A Letter from the Rev. Mr. Richard Dunthorne to the Reverend Mr. Richard Mason F. R. S. and Keeper of the Wood-Wardian Museum at Cambridge, concerning the Acceleration of the Moon

XIV. A Letter from the Rev. Mr. Richard Dunthorne to the Reverend Mr. Richard Mason F. R. S. and Keeper of the Woodwardian Museum at Cambridge, concerning the Acceleration of the Moon. SIR, Cambridge, Feb. 28, 1748-9. Read June 1. AFTER I had compared a good Number of modern Observations made in different Situations of the Moon and of her Orbit in respect of the Sun, with the Newtonian Theory, as in my Letter of Nov. 4, 1746;† I proceeded to examine the mean Motion of the Moon, of her Apogee, and Nodes, to see whether they were well represented by the Tables for any considerable Number of Years, and whether I should be able to make out that Acceleration of the Moon's Motion which Dr. Halley suspected. Vide Phil. Trans. n. 218. To this End I compared several Eclipses of the Moon observed by Tycho Brahe, as they are set down in his Progymnasmata p. 114, with the Tables *, and found them agree full as well as could be expected; considering the Imperfection of his Clocks, and the Difficulty there must commonly have been in determining the Middle of the Eclipse from the Facts observed, as published in his Historia Coelestis. Indeed the small Distance of Time between Tycho Brahe and † See these Transact. No. 482. p. 412. * My Tables corrected as in my former Letter; which is always to be understood of the Tables mentioned in this. and Flamsteed render'd Tycho's Observations but of little Use in this Enquiry. The next Observations that occurred to me were those of Bernard Walther and Regiomontanus, which being at double the Distance of Time from Flamsteed that Tycho's were, seemed to promise some Assistance in this Matter: Upon comparing such of their Eclipses of the Moon whose Circumstances are best related with the Tables, I found the computed Places of the Moon were mostly $5'$ too forward, and in some considerably more, which I could hardly persuade myself to throw upon the Errors of Observation; but concluded, that the Moon's mean Motion since that time, must have been something swifter than the Tables represent it; though the Disagreement of the Observations between themselves is too great to infer any thing from them with Certainty in so nice an Affair. Then I compared the four well-known Eclipses observed by Albategnius with the Tables, and found the computed Places of the Moon in three of them considerably too forward: This, if I could have depended upon the Longitude of Aracta, would very much have confirmed me in the Opinion, that the Moon's mean Motion must have been swifter in some of the last Centuries than the Tables make it; though the Differences between these Observations, and the Tables, are not uniform enough to be taken for a certain Proof thereof. I could meet with no Observations of Eclipses to be at all depended upon between those of Regiomontanus and Albategnius, except two of the Sun and one of the Moon made at Cairo in Egypt, related in the Prolegomena to Tycho Brahe's Historia Coelestis, p. 34; nor any between those of Albategnius and Ptolemy, besides the Eclipse of the Sun observed by Theon at Alexandria; notwithstanding I carefully searched all the Remains of Antiquity I could find with that View. These Eclipses of the Sun are the more valuable, because they were observed in Places the Longitudes and Latitudes whereof are determined by Monsieur Chazelles of the Royal Academy of Sciences, who was sent by the French King in the Year 1693, with proper Instruments for that Purpose. Du Hamel Hist. Acad. p. 309, 395. The solar Eclipse observed by Theon was in the 112th Year of Nabonassar the Day of Thoth, according to the Egyptians, but the 22d Day of Pauni, according to the Alexandrians: He carefully observed the Beginning of 2 temporal Hours and 50' Afternoon, and the End at 4½ Hours nearly Afternoon at Alexandria. Theonis Comment. in Ptol. mag. Construct. p. 332. This