Annotated Translations of Three of the Euler’s Papers on Celestial Mechanics

Annotated translations from Latin of three of the Euler’s papers on celestial mechanics are presented, which fall into the category of three-body problems. The first translation deals with an exact solution of three bodies that move around the common center of mass and always line up. This is considered the first work from which the three collinear Lagrange points could be obtained. The second translation deals with motions of Sun, Earth and Moon in syzygy and Moon libration as well, where, for the first time, Euler introduces an archaic form of a Fourier sine series expansion to describe the Moon’s wagging motion. The last translation relates to a paper that was written with the goal of alleviating astronomical computations of the perturbed motion of the Moon around the Earth by the Sun, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body. Despite showing great analytical skills, Euler gave no indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moon’s orbit.


Introduction
As earlier as the 1730's and until his death in 1783, Euler wrote more than 60 papers on astronomy, including the motion of planets and comets, astronomical perturbation, eclipses, tides and geophysics. Since many of these works often involve rather lengthy and intricate astronomical computations which, nowadays, are, perhaps, of limited interest, we have chosen to translate works of long lasting repercussion such as those related to theoretical and mathematical models to the motion of celestial bodies, and particularly to three-body problems in astronomy. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun, for which Euler gave a significant contribution.
Although some of Euler's papers on the subject had been written in French, most of them are written in Latin, which may represent a barrier to the modern reader. With the goal of disseminating the works of Euler on the subject, we present here annotated translations from Latin of three of Euler's papers related to three-body problems in astronomy.
"On the rectilinear motion of three bodies mutually attracting each other": in this publication, Euler considers three bodies lying on a straight line, which are attracted to each other by central forces inversely proportional to the square of their separation distance (inverse-square law). Although not explicitly mentioned by Euler, this is an exact solution of three bodies that move around the common center of mass and always line up.
"Considerations on the motion of celestial bodies": in this publication, Euler essentially focuses on the solution of two particular motions of a three-body problem consisting of Sun, Earth and Moon. The first motion, represents a hypothetical situation of these three celestial bodies in perpetual alignment in syzygy-the three-body problem on a straight line. The second motion considered by Euler was Moon libration, when these planets are aligned in regular syzygy.
"An easy method for calculating the motion of celestial bodies perturbed in any manner avoiding astronomical computations": as revealed by its title itself, the goal of this paper is to alleviate the astronomical computations in a typical celestial three-body problem represented by Sun, Earth and Moon. In this work, Euler's approach consists of two parts: geometrical and mechanical. The geometrical part contains most of the analytical developments, in which Euler makes use of Cartesian and spherical trigonometry as well.

On the Rectilinear Motion of Three Bodies Mutually Attracting Each Other (De Motu Rectilineo Trium Corporum se Mutuo Attrahentium, Euler, 1767)
I) Let A, B, C be the masses of three bodies such that their distances to a fixed point O at a given instant of time t is given by OA x = , OB y = and OC z = where, in fact, it is assumed that y x > and z y > . Hence, the principles of motion give these three equations: whence, because we lack a third integral equation, very little is possible to conclude about the movement.
2) Let us set x y p = − and z y q = + , such that p and q are positive quantities; and the first integral [Equation (5) In the original manuscript, these three last lines have been misplaced at the end of § 3.
which [by subtracting Equation (13) And then, each element ddp and ddq can be expressed separately in the following way 1) 4) Since the solution has been reduced to two differential equations involving p, q and t we should expect that significant advantage is to be obtained, if it were possible to reduce these equations to two others of first order only. This is a unique technique that I have discovered which can be applied in the following manner. I put q pu = , and the two differential equations [Equation (15) and Equation (16)] are represented as: ; because it will expose that for these substitutions, the two variables p and t can be eliminated from the calculations, such that only these three [variables] r, s and u are to be determined by their first differentials.
