Proposition 97

PROPOSITION 97. PROBLEM

795. With the sun at rest at S (Fig. 74) and with the earth T moving around it uniformly in the circle TD while the moon L is attracted to the earth T as to the sun S in the inverse square of the distances; with which put in place it is required to determine the motion of the moon, such as can be seen from the earth T.

SOLUTION

The distance of the earth from the sun ST is put equal to a and the force, and the force which attracts the earth to the sun is equal to ..

The distance of the moon from the earth is equal to y and the distance of the moon from the sun LS is equal to z. The force, by which the moon is attracted to the earth, is equal to h2 ; and indeed the y force, by which the moon is attracted to the sun f . [Paul Stackel’s note : In the formulas 2 and 2 , the letter f does not have along LS, is equal to the same value. On this account the solution to the problem would have to be modified. This translator’s note : Euler always deals with accelerations, or the forces per unit mass; thus, I am inclined to believe he has the accelerations of the bodies in mind here, rather than the actual forces, in which case the formulas are the same and so are correct. Euler talks about forces or strengths of forces where we would use the word acceleration.] Therefore from these forces acting, such lunar motion produced is to be investigated. But since it is agreed that such motion of the moon is to be viewed from the earth, then the earth is considered at rest ; when it is done, while the motion for the whole system is viewed relative to the earth, then equal and likewise opposite accelerations must be applied to that which the earth receives from the sun [and the moon], with the moon and the sun known to be carried round in the opposite direction to their true motion. [Euler is concerned with a kinematic problem involving relative accelerations; the basic physics has been attended to already in the setting up of the orbits. There is no question of an earth-centred dynamics problem being solved. He tries to fit a solution to this vexing problem.][p. 334] Moreover the speed of the earth in the orbit TD corresponds to the f height 2a , as can be gathered from the force f by which the earth is drawn towards the a2 sun. Therefore such a speed must be impressed both on the sun and the moon along the direction normal to TS. Besides, since the earth is drawn to the sun by the force f , it is a2

necessary that the effect of this force is the destruction of the original force, and if the sun is always attracted to the earth by such a force, then the moon truly is acted on by the same force along the line LN parallel to ST itself [meaning that they have a common acceleration which is the opposite of the centripetal acceleration of the earth towards the sun]. With this done the sun describes a circle SE around the earth T at rest with the same speed, which before the earth was carried around the sun. Truly the moon besides the forces pulling along LT and LS above is urged by a force equal to With LM drawn parallel to TS also, the force acting along LS f a2 in the direction LN . f is resolved into these z2 two forces, of which one is in the direction LT, and the other along LM. Hence by fy z considering the triangle LTS the force arises acting along LT equal to 3 and the force pulling along LM equal to af . Whereby with these forces combined, the moon is pulled z3 in the direction LT by a force equal to : and in the direction LM by a force equal to : from which forces the motion of the moon must be determined. Moreover it is to be noted that the direction LM is not constant but variable, clearly always parallel to the radius TS, which on account of the motion of the sun is carried along the periphery SE. Therefore with ST produced in A, in order that AB is the line of the conjunctions, and from L by sending the perpendicular LP to the line AB, TP is equal and parallel to LM. [p. 335] In the small interval of time dt the moon travels from L to l, and moreover the sun from S to s; and therefore meanwhile the line of conjunctions is carried to ab and the moon at l is acted on in part by a force along lT, and in part by a force pulling along the line parallel to Tp, clearly with the perpendicular lp sent from l to Ta. Moreover from these forces resolved, the normal and tangential force can be found, of which either gives the speed of the moon. Moreover these two equations can be solved to eliminate the speed, and present the equation of the curve ABL, in which the moon can be determined to move. Q.E.I.

Scholium 1.

