Monsieur Cassini His New and Exact Tables for the Eclipses of the First Satellite of Jupiter, Reduced to the Julian Stile, and Meridian of London

II. Monsieur Cassini his New and Exact Tables for the Eclipses of the First Satellite of Jupiter, reduced to the Julian Stile, and Meridian of London. Among the Books the Royal Academy of Sciences at Paris has lately gratified the World withal, there is one which has for Title, Recueil d'Observations faites en plusieurs Voyages pour perfectionner l'Astronomie & la Geographie, Avec divers traitez Astronomiques. In which those Sçavans have set a very commendable Example in ascertaining by undoubted Observations the true Geographical Site of all the Principal Ports of France, which it were to be wished other Nations would imitate. By this Survey they have demonstrated the Encroachments their Geographers, and particularly Sanson, had made on the Sea to enlarge their Kingdom, and have retrenched more of their Usurpations on the West, South, and North, than all their Acquirs on the East amount to twice told. The Method they have used to determine the Longitudes of their Places, is by the Observation of the Eclipses of the First Satellite of Jupiter, which they find almost instantaneous, and with good Telescopes discernable almost to the very Opposition of Jupiter to the Sun: And it may be said, that this Account of the Longitudes observed, has put it past doubt that this is the very best way, could portable Telescopes suffice for the Work. And could these Satellites be observed at Sea, a Ship at Sea might be enabled to find the Meridian she was in, by help of the Tables Monsieur Cassini has given us in this Volume, discovering with very great exactness the said Eclipses, beyond what we can yet hope to do by the Moon, tho' she seem to afford us the only means Practicable for the Seaman. However before Saylors can make use of the Art of finding the Longitude, it will be requisite that the Coast of the whole Ocean be first laid down truly, for which work this Method by the Satellites is most apposite: And it may be hoped that either the true Geometrick Theory of the Moon may be discovered, by the time the Charts are com- compleated; or else that some Invention of shorter Telescopes manageable on Ship-board, may suffice to shew the Eclipses of the Satellites at Sea, at least those of the Third Satellite, which fall at a good distance from the Body of Jupiter, being near three times as far from him as the first. The last but most considerable Treatise of this Collection gives the aforesaid Tables for computing the Motions of Jupiter's Satellites, but more especially those, for speedy finding the Eclipses of the first or innermost. Wherein Monsieur Cassini has employed his Skill to make easy and obvious to all Capacities the Calculation of them, which is otherwise operose to the Skilful, and not to be undertaken by the less knowing, who yet perhaps would be willing to find the Longitude of the Places they live in. These Tables have for Principles, That the innermost Satellite revolves to the Sun in $1^d\ 18^h\ 28'^m\ 36''$. so precisely, that in 100 Years the difference is not sensible; That in the time of the Revolution of Jupiter to his Aphelion, which he supposes in $4332^d\ 14^h\ 52'^m\ 48''$, this Satellite makes exactly 2448 Months or Revolutions to the Sun: and dividing the Orbite of Jupiter into 2448 parts, he has in a large Table of Equation shewn what is the inequality of the Motion of Jupiter in each Revolution reduced to Time, assuming Thirdly, the greatest Equation of Jupiter $5^o\ 30'$. whence the hourly Motion of the Satellite from Jupiter being $8^o\ 26' \frac{1}{4}$, it follows, that the greatest inequality (Jupiter passing the Signs of Cancer and Capricorn,) amounts to $39'.8''$. of time, to be added in Cancer, subtracted in Capricorn. Lastly, As to the Epoch or beginning of this Series of Revolutions, he has determined the Aphelion of Jupiter about 1° Degree forwarder