Chapter X: Part 10

I have three other coins of Sidon[169], of almost intirely the same type; only one of them exhibits a date in Greek numerals, and two bear Phœnician dates. The Greek numerals are EOT, CCCLXXV; and the Phœnician correspond with the numbers CXX, CXXVII, to both of which are prefixed the above-mentioned initial letters. We meet with draughts of two similar medals in[170] Arigoni, adorned with characters, expressing the numbers CXXVIII, CXXX. All these coins present to our view a turrited head and a branch of palm, pointing out to us the country to which they belong, and on the reverse the usual symbol of Sidon. The year handed down to us by the Greek date EOT, is the 375th of the æra of Seleucus; and those denoted by the Phœnician numerals answer to the 120th, 127th, 128th, and 130th, of the proper æra of Sidon, as will be hereafter more fully evinced. Hence we may certainly collect, that these pieces were struck at Sidon in the years of CHRIST 11, 18, 19, 21, and 64.

III.

Three coins of Sidon, different from the former, occur in[171] Sig. Haym, and seven[172] more in my little cabinet, whose type is altogether the same, with Phœnician dates, preceded by the two aforesaid initial letters, upon them. To which we may add five, preserved in the noble[173] cabinet bequeathed to Christ-Church, Oxon. by Archbishop Wake, and another in the valuable collection of the Rev. Dr. Barton[174], Canon of the said collegiate church, and a worthy member of this Society. On one side these medals all exhibit the head of Jupiter, and on the reverse the prow of a ship, the common symbol of Sidon. Most of them had various Phœnician letters at first imprest on the upper part of the reverse, and one of them (which is pretty remarkable) nearly the same characters there that appear in the exergue. The first of the coins mentioned here was struck in the year of Sidon 5. This has been perfectly well preserved, and is more curious than any of the rest; which were emitted from the mint at Sidon in various years of the proper æra of that city, _viz._ the 107th, 108th, 110th, 111th, 112th, 114th, 115th, 116th, 117th, and 119th. We meet on none of these medals with the figure denoting TWENTY, used by the Sidonians, during the period I am now upon. It not a little resembles that which prevailed at Tadmor[175] in the reign of the emperor Claudius, about forty-nine years after the birth of CHRIST. The most antient of the Phœnician coins I am now considering preceded the commencement of the Christian æra 104 years, and is consequently 153 years older than the earliest Palmyrene inscription that has hitherto come to our hands[176].

IV.

Some years since I published a small brass medal of Sidon[177], with the heads of Jupiter and Juno on one side, and the prow of a ship on the reverse; but did not accurately enough describe the numeral characters, and two initial letters, in the exergue. I therefore take the liberty to send[178] you a new draught, perfectly well done, of that inscription. Two more coins of the same type I have since acquired, and another may be seen in[179] Sig. Haym. These four pieces only exhibit the years of Sidon 125 and 132.

V.

My small collection likewise affords two[180] other Phœnician medals of Sidon,[181] and Archbishop Wake’s noble cabinet one, of the same type, with different Phœnician dates in the exergue. To these may be added five, with the publication of which the learned world has been obliged by Sig. Arigoni[182]. The anterior faces of these coins are adorned with a veiled head, representing the genius of the city wherein they were struck; and the reverses with a human figure leaning upon a pillar, and holding a branch of palm in its right hand. Several Phœnician letters also there appear, which may perhaps at first sight seem to render it somewhat doubtful, whether the medals belong to Sidon or not. But every suspicion arising from hence must immediately vanish, when we cast our eyes upon the two initial elements, and the numeral characters, in the exergue; which clearly enough indicate the pieces to have been struck at Sidon, in the 83d, 87th, 95th, 105th, 106th, 108th, 114th, and 116th years of the æra peculiar to that city. A Phœnician coin of Sidon likewise occurs in one[183] of Sig. Arigoni’s plates, and another[184] in my collection, with the turrited head and branch of palm visible on three of the[185] medals above described, which indisputably appertain to that city, together with the very Phœnician letters and symbol imprest on the Sidonian coins now before me. This, exclusive of other considerations, that might be offered, must set the point I am here insisting upon beyond dispute.

VI.

I have another brass Phœnician medal of Sidon[186], not a little resembling those above-mentioned, both in workmanship and size, presenting to our view on one side the head of Jupiter, and on the other a human figure with a lance in its right hand. This coin, which has never yet been published, is adorned with a Phœnician legend on the reverse, different from those of all the others that have hitherto appeared. I therefore judged that a draught of it would not be unacceptable, though the date imprest originally in the exergue (answering to the 26th year of Sidon) has a little suffered from the injuries of time.

VII.

The next Phœnician medal of Sidon, which I shall take the liberty here to describe, is a small brass one[187], now in my hands, with a veiled head on the anterior face, and the prow of a ship on the reverse. M. Bouterouë[188], who has published it, rightly asserts it to be a Phœnician coin. The year of Sidon, preserved in the exergue of mine, is 74; and that in the exergue of M. Bouterouë’s, 73, though the first numeral character of the latter is somewhat deformed.

VIII.

