The calculation of solar eclipses without parallaxes: With a specimen of the same in the total eclipse of the sun, May 11. 1724. ... By Will. Whiston, ...

Table of contents

- [halftitle]

- A Calculation of the Great Eclipse of the Sun, April 22d. 1715 in the Morning, from Mr. Flamsteed's Tables; as corrected accord ing to Sr. Isaac Newtons Theory of the Moon in the Astronomical Lectures, with its Construction for London Rome and stockholme. By W: Whiston MA.

- The Construction Explaind

- [publishersnotice]

- [authorsnotice]

- [titlepage]

- LEMMATA: OR, Preparatory Propositions.

- I.

- SCHOLIA.

- II.

- III.

- IV.

- V.

- VI.

- VII.

- VIII.

- IX.

- X.

- XI.

- XII.

- XIII.

- XIV.

- XV.

- XVI.

- XVII.

- XVIII.

- XIX.

- XX.

- XXI.

- XXII.

- XXIII.

- XXIV.

- XXV.

- XXVI.

- PROBLEMS.

- I.

- II.

- III.

- IV.

- V.

- VI.

- VII.

- VIII.

- IX.

- X.

- XI.

- XII.

- XIII.

- XIV.

- XV.

- XVI.

- XVII.

- XVIII.

- XIX.

- XX.

- XXI.

- XXII.

- XXIII.

- XXIV.

- XXV.

- A PROPOSAL For the Discovery of the LONGI TUDE of the several Places of the Earth, by Total Eclipses of the Sun.

- Some Account of Observations late ly made with Dipping-Needles, in Order to discover the LONGI TUDE and LATITUDE at Sea.

- [diagrams]

- ERRATA.

THE CALCULATION OF Solar Eclipses WITHOUT PARALLAXES.

A Calculation of the Great Eclipse of the Sun, April 22d. 1715 in the Morning, from Mr. Flamsteed's Tables; as corrected according to Sr. Isaac Newtons Theory of the Moon in the Astronomical Lectures, with its Construction for London Rome and stockholme. By W: Whiston MA.

NB. The Inquisitive are desir'd nicely to Observe whether in such Places where the Eclipse is plainly Total, there be not streaks of Red Light just before & after that Total Darkness; and how long it is visible; For if there be, it will imply that 'tis an Atmosphere about the Moon that is the occasion of it, & by its duration the height of the same Atmosphere - may in some measure be determined also.

The Sun's True Place and Anomaly

s

°

′

″

s

°

′

″

Anno Dom. 1701

9

20

43

50

3

7

40

10

Years—14

11

29

36

46

11

40

April, days—21

3

19

24

24

15

hours—21

51

45

3

7

52

5

minutes—36

1

29

Place of the Perihel.

Sun's Mean Motion

1

10

38

14

1

10

38

14

Equation Added

1

36

27

☉s Mean Motion

Sun's True Place

1

12

14

4•

10

2

46

9

☉s Mean Anomaly

The General Eclipse by the Calculation

From Dr. Halley

h

′

h

′

Beginning—

7.

30

7

21

Middle—

9.

51

9

42

End—

12.

12

12

3

The Eclipse at London from the Calculation.

From Dr. Halley

h

′

h

′

Beginning—

8.

18

8

7

Middle—

9.

24

9

13

End—

10.

35

10

24

The Construction Explaind

This Scheme represents one half of the Inlightened Disk of the Earth as seen from its Center Projected at the distance of the Moon. The Elliptick Parallels, with their Hours, represent the Cities of London, Rome, and Stockholme, as plac'd at those Hours at different Times. The Principal strait Line divided by dotts represents the Path of the Moons Center ever the Disk of the Earth: And by the Hours in Larger and those above and below in smaller Characters, the Position of the Center is determind at those Times for those Places respectively. So that if with a pair of Compasse• we take from the proper Scale the Semediameter of the Penumbra, and carry it along the Path till it first reaches to, and then leaves the same minute on any Parallel that is the very time of its Begin̄ing and Ending there. And if at any intermediate time in both you make Circles, one with the Moons Semidiameter on its Path; the other with the Suns on any of the 3 Parallels▪ the Intercepted part will shew the quantity of the Eclipse at that time in the Place to • which the Parallel you we does belong. And if you carry a Square along the Path, till the Perpendicular side cuts the same Hour and Minute there and in any Parallel, that is the Middle of the Eclipse there. Of all which you have examples in the Scheme. Only Note that the Center of the Penumbra at 21 after▪ 7 and at 3 after 12▪ which are the beginning and ending of the General Eclipse, extends beyond the▪ Copper Plate, and is to be supply'd by the Ren at the intersection of the proper Lines there to directed.

The Breadth of the intire Penumbra or paitial Eclipse upon this Perpendicular Plain, appears by the Construction to be no less than 1965 minutes or Geographical Mibes on each side of the Moons Path, or 3930 Miles in all; wch. correspond to many more on the Spherical Surface of the Earth: Nor is it all confind, as you may see here, to that Surface, but reaches▪ off a greatway into the empty Space beyond it Northward. The Lines which distinguish that breadth on each side into 12 parts denote so many Digits of the Suns Eclipses besides ⅓ for the Total shade) & the places both as to Long▪ 8▪ Lat: where the sun will at any Time be so much Eclipsed: And indeed I would willingly have procured a general Map here to have▪ shewd over what Countries and Places the intire Shadow would pass, as Doctor Halley has given us a particular Map of England for the Passage of the Total Shadow over it. But the nature of the Construction does not admit of that Projection (Such a Thing cannot be truly represented any other way than by the Copernicus; where there is a real Globe of the Earth, capable of a Diurnal motion, during the time of the Eclipse) the imposibility of which in all Perspective Projections of the Sphere renders that designs otherwise impracticable: Nor can I determin by this Construction whether the▪ Eclipse will be Total at London or not, because the Circles of the Sun and Moon at the Southern Limit seem here exactly coincident. But if we go by a Construction according to our Calculation the Digits Eclipsed at London will be hardly more than 11 ⅘ and the Shadow▪ will go full 30 Miles more Northward than in Dr. Halleys Map.

So that ye Middle of the General Eclipse in common or apparent Time▪ will be 50. 56. after Nine in the Morning▪ differing from Dr. Halley's Computation near 9 min. But Note that the Construction is accommodated to the Drs. Calculation.

Note also that hence the breadth of the Shadow of Total Darkness will be 98 Geographical Miles; and that its length on the Oblique Horizon of London will be near 150 Miles, as Dr. Halley's Description asserts.

But it must be here Observed that if in this Calculation the that and 6th New Equations of the Moon, taken from S. Isaac Newton's Theory, were neglected, this Calculation would be much nearer to Dr. Halley's, as it is now nearer to Mr. Flamsteed's. This Eclipse, if the Air prove clear for exact Observations, will go agreat •ay to determin how far those Equations are just; and how far they are necessary in the Calculation of the New and Full Moons, and of Eclipses, that happen only at those Times. St. Isaac Newton's third Equation, wch is no more than 13 to be Substr•ted, is here omitted, as very inconsiderable.

Moon's mean Motion

Motion of the Apogee

Motion of ye Nodr Retr.

s

°

′

″

s

°

′

″

s

°

′

″

Anno Dom.1701

10

15

19

50

11

8

18

20

4

27

24

20

Years—14

1

20

54

34

6

29

37

50

9

0

45

35

April▪ days—21

22

34

47

12

21

59

5

52

41

hours—21

11

31

46

5

51

2

47

minuted—36

1•

46

10

5

(Mean Time)

9

6

41

8

Moon's mean Motion

1

10

40

43

6

20

24

10

7

20

43

12

Suns mean Anomaly

10

2

40

9

An: Eq: Add.

16

8

An Eq: subst.

7▪

41

Physical parts Substracted

9

50

6

20

40

18

7

20

35

31

〈…〉 6th Equation Substra•

3

45

Mean Place Corrected

Mean Place Corrected

Sum

13

35

1

12

14

41

1

12

14

41

Moon's mean Place correcte•

1

10

27

8

Suns True Place

Suns True Place

True Place of the Apogee •ulot

6

27

5•

5

6

21

34

23

5

21

39

10

Moon's mean Anomaly

6

12

31

3

Annual Argument

☉• Distance from the Node

Equation Added

1

42

35

7

15

47

Equat: Subst.

25

19

Moon's Equat Place in its Orb

1

12

9

43

Equation Added

7

20

10

12

Sun's True Place

1

12

14

41

6

27

56

5

True Place of the Node

Moon's Distance from ye. sun

11

29

55

2

True Place of the Apogee

5

16

56

Variation substracted

2

Eccentricity

63

68

Inclination of ye Limit

Moon's True Place in it• Orbit.

1

12

9

41

☽ horary Mot.

38

0

Semidiam. ☉.

15

58

Node's True Place substract

7

20

10

12

☉•.

2

25

Semidiam. ☽.

16

47

Argument of Latitude

5

21

59

29

☽ from ☉

35

35

Semid. Penum.

32

45

Reduction Added

2

0

35⌊6′ 60′115:

8

22

Semid. Disk

61

30

Moon's True Place in the Eclip•

1

12

11

41

Eq. Time Add.

3

25

Ang• of ye ☽ way with ye Ecliptic

5°

35′

Moon's True North Latitude

44

10

Reduct Add.

3

9

Diff• of that ☉ & ☽s Diam 98″or 98 Miles.

Sum of 3 Add.

14

56

Engrav'd and Sold by Iohn Senex at ye Globe in Salisbury Court near Fleet street. And Will: Taylor at the Ship in Paternoster Row. Where are sold Mr Whiston's Astronomical Lectures, his Taquet's Euclid, and ye Scheme of ye Solar System. Also ye newest Globes and Maps.

The Copernicus, or Universal Astronomical Instrument, being now finish'd, is sold by ye Author Mr. Whiston, at his House in Cross street Hatton Garden

THE CALCULATION OF Solar Eclipses WITHOUT PARALLAXES. WITH A SPECIMEN of the same in the Total Eclipse of the Sun, May 11. 1724. Now first made Publick. To which is added, A PROPOSAL how, with the Latitude given, the Geographical Longitude of all the Parts of the Earth may be settled by the bare Knowledge of the Duration of Solar Eclipses, and especially of Total Darkness. WITH An ACCOUNT of some late Observations made with Dipping Needles, in order to discover the LONGITUDE and LATITUDE at Sea. By WILL. WHISTON, M. A. Sometime Professor of the Mathematicks in the University of Cambridge.

LONDON: Printed for J. SENEX in Fleetstreet; and W. TAYLOR in Pater-Noster-Row. 1724.

LEMMATA: OR, Preparatory Propositions.

I.

• HE most useful and most remarkable Cycle or Period for the Revolution of Eclipses, both Solar and Lunar, mentioned by Pliny, (Nat. Hist. II.) and by him only of all the Ancients, is the Interval of 223° Synodical Months = 6585 Days: or = 18 Julian Years: with 10 Days, when the Cycle •or Period contains 5 Leap-Years: and with 11 Days, when with 7 Hours 43′ ¼. In which Time the direct mean Motion of the Moon and her Apogee in the Ecliptick, is nearly so much more; and the Retrograde mean Motion of the Nodes nearly so much less than entire Revolutions, as the mean Motion of the Moon from the Sun, upon which all mean Conjunctions, Oppositions, and Eclipses properly depend, exceeds the like entire Revolutions Which Coincidences do therefore nearly restore the mean State of the Moon it self, its Apogee▪ Nodes, and Lunations: And produce an eminent Revolution of correspondent New Moons, •ull Moons, and Eclipses, after that Interval perpe •••lly.

This appears by the following Calculation of all these mean Motions from the Astronomical Tables.

Mean Motion of the Sun or Earth.

s

°

′

″

Years 18

11

29

28

32

Days 11

0

10

50

32

Hours 7

0

0

17

15

′ 43

0

0

1

46

″15

0

0

0

1

Sum

0

10

48

6

Mean Motion of the Moon in the Ecliptick.

s

°

′

″

Years 18

7

11

37

22

Days 11

4

24

56

25

Hours 7

0

03

50

35

′ 43

0

00

23

36

″ 15

00

00

00

08

Sum

00

10

48

06

Mean Motion of the Apogee.

s

°

′

″

Years 18

0

12

23

53

Days 11

0

1

13

32

Hours 7

0

0

1

57

′ 43¼

0

0

0

12

Sum

0

13

39

34

Mean Motion of the Node backward.

s

°

′

″

Years 18

11

18

7

39

Days 11

0

0

34

57

Hours 7

0

0

0

56

′ 43 ¼

0

0

0

6

Sum

11

18

43

38

Mean Motion of the Moon from the Sun.

s

°

′

″

Years 18

7

11

58

50

Days 11

4

14

5

53

Hours 7

0

3

33

20

′ 43

0

0

21

50

″ 15

0

0

0

7

Sum

0

0

0

0

°

′

″

From the mean Motion of the Apogee

13

39

34

Substract that of the Moon in the Eclipt.