796. The equations which hence are deduced for the motion of the moon, become so complex that from them neither the orbit of the moon nor the position of the apsides of this motion can be exactly determined. Moreover truly from the same calculation by neglecting very small quantities in a certain way approximate conclusions for the use of astronomy can be drawn, as the great Newton did in Book III of the Phil. Princ. Moreover even if this inconvenient calculation does not work, yet from this proposition without a great deal of rigor, the motion of the moon may soon be demonstrated. For we have put the sun forwards again as being at rest, which in a short while disagrees with the truth; then we consider the earth moving in a circle, and the orbit of the moon placed in the same plane with the earth, which likewise they have otherwise. Yet meanwhile it is certain, if the solution of this proposition can evolve and from that a table constructed, then it would be of the most use in astronomy. [p. 336]

Corollary 1.

797. Since the distance of the moon from the earth is very small with respect to the distance of the earth from the sun, it is possible to put z = a without sensible error and it is almost possible to put in that case the force acting along LM to vanish and the only force drawing the moon to the earth is equal to :

Corollary 2.

798. When the orbit of the moon does not differ much from a circle, it is possible in that case to consider the form of the moving ellipse, as we have done in Prop. 91 (747). Whereby a knowledge of the motion of the apsides is obtained from Coroll. 3 of this Proposition (750).

and the moon arrives at the apogee from the perigee from the absolute angular motion around the earth by turning through an angle equal to : where y, which does not change much, can be considered as a constant.

Scholium 2

799. Therefore the line of the apsides of the moon’s motion is continually regressing, since a 3 h + fy 3 is less than one, which is contrary to observation. Truly the reason for this a 3 h + 4 fy 3 error is that we have considered the quantity z as being constant. For although [the sun- moon distance] z is neither much increased or decreased in the ratio of this to itself, yet the increments and decrements with respect to the increments of y are small enough to be disregarded. Whereby, when the differential of P must be taken, in that equation we have wrongly considered z as constant and put a in its place. Moreover, since z cannot be given by y, the motion of the apsides cannot be determined in this way. Meanwhile nevertheless this is gathered, if it is the case that ady > ydz, then the line of apsides is as in the preceding, but if ady < ydz, as a consequence it is to be moving forwards, if indeed we stop thinking about a force acting along LM

Corollary 3

800. W ith LM = TP = x then we have approximately z = a + x , where x very small with respect to a. Therefore by ignoring x before a then the force, which pulls the moon to the fy 3 fx earth is equal to h2 + 3 and the force which is pulling along LM is equal to 3 . This y a a therefore vanishes when the moon is at right angles in its orbit to the earth, and is a maximum when the moon is in conjugation.

Scholium 3

801. Moreover since this is not the place to pursue considerations of the motion of the moon in more detail, which are clearly pertinent to the theoretical astronomy, we will proceed to piece together what remains from our present arrangement. For the principles are sufficiently well understood to the extent that tables of the movements can be constructed for any case of interest and the respective motions can be found from the tables.

Moreover, the motions of free bodies which are not made in the same plane remain to be included in this chapter [p. 338]. Indeed from the preceding it is evident that for a singe centripetal force present, the motion of the body always takes place in the same plane as the body was initially projected; and if there are several centres of force placed in the same plane in which the body is projected, then the curve described by the body is likewise completely in the same plane. What follows must therefore refer to the situation when a body is acted on by several forces, the directions of which lie in different planes; or also when the direction, along which the body is initially projected, is not in situated in that plane in which the directions of the forces are placed. Therefore in these cases, the motion of the body must be considered as if a certain curve is described on a certain convex or concave surface. Moreover the nature of the surface is expressed by an equation involving three variables, and from the nature of the line drawn on that surface that same equation is solved with another equation either in these three variables as well, or only two. For from these, the projection of the curved line onto a given plane can become known, and from the projection and the surface, likewise the curve described by the body placed on the surface is known. Again as in the co-planar case, the forces can be reduced to two, along the normal and the tangent, thus in this current business, the forces can be reduced to three (551), and what effect they exert on the body, we are about to find out. [p. 339]