u118And thus it would be if the moon was uniformly revolved in a circular orbit. But if the orbit is elliptical, the mean motion of the nodes will be diminished in proportion of the lesser axis to the greater, as we have shewn above; and the variation of the inclination will be also diminished in the same proportion.

COR. 1. Upon Nn erect the perpendicular TF, and let pM be the horary motion of the moon in the plane of the ecliptic; upon QT let fall the perpendiculars pK, Mk, and produce them till they meet TF in H and h; then IT will be to AT as Kk to Mp; and TG to Hp as TZ to AT; and, therefore, IT × TG will be equal to , that is, equal to the area HpMh multiplied into the ratio : and therefore the horary variation of the inclination will be to 33'' 10''' 33iv. as the area HpMh multiplied into to AT3.

COR. 2. And, therefore, if the earth and nodes were after every hour drawn back from their new and instantly restored to their old places, so as their situation might continue given for a whole periodic month together, the whole variation of the inclination during that [Pg 442]month would be to 33'' 10''' 33iv. as the aggregate of all the areas HpMh, generated in the time of one revolution of the point p (with due regard in summing to their proper signs + -), multiplied into to Mp × AT3; that is, as the whole circle QAqa multiplied into to Mp × AT3, that is, as the circumference QAqa multiplied into to 2Mp × AT2.

COR. 3. And, therefore, in a given position of the nodes, the mean horary variation, from which, if uniformly continued through the whole month, that menstrual variation might be generated, is to 33'' 10''' 33iv. as to 2AT2, or as to PG × 4AT; that is (because Pp is to PG as the sine of the aforesaid inclination to the radius, and to 4AT as the sine of double the angle ATn to four times the radius), as the sine of the same inclination multiplied into the sine of double the distance of the nodes from the sun to four times the square of the radius.

[Pg 443]

COR. 4. Seeing the horary variation of the inclination, when the nodes are in the quadratures, is (by this Prop.) to the angle 33'' 10''' 33iv. as to AT3, that is, as to 2AT, that is, as the sine of double the distance of the moon from the quadratures multiplied into to twice the radius, the sum of all the horary variations during the time that the moon, in this situation of the nodes, passes from the quadrature to the syzygy (that is, in the space of hours) will be to the sum of as many angles 33'' 10''' 33iv. or 5878'', as the sum of all the sines of double the distance of the moon from the quadratures multiplied into to the sum of as many diameters; that is, as the diameter multiplied into to the circumference; that is, if the inclination be 5° 1', as to 22, or as 278 to 10000. And, therefore, the whole variation, composed out of the sum of all the horary variations in the aforesaid time, is 163'', or 2' 43''.

PROPOSITION XXXV. PROBLEM XVI.

To a given time to find the inclination of the moon's orbit to the plane of the ecliptic.

Let AD be the sine of the greatest inclination, and AB the sine of the least. Bisect BD in C; and round the centre C, with the interval BC, describe the circle BGD. In AC take CE in the same proportion to EB as EB to twice BA. And if to the time given we set off the angle AEG equal to double the distance of the nodes from the quadratures, and upon AD let fall the perpendicular GH, AH will be the sine of the inclination required.

For GE2 is equal to GH2 + HE2 = BHD + HE2 = HBD + HE2 - BH2 = HBD + BE2 - 2BH × BE = BE2 + 2EC × BH = 2EC × AB + 2EC × BH = 2EC × AH; wherefore since 2EC is given. GE2 will be as AH. Now let AEg represent double the distance of the nodes from the quadratures, in a given moment of time after, and the arc Gg, on account of the given angle GEg, will be as the distance GE. But Hh is to Gg as GH to GC, and, therefore, Hh is as the rectangle GH × Gg, or GH × GE, that is, as , or ; that is, as AH and the sine of the angle AEG conjunctly. If, therefore, in any one case, AH be the sine of inclination, it will increase by the same increments as the sine of inclination doth, by Cor. 3 of the preceding Prop. and therefore will always continue equal to that sine. But when the point G falls upon either point B or D, AH is equal to this sine, and therefore remains always equal thereto. Q.E.D.