than Astronomia Carolina, and above 2 Degrees more than the Rudolphine Tables, viz. precisely in $9^o$ of Libra, in the beginning of this Century, which perhaps he finds the proper Motion of Jupiter about the Sun at this time to require; and the number of Revolutions since Jupiter was last in Perihelic, is here stated Num. I. A second Inequality is that which depends on the distance of the Sun from Jupiter, which he says Monsieur Romer did most ingeniously explain by the Hypothesis of the Motion of Light; to which yet Cassini by his manner of Calculus seems not to assent, though it be hard to imagine how the Earth's Position in respect of Jupiter should any way affect the Motion of the Satellites. This Inequality he makes to amount to two Degrees in the Satellites Motion, or $14' \cdot 10''$. of Time, wherein he supposes the Eclipses to happen so much sooner when Jupiter Opposes the Sun, than when he is in Conjunction with him. The distribution of this Inequality he makes wholly to depend on the Angle at the Sun between the Earth and Jupiter, without any regard to the Excentricity of Jupiter, (who is sometimes a Semidiameter of the Earth's Orb farther from the Sun than at other times) which would occasion a much greater difference than the Inequality of Jupiter and the Earth's Motion, both of which are accounted for in these Tables with great Skill and Address. But what is most strange, he affirms that the same Inequality of two Degrees in the Motion, is likewise found in the other Satellites, requiring a much greater time, as above two Hours in the fourth Satellite: which if it appeared by Observation, would overthrow Monsieur Romer's Hypothesis entirely. Yet I doubt not herein to make it demonstratively plain, that the Hypothesis of the progressive Motion of Light is found in all the other Satellites of Jupiter to be necessary, and that it is the same in all; there being nothing near so great an Annual Inequality as Monsieur Cassini supposes in their Motions, by his Table, pag. 9. and his Praecepta Calculi. The Method however used to compute this is very Curious; for having found that whilst the Sun revolves to Jupiter, there pass $398^d. \cdot 21^h. \cdot 13'$. wherein are made $225 \frac{1}{2}$ Revolutions of the Satellite to Jupiter, the Number of Revolutions since Jupiter was last in Opposition to the Sun, is what he calls Num. II. in which the Inequality of the Earth's Motion is allowed for in the Months, and that of Jupiter's Orb by a Table of the Equation of Num. II. amounting in all to $3 \frac{1}{2}$ Revolutions of the Satellite to Jupiter. This in the Tables following I have thought fit to leave out, shewing how to find it by help of the former Equation of Num. I. The Numbers are in effect the same with Monsieur Cassini's, only reduced to our Stile and Meridian, and the form of them abridged, and his hoped amended. See Philos. Transact. No. 136. Epocha Revolutionum primi Satellitis ad Jovis Umbrae sub Meridiano Londinensi. | Anno Jul Curr. | D. h. | Num I | Num II | Anno Jul Curr. | D. h. | Num I | Num II | |---------------|-------|-------|--------|---------------|-------|-------|--------| | 1660 | 0 11 5 48 | 968 | 2006 | 1690 | 0 16 8 24 | 2263 | 810 | | 61 | 0 17 24 | 1174 | 1812 | 91 | 0 6 20 | 21 | 616 | | 62 | 1 9 57 36| 1381 | 1629 | 92 | 0 15 12 | 228 | 433 | | 63 | 1 9 12 | 1587 | 1435 | 93 | 0 5 11 48| 434 | 239 | | 1664 | 1 8 49 24| 1794 | 1251 | 94 | 1 13 52 | 641 | 55 | | 65 | 0 23 1 | 2000 | 1057 | 1695 | 1 4 3 36 | 847 | 2115 | | 66 | 0 13 12 36| 2206 | 864 | 96 | 1 12 43 48| 1054 | 1931 | | 67 | 0 3 24 12| 2412 | 670 | 97 | 1 2 55 24| 1260 | 1737 | | 68 | 0 12 4 24| 171 | 486 | 98 | 0 17 7 | 1466 | 1544 | | 1669 | 0 2 16 | 377 | 292 | 99 | 0 7 18 36| 1672 | 1350 | | 70 | 1 10 56 12| 584 | 109 | 1700 | 0 15 58 48| 1879 | 1166 | | 71 | 1 7 48 | 790 | 2169 | 01 | 0 6 10 24| 2085 | 973 | | 72 | 1 9 48 | 997 | 1985 | 02 | 1 14 50 36| 2292 | 789 | | 73 | 0 23 59 36| 1203 | 1791 | 03 | 1 5 2 12 | 50 | 