Eclipse was June 16, in the Year of Christ 364: And the temporal Hour at Alexandria being at that time to the equinoctial Hour as 7 to 6, makes the Beginning at 3 equinoctial Hours and 18' Afternoon, and the End at 5 equinoctial Hours 15' nearly. The Eclipses observed at Grand Cairo were as follows. "Anno Hegirae 367, die Jovis, qui erat 28, rabie posterioris (is est ordine mensis quartus, et incipit ille annus Saracenicus die 19 Augusti, anno Christiano 977) observatum fuit Cahire in Aegypti metropoli initium eclipsis solaris, cum altitudo folis "effet "esse 15° 43'. quantitas obscurationis 8 digit. Ea finita, sol elevabatur 33½ gr. Ex Schickardo in MS." This Eclipse was Decemb. 13, in the Year of Christ 977, the Beginning at 8ʰ 25', and the End at 10ʰ 45' apparent Time in the Morning. "Anno eodem die Sabbathi, videiicet 29 mensis Sywal (numero decimi, qui Paschalis est corum) eclipsis Solis occupavit digitos 7½. In principio Sol altus fere 56°. In fine Sol occiduus elevaba- tur gradibus 26. Ex Schickardo in MS." —— This Eclipse was June 8, in the Year of Christ 978. The Beginning at 2ʰ 31', and the End at 4ʰ 50' apparent Time Afternoon. "Anno Hegiræ 368 (qui incepit die 9 Augusti, anno Christiano 978) die Jovis, 14 Sywal, Luna suit orta cum defectu, qui ad 5½ digitos accrevit; cum extaret supra horizontem gradibus etiam 26 subaudio finem tunc accidisse). Schickardus." —— This Eclipse was May 14, in the Year of Christ 979; but as the Middle cannot be known from what was observed of it, I made no use thereof in this Enquiry. The Account concludes with the following Para- graph: "Hæ tres observationes habita sunt ab Ibn-Junis, qui jusiu Regis Abu-Haly Almanzor, sapientis, Ægypto tunc Imperantis, rebus vacabat coelestibus. Hujus authoris tabulas habet Jac. Golius Profesor Lugdun. (qui mihi inde communicavit ista) in quibus plures alia, sui et superioris ævi ob- servationes extant. Locus observationis propinquus urbi Cahiro. Schickardus." That That the before-mention'd solar Eclipses might be applied to the Examination of the Lunar Motions, I contrived the following Method; which I think renders Eclipses of the Sun as useful at least as those of the Moon are in that Business. Let \(ABC\) in the annexed Figure represent half the Earth's enlightened Disk, \(AEC\) a Portion of the Ecliptic projected thereon \(FGH\) the Path of the Moon's Shadow over the Disk, \(EI\), the universal Meridian, \(\alpha\) the Situation of the Place at the Beginning of the Eclipse, \(\beta\) its Situation at the End thereof, \(\delta\) the Centre of the Shade at the Beginning, and \(\varepsilon\) its Centre at the End of the Eclipse. Draw \(EG\), \(\alpha\zeta\), and \(\beta\eta\), perpendicular to the Path of the Shadow, \(\beta\gamma\) parallel thereto; join \(\alpha\delta\) and \(\beta\varepsilon\), and through \(\alpha\) draw \(\theta\alpha\), perpendicular to \(AC\). Then (computing the true Places of the Sun and Moon at the observed Times of the Beginning and End of the Eclipse) we shall have given \(\delta\varepsilon\) the Motion of the Moon from the Sun in her Orbit during the Time of the Eclipse, and \(\alpha\delta = \beta\varepsilon\) the Semidiameter of the Penumbra; which are to be reduced into such Parts as the Semidiameter of the Disk contains tains 10000: The Angles $BEI$ and $BEG$, being found by Methods commonly known, $GEI$ their Sum or Difference will be likewise given. Also $E\alpha$ and $E\beta$ will be Sines of the Sun's Altitude at the Beginning and End of the Eclipse respectively; $IE\alpha$ and $IE\beta$ are the Angles at the Sun between the Vertex of the Place and the Pole of those Times; which being found, the Angle $\alpha E\beta$, their Difference will be known, from whence the Line $\alpha \beta$ and the Angle $E\alpha \beta$ may be computed. The Angle $GE\alpha$ is the Sum or Difference of the known Angles $GEI$ and $IE\alpha$: In the Figure before us, the Complement of this to a Semicircle is $Ex\gamma$; which being subtracted from $E\alpha \beta$ leaves the