The last Phœnician medals I shall at present produce, in order to settle the point in view, are[189] two in my possession, intirely agreeing both in type and form, as remarkable as any of the others here touched upon. A similar coin has been published by Sig. Arigoni[190], and another[191] by M. Bouterouë; both of which, on several accounts, merit the attention of the learned. They exhibit on one side the head of Jupiter laureated, with a beard; and on the reverse a double cornucopia, together with three or four Phœnician elements, one or two of which are in a great measure defaced. A brass medal of Sidon occurs in Archbishop Wake’s[192] collection, as well as one in[193] mine, with the head of Jupiter done exactly after the same manner as that on the pieces before me, and Europa carried by a bull on the reverse; which, exclusive of the inscriptions in the exergue, demonstrate the latter to belong to Sidon. The first of mine was struck in the 143d year of the proper æra of that city, and the second five years after. They correct the barbarous date assigned by Sig. Arigoni to his coin. M. Bouterouë has not favoured the learned world with an explication of the medal, of which he has given us a draught. Nor has M. l’Abbé Barthelemy, who likewise mentions this very coin, informed us to what place it appertains; but contented himself with barely[194] observing, that the letters preserved on the reverse are Phœnician. I flatter myself therefore that I shall not be charged with plagiarism by this celebrated antiquary, in case what is here submitted to the consideration of the Royal Society should be so happy as to meet with the approbation of that learned and illustrious body; not even by _only_ acquainting the public, with a sort of _politesse_ so peculiar to his countrymen, that it is now become one of the most distinguishing characteristics of their nation[195], “that a certain Oxford doctor has done him the honour to _adopt_ the explication he had given.”

IX.

For the farther illustration of what has been here advanced, it will be requisite to observe, that two æra’s were antiently followed at Sidon; the æra of Seleucus, and another peculiar to the inhabitants of that city[196]. On the Greek brass coins of Sidon, according to F. Frœlich[197], both these epochs seem to have been used. However, the supputation pointed out to us by the date on the Greek medal above-mentioned was undoubtedly made according to the æra of Seleucus; since otherwise the year exhibited by that date must have been nearly coincident with the 266th of CHRIST, which by those versed in this kind of literature will never be allowed. For had the piece presented to our view so recent a date, as Sidon first became a Roman colony in the reign of Elagabalus[198], above forty years before; the reverse ought to have been adorned with some other letters intimating this, as were those of the Sidonian[199] coins posterior to that event. As certain is it that all the Phœnician medals of Sidon, whose numeral characters have been interpreted here, acknowledge no other epoch than the proper one of that city, which commenced in the year[200] of Rome 643. This, I flatter myself, from the following considerations, exclusive of others that might, with equal facility, be offered, will even to demonstration appear.

1. The fifth year mentioned by the oldest of these coins cannot be the fifth year of the æra of Seleucus, because the Sidonians were then subject to Antigonus[201], in whose territories the supputation according to that epoch did not take place; and consequently the piece itself must have been struck in the fifth year of the proper æra of Sidon, nearly coincident with the 648th of Rome[202].

2. No dates ever occurred upon the medals of the Syrian kings presiding over the people of Sidon, either to F. Frœlich or Dr. Vaillant[203], who have so eminently distinguished themselves in this branch of literature, before the year of Seleucus 112; and therefore neither the Phœnician dates preserved on the aforesaid Sidonian coins whose numeral characters do not amount to 112, nor the Greek dates on others falling short of that number, can rationally be supposed to bear any relation to the æra of that prince. This certainly must be considered as a strong presumption, or rather an incontestable proof, that the last-mentioned Phœnician dates were deduced from the commencement of the proper Sidonian epoch, as from their genuine cardinal point. Which reasoning will by analogy extend, as the numeral characters exhibited by all the coins here explained are of the same kind, to every one of the rest.

3. None of the medals of the Syrian kings, with Phœnician letters upon them[204], hitherto published, bear any Phœnician dates. This, after what has been said, renders it extremely probable, that the pieces of Sidon I am considering were posterior to those coins; and even that their Phœnician dates referred to an æra different from that of Seleucus, followed by the Greek dates on the medals of the Syrian kings. Which if we admit, this æra could have been no other than the new one of the Sidonians, that commenced in the seventh century of Rome.

4. That the dates visible on these coins were supputed according to the latter epoch of Sidon, will be manifest from an examination of the Greek and Phœnician brass medals of that city explained, in[205] the beginning of this paper; whose type and workmanship are extremely similar, if not almost intirely the same. For this circumstance is to me an evident proof, that they could not have been struck at very distant times. Now if we take the Greek coin to have followed the æra of Seleucus, as was undoubtedly the case, and the others that peculiar to Sidon; the first of the Phœnician dates[206] will not be prior to the Greek one above fifty-three years, nor the last of them precede it above forty-three years. Whereas if we suppose the numeral inscriptions in the exergues of the Phœnician Sidonian coins to have been supputed according to the Seleucian epoch, the difference between the aforesaid dates will be five times as much; which with the similarity of workmanship and type, already observed, will be altogether incompatible.

5. As the Jews[207], about the time that the first of our medals was struck, denominated the æra of Seleucus, THE ÆRA OF THE KINGDOM OF THE GREEKS; we cannot well doubt but it went amongst the Sidonians, who were neighbours to the Jews, under the same denomination. From whence it will follow, that the epoch styled by them emphatically, THE ÆRA OF SIDON, must have been different from the æra of Seleucus; and consequently that which, after the 643d year of Rome, was peculiar to them.

PHŒNICIAN Numerals antiently used at SIDON, from _One_ to a _Thousand_.

_J. Mynde sc._]

The powers of the Phœnician numeral characters antiently used at Sidon, which I flatter myself are now discovered, having been for many ages unknown; the Society will perhaps not be displeased to see accurate draughts of the principal Phœnician medals, from whence they are deduced. I have therefore taken the liberty to transmit them[208] such draughts, which may be intirely depended upon. I have also constructed a table[209] of the numeral characters themselves, from UNITY TO A THOUSAND; which will demonstrate, in the clearest manner possible, the great affinity between them and those of the Palmyrenes.

1. From this table it plainly appears, that the people of Sidon had no particular character to denote Five, whilst the Phœnician numerals here explained were in vogue amongst them; that they expressed TWENTY by a character, during that period, not very different from the correspondent one used at Tadmor; and that in all other respects the Phœnician notation then prevailing at Sidon was, in a manner, the same with that of the[210] Palmyrenes.