595 | | 1674 | 0 14 11 12| 1409 | 1597 | 04 | 1 13 42 24| 257 | 411 | | 75 | 0 4 22 48| 1615 | 1403 | 1705 | 1 3 54 | 463 | 218 | | 76 | 0 13 3 | 1822 | 1219 | 06 | 0 18 5 36| 669 | 24 | | 77 | 0 3 14 36| 2028 | 1025 | 07 | 0 8 17 12| 875 | 2084 | | 78 | 1 11 54 48| 2235 | 841 | 08 | 0 16 57 24| 1082 | 1902 | | 1679 | 1 2 6 24 | 2441 | 647 | 09 | 0 7 9 | 1288 | 1706 | | 80 | 1 10 46 36| 200 | 464 | 1710 | 1 15 49 12| 1495 | 1523 | | 81 | 1 0 58 12| 466 | 279 | 11 | 1 6 0 48| 1701 | 1329 | | 82 | 0 15 9 48| 612 | 76 | 12 | 1 14 41 | 1908 | 1145 | | 83 | 0 5 21 24| 818 | 2136 | 13 | 1 4 52 36| 2114 | 951 | | 1684 | 0 14 1 36| 1025 | 1953 | 14 | 0 19 4 12| 2320 | 758 | | 85 | 0 4 13 12| 1231 | 1759 | 1715 | 0 9 15 48| 78 | 564 | | 86 | 1 12 53 24| 1438 | 1575 | 16 | 0 17 56 | 285 | 380 | | 87 | 1 3 5 0 | 1644 | 1381 | 17 | 0 8 7 36| 491 | 186 | | 88 | 1 11 45 12| 1851 | 1197 | 18 | 1 16 47 48| 698 | 03 | | 1689 | 1 1 56 48| 2057 | 1004 | 19 | 1 6 59 24| 904 | 2063 | | 1720 | 1 15 39 36| 1111 | 1879 | Tabula Tabula Revolutionum primi Satellitis Jovis in Anno. | Januarius | Num. I | Num. II | |-----------|--------|---------| | D. h. | | | | 0 0 0 0 | 0 | 0 | | 1 18 28 36| 1 | 1,0 | | 3 12 57 12| 2 | 2,1 | | 5 7 25 48 | 3 | 3,1 | | 7 1 54 24 | 4 | 4,1 | | 8 20 23 | 5 | 5,2 | | 10 14 51 36| 6 | 6,2 | | 12 9 20 12 | 7 | 7,2 | | 14 3 48 48 | 8 | 8,2 | | 15 22 17 24| 9 | 9,3 | | 17 16 46 | 10 | 10,3 | | 19 11 14 36| 11 | 11,3 | | 21 5 43 12 | 12 | 12,3 | | 23 0 11 48 | 13 | 13,4 | | 24 18 40 24| 14 | 14,4 | | 26 13 9 | 15 | 15,4 | | 28 7 37 36 | 16 | 16,5 | | 30 2 6 12 | 17 | 17,5 | | 31 20 34 48| 18 | 18,5 | | Februarius | Num. I | Num. II | |------------|--------|---------| | D. h. | | | | 0 20 34 48 | 18 | 18,5 | | 2 15 3 24 | 19 | 19,6 | | 4 9 32 0 | 20 | 20,6 | | 6 4 0 36 | 21 | 21,6 | | 7 22 29 12 | 22 | 22,6 | | 9 16 57 48 | 23 | 23,7 | | 11 11 26 24| 24 | 24,7 | | 13 5 55 0 | 25 | 25,7 | | Martius | Num. I | Num. II | |-----------|--------|---------| | D. h. | | | | 1 4 12 24 | 34 | 34,8 | | 2 22 41 | 35 | 35,8 | | 4 17 9 | 36 | 36,8 | | 6 11 38 | 37 | 37,9 | | 8 6 6 48 | 38 | 38,9 | | 10 0 35 | 39 | 39,9 | | 11 19 4 | 40 | 40,9 | | 13 13 32 | 41 | 41,9 | | 15 8 1 | 42 | 42,9 | | 17 2 29 | 43 | 43,9 | | 18 20 58 | 44 | 44,9 | | 20 15 27 | 45 | 45,9 | | 22 9 55 | 46 | 46,9 | | 24 4 24 | 47 | 47,9 | | 25 22 52 | 48 | 48,9 | | 27 17 21 | 49 | 49,9 | | 29 11 50 | 50 | 50,9 | | 31 6 18 | 51 | 51,9 | Aprilis. Tabula Revolutionum primi Satellitis Jovis in Anno. | Aprilis | Maius | Iunius | |---------|-------|--------| | D. h. | Num. | Num. | | 0 6 18 36 | 51 51,9 | 14 12 13 36 | 76 76,4 | | 2 0 47 12 | 52 52,9 | 16 6 42 12 | 77 77,4 | | 3 19 15 48 | 53 53,9 | 18 1 16 48 | 78 78,4 | | 5 13 44 24 | 54 54,9 | 19 19 39 24 | 79 79,3 | | 7 8 13 0 | 55 55,9 | 21 14 8 0 | 80 80,3 | | 9 2 41 36 | 56 56,9 | 23 8 36 36 | 81 81,2 | | 10 21 10 12 | 57 57,9 | 25 3 5 12 | 82 82,2 | | 12 15 38 48 | 58 58,9 | 26 21 33 18 | 83 83,2 | | 14 10 7 24 | 59 59,9 | 28 16 2 24 | 84 84,2 | | 16 4 36 0 | 60 60,8 | 30 10 31 0 | 85 85,2 | | 17 23 4 36 | 61 61,8 | | | | 19 17 33 12 | 62 62,8 | | | | 21 12 1 48 | 63 63,8 | 1 4 59 36 | 86 86,1 | | 23 6 30 24 | 64 64,8 | 2 23 28 12 | 87 87,1 | | 25 0 59 0 | 65 65,7 | 4 17 56 48 | 88 88,0 | | 26 19 27 36 | 66 66,7 | 6 12 25 24 | 89 89,0 | | 28 13 56 12 | 67 67,7 | 8 6 54 0 | 90 90,0 | | 30 8 24 48 | 68 68,6 | 10 1 22 36 | 91 90,9 | | | | 11 19 51 12 | 92 91,9 | | | | 13 14 19 48 | 93 92,9 | | | | 15 8 48 24 | 94 93,8 | | Maius | | | | 0 8 24 48 | 68 68,6 | 17 3 17 0 | 95 94,8 | | 2 2 53 24 | 69 69,6 | 18 21 45 36 | 96 95,7 | | 3 21 22 0 | 70 70,6 | 20 16 14 12 | 97 96,7 | | 5 15 50 36 | 71 71,6 | 22 10 42 48 | 98 97,7 | | 7 10 19 12 | 72 72,5 | 24 5 11 24 | 99 98,6 | | 9 4 47 48 | 73 73,5 | 25 23 40 0 | 100 99,6 | | 10 23 16 24 | 74 74,5 | 27 18 8 36 | 101 100,6 | | 12 17 45 0 | 75 75,5 | 29 12 37 12 | 102 101,5 | Julius. Tabula Revolutionum primi Satellitis Jovis in Anno. | Julius | Num. I | Num. II | Augustus | Num. I | Num. II | |--------|--------|---------|----------|--------|---------| | D. h. | | | D. h. | | | | 1 | 7 | 48 | 103 | 102,5 | | | 3 | 1 | 34 | 24 | 104 | 103,5 | | 4 | 20 | 3 | 105 | 104,4 | | | 6 | 14 | 31 | 36 | 106 | 105,4 | | 8 | 9 | 0 | 12 | 107 | 106,4 | | 10 | 3 | 28 | 48 | 108 | 107,3 | | 11 | 21 | 57 | 24 | 109 | 108,3 | | 13 | 16 | 26 | 0 | 110 | 109,3 | | 15 | 10 | 54 | 36 | 111 | 110,2 | | 17 | 5 | 23 | 12 | 112 | 111,2 | | 18 | 23 | 51 | 48 | 113 | 112,2 | | 20 | 18 | 20 | 24 | 114 | 113,1 | | 22 | 12 | 49 | 0 | 115 | 114,1 | | 24 | 7 | 17 | 36 | 116 | 115,1 | | 26 | 1 | 46 | 12 | 117 | 116,0 | | 27 | 20 | 14 | 48 | 118 | 117,0 | | 29 | 14 | 43 | 24 | 119 | 118,0 | | 31 | 9 | 12 | 0 | 120 | 119,0 | | Augustus | Num. I | Num. II | |----------|--------|---------| | 0 | 9 | 12 | | 2 | 3 | 40 | | 3 | 22 | 9 | | 5 | 16 | 37 | | 7 | 11 | 6 | | 9 | 5 | 35 | | 11 | 0 | 3 | | 12 | 18 | 32 | | 14 | 13 | 0 | | September | Num. I | Num. II | |-----------|--------|---------| | 1 | 5 | 46 | | 3 | 0 | 15 | | 4 | 18 | 44 | | 6 | 13 | 12 | | 8 | 7 | 41 | | 10 | 2 | 9 | | 11 | 20 | 38 | | 13 | 15 | 7 | | 15 | 9 | 35 | | 17 | 4 | 4 | | 18 | 22 | 32 | | 20 | 17 | 1 | | 22 | 11 | 30 | | 24 | 5 | 58 | | 26 | 0 | 27 | | 27 | 18 | 55 | | 29 | 13 | 24 | Tabula Revolutionum primi Satellitis Jovis in Anno. | October | November | |---------|----------| | D. h. | D. h. | | Num. I | Num. II | | Num. I | Num. II | | October | |---------| | D. h. | | Num. I | | Num. II | | November | |----------| | D. h. | | Num. I | | Num. II | | December | |----------| | D. h. | | Num. I | | Num. II | Num. Tabula Primae Æquationis Conjunctionum primi Satellitis cum Jove. | Num | Æquat. | Num | Æquat. | Num | Æquat. | Num | Æquat. | |-----|--------|-----|--------|-----|--------|-----|--------| | 0 | 0 | 200 | 28 | 400 | 34 | 600 | 39 | | 10 | 1 | 210 | 28 | 410 | 34 | 610 | 39 | | 20 | 2 | 220 | 29 | 420 | 35 | 620 | 39 | | 30 | 3 | 230 | 30 | 430 | 35 | 630 | 38 | | 40 | 4 | 240 | 30 | 440 | 36 | 640 | 38 | | 50 | 5 | 250 | 31 | 450 | 36 | 650 | 38 | | 60 | 6 | 260 | 32 | 460 | 36 | 660 | 38 | | 70 | 7 | 270 | 32 | 470 | 37 | 670 | 38 | | 80 | 8 | 280 | 33 | 480 | 37 | 680 | 38 | | 90 | 9 | 290 | 33 | 490 | 37 | 690 | 37 | | 100 | 10 | 300 | 34 | 500 | 37 | 700 | 37 | | 110 | 11 | 310 | 34 | 510 | 38 | 710 | 37 | | 120 | 12 | 320 | 35 | 520 | 38 | 720 | 37 | | 130 | 13 | 330 | 35 | 530 | 38 | 730 | 36 | | 140 | 14 | 340 | 36 | 540 | 38 | 740 | 36 | | 150 | 15 | 350 | 36 | 550 | 38 | 750 | 36 | | 160 | 16 | 360 | 36 | 560 | 38 | 760 | 35 | | 170 | 17 | 370 | 37 | 570 | 37 | 770 | 35 | | 180 | 18 | 380 | 37 | 580 | 37 | 780 | 34 | | 190 | 19 | 390 | 37 | 590 | 37 | 790 | 34 | | 200 | 20 | 400 | 34 | 600 | 39 | 800 | 33 | | 210 | 20 | 410 | 34 | 610 | 39 | 810 | 33 | | 220 | 21 | 420 | 35 | 620 | 39 | 820 | 32 | | 230 | 22 | 430 | 35 | 630 | 38 | 830 | 32 | | 240 | 23 | 440 | 36 | 640 | 38 | 840 | 31 | | 250 | 24 | 450 | 36 | 650 | 38 | 850 | 31 | | 260 | 25 | 460 | 36 | 660 | 38 | 860 | 30 | | 270 | 25 | 470 | 37 | 670 | 38 | 870 | 29 | | 280 | 26 | 480 | 37 | 680 | 38 | 880 | 29 | | 290 | 27 | 490 | 37 | 690 | 37 | 890 | 28 | | 300 | 28 | 500 | 37 | 700 | 37 | 900 | 27 | | | | | | | | | | | | | | | | | | | P p Tabula Tabula Secundae Aequationis Conjunctionum primi Satellitis cum Jove. | Num. | Æquat. add. | Æquat. add. | Æquat. add. | Æquat. add. | |------|-------------|-------------|-------------|-------------| | 0 | 0' 0'' | 28 | 56 | 84 | | 1 | 0 | 29 | 57 | 85 | | 2 | 1 | 30 | 58 | 86 | | 3 | 2 | 31 | 59 | 87 | | 4 | 3 | 32 | 60 | 88 | | 5 | 4 | 33 | 61 | 89 | | 6 | 6 | 34 | 62 | 90 | | 7 | 8 | 35 | 63 | 91 | | 8 | 10 | 36 | 64 | 92 | | 9 | 14 | 37 | 65 | 93 | | 10 | 17 | 38 | 66 | 94 | | 11 | 20 | 39 | 67 | 95 | | 12 | 23 | 40 | 68 | 96 | | 13 | 27 | 41 | 69 | 97 | | 14 | 32 | 42 | 70 | 98 | | 15 | 37 | 43 | 71 | 99 | | 16 | 42 | 44 | 72 | 100 | | 17 | 47 | 45 | 73 | 101 | | 18 | 53 | 46 | 74 | 102 | | 19 | 58 | 47 | 75 | 103 | | 20 | 1 | 48 | 