Then, in particular, the equation that the integral was found above [Equation whose usefulness it will be possible to assess. or: Moreover, in particular, it will be considered that which when combined give: 6) We see that we have two first-order differential equations [Equation (26) and Equation (27) Thus, it is possible to determine the quantity ∝ from this equation of the fifth degree: Thence, truly from the relation between r and p [Equation (21) [where ζ is a constant of integration], and then [since q pu p and then, B would be a much more negative quantity, since it is necessary that the value of ∝ itself be positive. 9) Then, since it is necessary that the quantity D be positive, it can be assumed that D aa = , and if also the number ∝ is considered as given, therefore the masses of the three bodies will be obtained as [from Equation (43) (37)], and by putting the constants E and F equal to zero, the locations of the three bodies A, B, C, which center of gravity is now located at O, are defined by r such that: Yet, the relation between r and the time t is [Equation (38)] 3 3 dr t n r rr aa n aa rr aa or 3 3 ln r rr aa t n r rr aa n aa Assuming that the constant a ∆ = , then, for the time 0 t = , r a = , meaning that all bodies are concentrated in the center of gravity [O], whence they will be driven out with an almost infinite velocities, and then, these [distances] are similar to each other as the quantities of: On the other hand, for the time t to be obtained, , then the distances will be:  The conclusion is that for these particular cases, the configuration of the masses during the motion is such that one of the masses occupies the center of gravity, with the two other masses remaining on the same straight line, and moving symmetrically around the center of gravity of the system. This is considered the first work from which the three collinear Lagrange points could be obtained, where the parameter that controls the distances among the bodies was found to be given by a quintic function.
A practical application of these results is to find, for instance, the Earth-Moon

Considerations on the Motion of Celestial Bodies (Considerationes de Motu Corporum Coelestium, Euler, 1766)
I) Although there is no doubt that the laws of motion of celestial bodies observed by Kepler and confirmed by Newton have brought very great gains to [the discipline of] Astronomy, nevertheless it is certain that no body in the heavens is met with that in its own motion follows these laws perfectly, since, instead, in all [of the motions of these bodies] deviations from these laws, that are by no means slight, are detected. Of course, it is true that the cause of all heavenly motions resides in the mutual attraction of these bodies, by which each and every [body] is attracted toward each of the others singly by forces consisting of a ratio composed directly by the un-squared [amount] of the masses [of the bodies], and inversely by the squared [amounts] of the distances. However, it is always convenient to consider that one force stands out among the remaining, and thus, the motion would approximately follows Kepler's rules; and then the relatively very small effect arising from the others can be determined by methods of approxi-mation. Without this simplification, we would be at the utmost ignorance about the celestial motion, since to date no method has been discovered by the application of which the motion, of three or more bodies mutually attracting each other, might require to be ascribed; unless, perchance, one force surpasses the others.
2) Yet indeed this case-in which [case] alone Geometers do not squander their work at all points in vain-cannot be taken as conclusive since the method of approximation itself, which Geometers are accustomed to use, is bound up with a great many difficulties besides, and an unlimited multitude of small perturbations is neglected, by which [fact] it becomes so that this approximation by itself [only] minimally carries through the business [of determining the motions], but on the contrary, for it to be completed, still more supports are desired. Wherefore, although from this Theory the motion of the Moon is determined accurately enough, that [fact of sufficient accuracy] ought to be ascribed more to special circumstances that obtain for the Moon than to any perfection to which [perfection] a general Theory would be required to measure up. For if the Moon were two or three times as far from the Earth, or if its orbit were more eccentric, then all the labors endured to this point would be lacking in all fruit [because they would be inapplicable], and by the way, not even its motion could be recalled to any fixed rule.
3) Therefore, much has stood before the Theory of Astronomy to be considered, for instance, if under the fictitious hypothesis that in case the Moon were much away from the earth, it would be certainly an excess to think that its motion could be evaluated with the maximum aid of this science. If, for instance, the Moon would have been a hundred times more distant from the earth, there is no doubt that the laws of motion of the main planet would no longer be followed as if it were a satellite of the earth, as one would expect. But if, on the other hand, the distance were ten times greater, its motion could then be compared, so that no doubt would remain, even with primary or secondary planets being added.