Angle $\gamma \alpha \beta$, from whence and the Line $\alpha \beta$, $\alpha \gamma$, and $\gamma \beta = \zeta n$ may be found. Let $a = \delta \varepsilon - \zeta n$, $b = x \delta = \beta \varepsilon$, $c = x \gamma$, and $x = \beta n = \gamma \zeta$. Then $\sqrt{bb - xx} = n\varepsilon$, and $\sqrt{bb - cc - 2cx - xx} = \zeta \xi$, by Eucl. I.47. Consequently $a - \sqrt{bb - xx} = \sqrt{bb - cc - 2cx - xx}$ which being reduced, gives us the quadratic Equation $xx - cx = \frac{4a^2b^2 - a^2 - 2a^2c^2}{4aa + 4cc}$. This Equation solved, gives us the Value of $x$, from which $\delta \xi$ and $n\varepsilon$ will be likewise had. In the Triangle $\alpha \xi b$ we have $\alpha \xi$ and the Angle $\xi \alpha b = GEB$ given, whence $\alpha b$ and $\xi \theta$ may be found: Consequently $\delta \theta$ will be known; and from the observed Time of the Beginning of the Eclipse, and hourly Motion of the Moon from the Sun, the Time when the Centre of the Shade is at $\theta$ will be had. Lastly, in the Triangle $E\alpha \alpha$, we have given the Side $E\alpha$, and the Angle $E\alpha = BE\alpha$. BEα (the Sum or Difference of the Angles BEI and IEα); therefore the Sides E₁ and α₁ may be found. But E₁ is the Distance of the Moon from the Sun in the Ecliptic, and α₁ + αθ the Moon's Latitude at the Time when the Centre of the Shade is at θ; which may be compared with the Computation from the Tables for that Time. By this Means I compared the aforesaid Solar Eclipses with the Tables, and found the Difference in Longitude and Latitude, as follows. | A.D. | Apparent Time at Greenwich. | Diff. by a ⊙ from E₁. | Lat. by Tab. | Diff. from Observed | Diff. in Lat. from Digits observed | |------|-----------------------------|---------------------|--------------|--------------------|----------------------------------| | | | | | | | | 364 | June 16. | 4 20 39 41 in conseq.| 34 37 Nor | -4 16 | +2 49 | | 977 | Dec. 12. | 19 12 30 43 39 in antec.| 30 23 Nor | +7 36 | +1 27 | | 978 | June 8. | 1 16 10 29 3 in conseq.| 8 24 Sou | +8 45 | -5 3 | The Agreement there is between the two last of these Differences in Longitude, shows that the Tables represent the mean Motion of the Moon's Apogee very well for above 700 Years, the Moon being very near her Perigee at the Time of one of those Eclipses, and near her Apogee at the Time of the other. By the same Method I also compared the Sun's Eclipse, July 29, 1478. (which appears, from what is related of it, to have been carefully observed by Bernard Walther at Nuremberg), with the Tables, and found the Difference in Longitude to be ± 10' 29'', and in Latitude ± 9' 12''. This wide Difference in Latitude, from the Tables, that agree so well with the former ancient Observations, confirmed me in the Opinion, that the Nuremberg Obser- Observations are too inaccurate to determine anything from them in this Affair. The Eclipses recorded by Ptolemy in his Almagest, are most of them so loosely described, that, if they shew us the Moon's mean Motion has been accelerated in the long Interval of Time since they happened, they are wholly incapable of shewing us, how much that Acceleration has been. There are indeed two or three of them attended with such lucky Circumstances as not only plainly prove, that there has been such an Acceleration, but also help us to guess at its Quantity. One of these is the Eclipse, said by Hipparchus to have been observed at Babylon, in the 366th Year of Nabonassar, the Night between the 26th and 27th Days of Thoth, when a small Part of the Moon's Disk was eclipsed from the North East, half an Hour before the End of the Night, and the Moon set eclipsed. This was in the Year before Christ 313, Decemb. 22. The Middle of this Eclipse at Babylon (supposing with Ptolemy the Meridian of that Place to be 50' in Time East of the Meridian of Alexandria), by my Tables was Dec. 22. 