and 112,590, the Log. of the perihelion Distance 9,883344, and the Time reduced employed in defcribing the Angle contained by the two vector Rays =50,729 agreeable to Obfervation. Having found two Parabolas anfwering the two firft Obfervations, we are next to examine which approaches nearest the Orbit of the Comet fought, by calculating the third Obfervation in each; for which Purpose I calculate the Place of the Perihelion, the Time of the Paffage at the Perihelion, the Inclination to the Ecliptic, and the Place of the Nodes of each Parabola. calculated in To determine thofe Elements in the firft Parabola, I find the Angle R=23° 40′ 15′′, then the Distance of the Comet reduced to the Ecliptic Elements of at the first Observation from the afcending Node 5° 25′ 45′′, which added the Comet to the heliocentric Longitude of the Comet, the 4th of March, which the first and is 6% 0° 17′ 52", because its heliocentric Motion is retrograde, gives the fecond HyPlace of the Node, in 6s 5° 43' 37". The Distance of the Comet pothefis. from the Node measured on its Orbit, which I find to be 130 38 14′′, fubtracted from the Place of the Node, gives the Place of the Comet in in its Orbit, at the Time of the firft Obfervation: and because it had then 45° 34' 28" true Anomaly, I add them to its Place in its Orbit to obtain the Place of the Perihelion in 7s 7° 39' 51". I add of the Log. of the perihelion Distance to that of 36,568 Days, Time correfponding to the least true Anomaly 45° 34′ 28", which gives 24,486 Days, for the Interval of Time elapfed between the first Obfervation, and the Inftant of the Paffage of the Comet at the Perihelion, which being fubtracted from the 4th of March at 16h 9' 50", or at 0,673, the Time of the firft Obfervation, fixes the Inftant of the Paffage at the Perihelion to the 8th of February at 0,188. In fine, I find the Angle of Inclination of the Plane of the Ecliptic, and that of the Comet to be 66° 56' 14". The fame Elements in the second Parabola are, the afcending Node in 65° 59' 6", the Place of the Perihelion in 75 7 53' 42, the Inclination, 66° 47' 14", and the Time of the Paffage at the Perihelion, February the 8th, 151. From thofe Elements I calculate the geocentric Longitude for the 28th of March, at 0,569 of the Day, in each Parabola. The Interval of Time elapfed between the Paffage at the Perihelion in the first Parabola, and the Time of the Obfervation 28th March 0,569 is 48,381 Days. The Log. of the perihelion Distance, 9,883870, its Triple is, 9,651610, its Half, 9,825805, which being fubtracted from 1,684675, Log. of 48,381 gives 1,858870, Log of 72,255 Days, which correlponds to 73° 11' 7", or 25 13° 11' 7" Anomaly, which fubtracted from the Place of the Perihelion 7 7° 39' 51", because the Comet being retrograde, the given Inftant follows, that of the Paffage at the Perihelion, which gives the true heliocentric Place of the Comet in its Orbit, and fecond 4° 24° 28' 44", from 45 24° 28' 44", fubtracting 6 5, 43' 37", the Geocentric Place of the afcending Node, the Argument of Latitude 10s 18° 45' 7" Long tu ie is obtained, which meafured on the Ecliptic is 11s 11° 2' 47"; confeof the Comet calculat quently the heliocentric Longitude of the Comet is 5s 16° 46′ 24′′, and ed in the first the heliocentric Latitude, 37° 20' 41" North because the Argument of Hypothefis. Latitude of the Comet, which is retrograde, is greater than fix Signs. The true Place of the Sun the 28 of March, at 13h 39m is os 8° 11′ 28′′, and the Log. of its Distance from the Earth, is 9,999841; therefore the true Place of the Earth feen from the Sun, is 6s 8° 11′ 28", which exceeds 5 16° 46' 24" by 21° 25' 4", which is the Angle of Commutation. I find the Log. of the curt Distance, correfponding to the third Obfervation 9,974915, I fubtract 9,974915 from 9,999841, Log. of the Distance of the Sun from the Eearth: The Remainder is 0,024926, which by adding 10 to its Characteraftic, gives 10,024926, Log. of the Tangent of 46° 38' 42", from which fubtracting 45, the Log. of Tan. of Remainder, 1° 38' 42", added to that of the Tangent of 79, 17' 28', (Complement of 10° 42' 32", half of the Angle of Commutation. 