The Motions of the Moon

Proposition 25 Problem 6: Find the forces with which the sun disturbs the motions of the moon

Let S represent the sun, T the earth, P the moon, CADB the moon’s orbit.

In SP take SK equal to ST; and let SL be to SK in the duplicate proportion of SK to SP: draw LM parallel to PT; and if ST or SK is supposed to represent the accelerated force of gravity of the earth towards the sun, SL will represent the accelerative force of gravity of the moon towards the sun.

But that force is compounded of the parts SM and LM, of which the force LM, and that part of SM which is represented by TM, disturb the motion of the moon, as we have shewn in Prop. LXVI, Book I, and its Corollaries.

For asmuch as the earth and moon are revolved about their common centre of gravity, the motion of the earth about that centre will be also disturbed by the like forces; but we may consider the sums both of the forces and of the motions as in the moon, and represent the sum of the forces by the lines TM and ML, which are analogous to thorn both.

The force ML (in its mean quantity) is to the centripetal force by which the moon may be retained in its orbit revolving about the earth at rest, at the distance PT, in the duplicate proportion of the periodic time of the moon about the earth to the periodic time of the earth about the sun (by Cor. 17, Prop. LXVI, Book I); that is, in the duplicate proportion of 27d.7h.43′ to 365d.6h.9′; or as 1000 to 178725; or as 1 to 17829⁄40.

But in the 4th Prop. of this Book we found, that, if both earth and moon were revolved about their common centre of gravity, the mean distance of the one from the other would be nearly 60½ mean semi-diameters of the earth; and the force by which the moon may be kept revolving in its orbit about the earth in rest at the distance PT of 60½ semi-diameters of the earth, is to the force by which it may be revolved in the same time, at the distance of 60 semi-diameters, as 60½ to 60: and this force is to the force of gravity with us very nearly as 1 to 60 × {\displaystyle \scriptstyle \times } 60. Therefore the mean force ML is to the force of gravity on the surface of our earth as 1 × {\displaystyle \scriptstyle \times } 60½ to 60 × {\displaystyle \scriptstyle \times } 60 × {\displaystyle \scriptstyle \times } 60 × {\displaystyle \scriptstyle \times } 17829⁄40, or as 1 to 638092,6; whence by the proportion of the lines TM, ML, the force TM is also given; and these are the forces with which the sun disturbs the motions of the moon. Q.E.I.