To such an extent that it certainly would disagree from all the motions observed in the sky, such that it can hardly give an idea even on how the average motion can be resolved. Perhaps, innumerable observations could reveal a certain law, from which, in a subsequent application, it can somehow give a clear prediction; however, by no means so evident, to such an extent that the Theory that should explain this type of motion may not be adapted. The very wise creator is seen to have being mindful of our weakness, because none of the bodies placed in the sky are such that their motion could be described neither by the law of the main planets nor of the satellites. 4) This sort of research, which is seen not to surpass the strength of the human mind, is certainly not suited to be undertaken hastily, but on the contrary, it will require that our efforts be undertaken step by step. Then, the general problem of three bodies mutually attracted to each other will be conveniently restricted to the case where one of the masses almost vanishes in front of the two remaining, where it is agreed that certainly it will be convenient that the two to venture an approach towards this problem before fatiguing in vain to unfold it, as I am forced to admit; however, in fact, this is a complete singular case as I have already observed, and with a remarkable simplification, in which the motion of the Moon would appear constantly connected or in opposition to the Sun, which is the case to be considered, with great utility in this very difficult matter should not be abandoned, and by no means to be seen with indifference. 5) Hence, the motion of the Sun and the Moon seen from Earth is assumed to take place in the ecliptic plane 6 , with the earth resting in T, and after a certain time has elapsed, I place the Sun in S, the Moon in L, and after laying a fixed straight line TA, directed to the First Star of Aries 7 , I ascribe the following angles: Be further the mean longitude of the Sun 8 = ζ, and its mean distance from the Earth = a, and from these we have for the motion of the Sun in its canonical form 9 : and for the motion of the moon 10 : where c is the mean distance which the Moon is solicited by the force of the Earth, and for the mean motion of revolution, there is a :1 n relation between the mean motion of the Moon and the mean motion of the Sun. Besides, regarding the differentials of the second degree, it should be noticed that the element dζ is assumed to be constant. 6 The ecliptic plane contains most of the objects which are orbiting the sun, and is tilted with respect to the Earth's spin axis at 23.5˚.

7
The First Star of Aries (or First Point of Aries), also known as the Cusp of Aries, is the location of the vernal equinox. 8 The Sun's ecliptical longitude is defined as the angle subtended at the earth between the vernal equinox and the Sun. The mean longitude is the ecliptical longitude that the planet would have if the orbit were a perfect circle. 9 The development of these equations can be found in E112-Recherches sur le mouvement des corps célestes en general. 10 The development of these equations can be found in L. Euler, Considerationes de theoria motus lunae perficienda et imprimis de eius variatione, Novi Commentarii Academie Scientiarum Imperialis Petropolitanae, Tom. XIII, pro Anno 1768. Advances in Historical Studies 11 6) All the difficulties in the resolution of both equations, consist on finding at any instant of time, the mean longitude of the Sun ζ , as well as the distance v, and the angle φ . Since so far, in general, the Geometers cannot proceed in their work, unless in case in which the distance v is much smaller than u and the number n is rather big, suitable approximations have been found, yet much is still justly needed in this matter. However this pair of equations in a general aspect, without any consideration to the Moon dwelling, which certainly has to be explained, can solve the problem completely. Such motion can take place in the sky, and it is in our ability to know it completely, even if its reasoning does not agree at all with the regular motion. 7) First of all, I should observe that these two equations admit a closed form solution for the case in which 0 η = , or φ θ = , when the Moon is seen to be in continuous communication with the Sun. Since sin 0 η = , and cos 1 η = , then z u v = − , and our equations will assume the following forms: which can be immediately compared with the given formulas for the motion of the Sun considering that v u α = , which it is certainly satisfied by the prior established equations. Hence the other equation for the Moon will be transformed into And since the other equation for the Sun is   12   8) Then, it will be necessary the determination of the number α or x from an equation of the fifth degree, which for its resolution, it is necessary to first observe that m should be a rather small fraction, and then ( ) likewise, it is evident that α will also have a small value, which can be approximated by or if the Moon were four times more distant from us, a motion of this kind would have been possible to exist, such that it [the Moon] would appear always connected to the Sun. It would then be possible to regard a Satellite of the Earth as if it were the Moon, and its motion would be most regular, however, deviating from the rules of Kepler the more close to the Sun than to the Earth, though it may revolved with the same time, because the force of the Earth in relation to the force of the Sun is reduced in the same proportion, although it may linger with a longer periodic time. Because the distance to the earth would be almost four times greater than the distance that Moon actually stands apart, as much as a limit would permit, so that bodies far more removed from the principal planets, in fact closer to the satellites of the Earth should permit. Similar limits in relation to other planets will be possible to be established. 