21° 25' 4") the Sum is the Log. of the Tangent of 8° 37' 39", which fubtracted from 79, 17′ 28′′; because the Distance of the Comet from the Sun, is less than that of the Earth from the Sun, gives 70° 39′ 49′′, or 2 10° 39′ 49′′, for the Angle of Elongation. By Means of a Figure reprefenting the Ecliptic divided into 12 Signs, in which I place the Sun, the Earth, and the Comet, according to their Longitudes found by the above Calculations, I perceive that the Comet feen from the Earth, is to the Eaft of the Sun. I therefore add the Angle of Flongation to the true Place of the Sun, which gives the true geocentric Longitude of the Comet, in 25 18° 51' 17", which is lefs than the obferved Longitude 2° 18° 52′ 45′′ by 1' 28"; by a like Process I find the geocentric Longitude of the Comet in the fecond Parabola, the 28 of March, in 2s 18° 45' 14", which is less than the observed Longitude, by 7' 31"; confequenly neither of the two Parabolas, is the Orbit of the Comet. THIRD HY- But because the Variations of the Orbits, are fenfibly proportional to those made in the curt Distances, to obtain the two curt Distances which correfpond to the Orbit fought. I make thofe two Proportions; (©) As 6' 3" Difference of the two Errors -1′ 28′′ and -7′ 31", Is to the leaft of the two 1' 28" So is 0,0007 and 0,0002, Corrections made to the two curt Distances Y and Z, to obtain two Parabolas anfwering the two firft Obfervations, to 0,000235 and 0,000065, Corrections to be made to thofe Diftances Y and Z, to obtain the Orbit required. = to To apply thofe Corrections, I obferve, that fince Y, fuppofed 0,879, gives an Error of 128", and Y fuppofed to 0,8783, gives an Error of 731", by diminishing Y, the Error is increased; from whence I conclude, that 0,000235 is to be added to 0,879, to obtain (c) I would have faid as the Sum of the Errors &c. if the one was by excefs and the other by defect. the true Value of Y, which confequently will be 0,879235; in like Manner, I find that Z fhould be suppofed -0,956735. ed in the VI Suppofition, Y=0,879235, and Z-0,956735, and I find the heli- Geocentrie ocentric Latitudes, E-12229 17", E-52° 3' 10; the Log, of the date Longiande vector Rays, V 9,954504, aud Vo, 191963, the heliocentric Lon- o the Cogitudes, L60° 14' 37", and L'5' 2° 38′ 19"; the true Anomalies, met calculat 45' 32' 0" and 90° 54' 4'; the corcfponding Times 36,528 and 112,243 third HypoDays; the Log. of the perihelion Distance 9,884049; and the Time thefis. employed in defcribing the Angle contained by the two vector Rays, 50,729; the Place of the Node in 6 5° 38' 29"; the Place of the Ferihelon, 77°35′ 13", the Inclination of the Orbit, 66° 59′ 14′′; and the Time of the Paffage at the Perihelion the 8th of February, at 4h 48': In fine, from thofe Elements, I calculate the geocentric Longitude and Latitude the 28th of March, at 13h 39', which I find, the one in 25 18° 53′ 18", the other 63° 3' 57" North, agreeable to Obfervation. By thefe Rules the following Table was calculated, containing the Elements of all the Comets which have been obferved with any Degree of Accuracy. 30. 7.00dir. 9.19. 3.000,5800 March. 11.12.coretr. 17. 6.10dir. 22. 0.00 [retr. 1. 1.00 retr. 28.22.32 retr. 1231 0.13.30.00 6. 5.00 7.19.29 dir. 8. 3.54 retr. 18.13.47 dir 10.20. 4 retr. 17. 3.12dir. Table of the Elements of the Comets. d h 6. 0.46 retr. 26.14.12 dir. 18.00.15 dir. 13. 2.59 retr. 8.10.25 dir. 16.14.42 dir. 1.15. 5 retr. 18.17. 6 retr. 1677 7.26.49.1079. 3.15 4.17.57. 50,2806 May. 1678 5.11.40.00 3.4.2010.27.46.00 1,1238 Aug. 1680 9. 2. 2.0060.56.00 8.22.39.300,0061 Decem. 1683 5.23.23.0083.11.00 2.25.29.300,5602 July. 1684 8.28.15.0065.48.40 7.28.52.000,9601 June. 1686 11.20.34.4031.21.40 2.17.00.300,3250 Sept. 1689 10.23.45.2069.17.00 8.23.44.450,0168 Decem. 1698 8.27.44.15 11.46.00 9.00.51.150,69130&. 1699 10.21.45.3569.20.00 7. 2-31. 60,7440 Jan. 13. 8.32 retr. 1702 6. 9.25.15 4.30.00 4.18.41. 30,645 9 March. 13.14.22 dir. 1706 0.13.11.4055.14.10 2.12.29.100,4258 Jan. 30. 4.52dir. 1707 1.22.46.35 88.36.00 2.19.54.560,8597 Decem.11.23.39dir. 1718 4. 8.43.00 30.20.00 4.01.30.00 1,1027 Jan. 1723 0.14.16.00 19.59.00 1.12.52.200,9876 Sept. 1729 10.10.32.37 76.58. 410.22.40.001,4261|June. 1737 7.16.22.0 18.20.45 10.25.55.000,2229 Jan. 1739 6.27.25.14 55.42.44 3.12.38.400,6736 June. 1742 6. 5.38.2956.59.14 7. 