THE CURVILINEAR MOTION OF FREE POINTS ACTED ON BY ABSOLUTE FORCES OF ANY KIND

PROPOSITION 92. PROBLEM

763. To find the centripetal forces acting towards two centres C and D (Fig. 69), which can be composed, in order that the body moves on a given curve AMB and with the speed given at individual points M.

SOLUTION.

764. The body is moving from A through M to B and its speed at M corresponds to the height v; and putting CM = y, DM = z. Truly with the tangent TV drawn at that point, and with perpendiculars CT and DV sent from C and D , called CT = p and DV = q. Again the centripetal force, that attracts towards C is equal to P, and that which pulls towards D is equal to Q. Therefore the normal force arising from each is equal to and the tangential force accelerating the motion of the body is equal to for falling tangents as previously. Therefore with the radius of osculation at M equal to r and with the element of the curve equal to ds then [p. 317] (561) and (559). Truly we have : and hence dv = − Pdy − Qdz .

From these two equations joined together [on eliminating ds] we have : and For indeed: hence Finally, calling CD = k then from which P and Q can be determined as you please. Q.E.I. Corollary 1. 764. If the body should be moving with uniform motion on the curve, thus in order that v = c and dv = 0, then we have : And

Corollary 2. 2 765. If we have v = ch2 or the speed of the body varies inversely as the perpendicular p sent, then dv = 2 2ch dp . With these put in place, then Q = 0; for p3 And, (714). For this force alone is acting, in order that the body moves on this curve in this way.

Example.

766. Let the given curve AMB be an ellipse and the centres C and D its foci. The transverse axis AB = A and the latus rectum = L , and from the nature of the ellipse [p. 318] But in addition : From which there is produced ; and thus

PROPOSITION 93. PROBLEM

767. A body is moving with a given speed on a curve also given AMB (Fig. 70), and it is required to find the centripetal force acting towards the centre C together with a force always acting normally to the line AB pulling the body in the direction MP, which two forces have the effect that the body is free to move on this curve with the prescribed speed.

SOLUTION.

Let the speed of the body present at the point M correspond to the height v and the distance MC = y, and the perpendicular MP = z. The force pulling the body towards the centre C is put equal to P and the force acting along MP is equal to Q. With the tangent MV at M drawn, the perpendiculars CT and PG are sent to that, which are called p and q. With these put in place, the normal force arising from both these forces is equal to and the tangential force is equal toEULER’S MECHANICA VOL. 1. Chapter Five (part d). Translated and annotated by Ian Bruce. page 451 Hence with the radius of curvature at M = r put in place : (561) and (559). From these it is thus found that : Moreover, on putting CP = x, we have : [p. 319] and with dx made constant, we have : Moreover, again we have : With these substituted, we have : and Q.E.I.

Corollary 1.

768. If the body is required to move uniformly on the curve, thus so that v = c and dv = 0, then we have :

Corollary 2.

769. If the curve is a circle, the centre of which is present at C, and the radius is called a, 2 then r = a, y = a, p = a and q = za . Whereby the forces are produced: Qz Therefore with Q known, then P = 2av − a . And if v = c and dv = 0, then Q = 0 and P = 2ac .

Scholium.

770. Observe that from this proposition, in which the curve described by the body is given, [conversely] a little of the usefulness follows in the determination of curves which bodies describe, acted on by composite forces. Indeed it is true that it is permitted to progress from this to other propositions, in which the curves described by bodies are not themselves given, but they are generated under the motion of one or more given forces, as in the above propositions in which the motion of the apsides has been treated.

PROPOSITION 94. [p. 320] PROBLEM.

771. If the body is moving on the curve AMB (Fig. 71) in whatever manner, while the curve itself is revolving around the central fixed point C, it is required to find two forces, one of which is always acting towards the fixed point C, and the other is directed normally to the line in the given position PC, which two forces have the effect that the body is free to move in this orbit.

SOLUTION.

