Of the Irregularities in the Motion of a Satellite Arising from the Spheroidical Figure of Its Primary Planet: In a Letter to the Rev. James Bradley D. D. Astronomer Royal, F. R. S. and Member of the Royal Academy of Sciences at Paris; By Mr. Charles Walmesley, F. R. S. and Member of the Royal Academy of Sciences at Berlin, and of the Institute of Bologna

CX. Of the Irregularities in the Motion of a Satellite arising from the Spheroidal Figure of its Primary Planet: In a Letter to the Rev. James Bradley D.D. Astronomer Royal, F.R.S. and Member of the Royal Academy of Sciences at Paris; by Mr. Charles Walmsley, F.R.S. and Member of the Royal Academy of Sciences at Berlin, and of the Institute of Bologna. Reverend Sir, Read Dec. 14, 1758. Since the time that astronomers have been enabled by the perfection of their instruments to determine with great accuracy the motions of the celestial bodies, they have been solicitous to separate and distinguish the several inequalities discovered in these motions, and to know their cause, quantity, and the laws according to which they are generated. This seems to furnish a sufficient motive to mathematicians, wherever there appears a cause capable of producing an alteration in those motions, to examine by theory what the result may amount to, though it comes out never so small: for as one can seldom depend securely upon mere guess for the quantity of any effect, it must be a blameable neglect entirely to overlook it without being previously certain of its not being worth our notice. Finding therefore it had not been considered what effect the figure of a planet differing from that of a sphere sphere might produce in the motion of a satellite revolving about it, and as it is the case of the bodies of the Earth and Jupiter which have satellites about them, not to be spherical but spheroidal, I thought it worth while to enter upon the examination of such a problem. When the primary planet is an exact globe, it is well known that the force by which the revolving satellite is retained in its orbit, tends to the center of the planet, and varies in the inverse ratio of the square of the distance from it; but when the primary planet is of a spheroidal figure, the same rule then no longer holds: the gravity of the satellite is no more directed to the center of the planet, nor does it vary in the proportion above-mentioned; and if the plane of the satellite's orbit be not the same with the plane of the planet's equator, the protuberant matter about the equator will by a constant effort of its attraction endeavour to make the two planes coincide. Hence the regularity of the satellite's motion is necessarily disturbed, and though upon examination this effect is found to be but small in the moon, the figure of the earth differing so little from that of a sphere, yet in some cases it may be thought worth notice; if not, it will be at least a satisfaction to see that what is neglected can be of no consequence. But however inconsiderable the change may be with regard to the moon, it becomes very sensible in the motions of the satellites of Jupiter both on account of their nearer distances to that planet when compared with its semidiameter, as also because the figure of Jupiter so far recedes from that of a sphere. This I have shewn and exemplified in the fourth satellite; in which case indeed the computation is more exact exact than it would be for the other satellites: for as my first design was to examine only how far the moon's motion could be affected by this cause, I supposed the satellite to revolve at a distance somewhat remote from the primary planet, and the difference of the equatorial diameter and the axis of the planet not to be very considerable. There likewise arises this other advantage from the present theory, that it furnishes means to settle more accurately the proportion of the different forces which disturb the celestial motions, by assigning the particular share of influence which is to be ascribed to the figure of the central bodies round which those motions are performed. I have added at the end a proposition concerning the diurnal motion of the earth. This motion has been generally esteemed to be exactly uniform; but as there is a cause that must necessarily somewhat alter it, I was glad to examine what that alteration could amount to. If we first suppose the globe of the earth to be exactly spherical, revolving about its axis in a given time, and afterwards conceive that by the force of the sun or moon raising the waters its figure be changed into that of a spheroid, then according as the axis of revolution becomes a different diameter of the spheroid, the velocity of the revolution must increase or diminish: for, since some parts of the terraqueous globe are removed from the axis of revolution and others depressed towards it, and that in a different proportion as the sun or moon approaches to or recedes from the equator, when the whole quantity of motion which always remains the same is distributed through the spheroid, the velocity of the diurnal rotation cannot be constantly the same. This variation variation however will scarce be observable, but as it is real, it may not be thought amiss to determine what its precise quantity is. I am sensible the following theory, as far as it relates to the motion of Jupiter's satellites, is imperfect and might be prosecuted further; but being hindered at present from such pursuit by want of health and other occupations, I thought I might send it you in the condition it has lain by me for some time. You can best judge how far it may be of use, and what advantage might arise from further improvements in it. I am glad to have this opportunity of giving a fresh testimony of that regard which is due to your distinguished merit, and of professing myself with the highest esteem, Reverend Sir, Your very humble Servant, Bath, Oct. 21. 