222

Sir Isaac Newton’s BookII.

fcribed. But if the moon be a&ed upon by the fum of theforementioned powers, and the velocity of the moon in thepoint A be augmented in a certain proportion a ; or if thatvelocity be diminished in a certain proportion, and the moonbe aCted upon by the difference of thofe powers; in boththefe cafes the line AE, which fhall be deferibed by themoon, is thus to be determined. Let the point M be that,into which the moon would have arrived in any given {paceof time, had it moved in the ellipfis A MB. Draw MT,and likewife C T D in fuch fort, that the angle under ATMfhall bear the fame proportion to the angle under AT C, asthe velocity, with which the ellipfis A M B muff have been de-feribed, bears to the difference between this velocity, and thevelocity, with which the moon muff fet out from the point Ain order to deferibe the path A E. Let the angle A T C be ta-ken toward the moon (as in fig. I o y.) if the moon be attract-ed by the fum of the powers; but the contrary way (as infig. 106.) if by their difference. Then let the line AB bemoved into the pofition C D, and the ellipfis A M B into theSituation C N D, fo that the point M be tranflated to L: thenthe point L fhall fall upon the path of the moon A E.

47. The angular motion of the line A T, wereby it isremoved into the Situation C T, reprefents the motion of theapogeon ; by the means of which the motion of the moonmight be fully explicated by the ellipfis A M B, if the a&ion ofthe fun upon it was directed to the center of the earth, and

a Y^hat this proportion i’, maybe known from Coroll, z> prop 44. Lib.I.Princ, philof. Newton.

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