12 A fifth degree polynomial was also obtained by Euler in E327-De motu rectilineo trium corporum se mutuo attrahentium (see Section 2). In this publication, Euler considers three bodies lying on a straight line, which are attracted to each other by central forces inversely proportional to the square of their separation distance (inverse-square law). 13 The Moon completes an orbit around the Earth once every 27.32 days. The Earth takes a year (365 days) to revolve around the Sun. Therefore, in a year period, the Moon completes and also z u v = + . Thus, the equations for the motion of the Moon will assume the following forms: , and the latter one reducing to the following: First, considering also the motion of the Sun, gives at once v u Yet, for the motion 10) These cases are most worthy to be commented, since they could be worked out absolutely without any approximation, even if both forces of the Sun and of the Earth concur in producing motion, since there is no other case which this can happen. However, the body would, in fact, move with such a simple motion, provided it would be at the assigned distance, and while it would appear from the Earth in conjunction or in opposition with the Sun, a motion of this type would be impressed, when it had began to advance in the same pace with the Earth in the ecliptic plane. However, on the other hand, if the impressed motion differs from this law, it would not, in fact, remain in continuous conjunction or in opposition with the Sun, but it would perform tiny excursions and, because of this, as almost oscillating. In the case where the motion had minimally differed from the formulation that was found, in the usual way, also by approximation, it will be possible to define such motion; in this case with the threshold of irregular motions, which still cannot be approached by any calculation, certainly it is seen with no lack of usefulness, if I will seek more carefully the nature of such motion. 11) However, although this investigation is by no means involved by trivial difficulties, however, our equations can be made conveniently easier to handle, when the distance v is much smaller than u, when it is possible to produce a convenient approximation. Of course, because 2 cos z uu uv vv , from which our equations that were found for the motion of the Moon will transform into the following forms: Then, also, the calculation can be made easier, if we consider the mean motion of the Sun, then u a = , and θ ζ = , and thus η φ ζ = − , or φ η ζ = + , whence arising the following equations where the last terms in these expressions can be omitted, since the fraction v a is very small, even establishing that the distance of the Moon is four times larger. 12) Now, be reminded the case where the Moon will be seen hesitating in almost oscillating motion around the Sun, and let us assume that the angle η is as small as possible, such that sinη η = , and 2 1 cos 1 2 η η = − , and then we have: Then, because the distance v is little changed, let us put ( ) whence, it is necessary to define the values of the quantities x and η for every angle ζ . 13) Since the angle η is minimum, and which alternates between negative and positive values, as the Moon is seen passing to and fro the Sun: it is easily allowed to conceive the existence of a relation between a certain angle ω and ζ , and thus to define the following The reduced form of The following expression is the result of equating the constant terms to zero. 19 The following expression is the result of equating the coefficients of cosω to zero. 20 The following expression is the result of equating the coefficients of cos 2ω to zero. Then, accordingly, the value of A is at our discretion, which depends on the digressions from the syzygy line 21 , which it is proper to assume as being a very small fraction, such that the quadratic terms can be considered of second order, being sufficient only the first terms. Then, for the distance ( )

An Easy Method for Calculating the Motion of Celestial Bodies Perturbed in Any Manner Avoiding Astronomical Computations (Methodus Facilis Motus Corporum Coelestium Utcunque Perturbatos Ad Rationem Calculi Astronomici Revocandi, Euler, 1768)
I) Although I have often attacked the investigation on how the motions of celestial bodies are perturbed due to their mutual action, most of the time, I have incurred in rather lengthy and laborious calculations, which, however, after many digressions, could be reduced to simpler formulas. However, the cause of this prolixity is due to a multiplicity of elements, which are necessary to introduce into the calculation, that is: not only for all the determinations, which are related to the motion of the perturbing bodies, should be examined, but also the perturbation of its own motion, in so far as if it is not in the same plane, it demands various elements, to which is a custom among Astronomers to consider variations originated in the nodal line 23 and inclination of the orbit. But in case all these considerations are simultaneously included into the calculation, it is really not worthwhile, because they will give rise to much trouble and confusion, to which no other remedy is seen to exist, unless all the elements are carefully distinguished, and all the operations are established so that no more elements are admitted into them, than those that are necessary to consider.