1758. C. Walmesley. Lemma I. Invenire gravitatem corporis longinquui ad circumfere- rentiam circuli ex particulis materiae in duplicata- ratione distantiarum inversè attrahentibus constan- tem. Esto NIK (Vid. Tab. xxxiii. Fig. 1.) circumferen- tia circuli, in cujus puncta omnia gravitet corpus longinquum S locatum extra planum circuli. In hoc planum agatur linea perpendicularis SH, et per cir- culi centrum X ducatur recta H X K secans circulum. in I et K, et SR parallela ad H X : producatur au- tem SH ad distantiam datam SD, et agantur rectæ D.C., DC, XC, ipsis HX, SD, parallelæ. Tum ductâ chordâ quavis MN ad diametrum IK normali eamque secante in L, ex punctis M, N, demittantur in SR perpendiculares MR, NR, concurrentes in R; junctisque SM, SN, erit SM = SN, MR = NR, SR = HL. Dicantur jam SD, k; HX five DC, b; XL, x; CX, z; XI, r; eritque HL = b — x, et SH = k — z. Est autem SM ad SH ut attractio \( \frac{1}{SM^2} \) corporis S versus particulam M in directione SM ad ejusdem corporis attractionem in directione SH, quae proinde erit \( \frac{SH}{SM^3} \): sed est SR = HL, et \( SM^2 = SR^2 + MR^2 = SR^2 + SH^2 + ML^2 \); unde fit \( \frac{SH}{SM^3} = \frac{SH}{HL^2 + SH^2 + ML^2} \), et ductâ mn parallelâ ad MN, vis qua corpus S attrahitur ad arcus quàm minimos Mm, Nn, exponitur per \( \frac{SH \times 2Mm}{SM^3} = SH \times 2Mm \times \frac{HL^2 + SH^2 + ML^2}{SM^3} \). Est autem \( HL^2 + SH^2 + ML^2 = kk - 2kz + zz + bb - 2bx + rr \), hincque ponendo kk + bb = ll, \( HL^2 + SH^2 = ML^2 = \frac{r}{l^3} + \frac{3kz}{l^5} + \frac{3bx}{l^5} - \frac{3rr}{2l^5} - \frac{3zz}{2l^5} + \frac{15kkzz}{2l^7} + \frac{15kbzx}{2l^7} + \frac{15bbxx}{2l^7} \), neglectis terminis ulterioribus ob longinquitatem quam supponimus corporis S. Quare, si scribatur d pro circumferentia IMKN, gravitas corporis S ad totam illam circumferentiam secundum SH, five fluens fluxionis \( SH \times 2Mm \times \frac{HL^2 + SH^2 + ML^2}{SM^3} \) evadit. \( k - z \times d \) in \( \frac{1}{l^3} + \frac{3kz}{l^5} - \frac{3rr}{2l^5} - \frac{3zz}{2l^5} + \frac{15kkzz}{2l^7} \). Simili modo obtinebitur gravitas ejusdem corporis S secundum SR. Q.E.I. **Lemma II.** *Corporis longinqui gravitatem ad Sphæroidem oblatam determinare.* Retentis iis quæ sunt in lemmate superiori demonstrata; esto C centrum sphæroidis, cujus æquatori parallelus sit circulus IMK. Sphæroidis hujus semiaxis major sit \(a\), semiaxis minor \(b\), eorum differentia \(c\), quam exiguam esse suppono; et dicatur D circumferentia æquatoris. Centro C et radio æquali semiaxi minori describi concipiatur circulus qui fecet IK in \(i\), eritque gravitas in directione SD, qua urgetur corpus S versus materiam sitam inter circumferentiam IMKN et circumferentiam centro X et radio Xi descriptam, æqualis gravitati in lemmate præcedenti definitæ ductæ in rectam Ii. Sed est \(Ii \cdot c :: IX \cdot a\), atque \(d \cdot D :: IX \cdot a\); unde \(Ii \times d \cdot D \times c :: IX^2 \cdot aa\), hoc est, ex naturâ ellipsoïds, ob \(CX = z\), et \(IX = r\), \(Ii \times d \cdot D \times c :: bb - zz \cdot bb\), adeoque \(Ii \times d = \frac{D \times c}{bb} \times bb - zz\), atque \(rr = aa - \frac{aa \times zz}{bb}\); scribi autem potest in sequenti calculo \(bb - zz\) pro \(rr\) ob parvitudinem differentiæ semiaxium in quam omnes termini ducuntur. Gravitas igitur corporis S in materiam inter circumferentias supradictas consistentem exprimetur per \(\frac{D \times c}{bb} \times bb - zz \times k - z\) in \(\frac{1}{l^3} + \frac{3kz}{l^5} - \frac{3bb}{2l^5} - \frac{15zz}{4l^5} + \frac{15bbbb}{4l^7} + \frac{45kkzz}{4l^7}\). Et si addatur gravitas in similem materiam ex ex alterâ parte centri C ad æqualem à centro distan- tiam, quia tunc CX sive z evadit negativa, gravitas corporis S in hanc duplicem materiam erit \( \frac{D \times c}{bb} \times \) \( \frac{2k}{l^3} - \frac{6kzz}{l^5} - \frac{3kbb}{l^5} + \frac{15k^3zz}{l^7} + \frac{15bkkbb}{2l^7} - \) \( \frac{15bkkzz}{2l^7} \). Ducatur jam gravitas hæc in \( \dot{z} \), et sumptâ gravitatum omnium summâ, factâ \( z = b \), gravitatio tota corporis S in totam materiam globo interiori su- periorem secundum directionem SD æquatori per- pendicularem prodit \( D \times c \times \frac{4kb}{3l^3} - \frac{4kb^3}{5l^5} + \frac{2bkkbb^3}{l^7} \). Simili ratiocinio gravitatio corporis S in eamdem materiam secundum directionem SR æquatori pa- rallelam invenitur æqualis \( D \times c \times \frac{4bb}{3l^3} + \frac{2bb^3}{5l^5} - \) \( \frac{2bkkbb^3}{l^7} \). Tum si addatur gravitatio corporis S in globum interiorem, ex unâ parte scilicet \( \frac{2b^3kD}{3al^3} \), et ex alterâ \( \frac{2b^3bD}{3al^3} \), habébitur gravitas corporis S in to- tum sphæroidem. Q. E. I. Coroll. Igitur gravitas corporis S secundum SD est ad ejuf- dem gravitatem secundum SR sive DC in materiam sphæroidis globo interiori incumbentem ut \( \frac{2k}{3} - \frac{2kb^2}{5l^2} \) \( + \frac{kbb^2}{l^4} \) ad \( \frac{2b}{3} + \frac{bb^2}{5l^2} - \frac{bkkb^2}{l^4} \), adeoque si gravitas prior exponatur per \( k \), posterior exprimetur per \( b - \frac{3bb^3}{5l^2} \) quamproximè. Unde cum sit DC = \( b \), patet gravi- tatem corporis S in sphæroidem oblatam non tendere ad ad centrum C, sed ad punctum c rectae DC in plano æquatoris jacentis vicinius puncto D. PROPOSITIO I. PROBLEMA. Vires determinare quibus perturbatur motus Satellitis circa Primarium suum revolventis. Exhibeat jam sphærois prædicta planetam quemvis figurâ hac donatum, et corpus S satellitem circa planetam tanquam primarium gyrantem. Quantitas materiæ globo sphæroidis interiori incumbentis æqualis est $\frac{4b^3cD}{3a}$ sive $\frac{4bcD}{3}$ proximè, et si materia illa locaretur in centro sphæroidis C, attraheret satellitem S secundum SC vi $\frac{4bcD}{3l^2}$, quæ reducta ad directionem SD fit $\frac{4bckD}{3l^3}$, et ad directionem DC fit $\frac{4bchD}{3l^3}$. Cum igitur vis $\frac{4bcD}{3l^2}$ non turbat motum satellitis, utpote quæ tendat ad centrum motûs et quadrato distantiae ab eodem centro sit reciprocè proportionalis, vires illæ $\frac{4bckD}{3l^3}$, $\frac{4bchD}{3l^3}$, in quas resolvitur, etiam motum non turbabunt. Itaque ex vi $D \times c \times \frac{4kb}{3l^3} - \frac{4kb^3}{5l^5} + \frac{2khhb^3}{l^7}$ auferatur vis $\frac{4bckD}{3l^3}$, et ex vi $D \times c \times \frac{4bb}{3l^3} + \frac{2hb^3}{5l^5} - \frac{2hhkb^3}{l^7}$ auferatur $\frac{4bchD}{3l^3}$, et remanebunt vires $D \times c \times \frac{-4kb^3}{5l^5} + \frac{2khhb^3}{l^7}$, $D \times c \times \frac{2hb^3}{5l^5} - \frac{2hhkb^3}{l^7}$, motuum satellitis S perturbatrices. Designetur vis $D \times c \times$ \[ \frac{2bb^3}{5l^5} - \frac{2bbkb^3}{l^7} \text{ per rectam } Sr \text{ (Fig. 2.) ac resolvatur in vim } Sq \text{ tendentem ad centrum planetæ primarii } C \text{ et ob triangula similia } Srq, SDC, æqualem } D \times c \times \frac{2b^3}{5l^4} - \frac{2kkb^3}{l^6}, \text{ existentibus ut priùs, } SD = k, DC = b, SC = l; \\ \text{et in vim } rq \text{ rectæ } SD \text{ parallelam et æqualem } D \times c \times \frac{2kb^3}{5l^5} - \frac{2k^3b^3}{l^7}; \text{ atque hæc vis posterior subducta ex vi } \\ D \times c \times \frac{4kb^3}{5l^5} + \frac{2kkbb^3}{l^7} \text{ relinquet } D \times c \times \frac{4kb^3}{5l^5} \text{ pro vi perturbatrice in directione } SD. \text{ Unde cum massa tota planetæ fit } \frac{2abD}{3}, \text{ gravitas satellitis tota in plane-} \\ \text{tam erit } \frac{2abD}{3l^2} \text{ proximé, vel etiam } \frac{2bbD}{3l^2}, \text{ et hæc gravi-} \\ \text{tas est ad vim } D \times c \times \frac{4kb^3}{5l^5} \text{ ut } 1 \text{ ad } \frac{6kbc}{5l^3}. \\ \text{Deinde vis illius } D \times c \times \frac{4kb^3}{5l^5} \text{ secundum } SD \text{ pars ea quæ agit in directione } SC \text{ est } D \times c \times \frac{4kkb^3}{5l^6}, \text{ quæ addita vi } Sq \text{ dat } D \times c \times \frac{2b^3}{5l^4} - \frac{6kkb^3}{5l^6} \text{ vim perturbatri-} \\ \text{cem tendentem ad centrum planetæ primarii, atque hæc vis est ad satellitis gravitatem } \frac{2bbD}{3l^2} \text{ in primarium ut } \\ \frac{3bc}{5l^2} - \frac{9kkbc}{5l^4} \text{ ad } 1. \quad Q.E.I. \\ \text{Coroll.} \\ \text{Designet } CK \text{ (Fig. 3.) lineam intersectionis plano-} \\ \text{rum æquatoris planetæ et orbitæ satellitis, et resolva-} \\ \text{tur vis } SD = \frac{6kbc}{5l^3}, \text{ quæ agit perpendiculariter ad } \] planum æquatoris, in vim DR perpendicularem ad planum orbitæ satellitis, et in vim SR jacentem in eodem plano. Producatur SR donec occurrat CK in K, eritque SK normalis ad CK, et planum SDK normale ad planum orbis satellitis; ac propterea ob similia triangula SDK, SRD, si \( m \) denotet sinum ad radius 1 et \( n \) cosinum anguli SKD, inclinationis scilicet orbitæ satellitis ad æquatorem planetæ, erit \[ DR = SD \times n = \frac{6kbcm}{5^{3/2}}, \quad \text{et} \quad SR = SD \times m = \frac{6kbcm}{5^{3/2}}, \] existente i gravitate totâ satellitis in primarium suum. Jam quoniam vis SR jacet in plano orbitæ satellitis, hujus plani situm non mutat; accelerat quidem vel retardat motum satellitis revolventis, sed hæc acceleratio vel retardatio ob brevitatem temporis ad quantitatem sensibilem non exurgit: vis DR eidem plano perpendicularis continuò mutat ejus situm, et motum nodi generat, quem sequenti propositione definiemus. **PROPOSITIO II.** **PROBLEMA.** *Invenire motum nodi ex praedictâ causâ oriundum.* Per motum nodi in hac propositione intelligo motum intersectionis planorum æquatoris planetæ et orbitæ satellitis; orbitam autem satellitis quamproximé circularem suppono. Esto S locus satellitis in orbe suo SN cujus centrum C, (Fig. 4.) SF arcus centro C descriptus perpendicularis in circulum æquatoris planetæ FN; SB arcus eodem centro descriptus perpendicularis ad orbem SN, atque in SB sumatur lineola Sr æqualis duplo spatio, quod satelles percurrere posset impellente vi DR in Coroll. præced. determinatâ, quo tempore in orbe suo descripteret arcum quam minimum \( pS \): per puncta \( r, p \), describatur centro \( C \) circulus \( rpn \) secans equatorem in \( n \), qui exhibebit situm orbitae satellitis post illam partculam temporis, nodo \( N \) translato in \( n \). Agantur \( SC, CN, \) et \( SH \) perpendicularis in lineam nodorum \( CN, \) et \( Nm \) perpendicularis in \( rpn \). Jam cum sint lineolae \( Sr, Nm \), ut sinus arcuum \( Sp, SN \), erit \( Sp \cdot Sr :: SH \cdot Nm \); deinde in triangulo rectangulo \( Nmn \) habetur \( m \cdot i :: Nm \cdot Nn \); unde per compositionem rationum \( Sp \times m \cdot Sr :: SH \cdot Nn = \frac{Sr \times SH}{Sp \times m} \): dato igitur arcu \( Sp \), est \( Nn \) sive motus nodi ut \( Sr \times SH \). In triangulo sphærico rectangulo \( SFN \) est sinus anguli \( N \), hoc est, anguli inclinationis orbitae satellitis ad æquatorem planetæ, ad sinum arcûs \( SF \), ut radius ad sinum arcûs \( SN \), id est, \( m \cdot \frac{k}{l} :: i \cdot SH \), adeoque \( \frac{k}{l} = m \times SH \); igitur \( \frac{k}{l} \) ut \( SH \). Vis autem \( Sr \) per Coroll. Prop. præced. est ut \( \frac{k}{l} \), adeoque ut \( SH \); quamobrem est \( Sr \times SH \), proindeque et \( Nn \), ut \( SH^2 \), hoc est, motus horarius nodi vi præfatâ genitus est in duplicatâ ratione distantiae satellitis à nodo. Et quoniam summa omnium \( SH^2 \), quo tempore satelles periodum suam absolvet, est dimidium summæ totidem \( SC^2 \), ideò motus periodicus est subduplus ejus qui, si satelles in declinatione suâ maximâ ab æquatore planetæ continuò persartet, eodem tempore generari posset. Sit igitur satelles in maximâ suâ declinatione sive in quadraturâ cum nodo, eritque \( SN \) quadrans circuli, et \( Nm \) mensura anguli \( Npm \) sive \( Spr \), eritque in hoc casu \( Nn \) sive motus horarius nodi ad \( Nm \), hoc est, ad angulum \( Spr \), ut \( i \) ad \( m \); est autem angulus $S_p r$ ad duplum angulum, quem subtendit sinus versus arcûs $S_p$ satellitis gravi- tate in primarium eodem tempore descripti, id est, ad angulum $SC_p$ qui est motus horarius satellitis circa primarium, ut vis $Sr$ ad gravitatem satellitis in primarium, hoc est (per Coroll. Prop. I.), ut $\frac{6kbcn}{5l^3}$ ad 1, sive, quia est in hoc casu $\frac{k}{l} = m$, ut $\frac{6bcmn}{5l^2}$ ad 1. Unde conjunctis rationibus est motus horarius nodi ad motum horarium satellitis ut $\frac{6bcn}{5l^2}$ ad 1; et si $S$ denotet tempus periodicum solis apparens, et $L$ tem- pus periodicum satellitis circa primarium suum, cum fit motus horarius satellitis ad motum horarium solis ut $S$ ad $L$, erit motus horarius nodi ad motum hora- rium solis ut $\frac{6bcn}{5l^2} \times \frac{S}{L}$ ad 1, et in eadem ratione erit motus nodi annuus ad motum solis annum, hoc est, ad $360^\circ$. Quare, si satelles maneret toto anno in maximâ suâ declinatione ab æquatore primarii, vis prædicta ex figurâ sphæroidicâ planetæ primarii pro- veniens generaret eodem tempore motum nodi æqua- lem $\frac{6bcn}{5l^2} \times \frac{S}{L} \times 360^\circ$, et ex supradictis motus verus nodi annuus erit hujus subduplus, nempe $\frac{3bcn}{5l^2} \times \frac{S}{L} \times 360^\circ$. Q.E.I. Coroll. Si computatio instituatur pro lunâ, assumendo mediocrem ejus orbitæ inclinationem ad æquatorem terrestrem, erit $n$ cosinus anguli $23^\circ 28'4''$; et posito femiaxi terræ $b = 1$, erit distantia lunæ à centro terræ mediocris $l = 60$ circiter, indeque in hypothesi quod sit sit differentia semiaxium \( c = \frac{1}{229} \), erit \( \frac{3bcn}{5l^2} \times \frac{S}{L} \times 360^\circ = 11''\frac{1}{4} \); et si fuerit \( c = \frac{1}{177} \), manente terrâ uniformiter densâ, erit ille motus \( = 15'' \). Hic erit motus nodorum annuus lunæ regressivus in plano æquatoris terrestris, qui reductus ad eclipticam, uti poëtâ docebitur, pro vario nodorum situ evadet multò velocior. Notabilis multò magis erit motus intersectionis orbitarum satellitum Jovis in plano æquatoris Jovialis; et computabitur satis accuratè per formulam suprà traditam, modò satelles non sit Jovi nimis vicinus. Sic pro satellite extimo erit \( L = 16d 16h 32' \), \( b = 1 \), \( l = 25,299 \) circiter, semiaxium Jovis differentia \( c = \frac{1}{13} \); et posìtâ orbis hujus satellitis inclinatione ad æquatorem Jovis æquali \( 3^\circ \), erit \( n \) cosinus hujus inclinationis, atque inde prodibit \( \frac{3bcn}{5l^2} \times \frac{S}{L} \times 360^\circ = 34' \) circiter, motus scilicet nodorum annuus satellitis quarti in plano æquatoris Jovis in antecedentia. Si minùs vel magìs inclinatur orbis ad Jovis æquatorem, augeri vel minui debet hic motus in ratione cosinûs hujus inclinationis. Cæterûm patet motum hunc nodorum in plano æquatoris planetæ primarii, æstimando distantiam satellitis in semidiametriis primarii, generatîm esse, dato tempore, in ratione compositâ, ex ratione directâ differentiæ semiaxium planetæ et cosinûs inclinationis orbis satellitis ad planetæ æquatorem, conjunctîm; et ex ratione inversâ temporis periodici satellitis et quadrati distantiae satellitis à centro planetæ, item conjunctîm. P R O- PROPOSITIO III. PROBLEMA. Motum nodorum Lunæ supra determinatum ad Eclipticam reducere. Sunto NAD (Fig. 5.) æquator, AGE ecliptica secans æquatorem in A, E æquinoctium vernum, A autumnale, LGN orbis lunæ secans eclipticam in G et æquatorem in N, LD circulus maximus perpendicularis in æquatorem; et sunto DN, LN, quadrantes circuli. Tempore dato vi prædictâ transferatur intersectio N in n, et describatur circulus Lgn exhibens situm orbis lunaris post illud tempus, secetque eclipticam in g. Ut autem intersectiones N et G sine verborum ambagibus distinguantur, priorem in posterum vocabo Nodum Æquatorium, posteriorem Nodum Eclipticum. Duætis itaque Nm, Gd, perpendicularibus in orbem lunæ, est Nn : Nm :: 1 : fin. GNA, et Nm : Gd :: 1 : fin. LG, itemque