A Letter from Mr. Richard Dunthorne, to the Rev. Mr. Cha. Mason, F. R. S. and Woodwardian Professor of Nat. Hist. at Cambridge, concerning the Moon's Motion

in Excuse of this Trouble, farther than that I have the Honour to be with great Respect, SIR, London, Jan. 14. 1746-7. Your most obliged, and humble Servant, Richard Brocklesby. XIII. A Letter from Mr. Richard Dunthorne, to the Rev. Mr. Cha. Mason, F.R.S. and Woodwardian Professor of Nat. Hist. at Cambridge, concerning the Moon's Motion. SIR, Cambridge, Nov. 4. 1746. Read Feb. 5. In the Preface to my lunar Tables, I hinted, that one Use of publishing those Tables would be, the assisting of Persons desirous farther to rectify the lunar Astronomy, by enabling them more readily to compare the Newtonian Theory with Observations. Since the Publishing those Tables, I have spent some Time myself in that Comparison; and here send you the Result, that you may communicate it to the Royal Society, if you think it deserves to be made public. As the Motion of every secondary Planet must partake of the Errors in the Theory of its primary, I thought proper, before I undertook the Examination of the lunar Numbers, to compare those of the Sun with Observations. I compared several Sets of Mr. Mr. Flamsteed's Observations, after the Method he himself teaches, in *Prolegom. Hist. Cælest.*, p. 133, &c., which, for many Reasons, I think the best Method hitherto used; and, with the Concurrence of a Gentleman well skilled in these Matters, determined the mean Motion of the Sun at Greenwich, the last Day of December at Noon, Anno 1700, O. S. w 20° 43' 40'' of its Apogee, $7° 30' 0''$, and the greatest Equation of the Sun's Centre $1° 55' 40''$; which, I am fully persuaded, are very near the Truth. The Theory of the Sun being thus settled, I proceeded to examine the Elements of the lunar Astronomy. I began with Observations of lunar Eclipses about the Equinoxes, when the Apogee of the Moon was in the Sun's Quadratures; because at those Times I could conceive the Moon's Motion affected with no Inequality, but the annual one, called by Newton the first Equation, and the elliptic one, called *Prosthapharesis*: From a Comparison of such Observations I obtained the Moon's mean Longitude, which came out $1'$, at least, greater than in the Tables, and very nearly as Newton has it in the last Edition of his *Principia*. I went on to examine the Place and Motion of the Apogee, and Theory of the Increase and Decrease of the Eccentricity, as well as the greatest and least Eccentricities themselves (from the best Observations, and best situate that I could procure) all which agreed so well with the Tables, about the Sun's mean Distances, that I dare venture to make no Alteration therein: Indeed I think the 6th Equation does not so well account for the Variation of the Motion of the Apogee, and Change of the Eccentricity, tricity, according to the greater