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© J R Stockton, ≥ 2013-01-22

A translation of
Considerationes de motu corporum coelestium
E.304, by Leonhard Euler
.
(Considerations on the motion of celestial bodies)

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Of those, Src3 contains the full material and is legible; it is the same edition as the less legible Src1, and is now my preferred source. But Google's very legible Src2 edition, for what it shows, can be useful.

Source Wanted : A copy of p.XII of the preface to Src2, or of the relevant part, would be appreciated. See the foot of p.554 herein, centrally numbered 13.12.

Details of Work

I have  highlighted  the parts of the text of E.304 which seem to be of greatest importance for Lagrange Points. Footnotes by M.S., found in the later copy Src1, are boxed thus. Notes by me are labelled "JRS", and/or are like this. Sections of the "ENGLISH" column written by me, for which emendments are definitely solicited, are indicated thus. Hovering over other parts of it shows where credit is due.

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In the original Src1 and Src3, it can be hard to see any difference between italic 'a' and Greek 'α'; also, 'u' and 'v' are generally equivalent; 'i' and 'j' may be; and 'ſ' represents 's'. Where, for example, a in algebra represents a simple variable, Euler used aa rather than a2.

Between equations, Euler's "et" may here be transcribed as "&" to obviate translation.

Conclusions

This page was written in order to determine and demonstrate whether Leonhard Euler had, in E.304, proved the existence of any of what are now known as the Lagrange Points.

The paper is about the Sun-Earth-Moon system, dealing with the apparent motion of the Moon. Lagrange Points L1 and L2 appear as interesting places in the Sun-Earth system to put our Moon. Section 7 considers L1. The end of Section 8 extends the argument to planets other than the Earth. Section 10 refers to L1 and L2.

No reference to L3 has been identified (if the Moon were at L3, it could hardly be called our Moon).

Leonhard Euler - E.304 -
"Considerations on the motion of celestial bodies"

Eneström Index :- E304 Considerationes de motu corporum coelestium. Auctore L. Eulero.
Novi commentarii academiae scientiarum Petropolitanae 10, (1764), 1766, pp. 544-558 + 1 figure. According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on April 22, 1762; according to the records, it was presented to the Petersburg Academy on May 17, 1762.
Abstract: A. a. O., Summarium dissertationum, pp. 66-67.
Reviewed in Nova acta erud. 1766, pp. 178-179.

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CONSIDERATIONES DE MOTU CORPORUM COELESTIUM
CONSIDERATIONS ON THE MOTION OF CELESTIAL BODIES
 
1 2 3
TRANSCRIPTION
by JRS
Row # ENGLISH
Rather rough, by JRS
Where polished, by others
 
SUMMARY
66
V.
Considerationes de motu Corporum Coelestium
Auctore Leon. Eulero pag. 544.
V.
Considerations on the motion of Celestial Bodies
Author Leon. Euler, page 544.
Licet nullum sit dubium, quin leges, quibus corpora coelestia in motibus suis obediunt, a Keplero detectae, a Neutono vero in maximum Astronomiae incrementum demonstratae sint, minime tamen existimandum est theoriam Astronomiae ad summum gradum perfectionis evectam esse.

Possumus equidem motum duorum corporum in ratione directa massarum et inversa quadratorum distantiarum in sese agentium, perfecte definire: verum si iis accedat tertium, ut unum quodque in reliqua secundum illam legem agat, motui eorum enodando omnia hucusque inventa Analyseos artificia minime sufficiunt.

Quidquid autem de motu Lunae et reliquorum corporum coelestium a viris celeberr. JRS: I am told, = "celeberrimus" hucusque praestitum est, id non nisi approximationibus innititur, quae eatenus locum inveniunt, quod inter ternas vires una prae reliquis maxime semper emineat, ita ut effectus a reliquis oriundus veluti minimis ope approximationum definiatur, quae tamen negotium minime conficiunt.
0.00
Although there is no doubt that the laws which the celestial bodies obey in their motions, discovered by Kepler, explained by Newton in a great advance in Astronomy, one should not think that theoretical Astronomy has yet been brought to the highest level of perfection.

The motion of two bodies acting upon themselves in the direct ratio of the masses of the two bodies and the inverse square of their distance apart can be perfectly defined : but if a third body is added, so that each interacts with the rest by that law, all of the artifices of Analysis are scarely sufficient to clarify their motion.

But whatever about the motion of the Moon, and of the rest of the celestial bodies, has been so far achieved by the most famous men, this not only supports approximations, which still find a place, that of the three forces one was always dominant, so that the effects arising from the others can be defined with the help of minor approximations, yet the task is not complete.
¶   Quoniam Solutio problematis de tribus Corporibus, secundum memoratam legem in sese invicem 0.01
¶   For the solution of the problem of three bodies, acting on each other according to the above mentioned law,
 
67
agentibus, sensu generali accepti vires humanas Cel. Auctori JRS: I am told, = "famous Author" merito transcendere videtur, tentavit illud restrictum soluere posita massa unius prae binis reliquis evanescente, ut scilicet incipiendo a casibus particularibus viam sternat ad solutionem problematis sensu generali accepti ;

Verum restricto etiam sic problemate tantae difficultates in solutione eius sese obtulerant, ut ipse Cel. Auctor frustra in evolvendo eo se desudasse JRS: desudavisse has been suggested. fateatur.

Interea observavit casum per quam singularem et notabilem, quo Lunae eiusmodi motus imprimi potuisset, ut perpetuo vel in oppositione vel in Coniunctione cum Sole apparuisset, si sc. Luna quater fere longius a nobis remota , et ei eiusmodi motus impressus fuisset, ut pari passu cum Terra in plano Eclipticae ingredi inciperet.

Licet fictam hypothesin Cel. Auctor in calculum introduxerit, verum nihilo minus exinde et minime expectatam elicuit de limite Satellitum Terrae conclusionem ; corpora nempe quae quadruplo magis a nobis ac Luna distant in numerum planetarum primariorum, propiora vero in numerum Satellitum terrae referenda sunt.
0.02
agents, a general sense of accepted human powers of the famous Author seems to transcend reason, he tried to solve it with the mass of one of the two remaining close to zero, so as to pave the way for the solution of the problem starting from the particular cases to a general sense of the result;

But even so the problem is restricted to the solution of such problems, they had offered to the famous author of the pieces that would evolve from had sweated avows.

Meanwhile, a case is observed which is unique and noteworthy, that the Moon itself could be impressed with a motion, such that it would appear perpetually either in opposition or in conjunction with the Sun, if the moon were nearly four times further removed from us , and itself would have been impressed with motion of this kind, together with the Earth in the plane of the ecliptic were started.

Although the famous author brought a fictitious hypothesis into the calculation, but nevertheless soon thereafter concluded of the limit of Satellite of Earth; four times farther from us as the Moon is, in a number of primary planets, but nearer the number satellites of earth are referred.
¶   Ut Luna coniuncta vel opposita perpetuo Soli maneat, motus illi in plano Eclipticae idemque cum Tellure imprimatur necesse est; quodsi vero ille a lege hac discrepet, Luna exiguas excursiones hinc inde oscillando tanquam conficiet.

Cel.Auctor naturam quoque earum adhibitis approximationibus eo, quo pollet acumine, in dissertatione ista scrutatus est.
0.03
¶   So that the Moon conjunction or opposition to the Sun should remain constant, motion in the same plane of the ecliptic and itself with Earth must be impressed ; and if it does not comply with this, the Moon makes small oscillatory excursions.