76 | 104 | | 21 | 11 | 49 | 77 | 105 | | 22 | 18 | 50 | 78 | 106 | | 23 | 25 | 51 | 79 | 107 | | 24 | 32 | 52 | 80 | 108 | | 25 | 40 | 53 | 81 | 109 | | 26 | 47 | 54 | 82 | 110 | | 27 | 56 | 55 | 83 | 111 | | 28 | 4 | 56 | 84 | 112 | Tabula Tabula Dimidiae Morae primi Satellitis in Umbra Jovis. | Num. | H. ' " | |------|-------| | 0 | 1 4 56 | | 40 | 1 4 33 | | 80 | 1 4 12 | | 120 | 1 3 59 | | 160 | 1 3 48 | | 200 | 1 3 39 | | 240 | 1 3 38 | | 280 | 1 3 48 | | 320 | 1 4 1 | | 360 | 1 4 16 | | 400 | 1 4 36 | | 440 | 1 4 56 | | 480 | 1 5 18 | | 520 | 1 5 41 | | 560 | 1 6 1 | | 600 | 1 6 21 | | 640 | 1 6 39 | | 680 | 1 6 53 | | 720 | 1 7 3 | | 760 | 1 7 11 | | 800 | 1 7 15 | | 840 | 1 7 13 | | 880 | 1 7 9 | | 920 | 1 7 2 | | 960 | 1 6 54 | | 1000 | 1 6 39 | | 1040 | 1 6 22 | | 1080 | 1 6 5 | | 1120 | 1 5 45 | | 1160 | 1 5 26 | | 1200 | 1 5 6 | | Num. | H. ' " | |------|-------| | 1200 | 1 5 6 | | 1240 | 1 4 48 | | 1280 | 1 4 26 | | 1320 | 1 4 7 | | 1360 | 1 3 54 | | 1400 | 1 3 38 | | 1440 | 1 3 38 | | 1480 | 1 3 44 | | 1520 | 1 3 52 | | 1560 | 1 4 7 | | 1600 | 1 4 24 | | 1640 | 1 4 42 | | 1680 | 1 5 0 | | 1720 | 1 5 22 | | 1760 | 1 5 46 | | 1800 | 1 6 10 | | 1840 | 1 6 28 | | 1880 | 1 6 45 | | 1920 | 1 6 57 | | 1960 | 1 7 7 | | 2000 | 1 7 13 | | 2040 | 1 7 14 | | 2080 | 1 7 15 | | 2120 | 1 7 15 | | 2160 | 1 7 10 | | 2200 | 1 6 49 | | 2240 | 1 6 32 | | 2280 | 1 6 15 | | 2320 | 1 5 58 | | 2360 | 1 5 38 | | 2400 | 1 5 18 | | 2440 | 1 5 2 | Pp 2 TABULA TABULA AEQUATIONIS DIERUM cum Solis loco adeunnda. | G. | S'' | A'' | II | S'' | S'' | m | |----|-----|-----|----|-----|-----|---| | 0 | 7 | 45 | 1 | 11 | 4 | 3 | | 1 | 7 | 26 | 1 | 24 | 4 | 0 | | 2 | 7 | 7 | 1 | 37 | 3 | 56 | | 3 | 6 | 48 | 1 | 49 | 3 | 51 | | 4 | 6 | 29 | 2 | 1 | 3 | 45 | | 5 | 6 | 10 | 2 | 12 | 3 | 39 | | 6 | 5 | 51 | 2 | 23 | 3 | 32 | | 7 | 5 | 31 | 2 | 33 | 3 | 25 | | 8 | 5 | 11 | 2 | 43 | 3 | 17 | | 9 | 4 | 51 | 2 | 53 | 3 | 9 | | 10 | 4 | 31 | 3 | 3 | 3 | 0 | | 11 | 4 | 11 | 3 | 13 | 2 | 51 | | 12 | 3 | 52 | 3 | 22 | 2 | 41 | | 13 | 3 | 33 | 3 | 30 | 2 | 31 | | 14 | 3 | 14 | 3 | 37 | 2 | 21 | | 15 | 2 | 55 | 3 | 43 | 2 | 10 | | 16 | 2 | 37 | 3 | 48 | 2 | 0 | | 17 | 2 | 19 | 3 | 53 | 1 | 49 | | 18 | 2 | 1 | 3 | 57 | 1 | 37 | | 19 | 1 | 43 | 4 | 1 | 1 | 25 | | 20 | 1 | 26 | 4 | 5 | 1 | 13 | | 21 | 1 | 9 | 4 | 8 | 1 | 4 | | 22 | 0 | 52 | 4 | 10 | 0 | 49 | | 23 | 0 | 35 | 4 | 12 | 0 | 37 | | 24 | 0 | 19 | 4 | 13 | 0 | 24 | | 25 | 0 | 3 | 4 | 11 | 0 | 10 | | 26 | 0 | A | 12 | 4 | 9 | 0 | | 27 | 0 | 27 | 4 | 8 | 0 | 16 | | 28 | 0 | 42 | 4 | 6 | 0 | 29 | | 29 | 0 | 57 | 4 | 5 | 0 | 44 | | 30 | 1 | 11 | 4 | 3 | 0 | 59 | TABULAE | G. | A'' | m' | f' | v' | w' | S'' | S' | |----|-----|----|----|----|----|-----|----| | 0 | 7 | 44 | 15 | 34 | 13 | 25 | 0 | | 1 | 8 | 5 | 15 | 42 | 13 | 7 | 0 | | 2 | 8 | 25 | 15 | 48 | 12 | 48 | 0 | | 3 | 8 | 45 | 15 | 53 | 12 | 29 | 0 | | 4 | 9 | 5 | 15 | 57 | 12 | 10 | 1 | | 5 | 9 | 25 | 16 | 1 | 11 | 50 | 1 | | 6 | 9 | 44 | 16 | 5 | 11 | 30 | 2 | | 7 | 10 | 3 | 16 | 7 | 11 | 10 | 2 | | 8 | 10 | 22 | 16 | 8 | 10 | 49 | 3 | | 9 | 10 | 41 | 16 | 9 | 10 | 28 | 3 | | 10 | 11 | 0 | 16 | 9 | 10 | 6 | 3 | | 11 | 11 | 19 | 16 | 9 | 9 | 42 | 4 | | 12 | 11 | 38 | 16 | 8 | 9 | 17 | 4 | | 13 | 11 | 57 | 16 | 7 | 8 | 51 | 5 | | 14 | 12 | 15 | 19 | 5 | 8 | 25 | 5 | | 15 | 12 | 33 | 16 | 1 | 7 | 58 | 6 | | 16 | 12 | 50 | 15 | 56 | 7 | 31 | 6 | | 17 | 13 | 7 | 15 | 50 | 7 | 5 | 7 | | 18 | 13 | 22 | 15 | 44 | 6 | 38 | 7 | | 19 | 13 | 36 | 15 | 37 | 6 | 12 | 7 | | 20 | 13 | 49 | 15 | 30 | 5 | 45 | 8 | | 21 | 14 | 2 | 15 | 22 | 5 | 19 | 8 | | 22 | 14 | 14 | 15 | 13 | 4 | 52 | 9 | | 23 | 14 | 26 | 15 | 3 | 4 | 26 | 9 | | 24 | 14 | 37 | 14 | 52 | 3 | 58 | 9 | | 25 | 14 | 47 | 14 | 46 | 3 | 30 | 10 | | 26 | 14 | 57 | 14 | 27 | 3 | 1 | 10 | | 27 | 15 | 7 | 14 | 13 | 2 | 31 | 10 | | 28 | 15 | 16 | 13 | 58 | 2 | 1 | 11 | | 29 | 15 | 25 | 13 | 42 | 1 | 30 | 11 | | 30 | 15 | 34 | 13 | 25 | 0 | 59 | 11 | This last Table of the Æquation of Natural Days might have