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

Book III. its epicycle BDA, in the reciprocal duplicate proportion of the Sun's ditance from the Earth. Therefore that it may move yet fater in the reciprocal imple proportion of the ditance; uppoe 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 (et off), the angle EDF equal to the exces of the foreaid annual argument above the ditance of the Moon's apogee from the Sun's perigee in conequentia; or, which comes to the ame 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 ditance 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 whoe 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 decribe a certain curve line, about the centre C, with a velocity which will be almot reciprocally as the cube of the Sun's ditance from the Earth, as it ought to be.

The calculus of this motion is difficult, but may be render'd more eay by the following approximation. ASS undefineduming as above the Moon's mean ditance from the Earth of 100000 parts, and the eccentricity TC of 5505 uch parts, the line CB or CD will be found $11721 3⁄4$, and DF $35 1⁄5$ of thoe parts. And this line DF at the ditance TC ubtends 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 poition, at the ditance of the upper focus of the Moon's orbit