The famous author, who possesses shrewdness, has studied the nature of their application in the approximations, in this dissertation.
 
ARTICLE
544
CONSIDERATIONES
DE MOTU CORPORUM COELESTIUM
  CONSIDERATIONS
ON THE MOTION OF CELESTIAL BODIES
Auctore
L. EULERO.
Author
L. EULER.

1.

Etsi nullum est dubium, quin leges motus corporum coelestium a Keplero observatae atque a Neutono confirmatae, Astronomiae maxima incrementa attulerint, tamen nunc quidem certissimum est, nullum in coelo reperiri corpus, quod leges istas in motu suo perfecte sequatur, cum potius in omnibus haud leves aberrationes ab istis legibus deprehendantur.


Vera scilicet omnium motuum coelestium causa in mutua horum corporum attractione est posita, qua unumquodque ad singula reliqua urgetur viribus rationem compositam ex directa simplici massarum, et inversa duplicata distantiarum tenentibus.

Semper autem commode usu venit, ut inter has vires una prae reliquis maxime emineat, ideoque motus proxime regulis Keplerianis conformis, evadat ; sicque effectus a reliquis oriundus veluti minimus per methodos appropinquandi definiri possit.


Quod nisi eveniret, in maxima adhuc ignoratione motuum coelestium versaremur, cum nulla methodus adhuc sit inventa, cuius ope trium saltem corporum se mutuo attrahentium motus assignari queat ; nisi forte una vis caeteras plurimum superet.
1.00

1.

Although there is no doubt that the laws of motion of celestial bodies observed by Kepler and confirmed by Newton have brought enormous gains to astronomy, nevertheless it is now absolutely certain that there is no body to be found in the heavens which perfectly follows those laws in its own motion, since, on the contrary, non-trivial deviations from those laws are observed in all of them.

The true cause of all celestial motions lies, of course, in the mutual attraction of these bodies by which they are urged towards each other by forces holding a compound reckoning from the product of the masses and the inverse square of the distances.

However it always conveniently happens that, among these forces, one very greatly predominates over the others, so that the motion turns out to be closely conformable with Kepler's laws; and thus the effect, as if minimal, arising from the other [forces] can be determined through methods of approximation.

If this were not the case then we would still be extremely ignorant of the motions of the heavens, since no method has as yet been found by which the motion of three or more mutually attracting bodies can be assigned; unless perchance one force be vastly superior to the others.
 
545DE MOTU CORPORUM COELESTIUM
¶   2. Verum etiam hic casus, in quo solo Geometrae operam suam non omnino frustra consumserunt, neutiquam pro confecto haberi potest, cum ipsa methodus appropinquandi, qua Geometrae uti solent, plurimis difficultatibus adhuc sic involuta, atque infinita minorum perturbationem multitudo negligatur, quo fit ut haec ipsa approximatio negotium minime conficiat, sed ad eam perficiendam plurima adhuc adminicula desiderentur.

Quare etsi motus Lunae ex hac Theoria satis accurate est definitus, id tamen potius singularibus circumstantiis, quae in Luna locum inveniunt, est tribuendum, quam cuipiam perfectioni, ad quam Theoria evecta censeri queat, si enim Luna bis vel ter longius a terra abesset, vel eius orbita magis esset excentrica, omnes labores adhuc exantlati JRS: sic omni fructu caruissent, ac ne nunc quidem eius motum obiter saltem ad certam quandam regulam revocare liceret.
2.00
¶   2. But even in this case, in which alone his work of Geometry are not completely consumed in vain, in no way can be considered as finished, with that method of approach, which Geometers often use, yet so many difficulties are involved, and an infinite number of minor perturbations are to be neglected, as a result of these minimum of effort of approximation is required, but to complete it much support is still needed.

Wherefore, although the movement of the Moon is determined with sufficient accuracy by this Theory, but rather that the individual circumstances, which in the case of the Moon are found, are to be attributed, than any perfection to which the extended Theory can be assessed. For if the Moon were removed twice or three times farther from the earth, or its orbit were more eccentric,
then all the labours so far expended would have lacked all success, and incidentally it would not be possible even now to reduce its motion to any fixed rule.
¶   3. Plurimum igitur is in Theoria Astronomiae praestitisse esset censendus, qui in hypothesi ficta, qua Luna multo longius a terra abesset, eius motum assignare valuerit, cum inde maxima adiumenta in hanc scientiam certo essent redundatura.


Si quidem Luna centies longius a terra esset remota, nullum est dubium, quin leges motus planetae principalis esset secutura, neque amplius, tanquam satelles terraem spectari posset.


Sin autem decies tantum magis distaret, eius motus ita foret comparatus ut in dubio relinqueretur, utrum planetis primariis, an secundariis, esset accensenda.


Tantopere certe ab omnibus motibus in coelo observatis
3.00
¶   3. It would be considered a very great achievement in Theoretical Astronomy if, in the fictitious hypothesis that the Moon was much further from the Earth, its motion could be predicted, with the largest contribution to this knowledge is certainly abundant. JRS: redundo = abound

If, indeed, the Moon were a hundred times further from the Earth, there can be no doubt, that it would principally follow the laws of motion of the planets, and no more to be taken as a satellite of the earth.

But if it were ten times further away, the motion would be compared so as to be left in doubt whether the primary planets, or secondary, was dominant.
JRS: 10, 100 : I think Euler may be factually wrong here - see The Hill Radius.

Certainly all the motions observed in the heavens differed so much,
 
546  DE MOTU CORPORUM
discreparet, ut vix intelligi possit, quemadmodum saltem ideam motus medii constitui conveniat.

Innumerabiles forsitan observationes legem quandam aperuissent, ex qua in posterum eius loca quodammodo praedicere licuisset ; nequaquam autem patet, quomodo Theoria ad huiusmodi motum explicandum accommodari potuisset.

Imbecillitati nostrae sapientissimus creator consuluisse videtur, quod nulla corpora in coelo ita collacauerit JRS: sic; but, I am told, should be "collocaverit"., ut eorum motus, neque ad legem planetarum principalium, neque satellitum, referri posset.
3.01
that they could scarcely be understood, at least the idea of the mean motion became established.

Numerous observations would perhaps reveal some law, from which somehow it would be possible to predict future positions ; but it is by no means clear, how a theory could be adapted to explain such motion.

The all-wise Creator seems to have had regard for our weakness, for there are no bodies in the heavens which He has placed in such a manner that their motion could be referred neither to the law of the principal planets nor of satellites.
¶   4. Huiusmodi investigationem, quae vires ingenii humani tantum non transcendere videtur, certe non subito suscipi conveniet, sed potius conatus nostros pedetentim eo dirigi oportebit.


Generale ergo problema trium corporum se mutuo attrahentium ita commodissime restringetur JRS: sic, ut unius massa prae binis reliquis quasi evanescat, . quo pacto id commodi assequemur, ut duo corpora, maiora scilicet, secundum leges Keplerianas moveantur, omnisque perturbatio in motu tertii consumatur, cuius situs et motus si ab initio ita fuerit comparatus, ut ad ambo maioram aequa vi quasi attrahatur, habebimus eiusmodi casum, cuius investigatio novam plane methodum postulat.




Plurimum abest, ut hoc problema aggredi ausim, ut potius, frustra in eo evolvendo desudasse, fateri cogar ; verum tamen casum observavi omnino singularem, ac simplicitate memorabilem, quo Lunae eiusmodi motus imprimi potuisset, ut perpetuo Soli, vel coniuncta, vel opposita, apparitura fuisset, cuius casus consideratio, cum forte usu in
4.00
¶   4. An investigation of this sort, which seems to all but surpass the powers of the human mind, certainly should not be taken up all of a sudden, but rather we should direct our attempts thither one step at a time.