been spared, as being publish'd in several other places, but it was thought proper to have all the Elements of this Calculus together, that there might be no occasion of any other Book to perform it. The Use of the Tables. To any given Year, Month, and Day, to find the next Eclipse of the first Satellite of Jupiter. I. In the Table of Epochæ (pag. 240.) find the Year of our Lord, and set down the Day, Hours, Minutes, and Seconds, with the Num. I. and Num. II. thereto annext; and (in pag. 241 and the following) seek the Month, and day of the Month, with the Hours and Minutes, and Num. I. and II. affixt, and add them together: and the respective Sums shall shew the mean time of the middle of the Eclipse sought, with Num. I. and Num. II. required. But it must be observed, that in January and February in the Leap-Year one Day is to be added to the Day thus found. II. If Num. I. be found less than 1224 with Num. I.; or if greater than 2448, Subtracting 2448 therefrom, with the residue, enter the Table, pag. 245. and you will have the first Æquation to be added to the mean Time before found. But if Num. I. be less than 2448, but greater than 1224, Subtract it from 2448, and entering the same Table with the remainder, you shall have the first Æquation to be subtracted from the mean Time. Then Divide the Minutes of the said first Æquation by 11, or rather 11, and the Quote shall be the Æquation of Num. II. (answering to the Eccentric Motion of Jupiter) to be added thereto when the first Æquation Subtracts, and è contra subtracted when that adds. III. If III. If Num. II. thus æquated exceed 225.4, Subtract 225.4 therefrom; and if the remainder or Num. II. be less than 113, with the said remainder or Number; or if greater than 113, with the complement thereof to 225.4. Seek in Table pag. 246. the second Æquation, which being added to the Time before found, gives the true Time of the middle of the Eclipse. IV. With Num. I. in Tab. pag. 247, seek the half Continuance of the Total Eclipse, which is to be added for the Emerfion when the æquated Num. II. is less than 113, or if more than 225.4, it be less than 338. But if it exceed 113 or 338, then is the Semimora to be substracted for the Immersion. V. Lastly, with the Sun’s true Place take out the Æquation of Natural Days (in Tab. pag. 248.) which added or substracted according to the Title, gives the time of the Immersion or Emerfion sought. Now how few Figures serve for this Computation, will best appear by an Example or two. Anno 1677. September 17th 8h. 9'. 40''. at Greenwich, Mr. Flamsteed observed the first Satellite to begin to Emerge; that is 8h. 9'. 20''. at London. | Year | Month | Day | Hour | Minute | Second | Num. I. | Num. II. | |------|-------|-----|------|--------|--------|--------|---------| | 1677 | Sept. | 17 | 8 | 9 | 40 | 2028 | 102.5 | | | | | | | | 147 | 145.5 | | | | | | | | 2175 | 248.0 | | | | | | | | 2448 | 2.3 | | | | | | | | 273 | 250.3 | | | | | | | | | 225.4 | | | | | | | | | 24.9 | | | | | | | | | | | | | | | | | | | Again, Again, Anno 1683. November 30th. 16h. 48'. 40'' under the Meridian of London, the Immersion of this Satellite was observed by E. Halley. | Year | Month | Day | Hour | Minute | Second | Num. I | Num. II | |------|-------|-----|------|--------|--------|--------|---------| | 1683 | Novemb. | 30 | 12 | 5 | 24 | 818 | 213.6 | | | | | | | | 189 | 188.2 | | | | | | | | 1007 | 401.8 | | | | | | | | 1.8 | — | | | | | | | | 400.0 | | | | | | | | | 225.4 | | | | | | | | | 174.6 | | | | | | | | | 50.8 | | A Third Example shall be the Emerision observed at Paris by Monsieur Cassini Anno 1693. January 14th. 10h. 40'. 28''. that is, at London at 10h. 30'. 