Page:Newton's Principia (1846).djvu/457

 the co-sine of double this distance, or of an angle of 37 degrees; that is in proportion of 10000000 to 7986355; and, therefore, in the preceding analogy, in place of S we must put 0,7986355S.

But farther; the force of the moon in the quadratures must be diminished, on account of its declination from the equator; for the moon in those quadratures, or rather in 18½ degrees past the quadratures, declines from the equator by about 23° 13′; and the force of either luminary to move the sea is diminished as it declines from the equator nearly in the duplicate proportion of the co-sine of the declination; and therefore the force of the moon in those quadratures is only 0.8570327L; whence we have L + 0,7986355S to 0,8570327L - 0,7986355S as 9 to 5.

Farther yet; the diameters of the orbit in which the moon should move, setting aside the consideration of eccentricity, are one to the other as 69 to 70; and therefore the moon's distance from the earth in the syzygies is to its distance in the quadratures, cæteris paribus, as 69 to 70; and its distances, when 18½ degrees advanced beyond the syzygies, where the greatest tide was excited, and when 18½ degrees passed by the quadratures, where the least tide was produced, are to its mean distance as 69,098747 and 69,897345 to 69½. But the force of the moon to move the sea is in the reciprocal triplicate proportion of its distance; and therefore its forces, in the greatest and least of those distances, are to its force in its mean distance is 0.9830427 and 1,017522 to 1. From whence we have 1,017522L $$\times$$ 0,7986355S to 0,9830427 $$\times$$ 0,8570327L - 0,7986355S as 9 to 5; and S to L as 1 to 4,4815. Wherefore since the force of the sun is to the force of gravity as 1 to 12868200, the moon's force will be to the force of gravity as 1 to 2871400.

. 1. Since the waters excited by the sun's force rise to the height of a foot and 11$1/30$ inches, the moon's force will raise the same to the height of 8 feet and 7$5/22$ inches; and the joint forces of both will raise the same to the height of 10½ feet; and when the moon is in its perigee to the height of 12½ feet, and more, especially when the wind sets the same way as the tide. And a force of that quantity is abundantly sufficient to excite all the motions of the sea, and agrees well with the proportion of those motions; for in such seas as lie free and open from east to west, as in the Pacific sea, and in those tracts of the Atlantic and Ethiopic seas which lie without the tropics, the waters commonly rise to 6, 9, 12, or 15 feet; but in the Pacific sea, which is of a greater depth, as well as of a larger extent, the tides are said to be greater than in the Atlantic and Ethiopic seas; for to have a full tide raised, an extent of sea from east to west is required of no less than 90 degrees. In the Ethiopic sea, the waters rise to a less height within the tropics than in the temperate zones, because of the narrowness of the sea between Africa and the southern parts of America. In the middle of the open sea the waters cannot rise with-