II) The principal part of this research is related to mechanics, when the perturbation of the motion by the forces of the disturbing body should be defined; thus mechanical principles are provided, and from them the location of the body, whose motion is seeked, can be conveniently determined at any time by three mutually orthogonal coordinates; truly the other part scarcely requires a swifter development, with which the location is firstly determined, and should be reduced to the acceptable practice in Astronomy, in which it is common to naturally express the different locations in the sky by longitude and latitude. And also in this second part, in which will be allowed to recall the geometry, being correct to distinguish a priori, that all mechanics is due to it, and I should observe that these two parts can furthermore not only be conveniently separated, but also to be able to deal with both matters in a much easier way than if we wished to deal jointly with both of them. However, it is seen that the geometrical mechanical investigation should precede, nonetheless it is possible to begin with the geometrical part in a neat way, with none impediment, from the location of the body, which motion we seek to obtain, as it were known, and that we identify by three coordinates. This inversion of the methodology is thus seen to follow, so that the development of the geometrical part gives much important support, with which the work in the calculation of the mechanical part that follows next will be con- 23 The nodal line is a line that joins the ascending node and the descending node of an orbit. It marks the intersection of the orbital plane and some reference plane, usually the ecliptic. siderably alleviated. We follow this method certainly with great advantage, provided that we handle the geometrical part without introducing into the calculation quantities related to the disturbing forces.
GEOMETRICAL PART III) Therefore, I assume that the motion of the body Z is to be determined and that, as usual, readily defined by mechanical principles. Certainly, firstly the motion in relation to certain point A, which is considered fixed, even if it happened that the said point is used as reference to the circular motion, then, next, a certain plane is considered traversing through this point and equally fixed, which is represented by the plane of the figure itself, in which it is drawn a fixed line AB, and at any time, the location of the body Z is thus defined by the three mutually orthogonal coordinates AX, XY and YZ, so that first, from the Z location, the perpendicular ZY to that plane is drawn, further on, truly from Y, the normal YX is guided in the direction of the line AB. Then, let us call the following three coordinates: which values at any elapsed time = t are considered to be known. Then, consequently, the distance of the body Z to the fixed point A is immediately obtained, which, for brevity, it is indicated by AZ v = , and then, 2 Then, we conceive that during the time dt the body advances from Z to z, such in this way, this exposes how the elementary angle d∅ can be conveniently expressed by the coordinates, which will be soon succinctly shown.
IV) A certain plane is defined by the segment Zz and point A, in which the body Z is, in fact, considered to move: this plane will cut somewhere the fixed plane of came to the same point z in another way, it is necessary that the nodal line and the inclination be both considered variables, since the point z should also have a tendency to a diversified orbit, and then, the differential dσ should not be considered to be equal to d∅ itself, but it should attain its proper value, which at the same time depends on the variation of the orbit. Therefore, since this double differentiation should lead to the same equations that we will obtain next, for which certain relations between the variations originated in the orbit will be defined, which will provide a maximum usage in a subsequent calculation. In fact, it possesses not only the differentiations of the local coordinates themselves, but also of the quantities thence derived, such as:  ψ ω ω σ = ; or the increment in the inclination will be due to the promotion of the nodal line, as the sine of the inclination to the tangent of the argument of the latitude; whence the following consequences can be drawn from it: 1) If the argument of the latitude σ is zero or 6s where the latitude is zero, in the mean time the nodal line will tend to remain at rest the more the inclination is varied 26 .
2) If the argument of the latitude σ is 3s or 9s where the latitude is zero, or tanσ = ∞ where the latitude is maximum, then the inclination will not vary; regardless if in the mean time the nodal line progresses or regresses.
3) If the argument of the latitude σ is contained between the limits 0s and 3s or between 6s and 9s, that is, while the latitude increases, then the inclination ω increases, if indeed the nodal line advances, but if it retreats, the inclination diminishes. 4) If the argument of the latitude σ is contained between the limits 3s and 6s or between 9s and 12s, that is, while the latitude decreases, then the advancement of the nodal line inclination lessens, and in reality it ceases to be increased.
IX) Next, it should be observed that the increase of the argument of the latitude σ promoted in its own orbit is not equal to the element d∅ , unless the nodal line stays immovable; since we found that d d d cos σ ψ ω = ∅ − , with the exception in the case when the inclination ω were a right angle. These phenomena will be expressed more clearly by using the spherical trigonometry. If in fact, as before, the circle BNY represents a fixed plane, which the motion of the point Z is referred to, so that its present motion takes place according to the circle NZ, such that BN ψ = , YNZ ω = and the arch NZ σ = , whereas after the point Z has progressed through the element d Zz = ∅ , and with its motion taking place according to the circle , the promotion of the nodal line will be given by d Nn Z v σ ω = , we find these very fitting formulas: Next, we eliminate dZ , firstly multiplying by Y, and then by X, and once the product is taken, it will give However, as we saw in § 7, cos cos sin sin The application of the Law of Sines to the spherical triangle NnZ does not give this expression. 28 It is not known where this expression comes from. 29 The claimed division does not lead to this expression. 30 From the above observations, it appears that this result was forced by Euler. Nonetheless, this has no further consequences, since the same expression was obtained before by another method in § 7. 31 This figure was incremented with more elements by the Translator.