48''. | Year | Month | Day | Hour | Minute | Second | Num. I | Num. II | |------|-------|-----|------|--------|--------|--------|---------| | 1693 | Jan. | 14 | 3 | 48 | 48 | 434 | 23.9 | | | | | | | | 8 | 8.2 | | | | | | | | 442 | 32.1 | | | | | | | | 3.2 | — | | | | | | | | 28.9 | | | | | | | | | 50.40' | | After this manner I have compared these Tables with many good and certain Observations, and scarce ever find them err above three or four Minutes of Time; which proceeds, as may well be conjectured, from some small small Eccentricity in its Motion, and from the Oval Figure of Jupiter's Body, whose quick diurnal Rotation has by its Vis Centrifuga dilated his Equinoctial, and made his Meridians much Elliptical, so as to be discernable by the Telescope. Mr. Newton has shewn that his Polar Diameter is to that of his Equinoctial as 40 to 41 nearly. But we may hope future Observations may shew how to divide those compounded causes of Error, and correct them; which Errors are exceeding small in comparison of the short time that the Satellites have been discovered, and argue the Skill and Diligence of the deservedly Famous Author of these Tables. I had almost forgot the Construction of the Table, pag. 247. shewing the half continuance of these Eclipses: In this the Semidiameter of the shadow of Jupiter is made by Cassini just 10 Degrees, and that of the Satellite 30'; and the Satellites Ascending Node being supposed in 15° of Aquarius, at the end of this Century, (that is, 55° 20' before the Perihelion of Jupiter) it will thence follow, that Num. I. being 816 or 2102, Jupiter passes the Nodes of the Satellites Orb, and consequently these Eclipses are Central, and of the greatest Duration. But Num. I. being 215 or 1481, the Satellite passes the shadow with the greatest Obliquity, viz. 2° 55' from the Center, whence the Semimora becomes of all the shortest. This Table is not however so nicely computed, but that it may admit of Correction in the Seconds, if a small part of a Minute were considerable in this affair. The Tables of the other three Satellites not being so perfect or exact as those of the first, having greater inequalities, are here given in another form, requiring the assistance of the Tables of Jupiter's proper motion. The Periods of their Revolutions to Jupiter's thade are as follows: | Period | |--------| | | Period. Secundi. 3d. 23h. 17'. 54''. 3'' five 2 1/3 Rev.primi. Period. Tertii. 7 3 59 39 22 five 4 1/3 Rev.primi. Period. Quartii. 16 18 5 6 50 five 9 1/3 Rev.primi. Whence the Table of the first Equation of the First Satellite, pag. 245, or Monsieur Cassini's larger Table, may by an easy Reduction serve the other three; the Equation of the Second being 2 1/3, or twice the Minutes with half so many Seconds as there are Minutes in the Equation of the First, and the greatest Equation thereof of 1h. 18'. 35''. The Equation of the Third is 4 1/3 times greater than that of the First, and when greatest amounteth to 2h. 38'. 29''. And the Equation of the Fourth being 9 1/3 times that of the First, is had by Subtracting 1/2 and 1/4 from ten times the Equation of the First, whence the greatest becomes 6h. 10'. 28''. So that Num.I. and Num.II. as here collected for the First, may indifferently serve all the rest. As to the Second Equation of the other Satellites, Monsieur Cassini has, by his Praecepta Calculi (as is before mentioned) supposed the Minutes thereof to be increased in the same proportion; as instead of 14'. 10''. in the First, to be 28'. 27''. in the Second, 57'. 22''. in the Third, and no less than 2h. 14'. 7''. in the Fourth; whereas if this second Inequality did proceed from the successive propagation of Light, this Equation ought to be the same in all of them, which Monsieur Cassini says was wanting to be shewn, to perfect Monsieur Romer's Demonstration; wherefore he has rejected it as ill founded. But there is good cause to believe that his motive thereto, is what he has thought not proper to discover. And the following Observations do sufficiently supply the Defect complained of in the making out of that Hypothesis. Anno 1676. Octob. 2. Stil. Nov. Ch. 10'. 37''. app. but 5h. 59'. 37''. æq. time, Monsieur Cassini at Paris observed the Emerion of the Third Satellite from Jupiter's shadow. shadow. And again, Novemb. 