or lesser Distance of the Sun from the Earth; and therefore I set myself to compute what Change this Difference of the Sun's Action upon the lunar Orbit would introduce in the Moon's Place in every Situation of the Sun and lu- nar Orbit; and found, after many tedious Compu- tations, that the Sun being in Apogee, this Change, where greatest, would amount to about $4'$, and to $4'16''$, when the Sun is in Perigee. In other Di- stances of the Sun from the Earth, this greatest Change is proportional to the Difference of the Cubes of the mean and present Distances; and in every Situation of the Moon, and of her Orbit, the present is to the greatest Equation nearly as the Sine of the Ex- cess of the Moon's mean Anomaly above twice the annual Argument to Radius. It increases the Moon's Longitude, when the Sun is in his Apogee Perigee Semicircle, and that Excess lets greater than $180^\circ$; and diminishes it when otherwise *. In fine, I compared the Theory of the Moon, as to her Longitude, with several Observations, as well in the Octants and Semi-Octants, as in the Syzygies and Quadratures, and found such an Agreement when the above Corrections were made, as seemed rather to be wished than hoped for, considering the many Inequalities wherewith the Sun's Action disturbs the Motion * If this Equation be increased and diminished in a direct Ratio of the Moon's horizontal Parallax, it will become more exact. And I think, if it were always diminished by a fourth or perhaps a third Part, it would agree better with Observations. Motion of the Moon, and the Defects to which the best Observations I have hitherto met withal are liable. I have compared 100 observed Longitudes of the Moon with the Tables; viz. 25 Eclipses of the Moon, all, except the first, taken from Flamstead's Historia Coelestis, the Philosophical Transactions, and the Memoirs of the Royal Academy of Sciences; the two great Eclipses of the Sun in 1706 and 1715; 25 select Places of the Moon from Flamstead's Historia Coelestis, and 48 of those Longitudes of the Moon computed from Flamstead's Observations by Dr. Halley (as I suppose) printed in the first Edition of the Historia Coelestis. They are as follows; | Place of Observation | A.D. | Apparent Time at Greenwich | Observed | D's Place Computed | Diff. from Observat. | |---------------------|------|---------------------------|----------|-------------------|---------------------| | Danzick | 1652 | Septem. 7. 6 21 35 | 11 25 25 51 | 2 20 6 2 0 3 | 1 25 26 19 0 19 | | Greenwich | 1678 | Octob. 19. 8 21 54 | 1 6 47 9 | 4 1 7 16 2 2 | 1 6 46 30 0 31 | | Paris | 1684 | Dec. 11. 10 47 6 | 3 1 11 20 | 5 24 1 8 7 16 | 3 1 13 1 47 | | Danzick | 1685 | Nov. 30. 10 35 32 | 2 19 40 0 | 5 13 11 26 5 25 | 2 19 39 46 0 14 | | Dublin | 1686 | Nov. 19. 11 13 11 | 2 8 11 57 | 5 21 10 12 4 4 1/2 | 2 8 11 2 0 29 | | Greenwich | 1689 | Sept. 18. 14 30 37 | 0 6 32 8 | 3 1 4 24 10 8 | 0 6 34 16 2 8 | | | 1690 | March 12. 