Let the speed of the body present at M, in which it traverses the element of the curve itself Mm, correspond to the height v, and the angular speed of the body in the orbit to the true angular speed of the body be in the ratio about C as 1 to w, or the angular speed of the body in orbit to the angular speed of the orbit itself, while the body is at M, is as 1 to w –

1. [If ωa , ωo , and ωb / o are the absolute, orbital, and relative to the orbital, angular speeds of the body then ωa = ωo + ωb / o ; Euler sets ωb / o = ω1 , then ωa ωb / o ωb / o = = ω1−1 ] The radius ωa −ωb / o ωo CM is put equal to y and the perpendicular CT from C sent to the tangent of the orbit at M is equal to p, while the tangent itself MT is equal to q, thus in order that q = ( y 2 − p 2 ) . From M the perpendicular MP which is called z, is sent to the position of the given line DP, and CP is called x, thus in order that x = ( y 2 − z 2 ) . Now while the element Mm is traversed, meanwhile the orbital angle is put equal to mCμ of the circumference; as from the composite motion of the body it arrives at μ, with Cμ = Cm , Mμ is an element of the true curve in which the body is moving, in which produced the perpendicular CΘ is sent from C. With centre C the little arc Mnν is described, [p. 321] where we have : mn = μν = dy et Mn : Mv = 1 : w . Truly, we have ydy Mm = q ,EULER’S MECHANICA VOL. 1. Chapter Five (part d). page 454 Translated and annotated by Ian Bruce. hence Mv = wpdy , q from which is found : and again, [as CΘM and Mμν are similar triangles :] and The ratio is made Mm : Mμ = v : v( w 2 p 2 + q 2 ) v( w 2 p 2 + q 2 ) , the square of which shows y y2 the height corresponding the true speed of the body ; therefore the increment of this is equal to Moreover, the radius of osculation of the true curve, in which the body falls, is equal to : With the whole sine put as 1, the sine of the angle CMP is equal to xy and the cosine is equal to zy . But the sine of the angle CMΘ is equal to is equal to q ( w p +q ) 2 2 2 wp ( w p2 +q2 ) 2 and the cosine of this . From which the sine of the angle PMΘ is equal to and the cosine of this is equal to wpx + qz y ( w2 p 2 + q 2 ) qpz − qz y ( w2 p 2 + q 2 ) . By sending the perpendicular PQ from P to the tangent MΘ , then And with the body traversing the element Mν the increment of the line PM is From which equation the relation between w and x becomes known and likewise the position of the apsidal line AB with respect to the line CP can be found.EULER’S MECHANICA VOL. 1. Chapter Five (part d). Translated and annotated by Ian Bruce. page 455 Now put the force acting on the body towards MC equal to P and the force pulling along MP equal to Q, [p. 322] from which there arises the tangential force retarding the motion of the body, which is equal to : dy which therefore taken by q ( w2 p 2 + q 2 ) can be put equal to : thus in order to give : Moreover, the normal force arising from both is equal to : which must be equal to : (561), whereby we have : From which equations solved for P and Q, we have : and Truly the angle, that the apsis line AB makes with the line CP, is equal to thus the position of this is known for any time. Q.E.I.

Corollary 1.

wpxdy + qzdy 772. Since dz = , putting z = ty then there comes about qy From which equation, if w is given in terms of y, in which also on account of the given curve AMB, p and q are expressed, t can be found and likewise also z and x. Corollary 2. 773. If the speed of the body in the orbit v varies inversely as the perpendicular CT 2 sent from C to the tangent or v = a 2c , then we can write : p Corollary 3. [p. 323] 774. If in this case w is constant, the force Q vanishes and there remains alone pulling towards the centre C, which effects the motion, as the body progresses in the orbit AMB moving around C, evidently as has been found above (734) Corollary 4. 2 775. If v is not a 2c , but w is constant, thus as the angular motion of the orbit is p proportional to the angular motion of the body in the orbit truly as w – 1 to 1, then we haveEULER’S MECHANICA VOL. 1. Chapter Five (part d). Translated and annotated by Ian Bruce. page 457 Example. 2 776. On putting v = a 2c the curve AMB is an ellipse having C in either focus. Therefore p if the transverse axis of this ellipse is called A and the latus rectum L, then Whereby we have

Scholion 1.