10

48

6

Remains—(2°⌊9)

2

51

28

To the mean Motion of the Moon

0

10

48

6

Add that of the Node

11

18

43

38

Sum

11

29

31

44

Difference from 12 Signs (0°⌊47)

0

0

28

16

Whence it appears that the Difference of the mean Motions of the Apogee and of the Nodes from that of the Moon her self in the Ecliptick, in such a Period, is but small: Not more in the former Case than 2° 51′28″ = 2°⌊9, nor in the latter than 28′ 16″ = 0°⌊47. Whence also it appears that the Lunar Apogee does in every such Cycle differ but 1/•2 of the entire Difference 2⌊9) 180 (62. That the Lunar Node only differs in that Cycle 1/••• of the entire Difference 0°⌊47) 180 (383. And that the Anomaly of the Sun it self differs only 1/1••• of the entite Anomaly: 10 8) 180 (16⌊6. Which Quantities being generally small, cannot occasion any great Inequality in the Times and Circumstances of New and Full Moons, or of Eclipses▪ nor by consequence greatly disturb the regular Succession of the same in any single Period; nor indeed very greatly in several successive Periods. For since the mean Motion of the Moon from the Sun is within a very small Matter ever certain and invariable, that Revolution is always just; and always determines the mean Time of all Conjunctions, Oppositions, and Eclipses rightly; and since the other Anomalies are but small, and always come right again in Length of Time, they cannot ever produce any very great Anomalies in our Calculations from them. As will farther appear under the following Scholia.

N. B. This Period for Eclipses, has of late been called, both by Mr. Flamsteed, and Dr. Halley, the Saros, or the Chaldean Saros: As if it were known and us'd by the old Chaldeans, and thence called by that Name. For which I know no sufficient Foundation. There is indeed a gross Mistake of Pliny's Number in Suidas, (who thus applies this term) 222 for 223 Months, as almost all the Editions of Pliny still have it; and He calls that Period by this Chaldean Name Saros. Yet the Chaldeans never, that we find, apply'd it to any other Period than that of 3600 Years or Days; by which Period alone all the Antediluvian Reigns are determined both in Abydenus and Berosus themselves, from the anc•∣entest Records of that Kingdom. See my New Theory, the later Editions, Hypoth. X. Lem. to the third Argument; and Appendix to the Essay towards restoring the true Text of the Old Testament, p. 203,—213.

SCHOLIA.

(1.) We may here Observe, that since the Limit for Eclipses of the Moon is about 11° 40′ = 700′ on each Side of the Node; as is the Limit for Eclipses of the Sun, about 16° 40′ = 1000′. If we divide 700, the Limit of the Moon's Eclipses, by 28′ 16″ = 28′⌊3, which is the Difference between the Revolution of the Moon to the Sun and of the Node above given, we shall have nearly Twenty Five for the Number of Cycles,, after a Central Lunar Eclipse in one of the Nodes, before the Moon goes off the Shadow of the Earth entirely at the same Node, and 450 Years (25 {multiplier} 18 = 450,) or double that Number 900 Years for the Time that the Moon begins to enter the Ecliptick Limit on one Side, till it goes out of it on the other. During which long Interval there will still be Eclipses of the Moon each Period. And if we divide 1000′ the Limit of the Sun's Eclipses, by the same Number 28′⌊3, we shall have nearly 35 for the Number of Periods after a Solar Central Eclipse at the Middle of the Earth, in one of the Nodes, before the Penumbra goes off the North or South Parts of the Disk of the Earth entirely at the same Time; i. e. 630 Years. (35 {multiplier} 18 = 630,) or double that Number 1260 Years, from the Time that the Moon in any such Period begins to enter the Ecliptick Limit on one Side, till it goes out of on the other: During which longer Interval there will still be somewhere Eclipses of the Sun each Period. After which respective long Intervals of Time there will be no such Eclipses for much longer Intervals.

(2.) Since the utmost Latitude of the Moon that can permit any Lunar Eclipse, is about 62′, and the same utmost Latitude that can permit a Solar Eclipse is about 92′: If we divide the first Number by 25, or the last by 35, the Numbers of Revolutions for the Ecliptick Limits, we shall have about 2′⌊6 = 2′:36″ for the mean Alteration of the Moon's Latitude in each single Period all along; and this both for Solar and Lunar Eclipses. Which Latitude will be South during the one half of the long Period of the Ecliptick Limits before-mentioned; and North during the other half: Gradually increasing, and as gradually decreasing perpetually: As in the following Table.

A Table of the mean Latitudes of the Moon each single Cycle, either North or South; beginning at an Eclipse in one of the Nodes, without any Latitude at all.

Latitudes for 25 Cycles in Lunar, and 35 in Solar Eclipses.

Cycles

′

″

1

2

26

2

4

52

3

7

19

4

9

46

5

12

14

6

14

43

7

17

12

8

19

42

9

22

12

10

24

43

11

27

14

12

29

46

13

32

19

14

34

52

15

37

26

16

40

1

17

42

36

18

45

12

19

47

48

20

50

25

21

53

3

22

55

41

23

59

20

24

60

59

25

63

39

26

66

0

27

68

41

28

71

23

29

74

6

30

76

49

31

79

33

32

82

17

33

85

2

34

87

48

35

90

34

(3.) Since the principal Alteration in the Quantity and Duration of total Eclipses of the Sun, arises from the Difference there is at any Time between the real Distances, and apparent Diameters of the Sun and Moon, at the Time of such Eclipses, that Quantity and Duration must depend on the Difference of their mean Anomalies, which gives us that Difference of Distances and Diameters; and must therefore answer in each Cycle one with another, to the Differences of those mean Anomalies during that Interval; which in the Sun comes to 10°⌊8/180 or 1/16⌊6 of its entire Anomaly. And in the Moon to 2/•▪ 9/• or 1/•• of its entire Anomaly. And since the whole mean Excentricity of the Moon is somewhat above three Times as great as that of the Sun, or as ••/1000 to 17/100. The Differences of the Sun's Distances and Diameters will be but a little greater in each Period one with another, than those of the Moon.

(4.) When therefore the Anomalies of the Sun and Moon are of the same Species; I mean both ascending, or both descending; their Distances and Diameters will, one with another, increase or decrease nearly in the same Proportion; and the Quantity and Duration of total Darkness will alter but little in such a Period. But when those Anomalies are of the contrary Species; that is, the one ascending while the other descends; they will alter considerably. So that if the Sun be descending, and its apparent Diameter Increasing; while the Moon is ascending, and its apparent Diameter Decreasing, the Eclipse of the Sun will, each succeeding Cycle, afford a smaller total Shadow; till at last it afford no total Shadow at all; but the Eclipses become Annular. And if the Sun be Ascending while the Moon is Descending, the contrary will happen; and the total Shadow grow greater perpetually. From which Circumstances of the Sun and Moon in each Revolution of the Cycle duly considered, we may nearly determine whether any succeeding correspondent Eclipse will afford us a greater or less total Shadow, or whether the Eclipses will be only Annular.

(5.) From the like Circumstances we may also nearly determine whether such Eclipses will come somewhat sooner or later, than that of the mean Revolution of the Period before us. For if the Earth be much nearer its Aphelion, than the Moon its Apogaeon, at the end of any Cycle; and by consequence if the Earth then revolve comparatively slower, and the Moon swifter than ordinary; the meeting of the Luminaries will be accelerated. And if the Earth be much remoter from its Aphelion than the Moon from its Apogaeon, the contrary will happen; and the Moon will be later than ordinary e'er it overtake the Sun. So that in the former Case the Eclipse will come a little before, and in the latter a little after the proper Conclusion of that Period.

(6.) Since the Motion of the Node backward in one of these Periods does not quite reach to the Conjunction or Opposition, that Node must every Cycle go forward, with respect to the Lunations and Eclipses; and at the ascending Node the Moon will pass more Southward, and at the descending Node more Northward successively. Thus at the Solar Eclipse May 1. 1706. the Moon, near its ascending Node, had greater Northern Latitude than it will have at the next corresponding Solar Eclipse, May 11. 1724. And thus at the total Solar Eclipse, April 22. 1715. the Moon near its descending Node, had less Northern Latitude than it will have at its corresponding great Eclipse, May 2. 1733.

(7.) Since the Motion of the Moon's Apogee forward is greater in one of these Periods than that of the Lunations, that Apogee must also go forward every Cycle: And if at any one Solar Eclipse that Apogee be in quadrature with the Sun, after it had been in Conjunction, the Moon will the next Period descend by going backward in its Eclipses, towards the Perigee. And if at any one such Apogee it be in the quadrature, after it had been in Opposition, it will the next Cycle ascend: The Reverse of all which is true in Lunar Eclipses.

(8.) The Place and Motion of the Sun in its Ellipsis is so easily known, and that for many Ages, by bare Memory and Reflection, that a few Words will suffice. The Sun is now farthest from the Earth Eight Days after the longest Day; and nearest to it Eight Days after the shortest: And its Motion about 1 Degree in 72 Years. Whence it is evident, that it has, for all the past Ages of Astronomy, been about the Summer Solstice in our Apogee, and about the Winter Solstice in our Perigee; if I may use the T•rms of the Ptolemaick System. Nor is it therefore any Wonder that the greatest total Eclipses of the Sun have happened still in the Summer, and the g•eatest Annular ones in the Winter half Year: Since the farther the Sun is off in the Winter, the less must be its apparent Diameter, and by consequence the greater the Excess of the Moon's Diameter above it. On which Excess alone the Greatness of such Eclipses depends. And the Reverse is equally evident in the Case of Annular Eclipses in Winter: Nor indeed is it very strange, that Annular Eclipses are in these Parts of the World so rarely observed; since they most usually happen in Winter Days; which being short, must afford us a proportionably small Number of them. To the Inhabitants of the other Side of the Equator the Reverse must happen. But those Eclipses very rarely come to our Notice.

(9.) Since this Period reaches only from the middle of one general Eclipse to another, without regard to the Position of any particular Place on the Earth's Surface, arising from the diurnal Motion, we must remember that if an Eclipse of the Sun happens at any particular Place considerably before Noon, it will come sooner, and after Noon later than the proper Conclusion of this Period. Though it must be noted, that Eclipses of the Moon being absolute in their own Nature, are here wholly unconcerned; and no way subject to any Acceleration, Retardation, or Alteration on account of the diurnal Motion of the Earth.

(10.) The principal Alteration of the Time of the Day in all Eclipses depends on the Excess of this Period above an even Number of Days; which is 7 Hours and 43′¼. So that the Cycle does naturally put every correspondent Eclipse later than the foregoing, almost 8 Hours, or one third part of a Day; which thing, by reason of the intervening diurnal Motion, greatly alters all Eclipses, especially Solar, not only as to the bare Time of the Day when, but also as to the Places on the Earth where such correspondent Eclipse will be visible.

(11.) If therefore we join three of these Cycles together, those odd Hours and Minutes will amount nearly to a whole Day; and will therefore nearly bring the middle Point of the correspondent Eclipses to the same Time, in the same Place, and, in part, with the same Circumstances as before: Which a single Cycle cannot possibly do. Only with the Anticipation of 50′ in Time. Which three single Cycles therefore of 19,756 Days, or of 54 Years; with 32 or 33 Days; I call the Grand Cycle. And this will be, I think, of the greatest and readiest use in remote Eclipses of any other Period whatsoever.

Thus, for Example, there was a total Eclipse of the Sun on Black Monday; as it has thence been called ever since, March 29. 1652. about Ten a Clock in the Morning. Total, I say it was in the North of Ireland, and the Northwest of Scotland, tho' not so at London, ot the remoter Parts of England and Scotland. To this if we add one Cycle, the Time of the next correspondent Eclipse will thus be discovered:

y

d

h

′

″

To A. D. 1652. March,

00

28

22

00

00

Add

18

10

7

43

15

Sum 1670 April

00

8

5

43

15

So that the correspondent total Eclipse ought to have been A. D. 1670. April the 8th 43′ ¼ past 5 a-Clock in the Evening. And because the Earth was then a small Matter nearer its Aphelion, than the Moon its Apogaeon, the Time would be a little anticipated on that Account. But then, because this Eclipse was towards Evening, it would be much more retarded on that Account, than anticipated on the other; and the main Part of the Eclipse would happen after Sun-set, and be here invisible.

