VOL. 



LIV.] ' PHILOSOPHICAL TRANSACTIONS. 8^ 



in the earth and moon. For having - = y^ , and — = —^ ; also t be- 

 ing given from observation, and x computed as above; it is manifest that the 

 ratio of or -f 1 to x, and, by division, that o{ x to 1, or of the mass of the 

 earth to that of the moon, is given. For example, if it should be concluded 

 from good observations, that t, the moon's mean distance, is 6o^ semidiameters 

 of the equator; from the logarithm of this distance, which is 1.7817554, take 

 the logarithm of x, or 1.7787954, thrice the remainder will be 0.00888, the 

 logarithm of ^"^ , = 1.02066: and the masses of the earth and moon would, 



on this supposition, be as 48.4027 to 1. 



In all this, a small variation from the law of attraction, arising from the 

 spheroid figure of the earth, is neglected as inconsiderable; which it will be found 

 to be by whoever takes the trouble to compute its quantity and effects. 



Remarks. — 1. If p and d were taken of their just quantities, the moon's ho- 

 rizontal parallax for an immovable earth being, at the equator, 57' 12-f", is a 

 limit which the true mean parallax cannot exceed: and the correspondent dis- 

 tance 60.03906 is a limit which the distance cannot fall short of: both being 

 computed on the supposition that j? + 1 = x, or that the matter of the moon 

 is as nothing in comparison of the earth. Nor can the parallax and distance be 

 supposed to lie very near these limits, without leaving too little attractive force in 

 the moon to raise the tides. 



2. If the moon's mean apparent semidiameter be 15' 384-", and the distance 

 of the centres 60.5883 semidiameters of the equator, according to Sir Isaac's 

 estimate of the masses; the semidiameter of the moon will be O.275601 parts of 

 the semidiameter of the equator, or .2763 of a mean semidiameter of the earth. 

 And the magnitudes of the moon and earth being as the cubes of their semidia- 

 meters, if the inverse ratio of their magnitudes be joined to the direct ratio of 

 their masses (1 to 39.788) the sum will be the ratio of their densities, that of 

 1.19143 to 1, a little less than 6 to 5. 



3. Supposing still the same semidiameter of the orbit as before, the force of 

 gravity will be to the earth's attractive force on the moon, as 8670.94 to 1, and 

 to the moon's force on the earth as 40.788 times that number, or 149730.4, to 

 1 . Again, the force of the moon on that surface of the ocean to which she is 

 vertical, being to her force on the earth's centre, as the square of 60.5883 to 

 that of 59.5883 ; and the difference of these squares being to the latter as 1 to 

 29.54623 ; this difference of the forces will support the weight of - m ' sm part 

 of the water at the vertex. And because the earth's semidiameter is small in 

 comparison to the moon's distance, the like differences of force will decrease 

 from the surface to the centre, nearly in an arithmetical progression, as the 



VOL. XII. N 