4h 4' apparent Time; the Duration was 1h 37', Ptolemy makes it 1h 30' nearly; whence the Beginning should have been about 5h 15' after Midnight: According to Ptolemy, the Night at Babylon was at that Time 14h 24' long, and therefore Sun rise at 7h 12' after Midnight; and as the Moon had then South Latitude, and was not quite come to the Sun's Opposition, her apparent Setting must have been something sooner, i.e. more than an Hour before the Beginning of the Eclipse, according to the Tables; whereas the Moon was seen eclipsed eclipsed some Time before her Setting; which, I think, demonstrates, that the Moon's Place must have been forwarder, and consequently her Motion since that Time less than the Tables make it by about $40'$ or $50'$. But the computed Place of the Moon in each of the before-mentioned Solar Eclipses observed at Grand Cairo, being about $8'$ before her Place, from Observation shews us, that the mean Motion of this Luminary has been something greater in the last 700 Years than the Tables suppose it, and therefore must have been accelerated. This Acceleration is further confirmed by the Eclipse, which Hipparchus says was observed at Alexandria, in the 54th Year of the second Calippic Period, the 16th Day of Messori, when (he says) the Moon began to be eclipsed half an Hour before her Rising, and was wholly clear again in the Middle of the third Hour of the Night. This was in the Year before Christ 201. Sept. 22. The Middle of this Eclipse at Alexandria by the Tables was Sept. 22. 7$^h$ 44$^m$ apparent Time; and the Duration 3$^h$ 4$^m$, which makes the Beginning at 6$^h$ 12$^m$ apparent Time, that is, about $10'$ after the rising of the Moon at Alexandria, or $40'$ later than the Beginning from Observation. This Difference in Time makes a Difference of near $20'$ in the Moon's Place. The most antient Eclipse of which we have any Account remaining, namely that related by Ptolemy, to have been observed at Babylon the first Year of Mardokempad, in the Night between the 29th and 30th Days of Thoth, in which the Moon began to be eclipsed when one Hour after her Rising was fully past; if, by reason of the Latitude of the Expression, pression, it be not a direct Proof of the Acceleration, it may nevertheless help to limit its Quantity. This Eclipse was in the Year before Christ 721. March 19. The Middle whereof at Babylon, by the Tables, was March 19. 10\textsuperscript{h} 26' apparent Time; and the Beginning at 8\textsuperscript{h} 32', the apparent Rising of the Moon at that Place was about 5\textsuperscript{h} 46' Afternoon; so that the observed Beginning of the Eclipse was at least 6\textsuperscript{h} 46' Afternoon, i.e. not above 1\textsuperscript{h} before the Beginning, by the Tables: Wherefore the Moon's true Place could precede her Place by Computation but little more than 50' at that Time. If we take this Acceleration to be uniform, as the Observations whereupon it is grounded are not sufficient to prove the Contrary, the Aggregate of it will be as the Square of the Time: And if we suppose it to be 10'' in 100 Years, and that the Tables truly represent the Moon's Place about A.D. 700, it will best agree with the before-mentioned Observations; and the Difference between the Moon's Place by the Tables and her Place in the Heavens, will be as follows. | Years before Christ | Error of Tab. | Years of Christ | Error of Tab. | Years of Christ | Error of Tab. | |--------------------|---------------|----------------|---------------|----------------|---------------| | 70 | -56 | 200 | -12 | 1100 | +4 | | 600 | -49 | 300 | -9 | 1200 | +4 | | 500 | -44 | 400 | -6 | 1300 | +4 | | 400 | -38 | 500 | -4 | 1400 | +3 | | 300 | -33 | 600 | -1 | 1500 | +2 | | 200 | -28 | 700 | 0 | 1600 | +1 | | 100 | -24 | 800 | +1 | 1700 | 0 | | A.D. | -19 | 900 | +2 | | | | 100 | -16 | 1000 | +3 | | | I am, SIR, Your humble Servant, Richard Dunthorne. XV. Alberti Halleri, Archiatri Reg. Medicin. Prof. Gotting. & R. S. Lond. S. Fabricæ morbofæ in cadaveribus repertæ historiae aliquæ. O B S. I. Read, June 8. In femina quadragenaria reperi Venam cavam inter renalis sinistræ originem, et inter iliacas venas, enormiter angustatam, ut vix quid-