have the following proportion: A M': B C:: E M': E C, or, by substituting the figures, A M': .00000266168242648:: 240000: 1. Therefore A M' .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 239999.9999991498535 +, which, subtracted E M, gives A M .0000008501464 +, and this fractional part of a mile, reduced to inches, gives