2. It may not be improper to observe, that two of the Sidonian coins I have been considering[211] exhibit the Phœnician word מא, equivalent to the Hebrew מאה, and Syriac מאא, AN HUNDRED, instead of the centenary numeral character. This, in conjunction with the appearance of that character, occupying the very place of the term אמ, on others of those coins, first induced me to believe, that the inscription preserved by every one of them in the exergue could be nothing else but a date.

3. I shall beg leave farther to remark, that none of the indubitable medals of Tyre, adorned with Phœnician letters, as far as I have been able to discover, present to our view any Phœnician dates at all. This still more clearly evinces the second element prefixed to the Phœnician numerals in the exergue to point out to us the city of Sidon, and not that of Tyre; which[212], indeed, seems already to have been sufficiently proved.

4. From the foregoing observations we may likewise collect, that the coin assigned to Demetrius III. by Mr. Masson, F. Frœlich[213], and Sig. Haym, exhibiting a Phœnician legend, without a Phœnician date, in the exergue, ought in reality to be attributed to Demetrius I. Those three learned men therefore have been guilty of a mistake in this particular. Nor can the head on this medal be denied to bear some resemblance to that of Demetrius I.[214] with a moderate beard, as it appears on a coin published by Dr. Vaillant, and in one of F. Frœlich’s plates. That the letters A K, behind the head, indicate the piece to have been struck in the twenty-first year of the proper Sidonian æra[215], as Mr. Masson and F. Frœlich are pleased to assert, can never be proved. On the contrary, the improbability of such a notion may be inferred from two similar letters, behind the turrited head of the _Dea Syria_[216], on a Phœnician coin, which Mr. Masson makes to point out the forty-first year of the proper epoch of Sidon; whereas, in truth, that piece seems to have been struck either in the reign of Demetrius I. or Antiochus IV.[217] many years before. Nay, that it was actually struck when Demetrius I. sat upon the Syrian throne, is rendered almost incontestable by a medal of that prince now in my possession, with a _Beta_ behind the head on the anterior part, and the very reverse of the last-mentioned coin. From the former of which circumstances it farther appears, that the alphabetic characters MA, supposed by Mr. Masson to denote 41, are by no means to be taken for a date. To which we may add, that the head on a Phœnician medal, with the two Greek elements AK behind it, published by Mr. Reland[218], is apparently that of Demetrius I.; and that the posterior part of this coin is nearly the same, in all respects, with the reverse of that supposed to[219] appertain to Demetrius III. by Mr. Masson and Sig. Haym. But to wave all other considerations, relative to the point in view, that may occur, the features and turns of the face on the medals of Demetrius III. are so different[220], that no inference of any validity can be drawn from the pretended identity or similitude of them, in support of Mr. Masson’s opinion.

5. The Palmyrene and Phœnician numerals, deduced from coins and inscriptions, may perhaps be thought not unworthy a place amongst the arithmetical characters of various nations, formerly[221] collected by Bishop Beveridge; and consequently may be allowed to render somewhat more complete the chronological institutions, or rather the chronological arithmetic, of that learned and judicious author.

You will pardon the prolixity of this letter, which the novelty of the subject may perhaps render a little more excusable than it would otherwise have been; and believe me to be, with the most perfect consideration and esteem,

SIR,
Your most obedient humble Servant,
J. Swinton.

Christ Church, Oxon. Nov. 17. 1758.

_J. Mynde sc._]

CX. _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.

[Read Dec. 14, 1758.]

Reverend Sir,

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 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 spheroidical, 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 spheroidical 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 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 equatoreal 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 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,
C. Walmesley.

Bath, Oct. 21. 1758.

LEMMA I.

_Invenire gravitatem corporis longinqui ad circumferentiam circuli ex particulis materiæ in duplicatâ ratione distantiarum inversè attrahentibus constantem._