These formulas when combined with those then found at the beginning, namely will be used with maximum advantage in the mechanical part, to the coordinates derived from the calculations, such that it provides to the next quantities of this sort, which use are retaken in Astronomy.
XI) However, those reductions of the Geometry clearly show that it makes no difference to which point A and fixed plane BAY we wish to refer the motion of the point Z. In fact, if we envisage an astronomical use, it is of most interest, not only in which way the point A, as the center of motion is taken into account, but also that plane, to which the motion of the point Z is referred to by longitude and latitude, because from this will depend the chief simplicity of the determination. To this end, it is necessary to consider how that choice should be made, together with the artifices, which so far have been devised, and only then, it can be used with some success, since the motion of the body, which is sought, should not disagree very much from the laws of Kepler, on account that the perturbations hand been very small. Moreover, when the motion is thus compared, such that the areas described around any point are nearly proportional to time, then this point is most suitable to be considered as that fixed point A. When that happens, if among the forces driving the body, one far exceeds the remaining, to that point this force should be directed to, then point A will be suited to be accepted: therefore, if the question would be related to perturbations of a certain chief planet or of a comet, then point A will be most suitable taken in the center of the Sun: if however, the perturbations in the motion of the Moon, or those made in another secondary planet should be defined; then it is proper be considered point A in the center of the Earth or [in the center] of the primary planet, such that the force of the body declared in Z impelling it to A, much exceeds the remaining forces to which this body is simultaneously driven.
XII) So, If the body Z has been solicited by just one principal force, the body will be revolved regularly around point A in a conical section, perpetually in the same plane, such that no matter in what way the fixed plane BAY is chosen, neither on how the nodal line nor the inclination and any mutation has been ever originated; however, meanwhile, the calculation, without doubt, has turned out very simple, if the fixed plane is chosen in the same plane of the motion. Truly, if the motion is disturbed by another celestial body, which motion is indeed also necessary to be assumed known in this investigation, the fixed plane can most conveniently be assumed as being congruent with the orbit of that disturbing body. Thus, if the perturbations of the Moon originated by the Sun are sought, the ecliptic plane, in which the Sun is seen to move from the Earth, as it [the Earth] were the center of the motion A, will render the fixed plane BAY, and no matter how the perturbation from another body is brought about, this plane, in which this body is seen to move from the center of motion A, should be selected.
Yet, if this body itself is not moved in the same plane, then some medium plane can be most conveniently adopted; but the effort to adapt the calculation to this case is hardly considered, but, if its employment will become indeed necessary, it can be easily provided.
MECHANICAL PART XIII) For the handling of the mechanical part, three bodies should be considered. The first, is the one which is putted in the center of the motion A, which exerts the main force in the body Z, which motion we investigate, such as it appears to an observer located in the very point A, let us then call the mass of the body positioned in this point = A. The other body, by which action the motion of the body Z is perturbed, that we assume is moving in any manner in the fixed plane BAY itself, such that its location can be assigned at any time. Be the mass of this body = B, and that now, in fact, it dwells in S, such that its distance to the central body AS u = , and the longitude or angle BAS θ = , whence, from S, the perpendicular SP is drawn to the fixed line AB, and be The third body is the one in Z itself, which motion we are looking for, be its mass = C, and as before, we put its distance to the center of motion AZ v = , and calling the three orthogonal coordinates AX X = , XY Y = and YZ Z = , which we obtain from the calculation by introducing the following elements: 1) longitude of the nodal line or angle BAN ψ = 2) Inclination of the orbit to the fixed plane = ω 3) argument of latitude or angle NAZ σ = .
