■^ PHILOSOPHICAL TRANSACTIONS. [aNNO 17(34. 



second, in the duplicate ratio of the diameter of a circle to its circumference; 

 that fall therefore at the equator, and in vacuo, is 16.10185 feet; the logarithm 

 of which number is 1.2068645 = log. p. 



The toises in a degree of the equator, or, which is the same, in a degree of 

 the meridian at lat. 54^, being nearly 5/200, the logarithm of the number of 

 feet English in the semidiameter of the equator, 



that is log. D will be nearly 7.321 igoo, 



And the log. versed sine of the moon's arc in 1 being — 12.5492882, 



Their sum — 5.8704782 



taken from log. f, leaves + 5.3363863, a 3d of which is 1.7787954, the loga- 

 rithm of X = 60.O8906 semidiameters of the equator. And the arithmetical 

 complement of this last logarithm, which is — 2.2212046, is the log. tangent of 

 the moon's mean horizontal parallax at the equator; which therefore is 57' 12".34. 



III. Such would be the distance of the earth's and moon's centres, were the 

 earth immovable; but it is somewhat increased by their revolution round their 

 common centre of gravity. Writing x + 1 for that distance, divided by the 

 centre of gravity in the ratio of x to 1 ; imagine a sphere of the same dimensions 

 as our earth, placed at that centre, to exert the same attractive force on the 

 moon as our earth actually does, the periodic time remaining unaltered: then 

 must the density of this sphere be diminished in the ratio of x"^ to (ar + 1 y\ that 

 its nearer distance from the moon may be compensated by the defect of density 

 and attractive force. If now an inhabitant of the fictitious earth were supposed 

 to compute its distance from the moon, in the manner just now shown; the 

 quantities v and d would be the same as in the former calculation; but h\sf 

 wouM be to our f, as x^ to (x -i- 1)^, and thence his x would be to our x as 



x^ to {x + 1)-*, that is, x = (~-j)+ X x. 



This is the distance from the fictitious earth, or from the common centre of 



gravity; but (t) the distance from our earth, is — p X (^^rf)"^ X x, greater, 



as was supposed, in the ratio of a: + I to a; that is, t = v^ x x. 



Sir Isaac Newton, from the phenomena of the tides, estimated the ratio of 

 a^ + 1 to ^ to be that of 40.788 to 39.788. In that case, the cubic root of 

 ^-will have for its logarithm 0.0035934; which added to 1. 7787954, the loga- 

 rithm of X computed from an immovable earth, gives 1.7823888, the logarithm 

 of 60.5883 semidiameters of the equator. And the moon's horizontal parallax 

 for this distance, is 56' 44".07. 



IV. On the other hand, if we had observations of the moon's parallax, and 

 distance, which could be reckoned exact enough for the purpose, we might 

 thence determine the ratio of a: to 1 , that is, the ratio of the quantities of matter 