So then, the general problem of three bodies in mutual attraction will be most conveniently restricted in this way, that the mass of one might, as it were disappear, in comparison with the other two masses. By this means we will achieve the convenience that two bodies, the larger ones of course, move according to the laws of Kepler, and all perturbation in the motion of the third is brought to nought; if the position and motion of this third body is handled thus from the start so that it be, as it were, attracted to each of the larger ones by equal force, we shall have the sort of case for which investigation demands a completely new method.
JRS: I do not think that Euler ought to have meant all of that second sentence. The two masses will obey Kepler's Laws, but the particle will move in their combined field, without affecting their motion. There are no equal forces, except for "action and reaction".

I am very far from daring to tackle this problem, nay rather I am forced to confess that I have sweated in vain over trying to solve it. However I have noticed a totally special case of memorable simplicity in which motion of such kind could have been stamped on the Moon that it would have appeared permanently in either conjunction or opposition to the Sun. Consideration of this case will not, I think, be displeasing since perhaps it is not without usefulness in this most difficult business.
 
547COELESTIUM
hoc difficillimo negotio non destituatur, haud displicitura videtur. 4.01
JRS: see previous page.
#page/n676/mode/1up Section 5 Marginal Note :-
Tab. XX, Fig. 13.
 Tab.XX, Fig.13.
¶   5. Motum igitur tam Solis, quam Lunae, ex terra visum in plano ecliptico fieri assumens, terram quiescentem in $~T~$, et post aliquod tempus elapsum Solem in $~S~$, Lunam vero in $~L~$, versari pono, et ducta recta fixa $~TA~$, ad principium arietis the First Point of Aries, no doubt directa, statuo, angulos $ ATS = \theta, ~ ATL = \phi ~~~~ \text{&} ~~~~ STL = \phi - \theta = \eta ~, $ tum vero distantias $~TS=u, ~TL=v ~~ \text{et} ~~ LS = \sqrt{(uu-2uv\cos\eta + vv)} = z.~$ Sit porro longitudo Solis media = $~\zeta~$, eiusque distantia media a terra = $~a~$ hisque positis pro motu Solis utpote regulari habebimus : 5.00





5.02
¶   5. Therefore the motion both of the Sun and of the Moon as seen from the Earth will be assumed to be in the plane of the Ecliptic, the Earth being at rest at $~T~$, and after a certain time has elapsed the Sun is at $~S~$, the Moon at $~L~$, ?? put the focus ??, and draw a fixed straight line $~TA~$ directed to the First Point of Aries, I confirm, the angles $ ATS = \theta, ~ ATL = \phi ~~~~ \text{&} ~~~~ STL = \phi - \theta = \eta ~, $ then the distances $~TS=u, ~TL=v ~~ \text{and} ~~ LS = \sqrt{(u^2-2uv\cos\eta + v^2)} = z.~$ Again the mean longitude of the Sun = $~\zeta~$, and the mean distance from Earth = $~a~$, so these positions for the motion of the sun as we will have regulated :
$$ \frac{2dud\theta + udd\theta}{d\zeta^2} = 0 ~~~~ \text{&} ~~~~ \frac{ddu - ud\theta^2}{d\zeta^2} + \frac{a^3}{uu} = 0 $$
pro motu autem Lunae : 5.04
but for the movement of the Moon :
$$ \frac{2dvd\phi + vdd\phi}{d\zeta^2} - \frac{a^3}{uu} \left( 1 - \frac{u^3}{z^3} \right) \sin\eta = 0 \\ \frac{ddv - vd\phi^2}{d\zeta^2} + \frac{nnc^3}{vv} + \frac{a^3v}{z^3} + \frac{a^3}{uu} \left( 1 - \frac{u^3}{z^3} \right) \cos\eta = 0 ~, $$
ubi $~c~$ est distantia media, ad quam Luna, a sola vi terrae sollicitata, pari motu medio revolveretur, existente $~n:1~$ ratione motus medii Lunae ad motum medium Solis.



Caeterum circa differentialia secundi gradus hic est monendum ; elementum $~d\zeta~$ constans esse sumtum.
5.06
where $~c~$ is the mean distance, to which the Moon, influenced only by the force from the Earth, revolving by the mean motion, existing $~n:1~$ ratio between the mean motion of the Moon and the mean motion of the Sun.
JRS: cf. $n$ months per year.

But here is some advice about the second-order differentials ; element$~d\zeta~$ is to be taken as constant.
¶   6. Tota ergo difficultas in resolutione harum duarum aequationum consistit, ut scilicet inde ad quodvis tempus, seu longitudinem Solis mediam $\zeta$, tam distantia $v$, quam angulus $\phi$, definiatur.

Quod cum in genere fieri nequeat, Geometrae adhuc in eo laboraverunt, ut saltem pro casu, quo distantia $v$, prae $u$,
6.00
¶   6. The whole problem consists in the solution of the two equations, namely, so that at any time, the longitude of the Sun's centre $\zeta$, so the distance $v$, so the angle $\phi$, shall be defined.


Although a general solution is not possible, yet geometers have toiled at it, so that at least for the case in which the distance to $v$, in comparison with $u$,
 
548  DE MOTU CORPORUM COELESTIUM
est vehementur parva, simulque $n$ numerus mediocriter magnus, idoneas approximationes eruerent, in quo tamen negotio plurimum adhuc iure desideratur.

Hic autem binas istas aequationes in genere specto, sine ullo respectu ad Lunam habito, et quosdam casus sum evoluturus, quibus iis absolute satisfieri queat.


Eiusmodi scilicet motus in coelo locum habere posse ostendam, quos perfecte cognoscere in nostra sit potestate, etiamsi eorum ratio maxime a motu regulari abhorreat.
6.01
is very much less, and $n$ is a moderately large number, suitable approximations appear, in which, however, still a great deal of business is desired by the law.

Here we have two of these equations are in general expected, without any respect to the Moon having, and in some cases I have been developing, those which can be absolutely satisfied.

Such a motion can be shown to have a place in heaven, whom the power is ours to know them perfectly, even though most of the system differs from regular motion.
¶   7. Primum igitur observo, has duas aequationes absolutam resolutionem admittere casu $~\eta=0~$, seu $~\phi=\theta~$, ita ut tum Luna perpetuo in coniunctione cum Sole esset apparitura.


Cum enim sit $~\sin\eta=0~$ et $~\cos\eta=1~$, erit $~z=u-v~$, nostrae aequationes has induent formas:
7.00
¶   7. So I first note that these two equations allow the complete solution of the case $~\eta=0~$, or $~\phi=\theta~$, so that then the Moon would appear in perpetual conjunction with the Sun.
JRS: L1? Order = Sun-----Moon-Earth?

For since $~\sin\eta=0~$ and $~\cos\eta=1~$, then $~z=u-v~$, our equations take these forms:
$$ \begin{aligned} \frac{2dvd\theta+vdd\theta}{d\zeta^2} = 0 , ~~~~ \text{&} ~~~~ \frac{ddv-vd\theta^2}{d\zeta^2} + \frac{nnc^3}{vv} + \frac{a^3v}{(u-v)^3} & \\ ~~~~~~~~~~ + \frac{a^3}{uu} . \frac{-3uuv+3uv^2-v^3}{(u-v)^3} = 0 & \end{aligned} $$

seu

or
$$ \frac{ddv-vd\theta^2}{d\zeta^2} + \frac{nnc^3}{vv} - \frac{a^3v(2uu-3uv+vv)}{uu(u-v)^3} = 0 $$
quae cum formulis, pro motu Solis datis, comparatae statim dant $~v=\alpha u~$, quippe pro pacto prioribus aequationibus satisfit.