Finally, we consider that during the infinitesimal time dt, the elementary angle d ZAz = ∅ is completed by the body Z. On the other hand, the relationships of these elements will be taken from the geometrical part. : which should be proportional to the acceleration of the body Z in the same directions, and considering a constant element of time we have: where the constant ∝ depends on each particular type of motion, and can be defined from the apparent motion of the Sun.
XV) However, before we consider these formulas further, we should precisely define the distance SZ w = , to introduce it afresh into the calculation.
Hence, with the reminded calculation, the following equation will be obtained: where the formula d cos d sin X Y θ θ + can be conveniently expanded. Truly, considering the formulas in § 10.a above, we have that Therefore, the equation that we found transforms into: Or if in Z we draw in the direction of the arch NZ another normal arch, and on it, and from S, we draw the perpendicular to the spherical surface, which we call = ν, then sin sin cos XVIII) The two remaining determinations from the differential equations deducted from the principles of motion will be conveniently obtained by the following procedures: firstly, from the equations obtained in § 14, the subtraction of the first equation multiplied by Y from the second equation multiplied by X gives: or, when the values for X and Y are substituted into this expression results in which is the other determination required to be sought. XX) Let us multiply this last equation by 2 2 d v ∅ , and leaving one integral just indicated, we will have the following expression a certain constant value of v itself should be satisfactorily attributed to this differential equation, for which the denominator fades away, however, as I have exposed in another place, this approach is not possible to be admitted to the integral, unless the factor of denomination 42 fades away to be of a minimum dimension of one, whence it is necessary that under the radical sign the same factor appears in a pair or squared, such that it reduces to the same value, so that the differential of the quantity placed after the sign reproduces the same factor.
Therefore, let us place that differential = 0, and XXII) Then, since the knowledge of the motion at any given time t allows the determination of the mean motion τ , here the time variable in our calculation will be redefined, by writing in the place of considered an impediment. Nonetheless, I will expose soon the method to such an extent as to liberate the calculation of these integrals. XXIII) Meanwhile, for the sake of brevity, let us consider that: then, the first two previous equations are contracted to these forms: 1) ( ) when the quantity under the radical sign vanishes. However, not only in Astronomy that these places are of primary importance, wherever the body Z is said to move along a segment of an arch, but even so, the choice of this important fact is at our disposal, with which the disturbed motion could be very neatly compared with the regular motion, and thus be capable to assign the aberrations from it. However, in a convenient way, this will be provided by introducing into the calculation a new angle Υ, which in astronomy is called the true anomaly 43 , and it is chosen in such way that either by reducing or increasing the [angular] distance between two lines according to the maximum or minimum value of v.
Therefore, with the purpose of approximating the real motion to a regular motion made along an ellipse, we now define 1 cos p v q = + ϒ , so that now the motion conforms to the regular motion along such an ellipse, in which the semi-latus rectum is = p, the eccentricity = q, and thus the semi-major axis 2 1 p q = − , and the true anomaly arising from the [major] axis = Υ. Then, it can be easily perceived that because of the perturbations, the aspect of this ellipse changes continuously, whence not only the anomaly Υ but also the letters p and q are expected to vary, and these variations are now investigated.
XXV) For that investigation to be rendered easier, let us introduce, for the sake of brevity, the following: ( )

Conclusion
"On the rectilinear motion of three bodies mutually attracting each other": this is considered the first work from which the three collinear Lagrange points could be obtained, where the parameter that controls the distances among the bodies was found to be given by a quintic function.
"Considerations on the motion of celestial bodies": one of the conclusions of this paper is that if the Moon were four times more distant from the Earth (either in conjunction or in opposition), a motion of this kind would have been possible to exist, such that the Moon would appear always connected to the Sun.
In this paper, perhaps for the first time, Euler introduces an archaic form of a Fourier sine series expansion to describe the Moon's wagging motion. However, as Euler himself recognizes, the calculations turned out very tedious and led him to greatly simplify his model in order to obtain some numerical values for the phenomenon.
"An easy method for calculating the motion of celestial bodies perturbed in any manner avoiding astronomical computations": with few sketches to show the geometrical constructions envisaged by Euler-represented by several geometrical variables-it is hard to follow publication. The Translator, on trying to clear the way to the non-specialized reader, used the best of his abilities to add his own figures to the translation. In the latter part of the work, Euler particularizes his developments to the Moon, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body, which makes his claim of an "easy method" as being rather fallacious. Despite showing great analytical skills, Euler did not give indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moon's orbit.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.