9 56 30 | 6 4 37 11 | 8 25 9 18 3 15 1/2 | 6 4 37 2 8 0 17 | | | 1696 | May 6. 12 2 0 | 7 26 53 14 | 10 18 3 1/2 8 28 | 7 26 53 4 0 26 | | Paris | 1699 | March 5. 7 13 4 | 5 25 3 27 | 8 16 8 27 3 1 | 5 25 3 27 1 0 13 | | New England | 1703 | Decem. 11. 13 29 8 | 3 0 49 56 | 5 23 11 24 1/2 | 3 0 48 51 1 5 | | Marseilles | 1706 | May 6. 21 13 57 | 1 20 46 44 | 10 12 6 22 7 6 | 1 20 46 3 0 12 | | Greenwich | 1707 | April 5. 13 39 0 | 6 26 18 33 | 9 17 11 9 5 3 | 6 26 19 0 0 36 | | | 1708 | Sept. 18. 9 10 25 | 0 6 39 23 | 2 1 2 12 8 7 | 0 6 39 1 0 4 | | | 1712 | Jn. 12. 7 34 0 | 4 2 57 47 | 6 24 1 17 1/2 | 4 2 58 37 0 50 | | | 1713 | Nov. 20. 15 27 30 | 2 9 53 21 | 5 3 9 21 3 17 | 2 9 56 50 0 25 | | | 1715 | April 21. 21 11 5 | 1 12 0 22 | 10 3 6 12 1/2 | 1 11 59 36 0 46 | | | 1717 | March 15. 15 7 4 | 6 6 23 50 | 8 26 8 22 1/2 | 6 6 22 3 0 19 | | | 1719 | August 18. 8 22 46 | 11 5 42 8 | 1 29 10 23 4 20 | 11 5 42 39 0 31 | | | 1722 | June 17. 13 46 10 | 9 6 47 43 | 11 29 5 8 1/2 | 9 6 47 33 0 10 | | | 1724 | Octob. 20. 15 40 40 | 1 9 0 0 | 4 2 5 25 1/2 | 1 8 58 50 0 10 | | | 1729 | Febr. 2. 8 42 55 | 4 25 13 39 | 7 16 3 25 9 14 | 4 25 14 40 0 1 | | | | July 28. 13 0 0 | 10 16 15 28 | 1 9 8 3 2 15 | 10 16 16 26 0 58 | | | 1731 | June 8. 13 47 51 | 8 28 9 58 | 11 19 4 27 | 10 11 8 28 0 45 1 13 | | | 1732 | Nov. 20. 0 49 25 | 2 10 3 54 | 5 3 7 8 1 24 | 2 10 0 50 2 58 | | A.D. | Apparent Time at Greenwich | D's true Place | M. Anom. | Ann. Diff. from Observed | D's Place computed | |------|---------------------------|----------------|---------|--------------------------|--------------------| | | h | s° o' m'' | s° o' m'' | s° o' m'' | s° o' m'' | | 1684 | March 13 | 8 | 9 | 8 | 2 28 48 | | 1693 | March 6 | 7 | 22 | 48 | 2 25 11 | | 1694 | Octob. 11 | 18 | 12 | 34 | 3 16 44 | | | Febr. 27 | 10 | 29 | 16 | 3 28 34 | | | August 23 | 11 | 13 | 54 | 4 27 27 | | 1694 | Septem. 15 | 5 | 34 | 13 | 8 27 | | | Decem. 21 | 10 | 59 | 31 | 11 22 | | | Febr. 8 | 3 | 55 | 22 | 1 5 14 | | | July 9 | 8 | 56 | 14 | 7 2 2 | | | Septem. 8 | 8 | 30 | 26 | 10 3 0 | | 1696 | January 16 | 17 | 29 | 2 | 7 5 4 | | | March 4 | 9 | 29 | 43 | 12 2 24 | | | February 18 | 6 | 29 | 47 | 20 18 16 | | | Septem. 15 | 7 | 54 | 4 | 10 1 7 | | 1698 | Septem. 8 | 2 | 35 | 11 | 12 15 6 | | | Novem. 27 | 3 | 49 | 54 | 10 11 33 | | | March 5 | 12 | 8 | 23 | 5 28 15 | | | Septem. 28 | 6 | 55 | 23 | 9 27 13 | | | October 16 | 6 | 15 | 49 | 10 3 25 41 | | 1703 | Septem. 13 | 11 | 58 | 50 | 11 28 50 | | | Octob. 6 | 6 | 28 | 39 | 9 27 1 44 | | | October 10 | 11 | 11 | 10 | 11 15 3 | | | September 6 | 6 | 34 | 7 | 9 3 44 13 | [417] Places of the Moon, computed by Dr. Halley, from Flamstead's Observations, compared with the Tables. | A.D. | Apparent Time at Greenwich | D's true Place observed | M. Anom. ⊙ | M. Anom. ♈ | Ann. Arct | D's Place computed | Diff. from Observation | |------|----------------------------|-------------------------|------------|------------|-----------|-------------------|-----------------------| | | h | s | s | s | s | s | | | 1689 | Novem. 