To this Time if we add another Cycle, the next correspondent Eclipse will in like Manner be discovered:

y

d

h

′

″

To A. D. 1670. April

00

08

5

43

15

Add

18

11

7

43

15

Sum 1688. April

00

19

13

26

30

So that the next correspondent total Eclipse ought to have been April 20th, 26′ ½ after one in the Morning; and was therefore, to be sure, utterly invisible to us here. To this Time if we add another Cycle, we have the next correspondent total Eclipse thus:

y

d

h

′

″

To A. D. 1688 April

00

19

13

26

30

Add

18

10

07

43

15

Sum 1706 May

00

00

21

09

45

So that the next correspondent total Eclipse was to have been A. D. 1706. 9′ ¾ past Nine a-Clock in the Morning; which is not much before the Time when it was observ'd here; and was no other than that famous Eclipse which was total at Cadiz, Barcelona, Marseilles, Geneva, Bern, and Zurich; and became very remarkable for the raising of the Siege of Barcelona during that total Darkness: Though I have been inform'd by several there present, that it came to both Armies wholly unexpected, till the great Darkness of the Sky forced them to attend to it. Now this affords us also a remarkable Instance of the near Approximation of our Grand Cycle, both as to Time and Place. For if instead of tracing this correspondent Eclipse through the distinct Cycles we had at once taken our Grand Cycle of 54 Years, and 33 Days; we had come immediately to this Eclipse: And by allowing the Anticipation of 50′ had been within about 20′ of the Calculation at London.

To this Time if we add another single Cycle, we shall have the next correspondent Eclipse thus:

y

d

h

′

″

To A. D. 1706. May

00

00

21

09

45

Add

18

10

07

43

15

Sum 1724 May

00

11

04

53

00

So that we ought hence to expect the total Eclipse next May 11, 53′ past 4 in the Afternoon. And because both the Position of the Sun and Moon in their Ellipses, and the more considerable Alteration from the Time of the Day, which is here much farther in the Evening than the last was in the Morning; oblige us to suppose about 1h ¾ Retardation, we hence justly expect that this Eclipse will be the nearest Total at London about 40′ past Six in the Evening; as the exactest Calculations do determine.

(12.) If we would now trace a few Lunar Eclipses by this Cycle, we may do it according to the following Examples:

A. D. 1681/2, Feb. 11. about 59′ past 10 a-Clock at Night, Mr. Flamsteed observ'd the Middle of a great and total Eclipse of the Moon at Greenwich. Proceed therefore as is already directed to find the next correspondent Eclipse of the Moon thus:

y

d

h

′

″

To A. D. 1681/2 February.

00

11

10

59

00

Add

18

11

07

43

15

Sum 1699/1700 Febr.

00

22

18

42

15

So that this first correspondent total Eclipse of the Moon ought to have been Feb. 23. 1699/1700, 42′ ¼ past Six a-Clock in the Morning; or in the day-time, and so must needs have been in great Part to us invisible.

y

d

h

′

″

To A. D. 1699/1700, Feb.

00

22

18

42

15

Add

18

10

07

43

15

Sum 1717/18 March

00

05

02

25

30

So that the next correspondent total Eclipse of the Moon ought to have been March 5, 1717/1•. 25′ ½ after Two a-Clock in the Afternoon; which was in the day-time also; and so must equally with the former have been here invisible.

y

d

h

′

″

To A. D. 1717/18, March

00

05

02

25

30

Add

18

10

07

43

15

Sum 1736 March

00

15

10

08

45

So that the next total correspondent Eclipse of the Moon is hence to be expected A. D. 1736. March 15. 8h ¾ past 10 a-Clock at Night: which is about an Hour and half sooner than the Calculation. Which difference we shall presently find to be near the greatest Difference that can happen.

This may also be equally obtain'd by one entire Grand Cycle of 54 Years, and 33 Days; with the fore-mentioned Anticipation of 50′ which from Feb. 11. 1681/2 10h 59′, brings us directly to March 15, 1736, 9′ past 10 a-Clock at Night; or to somewhat above an Hour and half before the Calculation.

N. B. As to the proper Quantity of the several Alterations arising in each Period, which ought to be allowed for, they are nearly these: The Moon and Sun being about 31′ ⅔ in Diameter, and the Digits of their Observation being 12. while the Difference of the Moon's Latitude, as we have seen, is about 2′ 36″ or the Twelfth Part of those Diameters, it is plain that the mean natural Alteration of every Period in the same Circumstances is about one Digit; though less in the lesser, and greater in greater Latitudes: which in the Moon, whose Eclipse is to all Spectators the same, holds constantly: And though the diurnal Motion of the Earth removes all particular Places, so much each Period as to render this Rule less observable in Solar Eclipses, yet after each grand Period, which nearly restores their former Position, it will hold in a good Degree there also, I mean so as to alter about 3 Digits therein: But besides that of the Digits eclipsed, we ought also to see what Alteration in Time may happen to each Period. Now as to the Inequality of the Sun's Motion, it is as we have seen 10°⌊8, and its greatest Velocity is at the Earth's Perihelion, and its least at its Aphelion: Its greatest Alteration therefore must be in Aphelio and Perihelio, and is the Difference of the Equation belonging every where to the Addition of 10°⌊8, and is here 48″, which Space the Earth goes in about 20′ of Time. So that the Difference of Time on this Account, must each Period be some Quantity less than 20′. And as to the Inequality of the Moon's Motion, it is also greatest at the Perigee and Apogee, and its greatest Alteration at the extreme Eccentricities of its Orbit is the Difference of the Equations at 2° 51′ ½ in Perigee and Apogee, according to those extreme Eccentricities: = 9′ which the Moon usually goes in somewhat less than 20′. So that the Difference of Time, on this Account, must each Cycle be some Quantity less than 20 Minutes also.

And now we come to the principal Alteration in Time that can happen in Eclipses; though it belongs only to those of the Sun: And that is the Time of the Day when they happen in any particular Place. Now because the Center of the Moon usually goes over an entire Diameter of the Disk of the Earth, in about three Hours and an half, Part of which is almost always before, and Part after Noon: while the odd Hours of a small Cycle 7h 43′ ¼, may reach equally from a Forenoon to an Afternoon's successive Eclipse; 'tis possible, such an Eclipse may appear an Hour and three Quarters later than the Period it self would determine it. Tho' usually this Alteration will not be near so great; especially when the Latitude of the Moon is very considerable. But then it is so easy to allow very nearly for this Inequality, upon a little Consideration, that it ought not to be objected against the Accuracy of this Period. If for Six Hours from Noon we allow about an Hour and half; and for two Hours, three Quarters of an Hour; we shall not err very much from the true Time.

Corollary. If we would know what is the greatest Inequality in Digits and Time in a grand Period, made up of Three common ones, or of 54 Years, 32 or 33 Days, besides the constant Anticipation of 50′, we must say it may possibly, though it will very rarely, be almost thrice the Quantities already stated for a single Cycle: excepting the last and principal Difference, peculiar to Solar Eclipses: which is never much greater than that already mentioned.

N. B. If any are not contented to know these Matters by such Approximations, but desire the utmost Accuracy; they must either make use of Dr. Halley's Equations, fitted to this Cycle, when published; or rather make use of Mr. Flamsteed's or Dr. Halley's most accurate Astronomical Tables, when published; with that Trigonometrical Calculation afterward, which I publish and exemplify in this Paper. In the former Part of which Work, this Cycle, with its proper Equations, will, at least, save us the one half of our Calculation, if it will not bring us it self to that utmost Accuracy: which indeed is hardly to be expected from it.

So that, upon the whole, If we duly consider the particular Circumstances of the Sun and Moon, with those of the Aphelion, Apogee and Node, and with the Times of the Day or Night when the Cycle ends, and rightly apply them to this single Cycle and to this grand Cycle, we shall be able nearly to determine the correspondent Eclipses with very small Trouble or Calculation.

II.

The Plane in which the Center of the Moon moves in Eclipses, is not that of the Ecliptick, but of the Orbit of the Moon, consider'd with the Annual Motion: Or it is a Plane inclined to the Plane of the Ecliptick in an Angle of about 5° 36′. Which in the Calculation of Eclipses is usually stiled, The Angle of the Moon's visible Way. This principal Plane I call, The Lunar Plane.

III.

This Lunar Plane cuts the Sphere of the Earth, considered without its diurnal Motion, in a Circle whose Pole or Vertex is distant from the Pole of the Ecliptick in the same Angle. This Circle I call The Lunar Circle.

IV.

In Eclipses which happen at the Solstices, and in the Nodes of the Moon's Orbit, the Distance of these Poles is exactly Eastward or Westward. In those which happen at the Equinoxes and Nodes, the Distance is exactly North and South. But in all other Cases it is Oblique.

V.

The Angle of that Obliquity is always compos'd of the Distance of the Sun and Moon from the Solstitial Colure; with the Distance of the same from the Nodes: And is sometimes the Sum, and sometimes the Difference of those Quantities.

VI.

The Distance between this Vertex and the Pole of the Earth, when Eclipses happen at the Solstices, is the Sum or Difference of the two forementioned Angles of Inclination; the one, of the Pole of the Equator and of the Pole of the Ecliptick = 23° 29′; the other of the Pole of the Ecliptick, and of the Vertex of the Lunar Circle = 5° 36′ nearly. But in all other Cases a Spherical Triangle must be solv'd, in order to find that Distance: Of which hereafter.

VII.

Since the Solstitial Colure is a great Circle, that is also a Meridian, or passes through the Poles of the Earth and Ecliptick: And since besides the Distance between the Vertex of the Lunar Circle and the Pole of the Earth, we shall want the Angle included between the Colure and that Line; this also must be obtained by the like Solution of a Spherical Triangle: Of which hereafter.

VIII.

Since the great Circle that passes thro' the Poles of the Earth▪ and of the Lunar Circle, and that alone cuts both those Circles, and their Parallels at Right Angles, That Meridian, and that alone wherein that Distance lies, is perpendicular to the Path of the Moon's Center along the other; and will determine the Point in that Path wherein 〈◊〉 Center of the Shadow cuts that Meridian at R ght▪Angles and approaches nearest of all to the Pole of the Earth: And indeed lays the Found• o• of our future Calculations. This Meri •••• I call The Primary Meridian.

IX.

The angular Distance about the Pole of the Earth, of the Meridian that is directed to the Sun at the middle Point of the whole Eclipse from the primary Meridian, is composed of the Sum or Difference of the Angle made by the Solstitial Colure and the primary Meridian; and of the Complement of the Sun's Right Ascension at the middle of the general Solar Eclipse. Which Angle is of the greatest Consequence in our future Calculations. This Angle I call the Primary Angle.

X.

Since the Motion of the Center of the Shadow of the Moon, in Solar Eclipses, is nearly even, and nearly recti-linear; since it is also in the Plane of the Lunar Circle; and is all one as if it were along a Line that touched that Circle at the Middle of the general Eclipse, the Point of Contact; we must divide each Quadrant of 90 Degrees into 90 or 180 unequal Parts: but so that the Difference of the Sines of those unequal Angles may be equal, and 1/9• or 1/•80 of the entire Radius: That so the first Sine •/•0 may be 1111/100000; the second 2222/100000; the third ••••/100000, &c. and this from the Table of natural Sines, with their corresponding Arcs or Angles at the Vertex, as follows:

Arcs.

D ff. Sines Equal.

Arcs.

Diff. Sine• Equal.

Diff.

°

′

Diff.

