VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 471 



tial to be known, in order to determine with precision the exact distance and 

 magnitude of a star, can ever be obtained, where it is in the same circumstances, 

 or nearly the same, with those above supposed, yet the other elements, such as 

 perhaps may be obtained, are sufficient to determine the distance, &c. with a 

 good deal of probability, within some moderate limits ; for in whatever ratio the 

 real distance of the two stars may be greater or less than the distance supposed, 

 the density of the central star must be greater or less in the sixth power of that 

 ratio inversely ; for the periodic time of the revolving body being given, the 

 quantity of matter contained in the central body must be as the cube of their 

 distance from each other. (Newton's Prin. b. 3, pr. 8, cor. 3.) But the quan- 

 tity of matter in different bodies, at whose surfaces the velocity acquired by 

 falling from an infinite height is the same, must be, according to art. 12, di- 

 rectly as their semi-diameters ; the semi-diameters therefore of such bodies must 

 be in the triplicate ratio of the distance of the revolving body ; and conse- 

 quently their densities, by art. 11, being in the inverse duplicate ratio of their 

 semi-diameters, must be in the inverse sextuplicate ratio of the distance of the 

 revolving body. Hence, if the real distance should be greater or less than that 

 supposed, in the proportion of 2 or 3 to 1, the density of the central body mutt 

 be less or greater, in the first case, in the proportion of 64, or in the latter of 

 729 to 1 . 



24. There is also another circumstance, from which perhaps some little addi- 

 tional probability might be derived, with regard to the real distance of a star, 

 such as that we have supposed ; but on which however, it must be acknowledged, 

 that no great stress can be laid, unless we had some better analogy to go on than 

 we have at present. The circumstance I mean is the greater specific brightness 

 which such a star must have, in proportion as the real distance is less than that 

 supposed, and vice versa ; since, in order that the star may appear equally lumi- 

 nous, its specific brightness must be as the 4th power of its distance inversely ; 

 for the diameter of the central star being as the cube of the distance between 

 that and the revolving star, and their distance from the earth being in the simple 

 ratio of their distance from each other, the apparent diameter of the centra! star 

 must be as the square of its real distance from the earth, and consequently, the 

 surface of a sphere being as the square of its diameter, the area of the apparent 

 disc of such a star must be as the 4th power of its distance from the earth ; but 

 in whatever ratio the apparent disc of the star is greater or less, in the same ratio 

 inversely must be the intensity of its light, in order to make it appear equally 

 luminous. Hence, if its real distance should be greater or less than that sup- 

 posed in the proportion of 2 or 3 to 1, the intensity of its light must be less or 

 greater, in the first case, in the proportion of 16, or in the latter of 81, to 1. 



25. According to Mons. Bouguer (see his Traite d'Optique) the brightness of 



