Page:Newton's Principia (1846).djvu/412

 11$5/6$″. Hence an arc of one degree appears to be 57061 Paris toises. And from these measures we conclude that the circumference of the earth is 123249600, and its semi-diameter 19615800 Paris feet, upon the supposition that the earth is of a spherical figure.

In the latitude of Paris a heavy body falling in a second of time describes 15 Paris feet, 1 inch, 1$7/9$ line, as above, that is, 2173 lines $7/9$. The weight of the body is diminished by the weight of the ambient air. Let us suppose the weight lost thereby to be $1/11000$ part of the whole weight; then that heavy body falling in vacuo will describe a height of 2174 lines in one second of time.

A body in every sidereal day of 23h.56′ 4″ uniformly revolving in a circle at the distance of 19615800 feet from the centre, in one second of time describes an arc of 1433,46 feet; the versed sine of which is 0,05236561 feet, or 7,54064 lines. And therefore the force with which bodies descend in the latitude of Paris is to the centrifugal force of bodies in the equator arising from the diurnal motion of the earth as 2174 to 7,54064.

The centrifugal force of bodies in the equator is to the centrifugal force with which bodies recede directly from the earth in the latitude of Paris 48° 50′ 10″ in the duplicate proportion of the radius to the cosine of the latitude, that is, as 7,54064 to 3,267. Add this force to the force with which bodies descend by their weight in the latitude of Paris, and a body, in the latitude of Paris, falling by its whole undiminished force of gravity, in the time of one second, will describe 2177,267 lines, or 15 Paris feet, 1 inch, and 5,267 lines. And the total force of gravity in that latitude will be to the centrifugal force of bodies in the equator of the earth as 2177,267 to 7,54064, or as 289 to 1.

Wherefore if APBQ represent the figure of the earth, now no longer spherical, but generated by the rotation of an ellipsis about its lesser axis PQ; and ACQqca a canal full of water, reaching from the pole Qq to the centre Cc, and thence rising to the equator Aa; the weight of the water in the leg of the canal ACca will be to the weight of water in the other leg QCcq as 289 to 288, because the centrifugal force arising from the circular motion sustains and takes off one of the 289 parts of the weight (in the one leg), and the weight of 288 in the other sustains the rest. But by computation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matter of the earth was all uniform, and without any motion, and its axis PQ were to the diameter AB as 100 to 101, the force of gravity in the place Q towards the earth would be to the force of gravity in the same place Q towards a sphere described about the centre C with the radius PC, or QC, as 126 to 125. And, by the same argument, the force of gravity in the place A towards the spheroid generated by the rotation of