777. These formulae for the curve of the ellipse can be made simpler in various ways, if the curve in which the body is moving is approximately circular. And in this case it is of some use in the theoretical motion of the moon to be defined. [p. 324] For the earth is put at rest at C and the sun on the line CP perpendicular at C is considered equally as being at rest ; with which put in place and with these forces compared both with the forces of the sun and the earth, the synodal motion of the moon is elicited for some position of the apsidal line and likewise the motion of the apsidal line, which only differs slightly from the true motion of the moon.

Scholium 2.

778. This proposition certainly appears of greater extent than the above (729), in which all the force was directed towards the centre of rotation of the orbit; indeed the former is included with the force Q vanishing. Yet the quadrature cannot account perfectly for the motion of the moon because the proportional force in P varies inversely with the cube of the distance MC (773). Because of this we offer other orbits besides the gyrations in the middle, and which appear wider and agree more with the physics questions. Of this kind are the motions of orbits by which curves are always in orbits parallel to themselves, which on contemplation deserve to be preferred from others, since the forces acting are both easier to find and the formulas are simpler to understand. Moreover there is an outstanding need for this, and the following theorem is presented.EULER’S MECHANICA VOL. 1. Chapter Five (part d). Translated and annotated by Ian Bruce. page 458 PROPOSITION 95. THEOREM.
779. The body is moving along the curve AM (Fig. 72), by some force acting around the point C, and in addition [p. 325] both the body and the point C are acted on by a force in the same direction; there is the relative motion of the body M with respect to the point C or the motion of the body M such as is seen from C, and likewise, if this new force is not to be added..

DEMONSTRATION.

With the point C at rest in the element of time dt the body arrives at the point m from M in m. Therefore in this small finite time the body M is apart from C by the interval mC and makes the angle mCA with a certain fixed line AC expressed in the region. Now with the body put present at M , both the body M and the point C are to be acted on by a force towards the same place, thus in order that the point C can be advanced through Cc in the small time dt. Therefore in the same short time dt the body M, if at rest, by this force is carried through the distance Mn parallel and itself equal to Cc. But since the body M now has the motion in place, in which element of time dt it traverses the distance Mm, then with both motions connected together it describes the diagonal Mμ to complete the parallelogram Mmμn. Wherefore with the occurrence of this new force, the body M in the short finite time dt is at a distance from the point C, that meanwhile is translated to c by the interval μc, and with the line ac drawn parallel to AC in the fixed direction, it makes the angle μca. But since mCcμ is a parallelogram μc = mC and the angle mCA is equal to the angle μca. Consequently the force is acting equally at each point M and C equally and acting along the same direction, it does not change the motion of M relative to the point C. Q.E.D. [p. 326] Corollary 1. 780. Therefore whatever the force acting at the point C, if the same likewise acts on the body M along the same direction, then the relative motion of the body M with respect to the point C does not change.EULER’S MECHANICA VOL. 1. Chapter Five (part d). Translated and annotated by Ian Bruce. page 459 Corollary 2. 781. Therefore a force of this kind can be put into effect acting equally on M and C, in order that the body M is moving in the moving orbit AM in some manner. Moreover the orbit by its own motion is able to follow the motion of the point C, as its position always remains parallel to the position of that point. Corollary 3. 782. It is also seen from the demonstration of the proposition, if an equal speed is impressed on both the point C and the body M along the same direction, then the relative motion is not going to be disturbed. Corollary 4. 783. With such motion impressed on the point C, it continues to move indefinitely in some direction without the continuation of any force, and the body M follows moving uniformly around the point C, that can be equally progressing or at rest. For this situation is only maintained, provided the same direction and speed of the body C are added to the body M. Corollary 5. 784. Therefore the body is free to describe the same curve about the centre of force C that is progressing uniformly [p. 327], that it describes about C at rest, but to M must be added as well as much speed as has been accepted by the centre of force. Corollary 6. 785. Moreover, as the body left to itself is unable by itself to progress along the curved line, nor to move unequally along a straight line; thus the body moving around the moving centre of force, either in performing the required curved motion or changes in the direction, cannot describe the same curve that is does for the resting position, but it is always urged on by the above force, to the extent required to maintain the motion along its own path.EULER’S MECHANICA VOL. 1. Chapter Five (part d). page 460 Translated and annotated by Ian Bruce. PROPOSITION 96. PROBLEMA. 786. If the body M (Fig. 71) is revolving around the centre of force L at rest in the curve BM, to determine the force that is effective in order that the body in the same orbit following the same curve AL is moving in a direction always parallel to itself. SOLUTION. Since the body M is freely moving in the orbit BM about the centre of force L at rest, then, with the distance LM = y and with the perpendicular sent from L to the tangent at M equal to p, the height corresponding to the 2 speed at M = a 2c (589) and the centripetal p force acting towards L is equal to 2a 2 cdp p 3 dy (592). Now the centre L is put to be moving on the curve AL attracted to the centre of force C, and CL = c and the perpendicular sent from C to the tangent [p. 328] at L is equal to 2 w. With which put in place the height corresponding to the speed at L = b 2e and the force w 2 attracting the point L to C is equal to 2b 3edw (592). Moreover the speed impressed on the w ds first point L along the tangent is equal to b we and it is also the same for the body M following the direction parallel to this tangent; and it is evident, if no additional force acts on the point L, but only this impressed motion is conserved, then the motion of the body M gone through is thus obtained following the motion of the point L , moving freely on the same curve BM, in order that the axis BL always remains parallel to the motion of L (783). But when the body M is moving in the same way around the advancing point L on the curve AL, it is required, as such a force is itself always acting, that it is of such a size as is necessary to keep the point L on this curve AL (781). Therefore, with MN drawn parallel to LC, the body M besides the force by which it is 2 urged towards L, must also be acted on by a force equal to 2b 3edw along the direction w ds MN. Whereby this two-fold force has the effect of solving the problem of the motion. For which moreover it is apparent, since the body M is acted on by the force with respect to the point C and to the fixed line AC parallel to BL, that the forces acting along MN and ML on the body M are to be resolved in two others, of which one has the direction MC, and the other MP, which line MP has been drawn perpendicular to AC. To this outstanding line MP is drawn the parallel line KLN cutting the line MC at O and the distance LI = x, MI = z, CK = r, KL = t and CP = X, PM = Z and CM = Y. On this account,