Gd : Gg :: fin. Ggd : 1; unde conjunctis rationibus provenit Nn : Gg :: fin. Ggd : fin. GNA × fin. LG, adeoque Gg = Nn × \frac{\text{fin. GNA} \times \text{fin. LG}}{\text{fin. Ggd}}. Scribantur s pro sinu et t pro cosinu anguli Ggd, inclinationis scilicet orbitæ lunaris ad eclipticam, ad radium 1, v pro sinu et u pro cosinu arcûs EG, p pro sinu et q pro cosinu obliquitatis eclipticæ; atque per resolutionem trianguli sphærici GAN, habebitur cos. GNA = n = qt + psu, indeque fin. GNA = \sqrt{1 - qtt - 2pqstu - p^2s^2u^2}; sed scribi potest 1 pro t, et rejici terminus p^2s^2u^2 ob exiguïatem finûs s anguli 5° 8′ 1⁄2, proindeque erit sin. GNA = \sqrt{pp - 2pqsu}; præterea est sin. GNA : sin. GA sive \(v\) :: sin. GAN sive \(p\) : sin. GN, ideoque sin. GN sive cos. LG = \frac{pv}{\text{fin. GNA}}, et fin. LG = \(u - \frac{qsvv}{p}\), ac fin. GNA × fin. LG = \(pu - qs\) quamproximé. Quare fit \(Gg = Nn \times \frac{pu - qs}{s}\), atque hic est motus nodorum lunarium tempore dato in plano eclipticæ: quod si tempus illud datum sit annus solaris, habetur \(Nn = \frac{3bcn}{5l^2} \times \frac{S}{L} \times 360°\), unde motus ille eclipticus nodorum annuus, nullà habitâ ratione mutationis sitûs nodorum ex aliâ causâ per id temporis factæ, fiet \(\frac{3bc}{5l^2} \times qt + psu \times \frac{pu - qs}{s} \times \frac{S}{L} \times 360°\), vel etiam \(\frac{3bcq}{5l^2} \times \frac{pu - qs}{s} \times \frac{S}{L} \times 360°\) proximé. Q. E. I. Quo motum nodi lunaris in hac propositione ad eclipticam reduximus, eodem profúsi ratiocinio motus nodi satellitis cujusvis ad orbitam planetæ primarii reducetur. Coroll. I. Exinde liquet nullum esse hunc motum nodi, ubi fin. LG = 0, vel etiam ubi \(pu = qs\), quod contingit: ubi orbitæ lunaris arcus GN eclipticam et æquatorem æqualis est 90°, sive ubi nodi lunares versantur in punctis declinationis lunaris maximæ, sive ubi arcus AG, cujus cosinus est \(u\), evadit æqualis 78° 5′; id est, ubi nodus ascendens lunæ versatur in 11° 55′ Cancri, vel 18° 5′ Sagittarii. Eritque progressivus hic motus, id est, fiet secundum feriem signorum, dum nodus ascendens lunæ transit retrocedendo ab 18° 5' Sagittarii ad 11° 55' Cancri, regressivus autem in reliquâ parte revolutionis; et maximus evadit motus regressivus, ubi \( u = -1 \), id est, ubi nodus ascendens versatur in principio Arietis; et maximus progressivus, ubi \( u = 1 \), id est, ubi idem nodus occupat initium Librae. Itaque cum motus ille nodorum annuus, de quo hic agitur, universaliter sit æqualis \( \frac{3bcq}{5l^2} \times \frac{pu - qs}{s} \times \frac{s}{L} \times 360^\circ \), hoc est, per Coroll. Prop. 2. æqualis \( 11''\frac{1}{4} \times \frac{pu - qs}{s} \) vel \( 15'' \times \frac{pu - qs}{s} \) prout differentia semiaxium terræ fuerit \( \frac{1}{279} \) vel \( \frac{1}{177} \), existentibus scilicet \( p \) sinu et \( q \) cosinu anguli \( 23^\circ 28'\frac{1}{4} \), atque \( s \) sinu anguli \( 5^\circ 8'\frac{1}{4} \); eo anno, in cujus medio circiter nodus lunæ ascendens tenuerit principium Arietis, motus nodorum regressivus, qui et maximus, erit \( 1' 2'' \) vel \( 1' 20'' \); ubi vero idem nodus subierit signum Librae, motus maximus progressivus erit \( 41'' \) vel \( 53'' \). In aliis nodorum positionibus eodem modo computabitur. Coroll. II. Si desideretur excessus regressus nodi supra progressum in integrâ nodi revolutione, sequenti ratione investigabitur. Jungantur equinoctia diametro EA, in quam demittatur perpendicularum GK, et sumpto arcu \( Gb \) quem describit nodus eclipticus \( G \) quo tempore nodus equatorius \( N \) describit arcum \( Nn \), ducatur \( bc \) perpendicularis in \( GK \). Per hanc propositionem est \( Gg \cdot Nn : : \frac{pu - qs}{s} \cdot 1 \), sive, quia est \( 1 \cdot u \) : : \( Gb \cdot Gc \), fit \( Gg \cdot Nn : : \frac{p \times Gc}{s} - q \times Gb \cdot Gb \); adeoque summa omnium \( Gg \) erit ad summam omnium nium Nn, hoc est, motus nodi ecliptici in integrâ sui revolutione erit ad motum nodi æquatorii eodem tempore factum, ut summa omnium in circulo quantitatum \( \frac{p \times Gc}{s} - q \times Gb \) ad summam totidem arcuum Gb, hoc est, ut \(-q\) ad 1. Signum autem — denotat motum fieri in antecedentia sive regressum nodi excedere ejusdem progressum. Unde cum motus nodi æquatorii N fit \(11''\frac{1}{2}\) vel \(15''\) quo tempore nodus eclipticus describit \(19^\circ 20'\frac{1}{2}\), motus ille nodi æquatorii tempore nodi ecliptici periodico evadit \(11''\frac{1}{2}\) \( \times \frac{360^\circ}{19^\circ 20'\frac{1}{2}} = 3' 34'' \) vel \(15'' \times \frac{360^\circ}{19^\circ 20'\frac{1}{2}} = 4' 39'' \); quo pacto prodit motus nodi ecliptici praefatus æqualis \(q \times 3' 34''\) vel \(q \times 4' 39''\), proindeque est radius ad cosinum obliquitatis eclipticæ ut \(3' 34''\) vel \(4' 39''\) ad motum quæsitum, nempe \(3' 16''\), existente \(\frac{1}{2} \frac{1}{9}\) differentiâ axium terræ, vel \(4' 16''\) eâ existente \(\frac{1}{77}\): atque hic est excessus regressûs nodi supra progressum in integrâ nodi revolutione vi prædictâ genitus. Excessu igitur hoc minuatur motus nodi lunaris periodicus \(360^\circ\), et remanebit motus ille quem generat vis folis. **PROPOSITIO IV.** **PROBLEMA.