ESTO NIK (_Vid._ TAB. xxxiii. _Fig._ 1.) circumferentia circuli, in cujus puncta omnia gravitet corpus longinquum S locatum extra planum circuli. In hoc planum agatur linea perpendicularis SH, et per circuli centrum X ducatur recta HXK secans circulum in I et K, et SR parallela ad HX: producatur autem SH ad distantiam datam SD, et agantur rectæ 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 sive DC, _h_; XL, _x_; CX, _z_; XI, _r_; eritque HL = _h_ - _x_, et SH = _k_ - _z_. Est autem SM ad SH ut attractio (1 ⁄ (SM)²) corporis S versus particulam M in directione SM ad ejusdem corporis attractionem in directione SH, quæ proinde erit SH ⁄ (SM)³: sed est SR = HL, et (SM)² = (SR)² + (MR)² = (SR)² + (SH)² + (ML)²; unde sit SH ⁄ (SM)³ = SH ⁄ ((HL)² + (SH)² + (ML)²⁽³⁄²⁾), et ductâ _mn_ parallelâ ad MN, vis qua corpus S attrahitur ad arcus quàm minimos M_m_, N_n_, exponitur per (SH × 2M_m_) ⁄ (SM)³ = SH × 2M_m_ × ((HL)² + (SH)² + (ML)²⁽⁻³⁄²⁾). Est autem (HL)² + (SH)² + (ML)² = _kk_ - 2_kz_ + _zz_ + _hh_ - 2_hx_ + _rr_, hincque ponendo _kk_ + _hh_ = _ll_, ((HL)² + (SH)² = (ML)²)⁽⁻³⁄²⁾ = (1 ⁄ _l_³) + (3_kz_ ⁄ _l_⁵) + (3_hx_ ⁄ _l_⁵) - (3_rr_ ⁄ 2_l_⁵) - (3_zz_ ⁄ 2_l_⁵) + (15_kkzz_ ⁄ 2_l_⁷) + (15_khzx_ ⁄ 2_l_⁷) + (15_hhxx_ ⁄ 2_l_⁷), neglectis terminis ulterioribus ob longinquitatem quam supponimus corporis S. Quarè, si scribatur _d_ pro circumferentiâ IMKN, gravitas corporis S ad totam illam circumferentiam secundum SH, sive fluens fluxionis SH × 2M_m_ × ((HL)² + (SH)² + (ML)²)⁽⁻³⁄²⁾ evadit (_k_ - _z_) × _d_ in (1 ⁄ _l_³) + (3_kz_ ⁄ _l_⁵) - (3_rr_ ⁄ 2_l_⁵) - (3_zz_ ⁄ 2_l_⁵) + (15 _kkzz_) ⁄ (2 _l_⁷) + (15 _hhrr_) ⁄ (4 _l_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 secet IK in _i_, eritque gravitas in directione SD, qua urgetur corpus S versus materiam sitam inter circumferentiam IMKN et circumferentiam centro X et radio X_i_ descriptam, æqualis gravitati in lemmate præcedenti definitæ ductæ in rectam I_i_. Sed est I_i_. _c_∷ IX. _a_, atque _d_. D∷ IX. _a_; unde I_i_ × _d_. D × _c_∷ (IX)². _aa_, hoc est, ex naturâ ellipseos, ob CX = _z_, et IX = _r_, I_i_ × _d_. D × _c_∷ _bb_ - _zz_. _bb_, adeoque I_i_ × _d_ = (D × _c_) ⁄ (_bb_) × (_bb_ - _zz_), atque _rr_ = _aa_ - (_aazz_) ⁄ (_bb_); scribi autem potest in sequenti calculo _bb_ - _zz_ pro _rr_ ob parvitatem differentiæ semiaxium in quam omnes termini ducuntur. Gravitas igitur corporis S in materiam inter circumferentias supradictas consistentem exprimetur per (D × _c_) ⁄ (_bb_) × (_bb_ - _zz_) × (_k_ - _z_) in 1 ⁄ _l_³ + (3_kz_) ⁄ _l_⁵ - (3_bb_) ⁄ (2_l_⁵) - (15_zz_) ⁄ (4_l_⁵) + (15_bbhh_) ⁄ (4_l_⁷) + (45_kkzz_) ⁄ (4_l_⁷). Et si addatur gravitas in similem materiam ex alterâ parte centri C ad æqualem à centro distantiam, quia tunc CX sive _z_ evadit negativa, gravitas corporis S in hanc duplicem materiam erit (D × _c_) ⁄ _bb_ × (_bb_ - _zz_) in 2_k_ ⁄ _l_³ - 6_kzz_ ⁄ _l_⁵ - 3_kbb_ ⁄ _l_⁵ + 15_k_³_zz_ ⁄ _l_⁷ + 15_hhkbb_ ⁄ 2_l_⁷ - 15_hhkzz_ ⁄ 2_l_⁷. Ducatur jam gravitas hæc in _ż_, et sumptâ gravitatum omnium summâ, factâ _z_ = _b_, gravitatio tota corporis S in totam materiam globo interiori superiorem secundum directionem SD æquatori perpendicularem prodit (D × _c_) × (4_kb_ ⁄ 3_l_³ - 4_kb_³ ⁄ 5_l_⁵ + 2_khhb_³ ⁄ _l_⁷). Simili ratiocinio gravitatio corporis S in eamdem materiam secundum directionem SR æquatori parallelam invenitur æqualis D × _c_ × (4_hb_ ⁄ 3_l_³ + 2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷). Tum si addatur gravitatio corporis S in globum interiorem, ex unâ parte scilicet 2_b_³_k_D ⁄ 3_al_³, et ex alterâ 2_b_³_h_D ⁄ 3_al_³, habebitur gravitas corporis S in totum sphæroidem. _Q. E. I._

COROLL.