Hinc altera aequatio pro Luna erit
7.04
while the formulae given for the motion of the sun, comparably immediately give $~v=\alpha u~$, in the way that the former equations are satisfied.

Hence the other equation for the Moon will be
$$ \frac{\alpha(ddu-ud\theta^2)}{d\zeta^2} + \frac{nnc^3}{\alpha\alpha uu} - \frac{\alpha a^3(2-3\alpha+\alpha\alpha)}{(1-\alpha)^3uu} = 0 ~ . $$
Quaere cum altera aequatio pro Sole sit 7.06
So the other equation for the Sun is
$$ \frac{ddu-ud\theta^2}{d\zeta^2} + \frac{a^3}{uu} = 0 ~ $$

necesse est sit :

which must be :
$$ \alpha a^3 = \frac{nnc^3}{\alpha\alpha} - \frac{\alpha a^3(2-3\alpha+\alpha\alpha)}{(1-\alpha)^3} $$

seu

or
$$ \frac{nnc^3}{\alpha\alpha a^3} = \frac{3\alpha - 3\alpha\alpha +\alpha^3}{(1-\alpha)^2} $$
 
549COELESTIUM
ubi cum sit $~\frac{nnc^3}{a^3}~$ quantitas constans, ponatur brevitatis causa: $~\frac{nnc^3}{a^3}=m~$, eritque $~m(1-\alpha)^2=\alpha\alpha(3\alpha-3\alpha^2+\alpha^3)~$ seu $~ m(1-\alpha)^2 = \alpha\alpha - \alpha\alpha(1-\alpha)^3 ~$.

Posito ergo $~2-\alpha=x~$,  JRS: $?~1-\alpha=x~?$,   fit $~mxx=(1-x^3)(1-x)^2~$, seu
7.12
where $~\frac{n^2c^3}{a^3}~$ is a constant quantity, we put for brevity: $~\frac{n^2c^3}{a^3}=m~$, so that $~m(1-\alpha)^2=\alpha^2(3\alpha-3\alpha^2+\alpha^3)~$ or $~ m(1-\alpha)^2 = \alpha^2 - \alpha^2(1-\alpha)^2 ~$.

Hence with $~1-\alpha=x~$,  JRS: $?~2-\alpha=x~?$,   becomes $~mx^2=(1-x^3)(1-x)^2~$, or
$$ 1-2x+x^2-mx^2-x^3+2x^4-x^5=0 ~ . $$
¶   8. Pendet ergo determinatio numeri $~\alpha~$ vel $~x~$ ab aequatione quinti gradus, pro cuius resolutione notari oportet, esse $~m~$ fractionem quam minimam; quare cum sit 8.00
¶   8. It therefore depends on the determination of the number $~\alpha~$ or $~x~$ from that equation of the fifth degree, for the solution of which it must be noted, $~m~$ fraction than minimum whereby when
$$ m(1-\alpha)^2=3\alpha^3-3\alpha^4+\alpha^5 $$
evidens est quoque, $~\alpha~$ minimum esse habiturum valorem, et quam proxime fore $~\alpha = \sqrt[3]\frac{m}3 = \frac ca \sqrt[3]\frac{nn}3 ~$, accuratius autem $$ \alpha = \sqrt[3]\frac m3 - \frac13\sqrt[3]\frac{mm}9 - \frac1{27}m + \frac1{81}m\sqrt[3]\frac m3 ~ . ~~~~ \tag{1} $$
1) non accuratum est: recte $~\frac4{243}~$ loco $~\frac1{81}~$. M.S.

Primus autem terminus sufficit, sicque est $~v = \frac{cu}a\sqrt[3]\frac{nn}3 ~$, unde cum sit proxime $~u=a$, et $~nn=175$, erit circiter $~v=4c$; seu si Luna fere quater longius a nobis esset remota, eiusmodi motum habere posset, ut Soli perpetua iuncta apparet.








Talis Luna aequo iure tanquam Satelles terrae ac planeta primarius spectari posset, et uterque motus maxime foret regularis, hoc tantum a regulis Keplerianis recedens, quod Soli proprior, quam terra, pari tamen tempore revolvatur, ob vim scilicet terrae vis Solis tantum imminuitur, ut cum maiori tempore periodico consistere possit.


Hinc distantiam a terra quasi quadruplo maiorem, quam Luna revera inde distat, tanquam limitem spectare licet, ut corpora magis remota pro planetis primariis, propiora vero pro satel-
8.01
it is also evident is that $~\alpha~$ has a minimal value, and would be approximately $~\alpha = \sqrt[3]\frac{m}3 = \frac ca \sqrt[3]\frac{n^2}3 ~$, but to be exact $$ \alpha = \sqrt[3]\frac m3 - \frac13\sqrt[3]\frac{m^2}9 - \frac1{27}m + \frac1{81}m\sqrt[3]\frac m3 ~ . ~~~~ \tag{1} $$
(1) is not accurate, $~\frac4{243}~$ should replace $~\frac1{81}~$. M.S.
The first term is sufficient, so we have $~v = \frac{cu}a\sqrt[3]\frac{n^2}3 ~$, Hence, since approximately $~u=a$, and $~n^2=175$ JRS: $n$ : see sec 5.06., so approximately $~v=4c$; or if the moon were four times further from us, it would to have a motion of this kind, so that the Sun would appear perpetually joined. JRS: That means L1.

JRS: Does that mean four times more distant or another four times more distant, making 5 times? The correct figure is about $13^{2/3}$ times more distant, say 5½ times.

Question : Late in Page 549, "seu si Luna fere quater longius a nobis esset remota,", a minor question : does he mean multiplying the Moon's distance from Earth by 4, or adding an extra 4 times, making 5 times the distance? 4 is a reasonable fit to the true value (5.5), but 5 would be better.
Reply : Literally: "or if the moon were about four times more-distantly remote from us".

The following applies equally to the Moon as to other satellites of the Earth and of a planet which can be considered to be a primary, and most of the motion of the pair would be regular, only deviating from regular Keplerian, when the Sun is rightly, as the Earth, however, the same time it revolves, because the force between the Sun and the Earth is sufficiently diminished, that it can consist with the major periodic time.

JRS: Hill sphere?
Something like - If the Moon were four times further away there would be a difference: a boundary such that bodies more remote from the primary planets are more akin to the earth than to a satellite.
 
550  DE MOTO CORPORUM
litibus terrae sint habenda. Similes limites circa reliquos planetas constitui poterunt 8.02
Similar limits around the other planets can be established
¶   9. Quemadmodum casus evolutus in perpetua coniunctione cum Solae constat, ita etiam perpetua oppositio similem casum suppeditat.

Pro quo ponamus $~\eta=180°~$ ut sit $~\sin\eta=0$, $\cos\eta=-1 ~$ et $~ \phi=180°+\theta $, ideoque $~ d\phi=d\theta ~$, atque $z=u+v$.

Aequationes ergo pro motu Lunae sequentes induent formas:
9.00
¶   9. Just as is evident in the case of perpetual conjunction with the Sun, so also perpetual opposition provides similar consequences.