°

′

19⌊1

19⌊4

0

18

½

7

1

11

19⌊1

19⌊4

0

38

1

7

20

½

19⌊1

19⌊5

0

57

½

7

40

12

19⌊1

19⌊5

1

16

2

7

59

½

19⌊1

19⌊5

1

35

½

8

18

13

19⌊1

19⌊5

1

54

3

8

37

½

19⌊1

19⌊6

2

13

½

8

57

14

19⌊2

19⌊6

2

33

4

9

16

½

19⌊2

19⌊6

2

52

½

9

36

15

19⌊2

19⌊6

3

11

5

9

55

½

19⌊2

19⌊7

3

30

½

10

14

16

19⌊2

19⌊7

3

49

6

10

33

½

19⌊2

19⌊7

4

8

½

10

53

17

19⌊2

19⌊7

4

28

7

11

12

½

19⌊3

19⌊7

4

47

½

11

32

18

19⌊3

19⌊7

5

6

8

11

51

½

19⌊3

19⌊7

5

25

½

12

11

19

19⌊3

19⌊8

5

44

9

12

30

½

19⌊3

19⌊8

6

3

½

12

50

20

19⌊4

19⌊8

6

23

10

13

10

½

19⌊4

19⌊8

6

42

½

13

30

21

19⌊8

20⌊5

13

49

½

20

50

32

19⌊9

20⌊5

14

9

22

21

10

½

20

20⌊5

14

28

½

21

30

33

20

20⌊6

14

48

23

21

50

½

20

20⌊6

15

8

½

22

11

34

20

20⌊7

15

28

24

22

32

½

20

20⌊7

15

48

½

22

53

20⌊1

20⌊8

35

16

7

25

23

13

½

20⌊1

20⌊9

16

•7

½

23

34

36

20⌊1

21

16

47

26

23

55

½

20⌊2

21

17

7

½

24

17

37

20⌊2

21⌊1

17

27

27

24

38

½

20⌊2

21⌊1

17

47

½

24

59

38

20⌊2

21⌊2

18

8

28

25

20

½

20⌊3

21⌊2

18

28

½

25

41

39

20⌊3

21⌊2

18

48

29

26

2

½

20⌊3

21⌊3

19

8

½

26

23

40

20⌊3

21⌊3

19

28

30

26

44

½

20⌊4

21⌊4

19

48

½

27

6

41

20⌊4

21⌊4

20

9

31

27

27

½

20⌊4

21⌊6

20

29

½

27

49

42

21⌊7

24⌊2

28

11

½

36

4

53

21⌊8

24⌊3

28

32

43

36

27

½

21⌊9

24⌊4

28

54

½

36

52

54

22⌊0

24⌊6

29

16

44

37

16

½

22⌊1

24⌊7

29

38

½

37

40

55

22⌊2

24⌊8

30

0

45

38

4

½

22⌊3

24⌊9

30

22

½

38

29

56

22⌊4

25⌊1

30

44

46

38

54

½

22⌊6

25⌊2

31

6

•

39

18

57

22⌊8

25⌊4

31

51

47

39

43

½

23•1

25⌊5

31

29

½

40

7

58

23⌊2

25⌊7

32

14

48

40

32

½

23⌊4

25⌊8

32

36

½

40

57

59

23⌊5

25⌊9

32

59

49

41

22

½

23⌊6

26⌊1

33

22

½

41

48

60

23⌊6

26⌊3

33

45

50

42

14

½

23•7

26⌊4

34

9

½

42

40

61

23•8

26⌊5

34

31

51

43

6

½

23•9

26•7

34

54

½

43

32

62

23⌊9

26⌊9

35

18

52

43

58

½

24⌊1

27⌊3

35

41

½

44

25

63

27⌊6

34

44

52

½

55

19

74

27⌊8

34⌊4

45

19

64

55

52

½

28⌊2

34⌊8

45

46

½

56

26

75

28⌊4

35⌊0

46

14

65

57

1

½

28⌊5

35⌊4

46

42

½

57

37

76

28⌊6

35⌊8

47

10

66

58

13

½

28⌊7

36⌊3

47

38

½

58

49

77

28⌊8

37

48

6

67

59

16

½

28⌊9

37⌊8

48

36

½

60

4

78

29

38⌊6

49

4

68

60

42

½

30

39⌊8

49

33

½

61

22

79

30

40⌊9

50

3

69

62

2

½

30⌊2

41⌊8

50

33

½

62

44

80

30⌊4

42⌊5

51

3

70

63

26

½

30⌊6

43

51

33

½

64

9

81

31

44⌊5

52

4

71

64

54

½

31⌊5

46

52

35

½

65

40

82

32

48

53

7

72

66

26

½

32⌊5

50

53

39

½

67

15

83

33

52

54

12

73

68

5

½

33⌊5

53

54

45

½

68

58

84

54

(86)

69

52

½

77

54

88

56

(97)

70

49

85

79

31

½

59

(116)

71

48

½

81

27

89

63

(69)

72

51

86

82

36

¼

67

(81)

73

58

½

83

57

½

72

(105)

75

10

87

85

42

¾

78

(258)

76

28

½

90

00

90

Where the Angle made by the Lunar Circle, and the Paral•el of the Latitude is considerable, instead of the first Number 19′⌊1 you must take for 1/1•0 the Numbers following; being in a reciprocal Proportion to the Secants of those Angles.

Angl.

0

19⌊1

5

19⌊1

10

18⌊8

15

18⌊4

20

17⌊9

25

17⌊3

30

16⌊5

N. B. Where the Parallel is different from that at the very Middle of the Eclipse; as it usually is; you must increase or decrease the same Numbers 19′⌊1, &c. in the Proportion of the Cosines of the Latitude thus:

°

00

000

10

174

20

342

30

500

40

643

50

766

60

867

70

940

80

98⌊5

90

100⌊0

E. G. If the Co-latitude at the Middle of the Eclipse be 60°, and come to be 40°; say, As 867 to 643, So is 19⌊1 to 14⌊2, which is there to be taken in its stead.

N. B. The Hint that I had several Years ago, that in the Determination of Solar Eclipses the Equality of the Difference of Sines was made us• of by Dr. Halley, was the Occasion of the Discoveries in these Papers.

XI.

The perpendicular Distance of every Point of the Penumbra; and the like Distance of every Point of the total Shadow from the Path of the Moon's Center, may be discovered by Tables made from the natural Sines; where those Sines themselves, as before, differ equally, or in arithmetical Progression; according to the Duration of the whole Eclipse, or of total Darkness: and their Co-sines correspond to the Distances from that Path. Both which Tables here follow:

For entire Eclipses: Which are here suppos'd 108′¾ long in the Middle, and the Semidiameter of the Penumbra 2000 Miles.

Duration in Minut.

Distance in Miles.

Digts

1

2000

2

1999

3

1999

4

1999

5

1998

6

1997

7

1996

8

1995

9

1993

10

1991

11

1990

12

1988

13

1986

14

1984

15

1981

16

1979

17

1976

18

1973

19

1970

20

1966

21

1963

22

1959

23

1955

24

1951

25

1946

26

1942

27

1937

28

1932

29

1927

30

1922

31

1917

32

1911

33

1905

34

1899

35

1893

36

1886

37

1879

38

1872

39

1866

40

1859

41

1852

42

1844

1838

1

43

1835

44

1828

55

1820

56

1812

47

1803

48

1795

49

1786

50

1776

½

1771

51

1766

½

1761

52

1756

2/1

1751

53

1746

½

1741

54

1736

½

1731

55

1725

•

1720

56

1714

•

1708

57

1703

½

1697

58

1691

½

1685

59

1679

1676

2

½

1672

60

1667

½

1662

61

1655

½

1648

62

1642

½

1635

63

1628

½

1621

64

1614

½

1609

65

1603

½

1596

66

1589

½

1582

67

1575

½

1568

68

1561

½

1554

69

1546

½

1538

70

1534

½

1528

71

1521

½

1514

1513

3

72

1506

½

1498

73

1490

½

1482

74

1474

½

1456

75

1448

•

1439

76

1430

•

1424

77

1412

•

1403

78

1394

½

1384

79

1374

½

1364

80

1354

1350

4

½

1344

81

1333

½

1322

82

1312

½

1302

83

1291

½

1280

84

1269

½

1258

85

1246

½

1234

86

1222

½

1110

87

1098

1188

5

½

1186

88

1173

½

1160

•9

1147

½

1134

90

1121

½

1107

91

1093

½

1079

92

1065

½

1050

93

1034

1025

6

½

1017

94

1000

½

983

95

971

½

955

96

937

½

919

97

902

½

884

98

864

863

7

½

844

99

824

½

804

100

784

¼

773

½

762

¾

751

101

739

¼

728

½

716

¾

704

700

8

102

691

¼

679

½

666

¾

653

103

639

¼

625

½

611

¾

597

104

582

¼

566

½

550

537

9

¾

534

105

517

¼

499

½

482

¾

463

106

443

¼

421

½

399

¾

376

375

10

107

352

¼

320

½

288

¾

258

108

226

212

11

¼

180

½

119

50

12

¾

0••

For Total Darkness of 166″ ⅔.

Duration in Secds.

Distance in Miles.

″

1

50⌊0

2

50⌊0

3

50⌊0

4

50⌊0

5

50⌊0

6

50⌊0

7

50⌊0

8

50⌊9

9

50⌊9

10

50⌊9

11

49⌊9

12

49⌊8

13

49⌊8

14

49⌊8

15

49⌊7

16

49⌊7

17

49⌊7

18

49⌊6

19

49⌊6

20

49⌊6

21

49⌊6

22

49⌊5

23

49⌊5

24

49⌊5

25

49⌊4

26

49⌊4

27

49⌊4

28

49⌊3

29

49⌊3

30

49⌊3

31

49⌊2

32

49⌊2

33

49⌊2

34

49⌊1

35

49⌊1

36

49⌊1

37

49⌊0

38

49⌊0

39

49⌊0

40

48⌊9

41

48⌊9

42

48⌊8

43

48⌊7

44

48⌊6

45

48⌊5

46

48⌊4

47

48⌊2

48

48⌊1

49

47⌊9

50

47⌊7

51

47⌊6

52

47⌊5

53

47⌊4

54

47⌊3

55

47⌊2

56

47⌊1

57

47⌊0

58

46⌊9

59

46⌊8

60

46⌊6

61

46⌊5

62

46⌊4

63

46⌊3

64

46⌊2

65

46⌊0

66

45⌊9

67

45⌊8

68

45⌊6

69

45⌊5

70

45⌊3

71

45⌊2

72

45⌊1

73

45⌊0

74

44⌊8

75

44⌊6

76

44⌊5

77

44⌊3

78

44⌊1

79

44⌊0

80

43⌊8

81

43⌊7

82

43⌊5

83

43⌊3

84

43⌊2

85

43⌊0

86

42⌊9

87

42⌊7

88

42⌊5

89

42⌊3

90

42⌊1

91

41⌊9

92

41⌊7

93

41⌊5

94

41⌊3

95

41⌊1

96

40⌊9

97

40⌊7

98

40⌊5

99

40⌊3

100

40⌊0

101

39⌊8

102

39⌊6

103

39⌊4

104

39⌊1

105

38⌊8

106

38⌊6

107

38⌊4

108

38⌊1

109

37⌊9

110

37⌊6

111

37⌊4

112

37⌊1

113

36⌊8

114

36⌊5

115

36⌊2

116

35⌊9

117

35⌊6

118

35⌊3

119

35⌊0

120

34⌊7

121

34⌊4

122

34⌊•

123

33⌊8

124

33⌊4

125

33⌊0

126

32⌊7

127

32⌊4

128

32⌊1

129

31⌊7

130

31⌊3

131

30⌊9

132

30⌊5

133

30⌊1

134

29⌊7

135

29⌊3

136

28⌊9

137

28⌊5

138

28⌊0

139

27⌊5

140

27⌊1

141

26⌊6

142

26⌊1

143

25⌊6

144

25⌊1

145

24⌊6

146

24⌊0

147

23⌊5

148

23⌊0

149

22⌊4

150

21⌊8

151

21⌊1

152

20⌊5

153

19⌊8

154

19⌊1

155

18⌊3

156

17⌊6

157

16⌊7

158

15⌊9

159

14⌊9

160

14⌊0

161

12⌊9

162

11⌊7

163

10⌊4

164

8⌊9

165

7⌊0

166

4⌊4

166 ⅔

0⌊0

Semidiameter of the Umbra or Total Darkness = 50 Miles.

XII.

The Angles at the Vertex of the Lunar Circle, on each Side of the Point of Contact, by Reason of the perpendicular Situation of that Axis to its own Circle; are always right Angles: Only diminish'd in the Proportion of the Minutes describ'd by the Annual Motion during the Continuance of the Eclipse. Thus in our present Eclipse, which retains the Center of the Shadow near three Hours upon the Earth's Disk, in which Time the annual Motion amounts to about 8′; each of those right Angles in Strictness are to be esteem'd only 89° 56′, and both together 179° 52′. Only because the Refraction of the Rays of the Sun through our Atmosphere, requires a somewhat greater Increase of this Angle, than the annual Motion requires its Diminution, I shall wholly omit it, in all my Calculations hereafter.

