Page:Newton's Principia (1846).djvu/459

 about three times less than the accelerative gravity on the surface of the earth.

. 6. And the distance of the moon's centre from the centre of the earth will be to the distance of the moon's centre from the common centre of gravity of the earth and moon as 40,788 to 39,788.

. 7. And the mean distance of the centre of the moon from the centre of the earth will be (in the moon's octants) nearly 60$2/5$ of the great est semi-diameters of the earth; for the greatest semi-diameter of the earth was 19658600 Paris feet, and the mean distance of the centres of the earth and moon, consisting of 60$2/5$ such semi-diameters, is equal to 1187379440 feet. And this distance (by the preceding Cor.) is to the distance of the moon's centre from the common centre of gravity of the earth and moon as 40,788 to 39,788; which latter distance, therefore, is 1158268534 feet. And since the moon, in respect of the fixed stars, performs its revolution in 27d.7h.43$4/9$′, the versed sine of that angle which the moon in a minute of time describes is 12752341 to the radius 1000,000000,000000; and as the radius is to this versed sine, so are 1158268534 feet to 14,7706353 feet. The moon, therefore, falling towards the earth by that force which retains it in its orbit, would in one minute of time describe 14,7706353 feet; and if we augment this force in the proportion of 178$29/40$ to 177$29/40$, we shall have the total force of gravity at the orbit of the moon, by Cor. Prop. III; and the moon falling by this force, in one minute of time would describe 14,8538067 feet. And at the 60th part of the distance of the moon from the earth's centre, that is, at the distance of 197896573 feet from the centre of the earth, a body falling by its weight, would, in one second of time, likewise describe 14,8538067 feet. And, therefore, at the distance of 19615800, which compose one mean semi-diameter of the earth, a heavy body would describe in falling 15,11175, or 15 feet, 1 inch, and 4$1/11$ lines, in the same time. This will be the descent of bodies in the latitude of 45 degrees. And by the foregoing table, to be found under Prop. XX, the descent in the latitude of Paris will be a little greater by an excess of about ⅔ parts of a line. Therefore, by this computation, heavy bodies in the latitude of Paris falling in vacuo will describe 15 Paris feet, 1 inch, 4$25/33$ lines, very nearly, in one second of time. And if the gravity be diminished by taking away a quantity equal to the centrifugal force arising in that latitude from the earth's diurnal motion, heavy bodies falling there will describe in one second of time 15 feet, 1 inch, and 1½ line. And with this velocity heavy bodies do really fall in the latitude of Paris, as we have shewn above in Prop. IV and XIX.

. 8. The mean distance of the centres of the earth and moon in the syzygies of the moon is equal to 60 of the greatest semi-diameters of the earth, subducting only about one 30th part of a semi-diameter: and in the