16. 11 59 0 | 2 6 12 43 | 4 29 | 6 1 | 0 0 | 2 6 14 40 | + 1 57 | | | Decem. 9. 6 1 | 11 28 43 54 | 5 21½ | 3 21 | 0 20½ | 11 28 48 40 | - 0 14 | | | 10. 6 46 35 | 0 12 46 49 | 5 22½ | 4 4 | 0 21½ | 0 12 47 14 | + 0 25 | | | 12. 8 26 33 | 1 12 13 8 | 5 24½ | 5 0½ | 0 23 | 1 12 10 39 | - 2 29 | | | 13. 9 24 30 | 1 27 38 36 | 5 25½ | 5 14 | 0 24 | 1 27 36 16 | - 2 20 | | 1690 | January 16. 12 42 0 | 3 15 14 54 | 5 28½ | 6 24 | 0 27 | 3 15 15 48 | + 0 54 | | | 4. 3 3 46 | 11 11 12 59 | 6 17 | 2 24 | 1 14 | 11 11 13 8 | + 0 9 | | | 6. 4 30 15 | 0 8 13 31 | 6 19 | 3 20½ | 1 16 | 0 8 14 36 | + 1 5 | | | 10. 7 59 22 | 2 6 11 20 | 6 23 | 5 14½ | 1 19½ | 2 6 10 37 | - 0 43 | | | 12. 10 8 49 | 3 7 5 21 | 6 25 | 6 12½ | 1 21 | 3 7 3 59 | - 1 22 | | | February 13. 11 14 0 | 3 22 36 28 | 6 26 | 6 25½ | 1 22 | 3 22 35 19 | - 1 9 | | | 2. 2 25 39 | 0 3 57 24 | 7 15½ | 3 5 | 2 10 | 0 3 56 9 | - 1 15 | | | 5. 4 51 10 | 1 16 31 33 | 7 18½ | 4 26 | 2 12½ | 1 16 34 15 | + 2 42 | | | 7. 6 48 17 | 2 15 58 14 | 7 20½ | 5 24 | 2 14½ | 2 15 59 13 | + 0 59 | | | 8. 7 51 54 | 3 0 56 20 | 7 21½ | 6 8 | 2 15½ | 3 0 56 21 | + 0 11 | | | 10. 9 56 26 | 4 0 55 24 | 7 24 | 7 6 | 2 17 | + 0 55 1 | - 0 23 | | | 11. 10 52 31 | 4 15 42 16 | 7 25 | 7 20 | 2 18 | 4 15 41 49 | - 0 27 | | | 14. 13 19 31 | 5 28 12 4c | 7 28 | 9 2 | 2 21 | 5 28 11 44 | - 0 56 | | | 19. 17 3 55 | 8 2 23 2c | 8 3 | 11 12 | 2 25½ | 8 2 21 17 | - 2 9 | | | 21. 18 45 37 | 8 27 1 25 | 8 5 | 0 10 | 2 27½ | 8 26 59 6 | - 2 19 | | Year | Month | Date 1690 | Date 1691 | |------|-------|-----------|-----------| | | Febr. | 23. 19 | 23. 35 | | | March | 7. 5 | 7. 8 | | | | 11. 9 | 11. 14 | | | | 12. 10 | 12. 17 | | | April | 14. 12 | 14. 18 | | | | 25. 20 | 25. 26 | | | | 7. 8 | 7. 14 | | | | 9. 9 | 9. 15 | | | May | 15. 13 | 15. 30 | | | June | 22. 19 | 22. 27 | | | Seprem. | 8. 12 | 8. 20 | | | February | 23. 3 | 23. 5 | | | March | 18. 12 | 18. 18 | | | Seprem. | 22. 8 | 22. 24 | a, The Time of the Middle of this Eclipse here set down is from the Beginning and End; but Hevelius says he could not observe the Beginning exactly. Several intermediate Phases compared together shew the Middle to have been about $4'$ sooner; to which the Moon's Place computed is $0^\circ. 6^\circ. 14'. 3''$. and Diff. $+34''$. b, b, b, The Moon's Places, observed on Feb. 2. April 7. and May 22. are computed by myself, from the Observations; there being manifestly Errors, either of the Computation or Press, in those printed in the Hist. Cælestis. Several observed Latitudes of the Moon, which I have compared with the Tables, shew them to be very near the Truth, both in the Motion of the Nodes, and also in the Quantity and Variation of the Inclination. I am, SIR, Your humble Servant, Richard Dunthorne.