For this we assume $~\eta=180°~$ so that $~\sin\eta=0$, $\cos\eta=-1 ~$ and $~ \phi=180°+\theta $, so $~ d\phi=d\theta ~$, and $z=u+v$.

So the equations for the motion of the Moon take the following forms:
$$ \begin{aligned} \frac{2dvd\theta+vdd\theta}{d\zeta^2} = 0, ~~~ \text{&} ~~~ \frac{ddv-vd\theta^2}{d\zeta^2} + \frac{nnc^2}{v^2} + \frac{a^3v}{(u+v)^3}& \\ -\frac{a^3}{uu}(1-\frac{u^3}{(u+v)^3}) = 0& \end{aligned} $$
quae posterior reducitur ad hanc : 9.02
which then is reduced to this :
$$ \frac{ddv-vd\theta^2}{d\zeta^2} + \frac{nnc^3}{vv} - \frac{a^3}{uu} + \frac{a^3}{(u+v)^2} = 0 ~. $$
Prior cum motu Solis collata praebet statim $~v=\alpha u$, unde posterior sit 9.04
Comparing the previous with the motion of the Sun immediately gives $~v=\alpha u$, hence the latter is
$$ \frac{\alpha(ddu-ud\theta^2)}{d\zeta^2} + \frac{nnc^3}{\alpha\alpha uu} - \frac{a^3}{uu} + \frac{a^3}{(1+\alpha)^2uu} = 0 . $$
At ex motu Solis est $~ \frac{ddu-ud\theta^2}{d\zeta^2} = -\frac{a^3}{uu} ~$, ex quo fit $- \alpha a^3 + \frac{nnc^3}{\alpha\alpha} - a^3 + \frac{a^3}{(1+\alpha)^2} = 0 $, seu 9.06
But the motion of the sun is $~ \frac{d^2u-ud\theta^2}{d\zeta^2} = -\frac{a^3}{u^2} ~$, hence $- \alpha a^3 + \frac{n^2c^3}{\alpha^2} - a^3 + \frac{a^3}{(1+\alpha)^2} = 0 $, or
$$ \frac{nnc^3}{a^3} -\alpha\alpha(1+\alpha) + \frac{\alpha\alpha}{(1+\alpha)^2} = 0 $$
et posito brevitatis gratia $\frac{nnc^3}{a^3} = m ~$, erit 9.08
and putting for brevity $\frac{n^2c^3}{a^3} = m ~$, we have
$$ m(1+\alpha)^2 = \alpha\alpha(1+\alpha)^3 - \alpha\alpha $$
quae ex superiori nascitur, sumendis $m$ et $\alpha$ negativis. Quamobrem hinc colligitur 9.10
from the above, taking $m$ and $\alpha$ as negative. So we collect
$$ \alpha = \sqrt[3]\frac m3 + \frac 13 \sqrt[3]\frac {mm}9 - \frac1{27}m - \frac1{81} m \sqrt[3]\frac m3 ; \tag{1} $$
1) Non accuratum est : recte $~-\frac4{243} ~ \text{loco} ~ -\frac1{81} ~.$ M.S.
9.12
(1) is not accurate; $~-\frac4{243} ~ \text{should replace} ~ -\frac1{81} ~.$ M.S.
 
551COELESTIUM
satis autem exacte est $ \alpha = \sqrt[3]\frac m3 ~~ \text{et} ~~ v = \frac{cu}a \sqrt[3]\frac{nn}3 ~$ ut ante. 9.13
So approximately $~ \alpha = \sqrt[3]\frac m3 ~~ \text{and} ~~ v = \frac{cu}a \sqrt[3]\frac{n^2}3 ~$ as before.
¶   10. Casus hi eo magis sunt notatu digni, quod sine ulla approximatione absolute expediri possunt, etiamsi ambae vires Solis et terrae ad motum producendum concurrant, id quod nullo alio casu praestare licet.


Tali autem motu simplici corpus re vera moveretur, si ipsi in distantia assignata, dum Soli, vel coniunctum, vel oppositum, ex terra appareret, eiusmodi motus imprimeretur, ut cum terra pari passu in plano eclipticae ingredi inciperet.

Sin autem motus impressus tantillum ab hac lege discrepet, non quidem perpetuo Soli, vel coniunctum, vel oppositum, maneret, sed exiguas excursiones hinc inde quasi oscillando conficeret.

Quo casu cum motus minime ab inventa ratione esset discrepaturus, more solito, approximando etiam, eiusmodi motum definire licebit; in quo cum quasi initium motuum irregularium, quos nullo etiamnum modo ad calculum revocare licet, conspiciatur, usu certe non carebit, si in naturam istiusmodi motuum accuratius inquisivero.
10.00 ¶   10.
These cases are all the more worthy of mention since they can be worked out absolutely without any approximating, even if both forces of Sun and Earth act together in producing motion, something which cannot be maintained in any other case.

But by such a simple motion a body would really be moved, if to it at a fixed distance, while it appeared from earth either in conjunction or opposition to the Sun, a motion were impressed of such kind that it began to go at the same pace with the earth in the plane of the ecliptic.

But if the impressed motion varied a fraction from this law, it would remain not perpetually in conjunction or opposition to the Sun, but would effect tiny excursions as if swinging here and there.

In a case where the motion differed minimally from the discovered formula, in the usual way, also by approximating, it will be possible to define such motion; in which when an apparent beginning of irregular motions be observed, which motions cannot be aligned at all as yet with any calculation, that motion will not lack usefulness if I inquire deeper into the nature of such motions.

JRS: conjunction = L1, opposition = L2?
¶   11. Cum autem haec investigatio haud levibus difficultatibus implicetur, statim ab initio aequationes nostras tractatu faciliores reddi conveniet, quod, cum distantia $v$ prae $u$ vehementer sit exigua, commode per approximationem fieri potest.

Scilicet ob $~ z = \sqrt(uu-2uv\cos\eta + vv) ~$ eliciemus proxime $ \frac1{z^3} = \frac1{u^3} + \frac{3v\cos\eta}{u^4} - \frac{3vv}{2u^5} + \frac{15vv\cos\eta^2}{2u^6} ~, $ ideoque $ 1 - \frac{u^3}{z^3} = -\frac{3v}u \cos\eta + \frac{3vv}{2uu}(1-5\cos\eta^2) ~ , $ ex quo aequationes
11.00





11.02
¶   11. Without understating the difficulties involved in this investigation, from the beginning our equations facilitate appropriate treatment, which, when the distance between v and u is very small, is conveniently possible by approximation.