Igitur gravitas corporis S secundum SD est ad ejusdem gravitatem secundum SR sive DC in materiam sphæroidis globo interiori incumbentem ut 2_k_ ⁄ 3 - 2_kb_² ⁄ 5_l_² + _khhb_² ⁄ _l_⁴ ad 2_h_ ⁄ 3 + _hb_² ⁄ 5_l_² - _hkkb_² ⁄ _l_⁴, adeoque si gravitas prior exponatur per _k_, posterior exprimetur per _h_ - 3_hb_² ⁄ 5_l_² quamproximè. Unde cum sit DC = _h_, patet gravitatem corporis S in sphæroidem oblatam non tendere ad centrum C, sed ad punctum _c_ rectæ 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 tanquàm primarium gyrantem. Quantitas materiæ globo sphæroidis interiori incumbentis æqualis est 4_bbc_D ⁄ 3_a_ sive 4_bc_D ⁄ 3 proximè, et si materia illa locaretur in centro sphæroidis C, attraheret satellitem S secundum SC vi 4_bc_D ⁄ 3_l_², quæ reducta ad directionem SD fit 4_bck_D ⁄ 3_l_³, et ad directionem DC fit 4_bch_D ⁄ 3_l_³. Cum igitur vis 4_bc_D ⁄ 3_l_² non turbat motum satellitis, utpote quæ tendat ad centrum motûs et quadrato distantiæ ab eodem centro sit reciprocè proportionalis, vires illæ 4_bck_D ⁄ 3_l_³, 4_bch_D ⁄ 3_l_³, in quas resolvitur, etiam motum non turbabunt. Itaque ex vi D × _c_ × (4_kb_ ⁄ 3_l_³ - 4_kb_³ ⁄ 5_l_⁵ + 2_khhb_³ ⁄ _l_⁷) auferatur vis 4_bck_D ⁄ 3_l_³, et ex vi D × _c_ × (4_hb_ ⁄ 3_l_³ + 2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷) auferatur 4_bch_D ⁄ 3_l_³, et remanebunt vires D × _c_ × - (4_kb_³ ⁄ 5_l_⁵ + 2_khhb_³ ⁄ _l_⁷), D × _c_ × (2_hb_³ ⁄ 5_l_⁵ - 2_hkkb_³ ⁄ _l_⁷), motuum satellitis S perturbatrices. Designetur vis D × _c_ × (2_hb_³ ⁄ 5_l_⁵ - 2_hhkb_³ ⁄ _l_⁷) per rectam S_r_ (_Fig. 2._) ac resolvatur in vim S_q_ tendentem ad centrum planetæ primarii C et ob triangula similia S_rq_, SDC, æqualem D × _c_ × (2_b_³ ⁄ 5_l_⁴ - 2_kkb_³ ⁄ _l_⁶), existentibus ut priùs, SD = _k_, DC = _h_, SC = _l_; et in vim _rq_ rectæ SD parallelam et æqualem D × _c_ × (2_kb_³ ⁄ 5_l_⁵ - 2_k_³_b_³ ⁄ _l_⁷); atque hæc vis posterior subducta ex vi D × _c_ × - (4_kb_³ ⁄ 5_l_⁵ + 2_khhb_³⁄_l_⁷) relinquet D × _c_ × 4_kb_³ ⁄ 5_l_⁵ pro vi perturbatrice in directione SD. Unde cum massa tota planetæ sit 2_ab_D ⁄ 3, gravitas satellitis tota in planetam erit 2_ab_D ⁄ 3_l_² proximé, vel etiam 2_bb_D ⁄ 3_l_², et hæc gravitas est ad vim D × _c_ × 4_kb_³ ⁄ 5_l_⁵ ut 1 ad 6_kbc_ ⁄ 5_l_³.

Deinde vis illius D × _c_ × 4_kb_³ ⁄ 5_l_⁵ secundum SD pars ea quæ agit in directione SC est D × _c_ × 4_kkb_³ ⁄ 5_l_⁶, quæ addita vi Sq dat D × _c_ × (2_b_³ ⁄ 5_l_⁴ - 6_kkb_³ ⁄ 5_l_⁶) vim perturbatricem tendentem ad centrum planetæ primarii, atque hæc vis est ad satellitis gravitatem 2_bb_D ⁄ 3_l_² in primarium ut 3_bc_ ⁄ 5_l_² - 9_kkbc_ ⁄ 5_l_⁴ ad 1. _Q. E. I._

COROLL.

Designet CK (_Fig._ 3.) lineam intersectionis planorum æquatoris planetæ et orbitæ satellitis, et resolvatur vis SD = 6_kbc_ ⁄ 5_l_³, 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 proptereà ob similia triangula SDK, SRD, si _m_ denotet sinum ad radium 1 et _n_ cosinum anguli SKD, inclinationis scilicet orbitæ satellitis ad æquatorem planetæ, erit DR = SD × _n_ = 6_kbcn_ ⁄ 5_l_³, et SR = SD × _m_ = 6_kbcm_ ⁄ 5_l_³, existente 1 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 prædictâ 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 S_r_ æqualis duplo spatio, quod satelles percurrere posset impellente vi DR in Coroll. præced. determinatâ, quo tempore in orbe suo describeret arcum quàm minimum _p_S: per puncta _r_, _p_, describatur centro C circulus _rpn_ secans equatorem in _n_, qui exhibebit situm orbitæ satellitis post illam particulam temporis, nodo N translato in _n_. Agantur SC, CN, et SH perpendicularis in lineam nodorum CN, et N_m_ perpendicularis in _rpn_. Jam cum sint lineolæ S_r_, N_m_, ut sinus arcuum S_p_, SN, erit S_p_. S_r_∷ SH. N_m_; deinde in triangulo rectangulo N_mn_ habetur _m_. 1∷ N_m_. N_n_; unde per compositionem rationum S_p_ × _m_. S_r_∷ SH. N_n_ = ((S_r_ × SH) ⁄ (S_p_ × _m_)): dato igitur arcu S_p_, est N_n_ sive motus nodi ut S_r_ × SH. In triangulo sphærico rectangulo SFN est sinus anguli N, hoc est, anguli inclinationis orbitæ satellitis ad æquatorem planetæ, ad sinum arcûs SF, ut radius ad sinum arcûs SN, id est, _m_. (_k ⁄ l_)∷ 1. SH, adeoque _k ⁄ l_ = _m_ × SH; est igitur _k ⁄ l_ ut SH. Vis autem S_r_ per Coroll. Prop. præced. est ut _k ⁄ l_, adeoque ut SH; quamobrem est S_r_ × SH, proindeque et N_n_, ut (SH)², hoc est, motus horarius nodi vi præfatâ genitus est in duplicatâ ratione distantiæ satellitis à nodo. Et quoniam summa omnium (SH)², quo tempore satelles periodum suam absolvit, est dimidium summæ totidem (SC)², ideò motus periodicus est subduplus ejus qui, si satelles in declinatione suâ maximâ ab æquatore planetæ continuò perstaret, eodem tempore generari posset. Sit igitur satelles in maximâ suâ declinatione sive in quadraturâ cum nodo, eritque SN quadrans circuli, et N_m_ mensura anguli N_pm_ sive S_pr_, eritque in hoc casu N_n_ sive motus horarius nodi ad N_m_, hoc est, ad angulum S_pr_, ut 1 ad _m_; est autem angulus S_pr_ ad duplum angulum, quem subtendit sinus versus arcûs S_p_ satellitis gravitate in primarium eodem tempore descripti, id est, ad angulum SC_p_ qui est motus horarius satellitis circa primarium, ut vis S_r_ ad gravitatem satellitis in primarium, hoc est (per Coroll. Prop. I.), ut (6_kbcn_) ⁄ 5_l_³ ad 1, sive, quia est in hoc casu _k_ ⁄ _l_ = _m_, ut (6_bcmn_) ⁄ 5_l_² ad 1. Unde conjunctis rationibus est motus horarius nodi ad motum horarium satellitis ut (6_bcn_) ⁄ 5_l_² ad 1; et si S denotet tempus periodicum solis apparens, et L tempus periodicum satellitis circa primarium suum, cum sit motus horarius satellitis ad motum horarium solis ut S ad L, erit motus horarius nodi ad motum horarium solis ut (6_bcn_) ⁄ 5_l_² × S ⁄ L ad 1, et in eadem ratione erit motus nodi annuus ad motum solis annuum, hoc est, ad 360°. Quarè, si satelles maneret toto anno in maximâ suâ declinatione ab æquatore primarii, vis prædicta ex figurâ sphæroidicâ planetæ primarii proveniens generaret eodem tempore motum nodi æqualem (6_bcn_) ⁄ 5_l_² × S ⁄ L × 360°, et ex supradictis motus verus nodi annuus erit hujus subduplus, nempe (3_bcn_) ⁄ 5_l_² × S ⁄ L × 360°. _Q. E. I._