In this demonstration I have supposed that the angle BEG, representing double the distance of the nodes from the quadratures, increaseth uniformly; for I cannot descend to every minute circumstance of inequality. Now suppose that BEG is a right angle, and that Gg is in this case the horary increment of double the distance of the nodes from the sun; then, by Cor. 3 of the last Prop. the horary variation of the inclination in the same case will be to 33'' 10''' 33iv. as the rectangle of AH, the sine of the inclination, into the sine of the right angle BEG, double the distance of the nodes from the sun, to four times the square of the radius; that is, as AH,[Pg 444] the sine of the mean inclination, to four times the radius; that is, seeing the mean inclination is about 5° , as its sine 896 to 40000, the quadruple of the radius, or as 224 to 10000. But the whole variation corresponding to BD, the difference of the sines, is to this horary variation as the diameter BD to the arc Gg, that is, conjunctly as the diameter BD to the semi-circumference BGD, and as the time of hours, in which the node proceeds from the quadratures to the syzygies, to one hour, that is as 7 to 11, and to 1. Wherefore, compounding all these proportions, we shall have the whole variation BD to 33'' 10''' 33iv. as to 110000, that is, as 29645 to 1000; and from thence that variation BD will come out 16' .

And this is the greatest variation of the inclination, abstracting from the situation of the moon in its orbit; for if the nodes are in the syzygies, the inclination suffers no change from the various positions of the moon. But if the nodes are in the quadratures, the inclination is less when the moon is in the syzygies than when it is in the quadratures by a difference of 2' 43'', as we shewed in Cor. 4 of the preceding Prop.; and the whole mean variation BD, diminished by 1' , the half of this excess, becomes 15' 2'', when the moon is in the quadratures; and increased by the same, becomes 17' 45'' when the moon is in the syzygies. If, therefore, the moon be in the syzygies, the whole variation in the passage of the nodes from the quadratures to the syzygies will be 17' 45''; and, therefore, if the inclination be 5° 17' 20'', when the nodes are in the syzygies, it will be 4° 59' 35'' when the nodes are in the quadratures and the moon in the syzygies. The truth of all which is confirmed by observations.

Now if the inclination of the orbit should be required when the moon is in the syzygies, and the nodes any where between them and the quadratures, let AB be to AD as the sine of 4° 59' 35" to the sine of 5° 17' 20'', and take the angle AEG equal to double the distance of the nodes from the quadratures; and AH will be the sine of the inclination desired. To this inclination of the orbit the inclination of the same is equal, when the moon is 90° distant from the nodes. In other situations of the moon, this menstrual inequality, to which the variation of the inclination is obnoxious in the calculus of the moon's latitude, is balanced, and in a manner took off, by the menstrual inequality of the motion of the nodes (as we said before), and therefore may be neglected in the computation of the said latitude.

SCHOLIUM.

By these computations of the lunar motions I was willing to shew that by the theory of gravity the motions of the moon could be calculated from their physical causes. By the same theory I moreover found that the annual equation of the mean motion of the moon arises from the various[Pg 445] dilatation which the orbit of the moon suffers from the action of the sun according to Cor. 6, Prop. LXVI. Book I. The force of this action is greater in the perigeon sun, and dilates the moon's orbit; in the apogeon sun it is less, and permits the orbit to be again contracted. The moon moves slower in the dilated and faster in the contracted orbit; and the annual equation, by which this inequality is regulated, vanishes in the apogee and perigee of the sun. In the mean distance of the sun from the earth it arises to about 11' 50''; in other distances of the sun it is proportional to the equation of the sun's centre, and is added to the mean motion of the moon, while the earth is passing from its aphelion to its perihelion, and subducted while the earth is in the opposite semi-circle. Taking for the radius of the orbis magnus 1000, and for the earth's eccentricity, this equation, when of the greatest magnitude, by the theory of gravity comes out 11' 49''. But the eccentricity of the earth seems to be something greater, and with the eccentricity this equation will be augmented in the same proportion. Suppose the eccentricity and the greatest equation will be 11' 51''.