XIII.

The Angles made at the Poles of the Earth, which shew the Difference of the two extreme Meridians, and limit the Extent of the entire central Eclipse, by reason of the Obliquity of the Earth's Axis to that Lunar Circle, are usually unequal to one another; and more or fewer than twice 90°, as the Eclipse happens at different Latitudes of the Moon, and Times of the Year.

XIV.

The Meridian that passes through the Middle or Central Point of general Solar Eclipses, is the same with that which passes through the Center of the Sun at the same Time, when the Moon has no Latitude, and the Eclipses are Central in the Plane of the Ecliptick; as also when they happen in either Solstices. Otherwise the Moon's Latitude being taken perpendicular to the Plane of the Ecliptick; and the nearest Distance of the Moon's Motion being taken perpendicular to the Lunar Circle, while the Meridians always pass through the Poles of the Earth; these Two Meridians will, generally speaking, be different, and their included Angle no otherwise to be known than by Trigonometry, as will appear hereafter.

XV.

The Dimensions of the Penumbra, or entire Eclipse, and the Extent of the total Shadow on the Earth, are continually different, according to the different Elevations of the Sun and Moon above any particular Horizon. For as the Moon is about the same Distance from every Place, when it is in its Horizon, as it is from the Earth's Center it self; with regard to which Center alone our first Calculations are always made: So when it is in the Zenith of any Place, it is one Semidiameter of the Earth nearer it; which Semidiameter being usually 1/60, and at our next Eclipse 1/55⌊5 of its entire Distance, as will appear hereafter, will deserve an Allowance. Nor will any lesser Elevation of the Moon be wholly inconsiderable in Eclipses, but in all accurate Determinations thereof must be particularly computed, in order to the distinct Knowledge of the Extent of such Eclipses; especially of the Breadth of the Total Shadow therein. Accordingly we are to observe that this Breadth of the Total Shadow will certainly be at this Eclipse considerably greater over North America; where the Luminaries are greater elevated above the Horizon; than over Europe, where they are much nearer it; as this Calculation requires: Of which hereafter.

XVI.

The Figure of the entire Penumbra, or general Eclipse; and of the Umbra, or Total Darkness; as they appear upon every Country, is different, on account of the different Obliquity of every Horizon; and will make Ovals or Ellipses of different Species perpetually. This in the vast Penumbra is best understood by such an Instrument as my Copernicus; or by the Perusal of a very scarce Book written by P. Coursier, (Philos. Transact. No. 343. p. 259.) and cited by Dr. Halley: Which distinctly treats of the Intersection of a Conical and Spherical Surface. But in the smaller Umbra, or Total Darkness, which is confined to a much narrower Compass, it very nearly approaches to the Intersection of a conick Surface with a Plane, which is a true Ellipsis.

XVII.

The Species of that Ellipsis depends on the Sun's Altitude above the Horizon at the Time of Total Darkness; as does the Position of its longer Axis on the Azimuth of the Sun at the same Time. Nor is it at all necessary that the Direction of this or any other Ellipsis should be along either of the Axes, but may as well be along any other Diameter whatsoever.

XVIII.

The Direction of the Center of the Shadow is according to the Direction of the Moon's Motion, along the Plane of the Lunar Circle, as compounded with the diurnal Motion, or with the Direction and Velocity of those Parts of the Earth over which it passes; and will be hereafter brought to Calculation. And indeed this Angle may be had, either by finding the several Points of the Path of the Moon's Way upon the Earth, in as many Meridians as we please, and drawing a curve Line through those Points; or by solving a Spherical Triangle, whose Sides are the Complements of the Latitudes of two neighbouring Places equally distant, East and West, from the Place you work for; and whose included Angle is the Angle at the Pole suited to the Difference of their Meridians; and taking half the the two Angles at the Base, the one internal and the other external, for the Angle desir'd▪ Of which hereafter.

XIX.

Every Ellipsis, made by the oblique Section of a Cone, has the Intersection of the (Fig. 1.) Axis of the Cone C at some Distance from the Center of the Ellipsis D. And the Proportion of those unequal Divisions B C and C A are the same with that of the Sides of the Cone V B and V A: As appears by the (Elem. III. 6.) Elements of Euclid. Whence it is evident, that the proper Center of the total Shadow in Eclipses of the Sun, or that made by the Axis of the Cone, is not the same with the Center of the Elliptick Shadow; and that the Proportion of its Distance from that Center may be easily determin'd by the Proposition here refer'd to: Of which more hereafter.

Scholium. This Ellipsis, when the Sun is of a considerable Altitude, is almost an exact one; but when the Sun is near the Horizon, it will be very long, and so less exact; because the Spherical Surface of the Earth is at that Distance more remote from a Plane.

XX.

The perpendicular Breadth of the Shadow is neither that of the longer, nor that of the shorter Axis: But that of the two longest Perpendiculars (Fig. 2.) A B and C D drawn from the Tangents parallel to the Diameter D B, along which the Direction of the Motion is: The length of which Perpendiculars will be hereafter determined.

XXI.

The Velocity of the Motion of the Center of the Shadow is unequal; not only on account of the Difference of the Moon's own Motion, at the beginning and ending of the entire Eclipse; which indeed is very inconsiderable; but chiefly by reason of the Difference of the Obliquity of the Horizon all the Way of its Passage. However, since the several Points may, in all Meridians be distinctly found by Trigonometry, as we shall shew presently, this Inequality need create us no new Difficulty in the Determination of Eclipses.

XXII.

The Number of Digits eclips'd, which are twelfth Parts of the Sun's Diameter, with sexagesimal Parts of the same Digits, are always to be estimated as distinct from the total Shadow; and may be discovered by the help of the foregoing Table, p. 29, 30, 31. Where the Digits are already noted at every proper Distance from the Path of the Moons Center; and where the intermediate Fraction 1/1•2⌊6 is more exact than 1/6•; but which, by dividing that Number 1/16•⌊6 by 2⌊7 will give those Sexagesimals, without any farther Trouble. The Application of that Table will be taught hereafter.

XXIII.

The Distance of the Vertex of the conical Shadow of the Moon, which sometimes just reaches the Surface of the Earth, as in total Eclipses sine morâ, Sometimes does not reach it; as in Annular Eclipses; and sometimes would overreach it, if it were not intercepted, as in total Eclipses cum morâ; may be easily discovered at any time by the Analogy following: As P C the (Fig. 3.) Semidiameter of the Moon: = 941 geographical Miles is to CV the Distance of the Moon from that Vertex = 215000∷ So is R s = 45 the smallest Semidiameter of the total Shadow, which is the same as of the circular Shadow it self, to s V, the Distance of that Vertex therefrom = 10316, which is thus: 941: 215000∷ 45: 10316.

XXIV.

The Determination of the Circumstances of Solar Eclipses, for any given Distance from the Path of the Moon's Center, either way, has no new Difficulty in it; but is to be made just as is that for the Center of the Penumbra. Only the Quantity of the Distance of the Vertex of the Lunar Plane from that Circle will be different; as the Path of the Moon's Center it self might be at another Eclipse, of otherwise the same Circumstances.

XXV.

If Two Bodies A and B set out together, the one from A, the other from B: and move evenly forward in a known Proportion as to Velocity; the Point C will be determined where the swifter will overtake the slower, and they will be coincident. Thus if the Velocity of A, (Fig. 4.) be to that of B, as 5 to 1, the Proportion of the Lines A B to B C will be as 4 to 1, and if we add 1 to 4 = 5 we have the place C where the swifter will overtake the slower. Thus if their Velocities be to each other as 5⌊48 to 1, the Lines of their Motion A B and B C, will be as 4⌊48 to 1. So that if we take in the former Case ¼ of A B and in the other 1/4⌊4• of A B, and add it to A B, we gain AC the Distance of the Point C from A.

Corollary. If therefore A represent the Center of the Shadow of a Solar Eclipse, as it is plac'd at the Middle of the general Eclipse; and B Greenwich at the same Moment of absolute Time; and at a known Distance from the middle Point; and if the Velocity of the Center of the Shadow along the Circle be to the Velocity of Greenwich in its diurnal Motion as 5• 48 to 1, if we ad 4⌊48 of their Distance at their setting out to that known Distance, we obtain the Point or Place where the Center of the Eclipse will overtake Greenwich, or the Time when the Eclipse will be at the Meridian of Greenwich. And this whether the Center and Greenwich move along the same Line as A B C, or two different Lines, as A B C and a b c.

XXVI.

The Duration of Solar Eclipses is different, according as their Middle happe•s about Six in the Morning or Evening; or a•out Noon; or about any intermediate Time. If that happens about Six a-Clock, Morning or Evening, the diurnal Motion then neither much conspires with, nor opposes the proper Motion of the Center of the Shadow; and the Duration is almost the same as it would be if the Earth had no diurnal Motion at all. If that happens about Noon, the diurnal Motion most of all conspires with that proper Motion of the Center, and makes the Duration of the Eclipse the longest possible. If it happens in the intermediate Times, the diurnal Motion, in a less Degree, conspires with the other Motion, and makes the Duration of a me•n Quantity, between that of the other Cases. But if it happens considerably be•ore Six a-Clock in the Morning, or after Six a-Clock in the Evening, the diurnal Motion is backward, and short•ns that Duration proportionably. Of the Quantity of which Duration we shall enquire more hereafter.

PROBLEMS.

I.

To find the nearest Distance of the Path of the Moon s Center, to the Center of the Disk of the Earth, as seen at the Distance of the Mo•n in the total Eclipse of the Sun A. D. 1724. May 11°. P. M.

This is equal to the Moon's true Latitude at the Time of the Conjunction in her own Orbit; and is set down in the Calculation 32′ 19″.

II.

To find the Sun's Declination at the Middle of the Eclipse.

As the Radius of the Circle: is to the Sine of the Sun's Longitude at that Time=61′ 39″∷ So is the Sine of the Sun's greatest Declination, = 23° 29′: to the Sine of the Sun's Declination then.

Rad.

10.

000000

S.

61°

39′

9.

944514

S.

23°

29′

9.

600409

°

″

S.

•8.

544923

=

20

32

=

Declination

☉.

III.

To find the Sun's Right Ascension for the same Time.

As the Radius: to the Cosine of the Sun's greatest Declination∷ So is the Tangent of the Sun's Latitude: to the Tangent of the Sun's right Ascension.

Rad.

10.

Sin.

66

31

9.

962453

Tang.

61

39

10.

267952

°

′

Tang. Right Ascen.

10.

230405

=

59

31

Compl.

30

29

IV.

To find the Distance between the Vertex of the Lunar Circle and the Pole of the Earth.

Let (Fig. 5.) E P represent the Distance between the two Poles of the Earth, and of the Ecliptick: = 23° 29′. EV the Distance between the Pole of the Ecliptick, and the Vertex of the Lunar Circle: = 5° 36′. P EV the Angle made by the Solstitial Colure P E: (in which the two Poles of the Earth, and of the Ecliptick always are) and that Arc EV. And because the Sun is here 28° 21′ distant from that Colure, which is the Complement of its Longitude from Aries; and the Ascending Node, or Argument of Latitude is then 5° 49′ distant from the Sun backward; The Sum of these Numbers gives 34° 10′, whose Complement is the Angle V E R = 55° 50′. In order then to gain V P proceed thus:

R.

10.

CS VER.

55

50.

9.

749429

T. VE.

5

36

8.

991451

°

′

T. RE

•8.

740880

=

3.

10.

+

23.

29.

=

26.

39.

Then say,

°

′

CS.

R E

3

9

9.

992343

CS.

V E

5

36

9.

997922

CS.

P R

26

39

9.

951222

19.

949144

°

′

CS.

P V

=

9.

949801

=

27.

0

V.

In the same Triangle V P E, to find the Angle V P E, included between the Colure E P, and the Prime Meridian P V.

°

′

S. P V

27

0.

9.

657047

S. V E R

55

50.

9.

917719

S. V E

5

36.

8.

989374

18.

907098

°

′

S V P E

=

•9.

249961

=

10

15

Corollary. The Primary Angle, composed of the Complement of the Right Ascension, and of this Angle V P E, is = 40° 44′.

Compl.Rad.

30°

29′

+

10

15

=

40

44

VI.