COROLL.

Si computatio instituatur pro lunâ, assumendo mediocrem ejus orbitæ inclinationem ad æquatorem terrestrem, erit _n_ cosinus anguli 23° 28´½; et posito semiaxi terræ _b_ = 1, erit distantia lunæ à centro terræ mediocris _l_ = 60 circiter, indeque in hypothesi quod sit differentia semiaxium _c_ = ¹⁄₂₂₉, erit (3_bcn_) ⁄ (5_l_²) × S ⁄ L × 360° = 11´´ ½; et si fuerit _c_ = 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 posteà 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_ = ¹⁄₁₃; et positâ orbis hujus satellitis inclinatione ad æquatorem Jovis æquali 3°, erit _n_ cosinus hujus inclinationis, atque inde prodibit (3_bcn_) ⁄ (5_l_²) × S ⁄ L × 360° = 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.

Cæterùm patet motum hunc nodorum in plano æquatoris planetæ primarii, æstimando distantiam satellitis in semidiametris primarii, generatìm esse, dato tempore, in ratione compositâ, ex ratione directâ differentiæ semiaxium planetæ et cosinûs inclinationis orbis satellitis ad planetæ æquatorem, conjunctìm; et ex ratione inversâ temporis periodici satellitis et quadrati distantiæ satellitis à centro planetæ, item conjunctìm.

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 transferratur intersectio N in _n_, et describatur circulus L_gn_ 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_. Ductis itaque N_m_, G_d_, perpendicularibus in orbem lunæ, est N_n_: N_m_∷ 1: sin. GNA, et N_m_: G_d_∷ 1: sin. LG, itemque G_d_: G_g_∷ sin. G_gd_: 1; unde conjunctis rationibus provenit N_n_: G_g_∷ sin. G_gd_: sin. GNA × sin. LG, adeoque G_g_ = N_n_ × (sin. GNA × sin. LG) ⁄ sin. G_gd_. Scribantur _s_ pro sinu et _t_ pro cosinu anguli G_gd_, 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 sin. GNA = √(1 - _qqtt_ - 2_pqstu_ - _p_² _s_² _u_²); sed scribi potest 1 pro _t_, et rejici terminus _p_² _s_² _u_² ob exiguitatem sinûs _s_ anguli 5° 8´ ½, proindeque erit sin. GNA = √(_pp_ - 2_pqsu_); prætereà est sin. GNA: sin. GA sive _v_∷ sin. GAN sive _p_: sin. GN, ideoque sin. GN sive cos. LG = (_pv_ ⁄ sin. GNA), et sin. LG = _u_ - (_qsvv_ ⁄ _p_), ac sin GNA × sin. LG = pu - qs quamproximé. Quarè fit Gg = Nn × ((_pu_ - _qs_) ⁄ _s_), atque hic est motus nodorum lunarium tempore dato in plano eclipticæ: quod si tempus illud datum sit annus solaris, habetur N_n_ = (3_bcn_ ⁄ 5_l_²) × (S ⁄ L) × 360°, unde motus ille eclipticus nodorum annuus, nullâ habitâ ratione mutationis sitûs nodorum ex aliâ causâ per id temporis factæ, fiet (3_bc_ ⁄ 5_l_²) × (_qt_ + _psu_) × ((_pu_ - _qs_) ⁄ _s_) × (S ⁄ L) × 360°, vel etiam (3_bcq_ ⁄ 5_l_²) × ((_pu_ - _qs_) ⁄ _s_) × (S ⁄ L) × 360° proximé. _Q. E. I._

Quo motum nodi lunaris in hac propositione ad eclipticam reduximus, eodem prorsùs ratiocinio motus nodi satellitis cujusvis ad orbitam planetæ primarii reducetur.

COROLL. I.