Clearly since $~ z = \sqrt(u^2-2uv\cos\eta + v^2) ~$ we get approximately $ \frac1{z^3} = \frac1{u^3} + \frac{3v\cos\eta}{u^4} - \frac{3vv}{2u^5} + \frac{15vv\cos\eta^2}{2u^6} ~, $ so $ 1 - \frac{u^3}{z^3} = -\frac{3v}u \cos\eta + \frac{3vv}{2uu}(1-5\cos\eta^2) ~ , $ from which our equations,
 
552  DE MOTU CORPORUM COELESTIUM
nostrae, pro motu Lunae inventae, in sequentes formas transibunt: 11.04
found for the motion of the Moon, take the following forms:
$$ \begin{aligned} \text{I.}& ~~~~ \frac{2dvd\phi + vdd\phi}{d\zeta^2} + \frac{3a^3v}{u^3}\sin\eta\cos\eta - \frac{3a^3vv}{2u^4}\sin\eta(1-5\cos\eta^2) = 0 \\ \text{II.}& ~~~~ \frac{ddv-vd\phi^2}{d\zeta^2} + \frac{nnc^3}{vv} + \frac{a^3v}{u^3}(1-3\cos\eta^2) + \frac{3a^3vv}{2u^4}(3\cos\eta-5\cos\eta^3) = 0 ~. \end{aligned} $$
JRS: In II above, the divisor here shown as 2 is in fact 2 in Src2 and 3 in Src1, Src3.   In II above, Src2 authorises the first ")".
JRS: $\cos\eta^2$ probably means $(\cos(\eta))^2$ and not $(\cos(\eta^2))$, likewise throughout.
Deinde etiam calculus non parum sublevabitur, si motum Solis, ut uniformem, spectemus, ut fit $~u=a~$, et $~\theta=\zeta~$, ideoque $~\eta=\phi-\zeta~$, seu $~\phi=\eta+\zeta~$, unde sequentes emergunt aequationes : 11.06
Next, the calculation is not very difficult, if the motion of the Sun is uniform, we see, so that $~u=a~$, and $~\theta=\zeta~$, and so $~\eta=\phi-\zeta~$, or $~\phi=\eta+\zeta~$, from which the following equations emerge:
$$ \begin{aligned} \text{I.}& ~~~~ \frac{2dvd\eta + vdd\eta}{d\zeta^2} + \frac{2dv}{d\zeta} + 3v\sin\eta\cos\eta - \frac{3vv}{2a}\sin\eta(1-5\cos\eta^2) = 0 \\ \text{II.}& ~~~~ \frac{ddv}{d\zeta^2} - v(1+\frac{d\eta}{d\zeta})^2 + v(13\cos\eta^2) + \frac{nnc^3}{vv} + \frac{3vv}{2a}\cos\eta(3-5\cos\eta^2) = 0, \end{aligned} $$
ubi etiam postrema membra facile omitti possunt, quia fractio $~\frac va~$ est vehementer parva, etiamsi distantia Lunae quadruplo maior statuatur. 11.08
where the later terms can easily be omitted, for a fraction of $~\frac va~$ is very small, even if four times greater than the distance of the Moon is set.
¶   12. Ut iam hinc casum memoratum, quo Luna circa Solem motu quasi oscillatorio nutare videbitur, eliciamus, angulum $~\eta~$ quam minimum concipiamus, ut sit $~\sin\eta=\eta$, et $~\cos\eta=1-\frac12\eta\eta~$, et habebimus : 12.00
¶   12. As already mentioned for the case, in which the Moon will be seen in oscillatory motion around the Sun, we can make the angle $~\eta~$ as small as we wish, so that $~\sin\eta=\eta$, and $~\cos\eta=1-\frac12\eta^2~$, and we will have :
$$ \begin{aligned} \text{I.}& ~~~~ \frac{2dvd\eta+vdd\eta}{d\zeta^2} + \frac{2dv}{d\zeta} + 3v\eta\eta = 0 \\ \text{II.}& ~~~~ \frac{ddv}{d\zeta^2} -v(1+\frac{d\eta}{d\zeta})^2 + \frac{nnc^3}{vv} - 2v + 3v\eta\eta = 0 . \end{aligned} $$
Deinde quia distantia $~v~$ parum immutatur, ponamus $~v=b(1+x)~$, ut $~x~$ sit quantitas minima, tum vero sit brevitatis gratia $~\frac{nnc^3}{b^3}=m~$, eritque : 12.02
Moreover, because the distance $~v~$ changes little, we assume $~v=b(1+x)~$, so that $~x~$ is a small quantity, and is brevity is $~\frac{n^2c^3}{b^3}=m~$, and will be:
$$ \begin{aligned} \text{I.}& ~~~~ \frac{2dxd\eta+xdd\eta}{d\zeta^2} + \frac{ddn}{d\zeta^2} + \frac{2dx}{d\zeta} + 3\eta + 3x\eta = 0 \\ \text{II.}& ~~~~ \frac{ddx}{d\zeta^2} - 3 - 3x - \frac{2d\eta}{d\zeta} - \frac{2xd\eta}{d\zeta} - \frac{d\eta^2}{d\zeta^2} - \frac{xd\eta^2}{d\zeta^2} + 3\eta\eta \\ &+ 3x\eta\eta\ + m - 2mx + 3mxx = 0 ~, \end{aligned} $$
 
553COELESTIUM
unde pro quouis angulo $~\zeta~$ valores quantitatum $~x~$ et $~\eta~$ definiri oportet. 12.04
Hence, for any angle $~\zeta~$ the values of the quantities $~x~$ and $~\eta~$ must be defined.
¶   13. Cum angulus $~\eta~$ sit minimus, alternatimque positiuus euadat et negativus; quoniam Luna ultro citroque a Sole digredi conspicietur : facile colligere licet, eum per quempiam angulum $~\omega~$ ad $~\zeta~$ datam rationem tenente ita definiri, ut sit 13.00
¶   13. When the angle $~\eta~$ is smallestst, alternately escaping positive and negative, for the Moon digressing to and fro from the Sun, see : we may infer easily, him by any angle $~\omega~$ to $~\zeta~$ given ratio defined and maintained so as to be
$$ \eta = A\sin\omega + B\sin2\omega + c\sin3\omega ~~ \text{etc.} $$
atque $~d\omega = \alpha d\zeta.~$ Quo posito est 13.02
and $~d\omega = \alpha d\zeta.~$ Put
$$ \frac{d\eta}{d\zeta} = \alpha A\cos\omega + 2\alpha B\cos2\omega + 3\alpha C\cos3\omega ~~~~~~~~ \text{&} \\ \frac{dd\eta}{d\zeta^2} = - \alpha^2 A\sin\omega - 4\alpha^2 B\sin2\omega - 9\alpha\alpha C\sin3\omega . $$
Quare cum aequatio prima in hanc formam transfundatur 13.04
So, with the first equation converted to the form
$$ \frac{2dx}{1+x} + \frac{dd\eta+3\eta d\zeta^2}{d\eta+d\zeta} = 0 $$

fiet

to get
$$ 2l(1+x)+l \left( 1+\frac{d\eta}{d\zeta} \right) + 3 \int \frac{\eta d\zeta}{1+\frac{d\eta}{d\zeta}} = \text{Const.} $$
seu ob $x$ et $\frac{d\eta}{d\zeta}$ minima: 13.08
or since $x$ and $\frac{d\eta}{d\zeta}$ are small :
$$ 2x - xx + \frac23 x^3 + \frac{d\eta}{d\zeta} - \frac{d\eta^2}{2d\zeta^2} + \frac{d\eta^3}{3d\zeta^3} + 3 \int \eta d\zeta \\ - \frac 32 \eta\eta + 3 \int \frac{\eta d\eta^2}{d\zeta} - 3 \int \frac{\eta d\eta^3}{d\zeta^2} = \text{Const}. $$
Nunc vero ob $~ d\zeta = \frac{d\omega}\alpha ~$ est 13.10
But now, since $~ d\zeta = \frac{d\omega}\alpha ~$, we have
$$ \int \eta d\zeta = - \frac A{\alpha}\cos\omega - \frac B{2\alpha}\cos2\omega - \frac C{3\alpha}\cos3\omega \\ \eta\eta = \frac12AA + AB\cos\omega - \frac12AA\cos2\omega - AB\cos3\omega ~~~~~~~~~~~~~~ 1) \\ +\frac12BB ~~~~~~~~~~~~~~~~~~~~~~~~~ + AC ~~~~~~~~~~~~~~~~~~~~~~~~~ $$
1) Vide praefationis p.XIIM.S.
13.12
1) See preface p.XIIM.S.
 