To find the Distance of the Pole or Vertex of the Lunar Circle from the Circle it self.

As the Semidiameter of the Earth's Disk = 61′ 28″: To the Latitude of the Moon, or nearest Distance to the Path of the Moon's Center from the Center of the Disk; 32° 19′∷ So is the Radius: to the Sine of the Complement of that Distance. In Decimals thus: 61′⌊63: 32′⌊32∷ 10000: 5244 = S. 31° 38′, whose Complement is 58° 22′: equal to the Distance of the Lunar Circle from its Vertex.

VII.

To find the Angle included between the Meridian that passes through the Center of the general Eclipse, and that passing through the Center of the Sun at the same Time.

In the Triangle (Fig. 6.) M S P where the Side S P is already given, = Complement of the Sun's Declination; find the Angle M S P thus:

- R. 10.

- S. of the Sun's Distance from the Solstitial Colure: = 28° 21′ 9.676562

- S. of the Sun's greatest Declination= 23° 29′ 9. 600409

°

′

S. of an Angle •9. 276971 =

10.

54.

To which add the Angle of the Moon's Way: +

5

36

The Sum is the Angle M S P =

16.

30

Then, in the same Triangle P M S, we have two Sides: M S equal to the Latitude of the Moon, or the length of the Perpendicular to the Moon's way = 31° 38′. and P S = 69° 28′. and the included Angle M S P = 16° 30′. to find M P S thus:

R.

10.

°

′

CS.M S P.

16.

30.

9.

981774

T. S M.

31.

38.

9.

789585

T. R M.

=

•9.

771359

=

30

34

M R

+

31

38

M S

=

62

12

R S

Then say,

S.S P.

69.

28.

9.

971493

S.S R.

62.

12.

9.

946738

T.M S P.

16.

30.

9.

471605

•9.

418343

°

′

T. M P S

=

9.

446850

=

15

38

+

40

44

=

56

22

=

Angle with the Primary Meridian.

VIII.

To find the Longitude and Latitude of the Center of the Shadow at the Middle of the general Eclipse; or to solve the primary Triangle.

Under Problem IV. we have sound the Distance of the Vertex of the Lunar Circle from the Pole of the Earth = 27°. 0′. Under Problem VI. we have found the Distance of that Vertex from its Circle = 58°. 22′; and under the last Problem we have found the Angle at the Pole of the Earth, between the Primary Meridian, and that Meridian which passes through the Center of the Eclipse, = 56° 22′. From which data the Primary Triangle (Fig. 7.) is thus to be solv'd:

R.

10.

°

′

CS. C P Q.

56.

22.

9.

743412

T. V P.

27.

0.

9.

707166

T. P R. =

15.

46.

•9.

450578

Then say,

CS.V P

27.

0.

9.

949880

CS.P R

15.

46.

9.

983345

CS.V C.

58.

22.

9.

719730

19.

703075

°

′

CS.P C.

9.

753195

=

55.

30

Deduct PR

=

15.

46

Remt. = P C.

=

39.

44

Ergo Lat.

=

50.

16

To find the Angle at the Vertex CV P, proceed in this Manner:

°

′

S. CV.

58.

22.

9.

930145

S. C P Q.

56.

22.

9.

920436

S. PC.

39.

44.

9.

805647

19.

726083

°

′

S.CVP.=

9.

795938

=

38

41

The following Analogy will give V C P the Complement of the Angle which the Direction of the Center of the Eclipse makes with the Meridian, that Direction being perpendicular to V C.

°

′

S. CV.

58.

22.

9.

930145

S. CP Q.

56.

22.

9.

920436

S. V P.

27.

0.

9.

657047

19.

577483

°

′

S.V C P.=

9.

647338

=

26.

21

Compl.

=

63.

39

Corollary. Hence we also learn the most Northern Latitude, where the Center of the Shadow will cross the Meridian at Noon, and at right Angles: And this without any particular distinct Calculation. For V Q = 58° 22′ • V P 27° 0′ = P Q = 31° 22′. whose Complement = 58° 38′. is that very Northern Latitude.

IX.

To find the Longitude and Latitude of the Center of the Shadow, when it crosses the Meridian that passes through the Center of the Sun at the Middle of the Eclipse; or to solve the second principal Triangle.

If in the foregoing Triangle we suppose the Angle at the Pole to be equal to the Primary Angle, or 40° 44′. we may thus solve this Triangle:

R.

10.

°

′

CS. CP Q.

40.

44.

9.

879529

T. V P.

27.

0.

9.

707166

°

′

T.P R.

=

•9.

586695

=

21

7

Then say,

CS.V P.

27.

0.

9.

949880

CS.P R.

21.

7.

9.

969811

CS.V C.

58.

22.

9.

719730

19.

689541

°

′

CS. CR.

=

9.

739661

=

56

42

Deduct

21

7

Remt. C P.

35

35

Ergo, Lat.

=

54

25

To find the Angle at the Center of the Eclipse V C P, proceed thus:

°

′

S. V C.

58.

22.

9.

930145

S. C P Q.

40.

44.

9.

814607

S. V P.

27.

0.

9.

657047

19.

471654

°

′

S.CV P.

=

9.

541509

=

20

22

X.

To find the Longitude and Latitude of the Center of the Shadow at its Entrance on the Disk of the Earth: Or to solve the third Principal Triangle.

Add the vertical Angle already found = 38° 41′ to a right Angle at the Vertex; = 90 + 38°. 41′ = 128°. 41′ this is equal to the Angle at the Vertex CV P. Substract this Angle from two right Angles. 180°—128°. 141′ = 51°. 19′, in order to gain the Supplement, whose Sines, &c. are the same with the others. (Fig. 8.) Then say,

R.

10.

°

′

CS. CV P.

51.

19.

9.

795891

T. V P.

27.

0.

9.

707166

°

′

T.V R.

=

19.

503057

=

17

40=V R

+

58

22=V C

=

76

02=C R

Then say,

°

′

CS. V R.

17.

40.

9.

979019

CS. V P.

27.

00.

9.

949880

CS. C R.

76.

02.

9.

382661

19.

332541

°

′

CS. C P.

=

9.

353522

=

76:

57

Ergo, Lat.

13.

03

In order to find the Angle at the Pole V P C, whose Supplement is the Longitude of that Point where the Center of the Shadow enters the Disk of the Earth from the Primary Meridian, proceed thus:

°

′

S. C P.

76.

57.

9.

988636

S. CV P.

51.

19.

9.

892435

S. C V.

58.

22.

9.

930145

19.

822580

•80

°

S.V P C.

=

9.

833944

=

43

01

Suppl.

136:

59

To find the Angle VC P, proceed thus:

°

′

S. C P.

76.

57.

9.

988636

S. CV P.

51.

19.

9.

892435

S. V P.

27.

00.

9.

657047

19.

549482

°

′

S.VC P.

=

9.

560846

=

21.

20

XI.

To find the Longitude and Latitude of the Center of the Shadow at its Exit from the Disk of the Earth; or to solve the fourth Principal Triangle.

Substract the Angle already found 38° 41′ from a right Angle 90° − 38° 41′ = 51° 19 = Angle at the Vertex P V C (Fig. 9.) Then say,

R.

10.

°

′

CS. P V C.

51.

19.

9.

795891

T. P V.

27.

00.

9.

707166

°

′

T.V R.

•9.

503057

=

17

40

from

58

22

Remnt

40

42

=

R C.

Then say,

°

′

CS. V R.

17.

40.

9.

979019

CS. P V.

27.

00.

9.

949880

CS. R C.

40.

42.

9.

879746

19.

829626

°

′

CS. P C

=

9.

850607

=

44.

51.

Ergo, Lat.

45.

9.

In order to find the Angle Q P C, or the Longitude of that Point where the Center of the Shadow departs out of the Disk of the Earth, from the Primary Meridian, proceed thus:

°

′

S. CP.

44.

51.

9.

848345

S. PVC.

51.

19.

9.

892435

S. VC.

58.

22.

9.

930145

19.

822580

°

′

S.QPC

=

9.

974235

=

70.

27

+

136

59

=

207

26

And for the Angle VCP thus;

°

′

S. CP.

44.

51.

9.

848345

S. PVC.

51.

19.

9.

892435

S. VP.

27.

00.

9.

657047

19.

549482

°

′

S. VCP.

=

9.

701137

=

30.

10.

Corollary. Hence the Angular Motion of the Center of the Eclipse about the Pole of the Earth, if there were no diurnal Motion, is 207°. 26′.

XII.

To find the Time in which the Center of the Shadow will go over the Diameter of the Lunar Circle.

Say, first, 35′⌊3: 60′∷ 123⌊26: 209⌊6; i. e. As the Number of Minutes of a Degree pass'd over in an Hour: to an Hour∷ So is the entire Diameter of the Disk from the Calculation: to the Number of Minutes for that Passage.

Then say, 10000: 8514∷ 209′⌊6: 178′⌊4 i. e. As the Radius: to the Sine of 58°: 22′ = the Distance of the Lunar Circle from its Pole or Vertex∷ So are the Minutes of the Passage over the entire Diameter: to the Minutes of the Passage over this Chord = 178′ 24″.

XIII.

To find the Proportion of the Velocities of the Center of the Shadow and of the diurnal Motion of the corresponding Point of the Earth at the Time of the Eclipse:

Say thus; As 178′⌊4 to 829′⌊6 = 207° 26′, or as 1 to 4⌊64, so is the Time of the Center of the Eclipse's Motion over the Diameter of the Lunar Circle: to the Timeof the diurnal Motion's going from the entranceto the Exit of the Center.

Corollary. Hence the real Angular Motion of the Center of the Eclipse about the Pole of the Earth, is no more than 162° 40′. For 4⌊64: 3⌊64∷ 207° 26′: 162° 40′.

XIV.

To find the Latitude of any Place, over or near which the Center of any Shadow passes, to any known Longitude or Time given. And, vice versa, To find the Longitude or Time of the nearest Approach to any such Place to any known Latitude. This is no more than proceeding in the Calculations as hitherto; by taking any known Meridian or Time; or else any known Latitude for our Examples.

I shall therefore give three several Examples in both Cases; because of the great Dignity and Usefulness of the Problem: viz. For Greenwich the Meridian of the Tables; for Dublin more Westward; and for Paris more Eastward.

Now I here suppose, from the Calculation and Construction of Eclipses, that the Middle of this general Eclipse will happen May 11. 1724. 17′ past 5 a-Clock in the Afternoon; and that its Center will cross the Meridian of Greenwich 41′ past 6. Upon which Hypothesis I thus compute:

°

′

°

′

From the Angle

100

15

=

6h

41

Deduct the Primary Angle

40

44

There remains the Angle of the Pole Q P C

=

59

31

Then proceed thus:

R.

10.

°

′

CS. Q P C.

59

31

9.

705254

T. P V.

27

00

9.

707166

°

′

T.P R.

=

•9.

412420

=

14

29

Say then,

CS. P V.

27

00

9.

949880

CS. P R.

14

29

9.

985974

CS. V C.

58

22

9.

719730

19.

705704

°

′

CS. R C.

=

9.

755824

=

55

15

Deduct R P

=

14

29

Remains P C

=

40

46

Ergo, Lat.

49

14

N. B. If we take Dr. Halley's Time 6h 36′, and substract 40°▪ 44′ out of 99° there remains 58°▪ 16′; and the Calculation will stand thus:

R.

10.

°

′

CS. QP C.

58.

16.

9.

720958

T. P V.

27.

0.

9.

707166

°

′

T.P R.

=

•9.

428124

=

15

0

Then say,

°

′

CS.P V.

27.

0.

9.

949880

CS.P R.

15.

0.

9.

984944

CS.V C.

58.

22.

9.

719730

19.

704674

°

′

CS.R C.

=

9.

754794

=

55

21

Deduct R P.

15

0

Remt. P C.

40

21

Ergo, Lat.

=

49

39

Diff.

00

25

N. B. My Calculation differs from Dr. Halley's Scheme no less than a full Degree of a great Circle, in the Meridian; if our Difference of Time, which is about 5′, be allowed. And though we take the Doctor's own Time, yet do we differ in Latitude 25 Minutes or Miles; by which Quantity the Doctor's Scheme brings the Center of the Eclipse nearer to London and Greenwich than this Calculation. The reason of which Difference I by no means understand. Time will discover which Determination is most accurate.