Exinde liquet nullum esse hunc motum nodi, ubi sin. 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 seriem 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 Libræ. Itaque cùm motus ille nodorum annuus, de quo hîc agitur, universaliter sit æqualis (3_bcq_ ⁄ 5_l_²) × ((_pu_ - _qs_) ⁄ _s_) × (S ⁄ L) × 360°, hoc est, per Coroll. Prop. 2. æqualis 11´´ ½ × ((_pu_ - _qs_) ⁄ _s_) vel 15´´ × ((_pu_ - _qs_) ⁄ _s_) prout differentia semiaxium terræ fuerit ¹⁄₂₂₉ vel ¹⁄₁₇₇, existentibus scilicet _p_ sinu et _q_ cosinu anguli 23° 28´ ½, atque _s_ sinu anguli 5° 8´ ½; 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 verò idem nodus subierit signum Libræ, motus maximus progressivus erit 41´´ vel 53´´. In aliis nodorum positionibus eodem modo computabitur.

COROLL. II.

Si desideretur excessus regressûs nodi supra progressum in integrâ nodi revolutione, sequenti ratione investigabitur. Jungantur equinoctia diametro EA, in quam demittatur perpendiculum GK, et sumpto arcu G_h_ quem describit nodus eclipticus G quo tempore nodus equatorius N describit arcum N_n_, ducatur _hc_ perpendicularis in GK. Per hanc propositionem est G_g_. N_n_∷ ((_pu_ - _qs_) ⁄ _s_). 1, sive, quia est 1. _u_ ∷ G_h_. G_c_, fit G_g_. N_n_∷ ((_p_ × G_c_) ⁄ _s_) - _q_ × G_h_. G_h_; adeoque summa omnium G_g_ erit ad summam omnium N_n_, hoc est, motus nodi ecliptici in integrâ sui revolutione erit ad motum nodi æquatorii eodem tempore factum, ut summa omnium in circulo quantitatum ((_p_ × G_c_) ⁄ _s_) - _q_ × G_h_ ad summam totidem arcuum G_h_, 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´´ ½ vel 15´´ quo tempore nodus eclipticus describit 19° 20´ ½, motus ille nodi æquatorii tempore nodi ecliptici periodico evadit 11´´ ½ × (360° ⁄ 19° 20´ ½) = 3´ 34´´ vel 15´´ × (360° ⁄ 19° 20´ ½) = 4´ 39´´; quo pacto prodit motus nodi ecliptici præfatus æqualis _q_ × 3´ 34´´ vel _q_ × 4´ 39´´, proindeque _est radius ad cosinum obliquitatis eclipticæ ut_ 3´ 34´´ _vel_ 4´ 39´´ _ad motum quæsitum_, nempe 3´ 16´´, existente ¹⁄₂₂₉ differentiâ axium terræ, vel 4´ 16´´ eâ existente ¹⁄₁₇₇: atque hic est excessus regressûs nodi supra progressum in integrâ nodi revolutione vi prædictâ genitus. Excessu igitur hoc minuatur motus nodi lunaris periodicus 360°, et remanebit motus ille quem generat vis solis.

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: fit NGRM orbis lunæ secans eclipticam in G et æquatorem in N, in 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_, posterius 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: sin. LR; sed ob NL = GR, est NG = LR; unde conjunctis rationibus est N_n_: R_r_∷ 1: _m_ × sin. 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_h_ decremento arcûs NG facto quo tempore nodus æquatorius N describit arcum N_n_, agatur _hk_ parallela ipsi GK, eritque 1: GK sive sin. NG∷ G_h_. K_k_; proindeque jam erit N_n_: R_r_∷ G_h_: _m_ × K_k_, adeoque summa omnium variationum R_r_, quo tempore nodus eclipticus G descripsit 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_h_, 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 ((2_m_ × NH) ⁄ MGN). Unde cùm 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 mediocris orbitæ lunaris ad æquatorem, ad sinum variationis totius inclinationis ejusdem orbitæ ad eclipticam._

In hoc computo inclinationem mediocrem orbis lunaris ad æquatorem, nempe 23° 28´ ½, usurpo, cum in revolutione nodi tantum ex unâ parte augetur, quantum ex alterâ minuitur, et omnes minutias hîc expendere supervacaneum foret. Motus autem nodi lunaris ecliptici est ad motum nodi lunaris æquatorii ut 19° 20´ ½ ad 11´´ ½ vel 15´´, sive ut 6055 vel 4642 ad 1, unde per theorema supra traditum prodit variatio inclinationis tota æqualis 27´´ vel 35´´, prout differentia axium terræ statuitur ¹⁄₂₂₉ vel ¹⁄₁₇₇. Hac igitur quantitate augetur inclinatio 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; sive differentia inter semissem variationis totius et variationem quæsitam est ad ipsam semissem variationis totius ut cosinus arcûs MG ad radium, hoc est, ut _u_ - (_qsvv_ ⁄ _p_) ad 1. _Q. E. I._

PROPOSITIO V.

PROBLEMA.

_Motum apsidum in orbe satellitis quamproximé circulari, quatenùs ex figurâ planetæ primarii sphæroidicâ oritur, investigare._