554  DE MOTU CORPORUM COELESTIUM
$$ \frac{d\eta^2}{d\zeta^2} = \frac12\alpha\alpha AA + 2\alpha\alpha AB\cos\omega + \frac12\alpha\alpha AA\cos2\omega + 2\alpha\alpha AB\cos3\omega \\ + 2\alpha\alpha BB ~~~~~~~~~~~~ + 3\alpha\alpha AC \\ \frac{d\eta^3}{d\zeta^3} = + \alpha^3 AAB + \frac34\alpha^3A^3\cos\omega + \alpha^3 AAB\cos2\omega \\ + 4\alpha^3 ABB ~, $$
ubi ob literas $~A,~B,~C~$ minimas altiores potestates merito negligimus. 13.14
where for the letters $~A,~B,~C~$ smaller higher powers we rightly neglect.
¶   14. Cum ergo sit 14.00
¶   14. Since, then,
$$ \begin{aligned} \frac{\eta d\eta^2}{d\zeta^2} &= \frac14\alpha\alpha A^3\sin\omega + \frac32\alpha\alpha AAB\sin2\omega + \frac14\alpha\alpha A^3\sin3\omega \\ &+~ 3\alpha\alpha ABB + 2\alpha\alpha BBB + \frac32\alpha\alpha A^2C \\ &-\frac32\alpha\alpha A^2C + \alpha\alpha ABB \end{aligned} $$
ob $~ d\zeta = \frac{d\omega}\alpha ~$ habebimus integrando: 14.02
for $~ d\zeta = \frac{d\omega}\alpha ~$, integrating:
$$ \begin{aligned} \int \frac{\eta d\eta^2}{d\zeta} &= - \frac14\alpha A^3\cos\omega - \frac34\alpha AAB\cos2\omega - \frac1{12}\alpha A^3\cos3\omega \\ &- 3\alpha ABB - \alpha B^3 - \frac12\alpha A^2C \\ &+\frac32\alpha A^2C - \frac13\alpha ABB \end{aligned} $$
ubi cum series $~A,~B,~C~$ maxime decrescat, plura membra omitti possunt. Deinde cum sit 14.04
where the series $~A,~B,~C~$ rapidly decreases, further terms can be omitted. Next
$$ \frac{\eta d\eta^3}{d\zeta^3} = \frac78\alpha^3A^3B\sin\omega + \frac38\alpha^3A^4\sin2\omega + \frac78\alpha^3A^3B\sin3\omega $$
erit integrando 14.06
integration
$$ \int \frac{\eta d\eta^2}{d\zeta^2} = - \frac78\alpha^2A^3B\cos\omega - \frac3{16}\alpha^2A^4\cos2\omega - \frac7{24}\alpha^2A^3B\cos3\omega $$
atque ex his tandem conficitur haec aequatio omissis partibus constantibus: 14.08
and from these finally, dropping constant terms, this equation:
 
555COELESTIUM
$$ \begin{aligned} 2x-xx+\frac23x^3 &+ \alpha A\cos\omega +2\alpha B\cos2\omega + 3\alpha C\cos3\omega = 0 \\ &- \alpha\alpha AB - \frac14\alpha\alpha AA - \alpha\alpha AB \\ &+ \frac14\alpha^3A^3 - \frac32\alpha\alpha AC - \frac{3C}{3\alpha} \\ &- \frac3\alpha A + \frac13\alpha^3 AAB + \frac32AB \\ &- \frac32AB - \frac{3B}{2\alpha} - \frac14\alpha A^3 \\ &- \frac34\alpha A^3 + \frac34AA + \frac14\alpha^3 A^3 \\ &- \frac32AC \\ &- \frac98\alpha AAB ~ . \end{aligned} $$ JRS: Equation, line 4, term 1, sources differ.
¶   15. Ad valorem ipsius $~x~$ hinc definiendum ponamus brevitatis gratia 15.00
¶   15. For brevity we put the value of $~x~$ here
$$ \left(\alpha-\frac3\alpha\right)A - (\alpha\alpha+\frac32)AB + \frac14\alpha(\alpha\alpha-3)A^3 = {\frak A} \\ \frac{(4\alpha\alpha-3)}{2\alpha}B - \frac{(\alpha\alpha-3)}4AA = {\frak B} \\ \frac{(3\alpha\alpha-1)}\alpha C - \frac{(2\alpha\alpha-3)}2AB + \frac14\alpha(\alpha\alpha-1)A^3 = {\frak C} $$

ut sit

that is
$$ 2l(1+x) + {\frak A}\cos\omega + {\frak B}\cos2\omega + {\frak C}\cos3\omega = 0 $$

et

and
$$ 1+x = e^{- \frac12{\frak A}\cos\omega - \frac12{\frak B}\cos2\omega - \frac12{\frak C}\cos3\omega } $$
unde concludimus fore 15.06
whence we would conclude
$$ \begin{aligned} x = & - \frac12{\frak A}\cos\omega - \frac12{\frak B}\cos2\omega - \frac12{\frak C}\cos3\omega \\ & + \frac18{\frak AB} + \frac1{16}{\frak AA} + \frac18{\frak AB} \\ & + \frac1{64}{\frak A}^3 - \frac1{192}{\frak A}^3 . \end{aligned} $$
Verum ne in calculos nimis taediosos immergamur, rem aliquanto minus curate expediamus, neglectoque angulo triplo, ut sit $~ \eta = A\sin\omega + B\sin2\omega $, habebimus 15.08
In order not to be immersed too deeply in tedious calculations, we shall disregard small quanitites, so neglecting the tripled angle, $~ \eta = A\sin\omega + B\sin2\omega $, we have
$$ x = - \frac{(\alpha\alpha-3)}{2\alpha}A\cos\omega - \frac{(4\alpha\alpha-3)}{4\alpha}B\cos2\omega + \frac{3(\alpha\alpha-1)(\alpha\alpha-3)}{16\alpha\alpha}AA\cos2\omega ~, $$
The final $~\cos2\omega~$ is in Src2; not in Src1 / Src3.
 
556  DE MOTU CORPORUM COELESTIUM
ubi brevitas gratia scribamus: 15.10
where for brevity we write:
$$ x = E\cos\omega + F\cos2\omega, $$
ut sit 15.12
so that
$$ E = \frac{3-\alpha\alpha}{2\alpha}A ~~~~ \text{&} ~~~~ F = \frac{3-4\alpha\alpha}{4\alpha}B + \frac{3(\alpha\alpha-1)(\alpha\alpha-3)}{16\alpha\alpha}AA ~ . $$
¶   16. Hi iam valores in aequatione secunda substituantur, atque reperiemus: 16.00
¶   16. These values are substituted into the second equation, and we find:
$$ \begin{aligned} \frac{ddx}{d\zeta^2} &= -\alpha\alpha E\cos\omega - 4\alpha\alpha F\cos2\omega \\ -3 - 3x &= - 3 - 3E - 3F \\ - \frac{2d\eta}{d\zeta} &= - 2\alpha A - 4\alpha B \\ - \frac{2xd\eta}{d\zeta} &= -\alpha AE - 2\alpha BE -\alpha AE \\ &~~~~~~~~~~~~~~~~~~ -\alpha AF \\ - \frac{d\eta^2}{d\zeta^2} &= -\frac12\alpha\alpha AA + 2\alpha\alpha AB ~~~ ^1) ~~~ - \frac12\alpha\alpha AA \\ + 3\eta\eta &= + \frac32AA + 3AB\cos\omega -\frac32AA\cos2\omega \\ + m &= + m \\ - 2mx &= - 2mE - 2mF \\ + 3mxx &= +\frac32mEE + 3mEF + \frac32mEE \end{aligned} $$
That follows Src2; Src1 Src3 differ significantly.
  1) M.S. Recte $~ -2\alpha\alpha AB$.
unde primo concludimus:
16.02
  1) : M.S. That should be $~ -2\alpha^2 AB$.
from which we first conclude
$$ m(1+\frac32EE) = 3 + \alpha AE + \frac12\alpha\alpha AA - \frac32AA = 3 $$

ideoque

so
$$ m = 3 -\frac92EE $$
pro determinatione numeri $~m~$ indeque distantia $~b$.