** Variationem inclinationis orbis lunaris ad planum eclipticæ ex figurâ terræ spheroidicâ ortam determinare. Esto ANH (Fig. 6.) æquator, AG ecliptica, et A punctum æquinoctii autumnalis: sit NGRM orbis lunæ secans eclipticam in G et æquatorem in N, in Vol. 50. quo sumantur arcus NL, GR, æquales quadrantibus circuli. Jam si nodus æquatorius N per temporis particulam vi prædictâ transferri intelligatur in n, et per punctum L describatur circulus n L r, exhibebit hic situm orbis lunæ post tempus elapsum, et si in eumdem demittantur perpendicula N m et R r, posteriorus R r designabit variationem inclinationis orbitæ lunaris ad eclipticam eodem tempore genitam. Est autem N n : N m :: 1 : m, itemque N m : R r :: 1 : fin. LR; sed ob NL = GR, est NG = LR; unde conjunctis rationibus est N n : R r :: 1 : m × fin. NG; ex quo patet variationem inclinationis momentaneam esse proportionalem sinui distantiæ nodi lunaris ecliptici à nodo æquatorio. Ad diametrum NM demittatur perpendiculum GK, et existente G b decremente arcûs NG facto quo tempore nodus æquatorius N describit arcum N n, agatur b k parallela ipsi GK, eritque 1 : GK sine fin. NG :: G b . K k; proindeque jam erit N n : R r :: G b : m × K k, adeoque summa omnium variationum R r, quo tempore nodus eclipticus G descriptit arcum MG, genitarum erit ad summam totidem motuum N n, hoc est, ad motum nodi æquatorii N eodem tempore factum, ut summa omnium K k ducta in m, ad summam totidem arcuum G b, id. est, ut m × MK ad MG. Sit NH motus nodi N tempore revolutionis nodi G ab uno equinoctio ad alterum, eritque variatio inclinationis eodem tempore genita, hoc est, variatio tota æqualis \( \frac{2m \times NH}{MGN} \). Unde cum \( \frac{NH}{MGN} \) exprimat rationem motûs nodi æquatorii ad motum nodi ecliptici, prodit theorema sequens: Est motus nodi lunaris ecliptici ad motum nodi æquatorii, ut sinus duplicatus inclinationis medior- cris orbitae lunaris ad æquatorem, ad sinum variationis totius inclinationis ejusdem orbitæ ad eclipticam. In hoc computo inclinationem mediocrem orbis lunaris ad æquatorem, nempe $23^\circ 28' \frac{1}{4}$, usurpo, cum in revolutione nodi tantum ex una parte augetur, quantum ex alterâ minuitur, et omnes minutias hic expendere supervacaneum foret. Motus autem nodi lunaris eclipsitici est ad motum nodi lunaris æquatorii ut $19^\circ 20' \frac{1}{4}$ ad $11'' \frac{1}{4}$ vel $15''$, five ut $6055$ vel $4642$ ad $1$, unde per theorema supra traditum prodit variatio inclinationis tota æqualis $27''$ vel $35''$, prout differentia axium terræ statuitur $\frac{1}{2} \frac{1}{29}$ vel $\frac{1}{177}$. Hac igitur quantitate augetur inclinationis orbis lunaris ad eclipticam in transitu nodi ascendentis lunæ ab æquinoctio vernali ad autumnale, et tantumdem minuitur in alterâ medietate revolutionis nodi. In loco quolibet G inter æquinoctia variatio inclinationis est ad variationem totam ut sinus versus arcûs MG ad diametrum, ut patet; five differentia inter semifissim variationis totius et variationem quæsitam est ad ipsam semifissim variationis totius ut cosinus arcûs MG ad radium, hoc est, ut $u = \frac{qswv}{p}$ ad $1$. Q.E.I. PROPOSITIO V. PROBLEMA. Motum apsidum in orbe satellitis quamproximè circulari, quatenus ex figurâ planetæ primarii sphæroïdicâ oritur, investigare. Per propositionem primam vis perturbatrix, quâ trahitur satelles ad centrum planetæ primarii, est ad satellitis gravitatem in ipsum primarium, ut $\frac{3bc}{5l^2} - \frac{9kbc}{5l^4}$ ad 1, sive, quia per Prop. 2. est $\frac{k}{l} = m \times SH$ (Fig. 4.) ponendo scilicet $m$ pro sinu inclinationis orbitae satellitis ad æquatorem primarii, et scribendo $y$ pro $SH$, ut $\frac{3bc}{5l^2} \times 1 - 3m^2y^2$ ad 1; et summa harum virium in totâ circumferentiâ cujus radius est 1, est ad gravitatem satellitis toties sumptam ut $\frac{3bc}{5l^2} \times 1 - \frac{3m^2}{2}$ ad 1. Vis igitur mediocris, quæ uniformiter agere in satellitem supponi potest, dum revolutionem suam in orbitâ propemodùm circulari absoluit, est ad ejus gravitatem in primarium ut $\frac{3bc}{5l^2} \times 1 - \frac{3m^2}{2}$ ad 1; atque hac vi movebuntur apsides, si nulla habeatur ratio vis alterius quæ orbis radio est perpendicularis et per medietatem revolutionis satellitis in unum sensum tendit, per alteram medietatem in contrarium. Jam quia ex demonstratis in hac et primâ propositione sequitur gravitatem satellitis circa planetam, cujus figura est sphæroïs oblata, revolventis in distantiâ $l$ generaliter esse ad ejusdem gravitatem in majori distantiâ $L$, ut $\frac{1}{l^2} + \frac{B}{l^4} \times 1 - \frac{3m^2}{2}$ ad $\frac{1}{L^2} + \frac{B}{L^4} \times 1 - \frac{3m^2}{2}$, existente $B$ quantitate datâ exigui valoris, sive ut $\frac{1}{l^2}$ ad $\frac{1}{L^2} - \frac{B}{l^2L^2} \times 1 - \frac{3m^2}{2} + \frac{B}{L^4} \times 1 - \frac{3m^2}{2}$ quamproximé, ideò gravitas satellitis diminuitur in majori quam duplicatâ ratione distantiæ auctæ quoties $m$ minor est quantitate $\sqrt{\frac{2}{3}}$, id est, ubi inclinatio orbitæ satellitis ad planetæ æquatorem non attingit $54^\circ$. 