And again, [p. 329] Besides indeed X = r + x and Z = t + z . Whereby on account of the given curves AL and BM since z, y, and p in x, and likewise t, s, and w in r are given, all these quantities in X, Z, and Y are able to be shown. Truly the equation between X and Z can be deduced from that, because the increment of the time through BM is equal to the increment through AL. Hence we therefore have : and from the integrals taken, the area ACL ± some constant area is to the area BLM as b e to a c . Therefore the force pulling along ML is resolved into two forces pulling along MO and LO or MI, and in a similar way the force along MN is resolved into two forces pulling along MO and NO , of which the latter acts on the opposite direction to MI on the body M ; to which resolved forces put in place it is necessary to know the angles. For indeed the angle MLO is equal to the angle LMI and therefore the sine of this is equal to xy and the cosine zy , by taking 1 for the total sine. Similarly the angles MON and CMP are equal, therefore the sine of this is YX and the cosine YZ . Therefore the sine of − Zx . And finally the the angle LMO, which is the difference of these, is equal to XzYy angles MNQ and CLK are equal, whereby the sine is equal to rs and the cosine st . − Xt . Also, Consequently as NMO = NMO = MNQ – MON the sine of this angle is ZrYs Xx − Zx = Zr − Xt as X = r + x and Z = t + z . From these the ratio of MN:MO or YX : rs can be made and thus the ratio of the force 2 pulling along MN , 2b 3edw to the force along MC, which hence is equal to w ds − Xt , thus the force along MN , 2b edw , to the force along And the ratio MN:NO or YX : ZrYs 3 2 w ds ON, [p. 330] which is equal to