Manifestum autem est, esse proxime $~m=3~$, ideoque $~b = c\sqrt[3]\frac{nn}3 .~$ Porro autem fit
16.06
to give the value of $~m~$ and thence the distance $~b$.

Now it is clear that approximately $~m=3~$, and so $~b = c\sqrt[3]\frac{n^2}3 .~$ Moreover, the results
$$ - \alpha\alpha E - 9E -2\alpha A - 2\alpha BE - \alpha AF + (2\alpha\alpha+3)AB + 9EF = 0 ~, $$
unde neglectis terminis minimis ob $~ \frac EA = \frac{3-\alpha\alpha}{2\alpha} ~$ erit 16.08
thus neglecting the smallest terms for $~ \frac EA = \frac{3-\alpha^2}{2\alpha} ~$, get
$$ (\alpha\alpha+9)(3-\alpha\alpha) + 4\alpha\alpha = 0 ~, $$
seu $~ \alpha^4 + 2\alpha\alpha - 27 = 0 $, hincque $~ \alpha\alpha = \sqrt{28} - 1 $. 16.10
or $~ \alpha^4 + 2\alpha^2 - 27 = 0 $, hence $~ \alpha^2 = \sqrt{28} - 1 $.
 
557COELESTIUM
Tertia denique aequatio dat 16.11
Finally the third equation gives
$$ - (4\alpha\alpha+9)F - 4\alpha B -\alpha AE - \frac12\alpha\alpha AA - \frac32AA + \frac92EE = 0 $$
ideoque: 16.13
so:
$$ \left. \begin{array}{l} \frac{(4\alpha\alpha-3)(4\alpha\alpha+9)}{4\alpha}B - \frac{3(\alpha\alpha-1)(\alpha\alpha-3)(4\alpha\alpha+9)}{16\alpha\alpha}AA \\ -4\alpha B ~~~~~ +\frac{(\alpha\alpha-3)}2AA \\ -\frac{(\alpha\alpha+3)}2AA \\ +\frac{9(\alpha\alpha-3)^2}{8\alpha\alpha} AA \end{array} \right\} ~ = 0 $$
unde colligitur: 16.15
from this we conclude
$$ B(16\alpha^4+8\alpha\alpha-27) = \frac32AA\alpha(13-7\alpha\alpha+2\alpha^4) $$
seu ob $~ 27 = \alpha^4 + 2\alpha\alpha $ 16.17
or since $~ 27 = \alpha^4 + 2\alpha^2 $
$$ 3B(5\alpha\alpha+2) = \frac{3AA}{2\alpha}(13-7\alpha\alpha+2\alpha^4) $$
ideoque 16.19
so:
$$ B = \frac{13-7\alpha\alpha+2\alpha^4}{2\alpha(5\alpha\alpha+2)}AA = \frac{67-11\alpha\alpha}{2\alpha(5\alpha\alpha+2)}AA \\ F = \frac{291-94\alpha\alpha-23\alpha^4}{8\alpha\alpha(5\alpha\alpha+2)}AA = - \frac{165-24\alpha\alpha}{4\alpha\alpha(5\alpha\alpha+2)}AA ~~~~ ^1). $$
1) Recte : $~ F = \frac{291-94\alpha\alpha-\alpha^4}{8\alpha\alpha(5\alpha\alpha+2)}AA = \frac{66-23\alpha\alpha}{2\alpha\alpha(5\alpha\alpha+2)}AA ~.~$ M.S.
16.21
1) Right : $~ F = \frac{291-94\alpha^2-\alpha^4}{8\alpha^2(5\alpha^2+2)}AA = \frac{66-23\alpha^2}{2\alpha^2(5\alpha^2+2)}AA ~.~$ M.S.
¶   17. Quantitas ergo $~A~$ arbitrio nostro relinquitur, a qua digressiones a linea syzygiarum pendent, pro ea autem valde parvam fractionem assumi oportet, quae si fuerit tam exigua, ut eius quadratum nullius sit momenti, primi termini sufficiunt.

Pro distantia ergo $~ v=b(1+x) ~$ erit $~ b=c\sqrt[3]\frac{nn}3 ~$, et angulus $~\omega~$ ita definitur, ut sit $~ \omega=\alpha\zeta+\beta ~$ existente $~ \alpha\alpha=\sqrt{28}-1 $, hincque $~ \alpha\alpha=4,291502 ~$ et $~ \alpha=2,071594$.


Tum vero erit $~ \eta=A\sin\omega ~$ et $~ v=b \left(1-\frac{(\alpha\alpha-3)}{2\alpha}A\cos\omega \right) ~$ seu $~v=b(1-0,311717A\cos\omega).$
17.00
¶   17. The value of $~A~$, therefore, is left to our choice, on which the digressions from the line of syzygies depend, but for a very small fraction of it must be assumed, which if it was so small, so that its the square is of no importance, the first terms are sufficient.

Therefore, the distance between $~ v=b(1+x) ~$ will be $~ b=c\sqrt[3]\frac{n^2}3 ~$, and angle $~\omega~$ is so defined may be $~ \omega=\alpha\zeta+\beta ~$ with $~ \alpha^2=\sqrt{28}-1 $, hence $~ \alpha^2=4.291502 ~$ and $~ \alpha=2.071594$.

Then we have $~ \eta=A\sin\omega ~$ and $~ v=b \left(1-\frac{(\alpha^2-3)}{2\alpha}A\cos\omega \right) ~$ or $~v=b(1-0.311717A\cos\omega).$
 
558  DE MOTU CORPORUM COELESTIUM
Excursiones fiunt maximae, si angulus $\omega$ sit $90^\circ.270^\circ.$etc. ergo ab una digressione maxima ad sequentem est $\alpha\zeta=180^\circ$ et $\zeta=86^\circ53½'$: at in digressionibus maximis est $v=b$.

Verum etiam huiusmodi librationis, si maior existeret, determinatio insignibus laborat difficultatibus, ut, quo accuratius omnes variationes definire vellemus, eo minus certi de reliquis neglectis redderemur.
17.01
Excursions are greatest, if the angle $\omega$ is $90^\circ, 270^\circ,$ etc., therefore from one excursion, the greatest to the following is $\alpha\zeta=180^\circ$ and $\zeta=86^\circ53\small{^1\!\!/\!_2}'$: but in the greatest excursion $v=b$.

But even such a libration, if so great existed, the determination has remarkable difficulties, so that, to define all the variations more accurately, it is less certain about ignoring the rest.
THE END
Thanks to Thiluxshan for part of the transcription of Src2.

Partly hidden spare parts follow.
1
Left 1
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Something else.
Left 2
Right 2

Acknowledgements

To participants in Usenet news:alt.language.latin, Autumn 2012 ff.

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