Dublin is about 6° 22′ Westward in Longitude from Greenwich. Let us find the Latitude of the Center of the Shadow, when it crosses the Meridian of Dublin. We must proceed thus: As 3⌊64 to 4⌊64 so is 6′. 22″ to 8′. 7″ Deduct then from the Angle at the Pole used for Greenwich this Difference of these Angles 59° 31′ − 8°. 7′ = 51° 24′. which is our Angle at the Pole for Dublin. So that if we use the former Figure with that Angle, we compute as before;

R.

10.

°

′

CS. Q P C.

51.

24.

9.

795101

T. P V.

27.

0.

9.

707166

°

′

T. R P.

•9.

502267

=

17.

38

Then say,

°

′

CS. P V.

27.

0.

9.

949880

CS. V C.

58.

22.

9.

719730

19.

698829

°

′

CS. R C.

=

9.

748949

=

55

53

Deduct R P.

17

38

Remnt P C.

38

15

Ergo, Lat.

51

45

Paris is 2° 19′ more Easterly than Greenwich, Say therefore 3⌊64: 4⌊64∷ 2°. 19′: 2°. 57′ Now 59°. 31′ + 2°. 57′ = 62°. 28′ Then,

R.

10.

°

′

CS. Q P C.

62.

28.

9.

664891

T. P V.

27.

0.

9.

707166

°

′

T. R P.

=

•9.

372057

=

13

15

Say then,

CS. P V

27.

0.

9.

949880

CS. R P

13.

15.

9.

988282

CS. V C.

58.

22.

9.

719730

19.

708012

°

′

CS. R C.

9

758132

=

55.

03

Deduct R P

=

13.

15

Remt. = P C.

41.

48

Ergo, Lat.

=

48.

12

N. B. By such Calculations we may determine the Latitude of the Center of the Shadow's Way, from its entry upon, till its exit out of the Disk of the Earth, to every known Meridian. A Specimen of which I intend to give presently for the several East and West Longitudes from London in the Eclipse before us: And another Specimen in the Eclipse, Sept. 4. 1727.

If the Latitudes be given, as for the Meridian of Greenwich 49°. 14′; For Dublin 51°. 45′; For Paris 48°. 12′: The Case will be that of a Spherical Triangle, when all the Sides are given; and the Longitude or Time is an Angle sought. Thus in the foregoing Figure for Greenwich, V C = 58°. 22′ is the Side against the Angle sought. V P = 27°. 0′, and P C = 40°. 46′. From which data we thus discover the Angle Q C P.

°

′

V C

=

58

22

R.

10.

V P

=

27

00

S.

9.

657047

P C

=

40

46

S.

9.

814900

Sum of 3

=

126

08

Sum

19.

471947

½ Sum

63

4

S.

9.

950138

Diff. of V C

4

42

S.

8.

913488

Double Radius

20.

The Sum

38.

863626

The Remainder

19.

391679

°

′

½ Remainder

9.

695839

= CS.

60

14

double

120.

28

from

180.

00

as before remains

59.

32

Add the Primary Angle

40.

44

Sum

100.

16

Equal in Time to

6h.

41′.

The Time past Noon of the Center's Transit over the Meridian of Greenwich.

For Dublin thus:

°

′

V C

=

58

22

R.

10.

VP

27

00

S.

9.

657047

PC

38

15

S.

9.

791756

Sum of

3. 123

37

Sum

19.

44880•

Half Sum

61

48 ½

9.

945159

Diff. of V C

3

26 ½

8.

778383

Double Radius

20.

The Sum

38.

723542

The Remainder

19.

274739

°

′

Half the Remainder

9.

637369

C S.

=

64

18

double

128.

36

from

180.

00

remains as before

51.

24.

To

51.

24.

Add

40.

44.

Sum 92. 08

=

6h

8½

Which deducted from 6h. 41′, leaves 32′ ½ for the Difference of the Angle at the Pole in Time. Say then 46⌊4: 3⌊64∷ 32′⌊5: 25′⌊4 which is the Difference in time of the Meridians of Greenwich and Dublin.

For Paris thus:

°

′

VC

=

58

22

R.

10.

V P

=

27

00

S.

9.

657047

P C

=

41

48

S.

9.

823821

Sum of 3

127

10

19.

480868

Half Sum

63

35

9.

952105

Diff. of VC

5

13

8.

958670

Double Radius

20.

Sum

38.

910775

Remainder

19▪

429907

°

′

Half Remainder

9.

714953

=

C S.

58.

45

double

117

30

As before, Remainder

62

30

Add, Primary Angle

40

44

Sum

103

11

In Time

6h

52½

From which deduct 6°. 41′. the Remainder 11′ ½ is the Difference of the Angle at the Pole in Time. Say then, 4⌊64: 3⌊64∷ 11 •/2: 9′. which is the Difference in Time, of the Meridians of Greenwich and Paris.

Corol. (1.) The latter Branch of the Problem determines the Hour and Minute when the Centre of the Eclipse crosses the Meridian at any assign'd Latitude; and by a very small Allowance when the very middle of the Eclipse, or of Total Darkness happens in any Place very near the same.

Corollary (2.) The same Branch determines the true Difference of Meridians in all Places over or near which the Center of the Eclipse passes, and so the Distance East and West from any known Meridian whatsoever. Thus because the Meridians of Paris, Greenwich, and Dublin do hence appear to be different as to the Angles of the Pole from the Primary Meridian, in the foregoing Angles, 62°. 28′. 59°. 31′. and 51°. 24′. respectively; It is plain, that the respective Longitude of Paris and Greenwich, when reduc'd from Angles of the Pole to the Difference of Meridians, is 2° 19′. and that of Greenwich and Dublin 6°. 22′. and by Consequence of Paris and Dublin 8°. 41′. Which is no other than the Foundation of my grand Problem of Discovering the Geographical Longitude of Places by Solar Eclipses, from the Latitude given: Of which more hereafter.

XV.

To find the Duration of a Solar Eclipse, along or near the Path of the Moon's Center, in any Place whatsoever.

From the Motion of the Moon from the Sun gain the Duration of the entire Eclipse; or the Time of that Center's Passage over the Diameter of the Penumbra, if there were no diurnal Motion during that Time, thus:

As the horary Motion of the Moon from the Sun, which is in angular Measure 35°. 18′. and is given in the Calculation; to an Hour or 60′ in Time∷ So let the Diameter of the Penumbra there given, also in angular Measure = 65°. 10′. be to a fourth Number: which will be the Number of Minutes requir'd. In Decimals thus: 35′⌊3:60′∷65′⌊17:110′⌊8=1h:50 ⅘. But then, because the diurnal Motion of the Earth must be compounded with this rectilinear Motion of the Moon; and that as we have already stated it, this Eclipse will cross the Meridian of Greenwich 41′ past 6 a-Clock in the Evening; the one Part of the Duration of the Eclipse being before, and the other Part after that Time, both Intervals must be unequally affected by the diurnal Motion, and we must then take the former half-Duration 55′. 24″ distinctly. And since the diurnal Motion of Greenwich is in a larger Parallel than that of the Center of the general Eclipse; while its Obliquity to the Path of the Shadow's Center increases; their respective Motions will nearly keep the same Proportion all along; and so we may safely omit the Consideration of them both. We have also already discovered, that the Velocity of this Center's Motion is here, to that of the Velocity of the diurnal Motion, As 829⌊6 to 178⌊4. or as 4⌊64 to 1. And because the Eclipse begins about 14′ before 6, 14′ after 6 balances the same; and these 28 Minutes are almost all one, as if there were no diurnal Motion at all. So that we have only 27′ 24″ capable of Retardation in the first half: The middle Time of which will be about 30′ after 6. We must therefore look into the Table, p. 22,—26 for the Arc 97½, o• its Supplement 82 1/•, corresponding to 89¼ of the Sines. Where the Difference of half a Degree is instead of 19⌊1, as about Noon, no less than 160, which multiplied by 4⌊64 comes to 7424. So that the Motion is here retarded a 39th Part. Say then, As 39 to 38, so is 27⌊4 to 26⌊7 which 26⌊7 or 26′ 42″ is to be added to the 28 before excepted, for the Duration of the former Part of the Eclipse = 54′ 42″. The Middle of the second Part will be about 73 = 18°¼ backward beyond 90°. that is, 71° ¾ of the Arc, which corresponds to 85° ½ of the Sines, where the Difference of half a Degree is 61° 1/•. This multiplied by 4⌊64 gives 285. So that the Retardation is as 19⌊1 to 285, or 1 to 14⌊9. Say then, As 14⌊9 to 13⌊9: so is 55⌊4 to 51⌊7 = 51′ 42″, which is the Duration of the latter Part of the Eclipse, and added to the former Part of the Duration, gives us the whole Duration = 106′ 24″ = 1h46′ 24″, without the Consideration of the Elevation of the Luminaries above the Horizon, which a small Matter enlarges that Duration: Of which hereafter.

N. B. If we would be still more precisely nice, we may distinctly allow for the Difference of the Parallels, and the different Obliquity of the Direction; of which p 26, 27. before: Which yet are here omitted, as very inconsiderable.

XVI.

To find the Duration of the Total Darkness along the Path of the Moon's Center; if the Luminaries were in the Horizon.

Say, As the Moon's Motion from the Sun in an Hour, or 60′; in the Calculation = 35′ 18″ is to those 60′: So is the Difference of the Diameters of the Sun and Moon in the same Calculation, = 1′ 38″ to a fourth Number: Which will be the entire Duration of the Total Darkness if the Luminaries were not at all elevated above the Horizon. In Decimals thus: 35′⌊3:60′∷1′⌊62:2′⌊73=2′46″⅔.

XVII.

To find the Altitude of the Sun above the Horizon, when the Center of the Shadow crosses the Meridian at Greenwich.

In the Triangle (Fig. 10.) Z P S, Z P is = Distance between the Zenith and Pole of the Earth: 38° 30′. P S is = Distance between the Pole of the Earth and Center of the Sun: = Complement of the Sun's Declination, or to 69° 28′. And the Angle Z P S = interval of Time, from the Meridian = 6h 41′ = 100° 15′, whose Supplement is 79° 45′.

Say then,

R.

10.

°

′

CS.Z P S

79

45

9.

250280

T.Z P

38

30

9.

900605

°

′

T. P R

=

• 9.

150885

=

8

3 =

P R

+

69

28 =

P S

=

77

31 =

R S

And,

°

′

CS. P R

8

3

9.

995699

CS. Z P

38

30

9.

893544

CS R S

77

31

9.

334766

19.

228310

°

′

CS. Z S

=

9.

232611

=

80

10

=

Z S

Ergo, Altitude

=

9

50

XVIII.

To find the Azimuth of the Sun at the same Time and Place.

In the former Triangle the Complement of the Altitude being now found = Z S = 80° 10′. Proceed thus:

°

′

S. P S

80

10

9.

993572

S. Z PS

79

45

9.

993013

S. P S

69

28

9.

971493

19.

964506

°

′

S. PZS=

Suppl.

=

9.

970934

=

69

16

=

P Z

From the West

20

44

XIX.

To find the Number of Minutes that any Solar Eclipse extends to, in a plain Perpendicular to the Axis of the Penumbra.

When the Parallax of the Moon is 57 ′ 17″, which is near its mean Quantity; the Moon's Distance is 60 Semidiameters of the Earth: And one Second on the Disk of the Earth as viewed at the Distance of the Moon; or on the Disk of the Moon viewed at the Distance of the Earth, is exactly one geographical Mile. And 60″ or one Minute, is one Degree upon the Disk of the Earth or Moon. But when the Moon's Parallax, as in the Calculation of this Eclipse, is 61′. 48″ whose Sine is 17976/1000000 or 1/55⌊5 of the Radius nearly: One Second is less than a Mile; and that in the Proportion of 55⌊5 to 60. So that about 65 geographical Miles correspond to 60″, or one Minute: And accordingly, 1781 such Miles will correspond to 32′ 35″, which in that Calculation, is the Semidiameter of the Penumbra, or the utmost perpendicular Extent of the Eclipse.

N. B. We may always make use of the foregoing Tables, pag. 29,—34 for the perpendicular Distances from the Path of the Moon's Centre, at all Durations of the Eclipses, and of Total Darkness, by still calling the Radius of every Penumbra 2000 equal Parts or Miles, and the Radius of every Umbra or Total Shadow, 50 such Parts or Miles; and using them accordingly. And this without any other Inconvenience than the Supposition of other than geographical Miles: Which Inconvenience, by the Reduction of them to geographical Miles afterwards, will always, as here, come to nothing.