44'; diminuitur autem in minori ratione, quoties est \( m \) major quam \( \sqrt{\frac{2}{3}} \), id est, ubi illa inclinatio superat \( 54^\circ 44' \); adeoque in priore casu progrediuntur apsides orbis satellitis, in posteriori regrediuntur. Quantitas autem hujus progressus vel regressus sic innotescet. Per exemplum tertium prop. 45. lib. I. Princ. Math. Newt. si vi centripetæ, quæ est ut \( \frac{1}{r^2} \), addatur vis altera ut \( \frac{e}{r} \), hoc est, quæ sit ad vim centripetam \( \frac{1}{r^2} \) ut \( \frac{e}{r} \) ad 1, angulus revolutionis ab apside unâ ad eamdem erit \( 360^\circ \sqrt{\frac{1+e}{1-e}} \) vel \( \frac{360^\circ}{1-e} \) quamproximé, existente \( e \) quantitate valdé minutâ. Porrò cum fit motus satellitis in orbitâ suâ revolventis ad motum apsidis ut \( \frac{360^\circ}{1-e} \) ad \( \frac{360^\circ}{1-e} - 360^\circ \), hoc est, ut 1 ad \( e \), erit motus apsidis tempore revolutionis satellitis ad sidera æqualis \( 360^\circ \times e \), et hic motus apsidis erit ad ejusdem motum tempore alio quovis dato ut tempus periodicum satellitis ad tempus datum. Est autem in hac nostrâ propositione \( e = \frac{3bc}{5l^2} \times 1 - \frac{3m^2}{2} \); unde datur motus apsidum quæsitus. Q. E. I. Coroll. Si ad lunam referatur hæc determinatio, habebuntur \( b = 1 \), \( l = 60 \), \( m = \sin \text{ui anguli } 23^\circ 28' \frac{1}{4} \), et si fuerit \( c = \frac{1}{29} \), erit \( e = \frac{1}{1803203} \), atque motus apogæi lunæ spatio centum annorum æqualis 16' proximé in consequentia; si fuerit \( c = \frac{1}{77} \), erit \( e = \frac{1}{1393742} \), et motus apogæi æqualis 20', 7. Hac igitur quantitate minuendus est motus medius apogæi lunæ prout. prout observationibus determinatur, ut habeatur motus ille quem generat vis solis. Pro quarto autem Jovis satellite, erunt $b = 1$, $l = 25,299$, $c = \frac{1}{3}$, $m = \sin(\text{anguli} 3^\circ)$, $e = \frac{1}{39247}$; hincque motus apsidis spatio unius anni solaris prodit $33', 95$ vel ferè $34'$ in consequentia, qui tempore annorum decem fit $5^\circ 40'$. Insuper autem notandum est vi solis perturbari motum satellitis simili modo quo perturbatur motus lunæ; ideoque, quoniam vis solis, qua perturbatur motus lunæ est ad lunæ gravitatem in terram in duplicatâ ratione temporis periodici lunæ circa terram ad tempus periodicum terræ circa solem, hoc est, ut $1$ ad $178,725$; pariter vis solis, qua perturbatur motus satellitis Jovialis, est ad ipsius satellitis gravitatem in Jovem in duplicatâ ratione temporum periodicorum satellitis circa Jovem et Jovis circa solem, hoc est, ut $1$ ad $67394,6$: vires igitur, quibus perturbantur motus lunæ et satellitis, sunt ad se invicem, relativé ad eorum graviitates in planetas suos primarios ut $\frac{1}{178,725}$ ad $\frac{1}{67394,6}$, sive ut $37,708$ ad $1$. Unde cum viribus similibus proportionales sunt motus his viribus dato tempore geniti, si vis prior vel ejusdem vis pars quælibet motum apsidis generat æqualem $40^\circ 40'\frac{1}{4}$ in orbe lunari annuatim, vis posterior vel ejusdem pars similis et proportionalis motum apsidis eodem tempore generabit æqualem $6'\frac{1}{4}$ in orbe satellitis, atque decem annorum spatio $1^\circ 5'$ in consequentia. Addatur $1^\circ 5'$ ad $5^\circ 40'$, et motus apsidum totus in orbe satellitis extimi Joviali ex duabus prædictis causis oriundus spatio decem annorum erit $6^\circ 45'$ in consequentia. Observationibus Astronomicis collegit Ill. Bradleius hunc motum tempore prædicto esse quasi $6'$; differentia illa qualificumque liscumque $45'$ inter motum observatum et computatum actionibus satellitum interiorum debet ascribi. **SCHOLIUM.** Ex praecedentibus colligere licet motuum lunarium inaequalitates originem suam omnem non ducere ex vi solis, sed earum partem aliquam deberi actioni Telluris quatenus induitur figurâ sphæroidicâ. Sufficiat hic illarum computasse valorem, et legem, quâ generantur, demonstrasse: utrum autem hujusmodi correctiones tales sint ut tabulis Astronomicis inscribi mereantur, dijudicent Astronomi. Item manifestum est præter inaequalitates eas, quae in motibus satellitum Jovialium ex vi solis et actionibus satellitum in se invicem nascentur, oriri alias ex figurâ Jovis sphæroidicâ ita notabiles ut Observationes Astronomicas continuò afficere debeant. **De Variatione motûs Terræ diurni.** Si terra globus esset omnino sphæricus quicumque foret revolutionis axis, manente eadem in globo motûs quantitate, eadem maneret rotationis velocitas: secùs autem est, ubi ob vires solis et lunæ terra induit formam sphæroidis oblongæ per aquarum ascensum. Hic enim non considero figuram telluris oblatam ob materiæ in æquatore redundantiam, sed sphæricam suppono nisi quatenus per aquarum elevationem et depressionem in sphæroidicam mutatur. Jam verò in sphæroide hujusmodi, quamvis eadem maneat motûs quantitas, mutatâ inclinatione axis transversi ad axem revolutionis, mutabitur revolutionis velocitas, uti satis manifestum est: cum autem axis tranf- transversus transit semper per solem vel lunam, singulis momentis mutabit situm suum respectu axis revolutionis ob motum quo hi duo planetæ recedunt ab æquatore terrestrì et ad eum vicissim accedunt. **PROBLEMA.** *Variationem motûs terræ diurni ex prædictâ causâ oriundam investigare.