Per propositionem primam vis perturbatrix, quâ trahitur satelles ad centrum planetæ primarii, est ad satellitis gravitatem in ipsum primarium, ut (3_bc_ ⁄ 5_l_²) - (9_kkbc_ ⁄ 5_l_⁴) ad 1, sive, quia per Prop. 2. est (_k ⁄ l_) = _m_ × SH (_Fig._ 4.) ponendo scilicet _m_ pro sinu inclinationis orbitæ satellitis ad æquatorem primarii, et scribendo _y_ pro SH, ut (3_bc_ ⁄ 5_l_²) × (1 - 3_m_²_y_²) ad 1; et summa harum virium in totâ circumferentiâ cujus radius est 1, est ad gravitatem satellitis toties sumptam ut (3_bc_ ⁄ 5_l_²) × (1 - (3_m_² ⁄ 2)) ad 1. Vis igitur mediocris, quæ uniformiter agere in satellitem supponi potest, dum revolutionem suam in orbitâ propemodùm circulari absolvit, est ad ejus gravitatem in primarium ut (3_bc_ ⁄ 5_l_²) × (1 - (3_m_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 sphærois oblata, revolventis in distantiâ _l_ generaliter esse ad ejusdem gravitatem in majori distantiâ L, ut (1 ⁄ _l_²) + (B ⁄ _l_⁴) × (1 - (3_m_² ⁄ 2)) ad (1 ⁄ L²) + (B ⁄ L⁴) × (1 - (3_m_² ⁄ 2)), existente B quantitate datâ exigui valoris, sive ut (1 ⁄ _l_²) ad (1 ⁄ L²) - (B ⁄ _l_²L²) × (1 - (3_m_² ⁄ 2)) + (B ⁄ L⁴) × (1 - (3_m_² ⁄ 2)) quamproximé, ideò gravitas satellitis diminuitur in majori quam duplicatâ ratione distantiæ auctæ quoties _m_ minor est quantitate √⅔ id est, ubi inclinatio orbitæ satellitis ad planetæ æquatorem non attingit 54° 44´; diminuitur autem in minori ratione, quoties est _m_ major quàm √⅔, id est, ubi illa inclinatio superat 54° 44´; adeoque in priore casu progrediuntur apsides orbis satellitis, in posteriori regrediuntur. Quantitas autem hujus progressûs vel regressûs sic innotescet.

Per exemplum tertium prop. 45 lib. 1. _Princ. Math. Newt._ si vi centripetæ, quæ est ut 1 ⁄ _l_², addatur vis altera ut _e ⁄ l_⁴, hoc est, quæ sit ad vim centripetam 1 ⁄ _l_² ut _e ⁄ l_² ad 1, angulus revolutionis ab apside unâ ad eamdem erit 360° √((1 + _e_) ⁄ (1 - _e_)) vel 360° ⁄ (1 - _e_) quamproximé, existente _e_ quantitate valdé minutâ. Porrò cum sit motus satellitis in orbitâ suâ revolventis ad motum apsidis ut 360° ⁄ (1 - _e_) ad 360° ⁄ (1 - _e_) - 360°, hoc est, ut 1 ad _e_, erit motus apsidis tempore revolutionis satellitis ad fidera æqualis 360° × _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_ = (3_bc_ ⁄ 5_l_²) × (1 - (3_m_² ⁄ 2)); unde datur motus apsidum quæsitus. _Q. E. I._

COROLL.

Si ad lunam referatur hæc determinatio, habebuntur _b_ = 1, _l_ = 60, _m_ = sinui anguli 23° 28´ ½, et si fuerit _c_ = ¹⁄₂₂₉, erit _e_ = ¹⁄₁₈₀₃₂₀₃, atque motus apogæi lunæ spatio centum annorum æqualis 16´ proximé in consequentia; si fuerit _c_ = ¹⁄₁₇₇, erit _e_ = ¹⁄₁₃₉₃₇₄₂, et motus apogæi æqualis 20´, 7. Hac igitur quantitate minuendus est motus medius apogæi lunæ prout observationibus determinatur, ut habeatur motus ille quem generat vis solis.

Pro quarto autem Jovis satellite, erunt _b_ = 1, _l_ = 25,299, _c_ = ¹⁄₁₃, _m_ = sinui anguli 3°, _e_ = ¹⁄₁₃₉₂₄,₇; hincque motus apsidis spatio unius anni solaris prodit 33´, 95 vel ferè 34´ in consequentia, qui tempore annorum decem fit 5° 40´. Insuper autem notandum est vi solis perturbari motum satellitis simili modo quo perturbatur motus lunæ; ideoque, quoniam vis solis, quâ 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 gravitates in planetas suos primarios ut ¹⁄₁₇₈,₇₂₅ ad 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° 40´ ½ in orbe lunari annuatìm, vis posterior vel ejusdem pars similis et proportionalis motum apsidis eodem tempore generabit æqualem 6´ ½ in orbe satellitis, atque decem annorum spatio 1° 5´ in consequentia. Addatur 1° 5´ ad 5° 40´, et motus apsidum totus in orbe satellitis extimi Jovialis ex duabus prædictis causis oriundus spatio decem annorum erit 6° 45´ in consequentia. Observationibus Astronomicis collegit Ill. _Bradleius_ hunc motum tempore prædicto esse quasi 6°; differentia illa qualiscumque 45´ inter motum observatum et computatum actionibus satellitum interiorum debebit ascribi.

SCHOLIUM.

Ex præcedentibus colligere licet motuum lunarium inæqualitates originem suam omnem non ducere ex vi solis, sed earum partem aliquam deberi actioni Telluris quatenùs induitur figurâ sphæroidicâ. Sufficiat hîc 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 inæqualitates eas, quæ in motibus satellitum Jovialium ex vi solis et actionibus satellitum in se invicem nascuntur, oriri alias ex figurâ Jovis sphæroidicâ ita notabiles ut Observationes Astronomicas continuò afficere debeant.

_De Variatione motûs Terræ diurni._

Si terra globus esset omninò sphæricus quicumque foret revolutionis axis, manente eâdem 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. Hîc enim non considero figuram telluris oblatam ob materiæ in æquatore redundantiam, sed sphæricam suppono nisi quatenùs 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: cùm autem axis transversus transit semper per solem vel lunam, singulis momentis mutabit situm suum respectu axis revolutionis ob motum quo hi duo planetæ recedunt ab æquatore terrestri et ad eum vicissìm accedunt.

PROBLEMA.

_Variationem motûs terræ diurni ex prædictâ causâ oriundam investigare._