Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/339

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Book III.
of Natural Philoſophy
303

its epicycle BDA, in the reciprocal duplicate proportion of the Sun's diſtance from the Earth. Therefore that it may move yet faſter in the reciprocal ſimple proportion of the diſtance; ſuppoſe that from D the centre of the orbit a right line DE is drawn, tending towards the Moon's apogee once equated, that is, parallel to TC, and (ſet off), the angle EDF equal to the exceſs of the foreſaid annual argument above the diſtance of the Moon's apogee from the Sun's perigee in conſequentia; or, which comes to the ſame thing, take the angle CDF equal to the complement of the Sun's true anomaly to 360°. And let DF be to DC, as twice the eccentricity of the orbis magus to the Sun's mean diſtance from the Earth and the Sun's mean diurnal motion from the Moon's apogee to the Sun's mean diurnal motion from its own apogee conjunctly, that is, as 332 7/8 to 1000, and 52'. 27''. 16'''. to 59'. 8''. 10'''. conjunctly; or as 3 to 100. And imagine the centre of the Moon's orbit, placed in the point F, to be revolved in an epicycle whoſe centre is D, and radius DF, while the point D moves in the circumference of the circle DABD. For by this means the centre of the Moon's orbit comes to deſcribe a certain curve line, about the centre C, with a velocity which will be almoſt reciprocally as the cube of the Sun's diſtance from the Earth, as it ought to be.

The calculus of this motion is difficult, but may be render'd more eaſy by the following approximation. Aſſuming as above the Moon's mean diſtance from the Earth of 100000 parts, and the eccentricity TC of 5505 ſuch parts, the line CB or CD will be found 11721 3/4, and DF 35 1/5 of thoſe parts. And this line DF at the diſtance TC ſubtends the angle at the Earth, which the removal of the centre of the orbit from the place D to the place F generates in the motion of this centre; and double this line DF in a parallel poſition, at the diſtance of the upper focus of the Moon's orbit