x X In a like manner the ratio ML : MO = Y : y , and thus the ratio of the force along ML, 2 a 2 cdp , to the force along MC, which is hence equal to w3 dy − Xt , and thus the ratio of the force along ML, And ML : MO = YX : ZrYy along OL or MP, which is hence equal to 2a 2 cdp , to the force p 3 dy 2a 2 cdp( Zr − Xt ) . Xyp 3 dy From these collected together the body M is acted on by a force pulling in the direction of MC equal to: and by a force along MP : Which expressions can all be shown in terms of X, Z and Y and besides an equation can be assigned between these quantities which relate to the true curve described by the body M. Q.E.I. Corollary 1. 787. If the curve AL is the periphery of a circle, the centre of which is C and the radius AC = b, then s = w = b and r 2 + t 2 = b 2 . Therefore in this case the force pulling along MC is equal to : and the force along MP is equal to : And e is the height corresponds to the speed of the point L. Corollary 2. 788. Let the curve BM be an ellipse having the centre at L, and BL= a is the transverse semi-axis of this curve, thus in order that the height corresponding to the speed at B is equal to c, and indeed the other axis is set equal to h. And the equations arise : Whereby the force along MC becomes equal to :

[p. 331] And the force along MP is equal to : Corollary 3. 789. And if the curve AL is the circle as in Cor.1 and the curve BM the ellipse as in Cor. 2, then the force along MC is equal to : and the force along MP is equal to : where r and t and x are to be determined in terms of X, Z, and Y from these equations : Moreover the equation emerging rises to the fourth power. Corollary 4. 790. If the ellipse BM is put infinitely small or even extremely small with respect to the circle AL, yet thus, as the periodic time of the ellipse is a finite quantity, then and Corollary 5. 791. If the curve BM is also a circle having centre L, we can place h = a and thus from which on putting b 2 − a 2 = f 2 it is found that andEULER’S MECHANICA VOL. 1. Chapter Five (part d). page 464 Translated and annotated by Ian Bruce. And, Scholium 1. [p. 332] 792. Finally we can add this corollary, as it is apparent that such forces are required for a body to be turning in an epicycle about a centre of force, as the adherents of Ptolemy considered the planets to move. Corollary 6. 793. From the hypothesis of Cor. 3 it is understood, that if e2 = c2 , or the speed of the b h body L to the speed of the body M at B as the diameter of the circle AL is to the conjugate axis of the ellipse BM, then the true curve described by the body M is an ellipse with the centre at C, when the force acting towards C becomes equal to 2eY 2 and with the force b along MP vanishing. The semi-major axis of this ellipse is b + a, and the minor truly b + h. Scholium 2. 794. I have therefore especially reported on this Proposition, because in an appendix to the new edition of Newton’s Principia in English, the most distinguished Machin asserts [John Machin (1680-1751), The mathematical principles of natural philosophy translated into English, London 1729; to which book the Appendix has been added : The laws of the motion of the moon. P. St.] that the motion of the moon can be considered as in an ellipse, the transverse axis of which shall be in the ratio of 2 : 1 to the conjugate axis made around the centre of the ellipse, while meanwhile with that ellipse itself moving parallel to the periphery of a circle, on which it progresses freely, as I have explained in Cor. 3. For my part, I do not deny that this motion is extremely similar to the motion that the moon can show, but I would doubt very much that it was an exact ratio. Moreover in the following proposition I have decided to determine, what needs to be indicated to determine the motion of the moon, [p. 333]. Even if indeed this proposition pertains to astronomy, yet it is assumed that the method reported here can be used to examine fundamental questions of this kind that are to be resolved. [At this time, the laws governing angular momentum and the conservation of energy were not yet fully understood; in the following proposition the inverse square law solution is taken for the earth in a circular orbit, and the moon is assumed to rotate about the earth in an elliptical orbit according to the same law. The moon is given, along with the sun, the negative of the earth’s centripetal acceleration, resulting in the moon’s orbit being transported parallel to itself according to a person on the earth, Thus, the problem is reduced to one of kinematics.]EULER’S MECHANICA VOL. 1. Chapter Five (part d). Translated and annotated by Ian Bruce. page 465 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 f . The a2 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 z2 f f the formulas 2 and 2 , the letter f does not have a z 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. [p. 337] 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]