VOL. LIV.J PHILOSOPHICAL TBANSACTIONS. 87 



IF. Of the Moons Distance and Parallax. By P. Murdoch, D. D., and 



F. R. S. p. 29. 



Tlie following contains an easy rule for determining the moon's distance, from 

 the received theory of central forces. 



Sect. I. Sir Isaac Newton investigated the law of gravitation, in the duplicate 

 ratio of the distance of the central body inversely, from the following data. 1 . 

 The length of a simple pendulum which vibrates in one second of time, gave 

 him, by Huygens' theorem, a determinate measure of the force of gravity, at 

 the place of observation. And, by his own theory, he could thence infer the 

 like measure for any other place, of a given latitude. 2. The earth's semidia- 

 meter was computed from the Abbe Picard's measure of a degree of the terres- 

 trial meridian. 3. The moon's parallax as determined by the most skilful astro- 

 nomers, gave him the moon's distance in semidiameters of the earth. 4. The 

 time of a periodical month gave him the ratio of the versed sine of the arc of 

 the moon's orbit which she describes in one second, to the radius. 



And from these his conclusion was: that the gravitation at the earth's surface, 

 being diminished as the square of the distance from the earth's centre increases, 

 would, at the distance of the moon, produce a fall from rest, in one second, 

 precisely equal to that versed sine. Or, that the gravitation of the moon toward 

 the earth, being increased as the square of that distance is diminished, would at 

 the earth's surface, be of the same quantity as that of falling bodies is (by the 

 experiment of the pendulum) actually found to be. 



II. But the law of gravitation, thus deduced, being found to hold universally, 

 and reciprocally, among all the great bodies of our system, so that even the mi- 

 nute anomalies of their motions are explained from it; we may now assume it as 

 given, and make the moon's distance the quantity sought. Thus, writing p for 

 the number of feet which a body falling from rest describes, in vacuo, at the 

 equator, in one second ; v for the versed sine of the arc of the moon's orbit de- 

 scribed in the same time, to the radius unity; d for the semidiameter of the 

 equator in feet: and the ratio of the distance of the centres of the earth and 

 moon, to the semidiameter of the earth, that of x to 1 : we shall have, by the 

 general law, the moon's fall in 1', equal to — ; but the same fall is equal to v 



X D X x; whence x^ = , and x = \/ is the distance sought, in semi- 



VXD' "^VXD o' 



diameters of the equator. 



Now a simple pendulum which beats seconds, measuring at London 39.126 

 inches; if the usual allowance is made for the weight of the air, and for the 

 Newtonian figure of the earth, the weight (-j-f-j-) taken off by the centrifugal 

 force being likewise restored, a second pendulum at the equator would be39.154 

 inches long. And, by Huygens' rule, half this length is to the initial fall in one 