14 following, 6h. 20'. 55''. app. Time, but 6h. 5'. 55''. æq. T. he observed the like Emerion of the same Satellite. The observed Interval of Time between these Emerions was 43d. 0h. 6'. 18'' which is 8'. 22''. more than 6 mean Revolutions of this Satellite, of which 4'. 27''. arises from the difference of the first Æquations and the greater continuance of the latter Eclipse; so that the other 4 Minutes is all that is left to answer for the difference of the second Æquations; and Num. II. in that time increasing from 48 to 72, gives 4'. 36''. for the difference of the second Æquations of the First Satellite. So that here the second Æquation of the Third is found rather less than that of the First, but the difference is so small, that it may rather be attributed to the uncertainty of Observation. Whereas according to Monsieur Cassini's Method of Calculating, instead of four Minutes it ought to be 18'. 38''. and the Interval of these two Emerions 43d. 0h. 21''. exceeding the Time observed by a whole quarter of an hour; which that Curious Observer could not be deceived in. The like appears yet more evidently in the Fourth Satellite. By the Observation of Mr. Flamsteed at Greenwich, Anno 1682. Sept. 24o. 17h. 45'. T. app. but 17h. 32'½ T. æq. the Fourth Satellite was seen newly come out of the shadow, so that about 17h. 30'. T. æq. the first beginning of Emerion was conjectured; and after five Revolutions, viz. Decemb. 17d. 11h. 16'. or 11h. 18'. T. æq. he again observed the first appearance of the Satellite beginning to Emerge, that is, after an Interval of 83d. 17h. 48'; whereas this Satellite makes five mean Revolutions in 83d. 18h. 25'½. Here we have 37'½ to be accounted for by the several Inequalities. Of this 21' is due to the first Æquations, which is reduced to 19' by the greater continuance of the latter Eclipse, Jupiter then approaching to his descending Node: So that there remains only 18'½ for the difference of the Second Æquations, tions, whilst the Earth approached Jupiter by more than the Radius of its own Orb; and the difference of the second Equations of the First Satellite being according to Cassini $8'30''$, the said difference in the Fourth ought to be $1^h20'5$ instead of $18\frac{1}{2}$; whence the Interval of these two Emergences would be according to his Precepts, but $83^d16^h46'$, instead of $83^d17^h48'$. observed. And whereas $18'\frac{1}{2}$ may seem too great a difference; it must be noted, first, that Monsieur Romer had stated the whole second Equation $22'.00''$, (vide Phil. Trans. Num. 136.) which Monsieur Cassini has diminished to $14'.10''$; so that instead of $8'\frac{1}{2}$, Monsieur Romer allows above $13'$; and secondly, that in the first of these Observations, being about half an hour before Sun-rise, the brightness of the Morning might well hinder the seeing of this smallest and slowest Satellite, till such time as a good part thereof was emerged. But I have exceeded the Bounds of my intended Discourse, and shall only Advertise, That these Tables are not Printed with the usual Care of the Imprimerie Royale à Paris, That the Tabula Revolutionum primi Satellitis Jovis in Annis 100, pag. 13 & seq. is faulty in these Years, 16, 39, 55, 98 & 99; as is also the Epocha for the Year 1700, pag. 99. where pro Num. I. 1853 lege 1873, and pro Num. II. 1004, lege 110,4: And that the Number of Revolutions of the Second Satellite in 100 Years, pag. 60, 61; of the Third, pag. 76,77; and of the Fourth, pag. 90, 91, are by a gross mistake of the Calculator, all false and erroneous, and must be amended by whosoever would use them. Which yet ought not in the least to be attributed to the Excellent Author, but rather to the Negligence of those employed by him. The Reader hereof is desired to amend these following Errata, which were discovered when it was too late. ERRATA. Pag. 238. lin. 24. pro $5^o30'$. leg. $5^o31'40''$. lin. 25. pro $8^o26'\frac{1}{4}$. leg. $8^o28'\frac{1}{2}$.