* Exhibeat sphærois oblonga ADCd (Fig. 7.) terram fluidam, cujus centrum T, AC axis transversus jungens centra terræ et solis vel lunæ, Dd axis minor, EO diameter æquatoris, et XZ axis motûs diurni. Centro T et radio TD describatur circulus BDd secans axem transversum AC in B, et agatur BK perpendicularis in TE: tum ex quovis circuli puncto P ductâ PM ad axem XZ normali quæ secet TA in H, fit Pp r circumferentia circuli quam punctum P rota- tione suâ diurnâ descriptit, ad cujus quodvis punctum p ducatur Tp et producatur donec occurrat superficiei sphæroidis in q; deinde demissâ pG perpendiculari in PM, et GF perpendiculari in TA, si per puncta AqC transfire intelligatur ellipsis ellipsi ADC similis et æqualis, erit ex naturâ curvæ, quia sphærois nostra parûm admodûm differt à sphærâ, pq = AB × \(\frac{TF^2}{TP^2}\) quamproximé. Jam designet U velocitatem particulæ in terræ æquatore revolventis motu diurno circum axem XZ ad distantiam semidiametri TP, eritque \(U \times PM\) TP velocitas particulæ P circulum Pp r describen- tis, et cum sit TF = \(\frac{GM - HM \times TK}{TP} + TH\), erit motus motus totius lineolae \( pq \) æqualis \( pq \times \frac{U \times PM}{TP} = \frac{U \times AB \times PM}{TP^3} \times \frac{GM - HM \times TK^2}{TP} + TH \), adeoque summa horum motuum in circuitu circuli \( Ppr \), hoc est, motus superficiei inter circulum \( Ppr \) et sphæroidem in directione \( Tp \) contentæ, æquabitur circumferentia ejus circuli ductæ in \( \frac{U \times AB \times PM}{TP^3} \times \frac{TK^2 \times PM^2}{2 TP^2} + \frac{TK^2 \times HM^2}{TP^2} \) \[- \frac{2 TK \times HM \times TH}{TP} + TH^2 \] sive quia est \( HM \cdot TM \) :: \( TK \cdot BK \), et \( TH \cdot HM :: TP \cdot TK \), scribendo \( D \) pro circumferentia circuli \( BDd \), æquabitur ille motus quantitati \( \frac{U \times AB \times D}{2 TP^6} \times \frac{TK^2 \times PM^4 + 2 BK^2 \times TM^2 \times PM^2}{TP^2} \). Deinde horum motuum summa in toto circuitu globi collecta, hoc est, motus totius materiae globo \( BDd \) incumbentis prodibit æqualis \( \frac{U \times AB \times DD}{32} \times \frac{3 TP^2 - BK^2}{TP^2} \). Ubi planeta in plano æquatoris consistit, fit \( BK = 0 \), et motus praedictus æqualis \( \frac{U \times 3AB \times DD}{32} \). Motus autem globi \( QPR \) circa eundem axem est (uti facilé demonstratur) \( \frac{U \times TP \times DD}{16} \), adeoque motus terræ totius fit \( \frac{U \times TP \times DD}{16} + \frac{U \times AB \times DD}{32} \times \frac{3 TP^2 - BK^2}{TP^2} \), qui cum idem semper manere debeat, denotet \( V \) velocitatem in superficie æquatoris terrestris ubi planeta versatur in plano æquatoris, eritque \( \frac{U \times TP \times DD}{16} + \frac{U \times 3AB \times DD}{32} = \) \[ \frac{U \times TP \times DD}{16} + \frac{U \times AB \times DD}{32} \times \frac{3TP^2 - BK^2}{TP^2}; \text{ unde scribendo } 1 \text{ pro } TP \text{ quatenus est radius ad sinum } BK \text{ anguli BTK, habetur } V.U :: TP + \frac{3AB}{2} - \frac{AB \times BK^2}{2}. TP + \frac{3AB}{2}, \text{ indeque, quia minima est altitudo } AB \text{ respectu semidiametri } TP, U - V.V :: AB \times BK^2. 2TP, \text{ et } U - V = V \times \frac{AB \times BK^2}{2TP}: \text{ pro } V \text{ autem patet scribi posse velocitatem angularem terrae mediocrem quia ab ea differt quam minimae et ducitur in quantitatem perexiguam } \frac{AB \times BK^2}{2TP}, \text{ et quia tempora revolutionum terrae circa centrum suum sint reciproce ut motus angularis } U, V, \text{ fiet differentia revolutionum terrae ubi planeta aequatore tenet et ubi ab aequatore distat angulo BTK, aequalis } 23^\circ 56' \times \frac{AB \times BK^2}{2TP}. \text{ Quoniam igitur est acceleratio horaria ad motum terrae horarium mediocrem circa centrum suum ut } AB \times BK^2 \text{ ad } 2TP \text{ sive (quia est sinus inclinationis eclipticae ad aequatorem ad radium } 1 \text{ ut sinus } BK \text{ ad sinum distantiae planetae ab aequinoctio, quem sinum dico } K) \text{ ut } AB \times p^2 \times K^2 \text{ ad } 2TP; \text{ adeoque acceleratio horaria rotationis terrae crescit in ratione duplicata finis distantiae planetae a puncto aequinoctii, et summa omnium illarum acceleratarum, quo tempore transit planeta ab aequinoctio ad solstitium, est ad summam totidem motuum horariorum mediocrium, hoc est, acceleratio tota eo tempore genita est ad tempus illud ut summa quantitatum omnium } AB \times p^2 \times K^2 \text{ in circuli quadrante ad summam.} mam totidem \(2 \text{ TP}\), id est, quia summa omnium \(K^2\) in circuli quadrante dimidium est summæ totidem quadratorum radii, ut \(AB \times p^2\) ad \(4 \text{ TP}\). Quamobrem, si denotet \(P\) quartam partem temporis planetæ periodici circa terram, erit acceleratio tota motûs terræ circum axem suum in transitu planetæ ab æquinoctio ad solstitium genita æqualis \(\frac{AB \times P \times p^2}{4 \text{ TP}}\), atque eadem erit retardatio in transitu planetæ à solsticio ad æquinoctium. Unde sponte nascitur hoc Theoremum: Est quadratum diametri ad quadratum sinús obliquitatis eclipticæ ut quarta pars temporis periodici solis vel lunæ ad tempus aliud; deinde, est semidiameeter terræ ad differentiam semiaxium ut tempus mox inventum ad accelerationem quæsitam. Ascensus aquæ \(AB\) vi solis debitus est duorum pedum circiter, existente semidiametro terræ mediocri \(TP = 19615800\), unde prodit per theorema acceleratio terræ circa centrum suum gyrantis facta quo tempore incedit sol ab æquinoctio ad solstitium, æqualis \(1''55^{iv}\) in partibus temporis; et si vi lunæ ascendent aquæ ad altitudinem octo pedum, acceleratio revolutionis terræ inde orta, quo tempore luna transit ab æquatore ad declinationem suam maximam, erit \(34^{iv}\): et summa harum accelerationum, quæ obtinet ubi hi duo planetæ in punctis solstitialibus versantur, cum non superet duo minuta tertia temporis cum semisse fīve \(37\) minuta tertia gradûs, vix observabilis erit. Q.E.I. Cùm igitur tantilla sit hujusmodi variatio in hypothesi sphæricitatis terræ; qualis evaderet, terrâ existente sphæroide oblatâ, frustrà quis inquireret.