7.35.130,7657 Feb. 1743 2.18.21.15 2.19.33. 3. 2.41.450,8350 Jan. 1743 0. 5.16.25 45.48.20 8. 6.33.520,52 5 Sept. 1744 1.15.46.117. 5.18 6.17.10.000,2225 March. 1747 4.27.18.50179. 6.20′ 9. 7. 2.00 1,2198 March. 1748 7.22.52.16.85.26.57 7. 5. 0.502,8407 April. 1748 1. 4.39.4356.59. 39. 6. 9.240,6553 June. 1757 7. 4. 5.50 12.39. 6 4. 2.39 0033910&. 1758 7.20.50. 9l68.19.00 8.27,37.450,2154 June. 11. 3.27 dir. 1759 4.19.39.2478.59.22 1.23.24.200,7985 Novem.27. 2.28 dirr. 1759 2.19.50.45 4.51.32 4.18.24.35|0,9660 Decem. 16.21.13 ret. 1762 11.19.00.0085.20.00 3.14.00.001,0090 May. 28.00.00dir. 1763 11.26.17.00 72.42.00 2.24.43.00,4991 Novem. 1.18.39dir. 1764 4. 0. 7.0052.47.00 0.15.26.000,5567 Feb. 1766 8. 4.10.5040.52.2 4.23.15.250,5053 Feb. 14.23.48 retr. 27.16.20retr. 25.11. 6dir. 30. 8.30ldir. 17.10. 9retr. 8. 4.4Sretr. 10.20.35 dir. 20.21.26 retr. 1. 8.13 dir. 3. 7.21 retr. 28.19.34 retr. 18. 1.33 dir. 21. 9.42 dir. 12.13.40 retr. 17. 8.50retr. 1766 1.17.22.19 8.18.45 6.26. 5.13 0.6368 April. 17.10.26 dir. Elements of the COMET of HALLEY, in its different Revolutions. 1456 1.18.30.00 17.56.00 10.1.00.000,5856June 8. 22. 10.retr. 1531 1.19.25.00 17.56.00 10.1.39.000,5670¡ Aug. 24. 21.27. retr. 1607 1.20.21.00 17. 2.00 10.2.16.00 0,5868 O&. 26. 3. 59. retr. 1682 1.20.48.00 17.42.00 0.1.36.00 0,5825 Sept. 14. 21.31. retr. 1759 1.23.49.00 17.39.00 10.3.16.00 0,5835 March 12. 13.41. retr. VIII. Calculation Newton having thus folved the above-mentioned Problem, and applied Newton veit to all the Comets obferved, deduced from thence a complete Confir-rifies his mation of his Conjecture. For all the Places of the Comets calculated by the Obin the parabolic Orbits, whofe Elements were delivered in the foregoing fervations of Table, compared with thofe immediately deduced from Obfervation, ber of never differed fenfibly, which will appear fo much the more fur- Comets. prifing, when we confider how difficult it is to attain to Precision in Obfervations of this Nature. IX. a great Num tion of their from the As to the Duration of the Periods of the Comets, it cannot be de- The Duraduced from the fame Calculation, because as we have already hinted, Period cantheir Orbits being fo_excentric that they may be taken for Parabolas not be dewithout any fenfible Error, very great Differences in their Duration duced but would produce fcarce any Alteration in the Arc of their Orbit, which Hiftory of they defcribe during the Time they are visible. However, it no lefs the Apariconfirms the Theory of Newton, to have fhewn, that in this Por- tions of the tion of their Orbit, they obferve the Law of Kepler, that of the Areas the fame being proportional to the Times, and that the Sun attracts them in the Circumftanfame Manner as all the other celeftial Bodies, in the inverfe Ratio of ces, and at equal Interthe Squares of the Distances. X. Comets in vals. Halley, on examining the famous Comet of 1680, having found that Halley em the Obfervations of a Comet recorded in History, agreed with it in very period of ploys the remarkable Circumstances, and that they had appeared at the Distance of the Comet 575 Years from each other, conjectured, that it might be but one and the of 1680 to fame Comet, performing its Revolution about the Sun in this Period, he Orbit. rectify its therefore fuppofed the Parabola to be changed into an Ellipfe defcribed by the Comet in 575 Years, and having the fame Focus and Vertex with the Parabola. Calculating afterwards, the Places of the Comet in this elliptic Orbit, he found them to agree perfectly with those where the Comet was obferved; fo that the Variation did not exceed the Difference found between the calculated Places of the Planets, and what are immediately deduced from Obfervation, though the Motions of the Planets have been the Object of the Inquiries of Philofophers for thoufands of Years. XI. Besides the Comet of 1680, Halley found three others, which nearly agreed, thofe of 1531, of 1607, and of 1682, the three Parabolas were fituated after the fame Manner, the perihelion Distances were equal, and the Intervals of Time 75 or 76 Years; he conjectured that it might be but one and the fame Comet, and that the Differencce in their Inclinations and Periods, might arife from the Attractions of the fu |