XX.

To find the Alteration there is in the Extent and Duration of Eclipses, and of total Darkness, on Account of the Elevation of the Luminaries at that Time above the Horizon.

This is ever, as the Radius: to the Sine of the Sun's Altitude: As compared with the particular Distance of the Moon at that Time. Thus at the Middle of this Eclipse at Greenwich, the Altitude of the Sun has been found to be 9°. 50′, whose Sine is 1708/10000, which divided by 55′⌊5 is equal to ½ of the whole: And in the Semidiameter of the Penumbra, as well as Umbra, amounts to about 6 Miles every Way. Thus also at the Middle of the Eclipse in North-America, where 'tis Central at Noon, the Sun will be about 52°▪ above the Horizon: whose Sine is 788/1000, which divided by 55⌊5 is 1/•0 nearly; which in the same Semidiameters, amounts to about 29 Miles every way. And for the Duration at Greenwich add to the common Duration already found, or to 1h 46′ 24″ the 1/324 Part of the same, or about ⅓ of a Minute both ways: which is about 2/• in the whole; which will bring the entire Duration from 1h 46′ 24″to full 1h 47′ and for Total Darkness will add about 6/50 of the whole total Darkness = 20″, and increase the Duration from 2′ 46⅔ to 3′ 6″ •/3. But for the Alteration in North America, which is about 1/7• both ways, or 1/35 in the whole; this will there increase the entire Duration 4′ 12″, and from 1h 46′ 24″ bring it to 1h 50′ 36″, and for the total Darkness will add about 29/50 of the whole, or bring the 2′ 46″ ⅔ to 4′ 47″.

XXI.

To find the Species and Dimensions of the Ellipsis made by the Total Shadow at the same Places and Altitudes.

The shorter Axis of the Ellipsis, (which is the same with the Diameter of the Conick Shadow it self, in a Plane Perpendicular to the Axis;) is to the Longer, as the Sines before▪mentioned. Or at Greenwich, as 1708 to 10000; and at North America, as 788 to 1000: And since that •horter Axis is in the Calculation 98 Minutes or Miles, of 65 to a Degree; they must be equal to 90 Geographical Miles; which by the last Problem must be increas'd to 96 and to 119 respectively. The golden Rule will therefore soon shew the Length of the longer Axis of the Shadow in both Places.

sine

rad.

miles

miles

For Greenwich

1708:

10000∷

96:

562.

For North-America

788:

1000∷

119:

151.

XXII.

To determine the central Point in the Elliptick Shadow for Greenwich, at the Middle of the same Eclipse.

B• Lemma XIX. and in its Figure; As V B: to V A∷ So is the Sine of B A V = 9° 34′ = 166: to the Sine of A B V = 10° 6′ = 175, and so is B C: to A C. And componend•, As 166 + 175 = 341 to 175∷ So is B C + C A = 562 to C A, which therefore is equal to 289. Also As 166 + 175 = 341 to 166∷ So is B C + C A = 562 to B C; which therefore is equal to 274.

For 341 : 175 ∷ 562 : 289.

And 341 : 166 ∷ 562 : 274.

Whence the Parts of the longer Axis being found, the larger C A = 289 Miles: the lesser B C = 274 Miles. The half of which, D C is the Distance between the central Point of the Shadow D, and the Center of the Ellipsis C, = 7½ Miles.

XXIII.

To determine the Angle of Direction of the Total Shadow over the Meridian of Greenwich.

This to be done either by Construction, upon drawing the Curve Line of this Motion, through the several Points where, by Calculation, the Center of the Shadow crosses the Meridians of Paris, Greenwich▪ and Dublin; and measuring the Angle it makes with the several Meridians; or more exactly for Greenwich by solving the Triangle (Fig. 11.) P V O, where the Arc V O is equally distant West, as Paris is East from Greenwich, is to be found by the Method already made use of; thus:

Paris Angle at the Pole = 62° 28′. Angle for O V at the Pole is less by twice the Difference of the Angles at the Pole for Paris and Greenwich = 5° 54′, And 62° 28′ − 5° 54′ = 56°: 34. First then find the Arc V O = Compl. Latitude for O, as in Probl. XIV. by this Analogy.

R.

10.

°

′

CS. Q P C.

56.

34.

9.

741125

T. P V.

27.

00.

9.

707166

°

′

T.

P R.

•9.

448291

=

15

41

Then say,

°

′

CS. P V.

27.

00.

9.

949880

CS. P R.

15.

41.

9.

983523

CS. V C.

58.

22.

9.

719730

19.

703253

°

′

CS. P C

9.

753373

=

45.

29.

Deduct

15.

14.

Remt. V O.

39.

48.

V G.

40.

46.

V P.

41.

48.

Then in the Triangle V O P whose included Angle is 4° 38′, and whose two Sides are 39° 48′ and 41° 48′. Find the Arc P R as usual thus:

R.

10.

°

′

CS. O V. R.

04.

38.

9.

998577

T. V O.

39.

48.

9.

920733

°

′

T. V R.

•9.

919310

=

39

48

from

41

42

Remnt 2° 6′ = PR.

Then for the Angle VPO, say:

°

′

S. P R.

02.

06.

8.

563999

S. V R.

39.

42.

9.

805343

T. O V R.

04.

38.

8.

908719

18.

714062

°

′

T. V P O.

=

10.

150063

=

54.

42.

And for V O R;

°

′

S. V O.

39.

42.

9.

805343

S. V P O.

54.

42.

9.

911763

S. VP.

41.

48.

9.

823821

19.

735584

°

′

S. V O P =

9.

930241

=

58.

23

+

113

05

Half

56

32½

Compl.

33

27½

So that the Acute Angle V G O, which the Course of the Center of the Shadow makes with the Meridian of Greenwich, is 56° 32′. ½; And by consequence the Angle it makes with the Parallel of Latitude there is 33° 27′ ½. 5 or 6 Degrees mo•e than in Dr. Halley's Scheme.

Corollary. The longer Axis of the total Shadow at the Meridian of Greenwich, makes an Angle of 12° 43′ ½ with the Direction of the Shadow.

for if from the Angle that the Directi•n of the Center of the Shadow makes with the Parallel of Greenwich,

33

27½

We deduct the Angle that the longer Axis makes with the same Parallel, which is the Azimuth formerly discovered,

20

44

There remains this Angle

12

43½

XXIV.

To D termine the perpendicular Breadth of the Total Shadow, when that Shadow is nearest to London; or when the central Point is about the Meridian of Padsto• in Cornwall.

This Problem is, In an Ellipsis of 96 Miles broad, and 562 Miles long; whose longer▪ Axis makes an Angle with the Direction of the Shadow of 12° 43½; To determine the Length of the greatest Perpendicular that falls on that Diameter, along which the Direction of the Motion is. For twice that Perpendicular is the Breadth required. Now I have actually drawn this Ellipsis, or total Shadow, for my Map of this Eclipse. And I find by that Construction, that the Breadth of the total Shadow over England is about 155 Miles; or 30 Miles broader than in Dr. Halley's Scheme. Time will probably discover which is nearest the Truth.

N. B. This total Shadow increases so greatly in Length as it goes Eastward, that it will reach from Paris to the utmost Boundary of the Sunsetting at once. And though the Center of the Shadow, by my Calculation, will end 8° ⅔ East of the Meridian of London, in the Latitude of 45° 9′, or in the Alps, not far from Milan and Turi•; Yet by reason of the Extension and Breadth of the total Shadow, and the Refraction of the Rays of Light near the Horizon, it seems probable to me, that all Switzerland and Lombardy, as far as Trent, Mantua. Cremona, and Parma; nay, perhaps, as far as Venice, Padua, Bononia, Ferrara, Ravenna and Florence; may at the same Time be invelop'd in the Total Shadow; and that the Sun may set eclipsed at all or most of those Places at once.

XXV.

To determine the Digits eclipsed, and the Duration of the Eclipse, or of total Darkness, at any given Distance from the Path of the Moon's Center; and vice versâ. These Digits and Durations, if we do not consider the diurnal Motion, are immediately found to any given Distance in the Tables, P. 29,—34. Thus, if the Digits be 4, or Duration 80′ 24″, the first Table gives us the Distance of 1350 Miles. And, vice versâ if the Distance be known to be 1350 Miles; the same Table gives us 4 Digits and 80′ 24″ Duration. And the like is true of the total Shadow, and the Distance or Duration in the second Table. Thus also we learn from the Map of this Eclipse, that the nearest Approach of the total Shadow to London will be about 40 geographical Miles: Which 40 Miles, in this Obliquity of the Motion, is about 31 perpendicular Miles. So that 32/16•• = 12/6• of a Digit, is the Quantity of the Rim of Light that will be seen at London, when the Eclipse is greatest: And by consequence the Digits there eclipsed will be 11 48/60. Nor does it appear to me, that the total Darkness will come nearer to London, than 40 Miles; although Dr. Halley's Scheme brings it within 7 or 8 Miles. As for the Duration of the entire Eclipse here, it will be much the same as if it were Central: And has been already determined to 1h 47′. And for the total Darkness along the Path of the Center between Exeter and Plimouth, its Duration will be, as before stated, 3′ 6″½.

N. B. As to the Alterations which will arise in more considerable Distances from the Path of the Moon's Center, proceed thus: First, find the Minutes of Duration corresponding to any given Distance in Miles, e. g. 1350, thus; 2000: 1350∷ 32′⌊63: 22′⌊02. Or, As 2000 Miles, the Radius: to 1350 the Sine∷ So are the Number of Minutes for the Semidiameter of the Penumbra: to the Number of Minutes for that Sine. Now 32′ 19″ the present Length of the greatest Distance − 22′ 01″ = 10′ 18″. Then compute every Thing as if the Center of the Shadow were 10′ 18″, Latitude or nearest Distance; and as if the Chord of that Arc over the Earth's Disk were the Path of the Moon's Center: and the Chord of the Distance from the Center of the Penumbra were the Diameter of the Penumbra: And all will be discovered by the Rules before going; without any new Directions whatsoever. Only if this Distance be the contrary way to the present Example, we must here make use of Addition instead of Substraction.

N. B. Upon confulting the French Ephemerides of Des Places, recommended, in some Degree, by Cassini himself, I find that he determines the Eclipse at Paris thus.

h

′

″

Beginning, May 22. (N. S.)

06

03

00

Middle

06

58

38

End

07

54

00

Whence the Duration there is

01

51

00

Digits eclipss'd

10

17/60

Latitude of the Moon at the Middle

39′

½

Difference from me in Digits

01

31/••

In Latitude

07

10

If these Digits and this Latitude be right, Dr. Halley, and I, and those English Astronomers that have computed and constructed this E•lipse, who all, I think, do agree, that it will be to••• at Paris, and that for two or three Miuutes also, are prodigiously mistaken. For if Des Places be in the right, Copenhagen and Stockholm will stand much fairer for the Pretence to a total Eclipse, than either Dublin, London, or Paris. Time will certainly determine who are in the right.

N. B. If we add the Duration of the Eclipse, consider'd without Regard to the diurnal Motion, to the Time of the Center of the Shadows passing over the entire Disk, we gain the entire Duration of the Eclipse in general, thus:

h

′

″

To

1

50

48

Add

2

58

24

Sum

4

49

12

Therefore the general Eclipse by the Meridian of London,

h

′

″

Begins

II.

52

24

Middle

V.

17

00

End

VII.

41

36

Duration

4h

49

12

Eclipse at London,

Begins V.

45

00

Middle VI.

41

00

End VII.

32½

00

Duration 1h

47½

00

Digits eclipsed

11

48/60

Duration of Total Darknes• between Exeter and Plimouth,

03

6½

N. B. Dr. Halley's Times are still about 5′ sooner than mine, and his general Duration about 7′, and his Duration at London about 1′ longer.

N. B. I cannot tell the Reason why my Original Calculation of this general Eclipse, which has been carefully made according to Sir Isaac Newton's famous Theory of the Moon, does here so much differ from Dr. Halley's Determination, as 5′ in Time; especially since both those Methods did very well agree, in the last celebrated total Eclipse, Apr. 22. 1715. 'Tis Time alone that can determine between these two Methods of Calculation.

N. B. I have lately been shewed an exact Scheme of this next Eclipse, according to Mr. Flamsteed's own Tables and Determination, and made in his Life-time; wherein the Digits eclipsed are 11 18/60 exactly according to my Determination in this Paper.

A PROPOSAL For the Discovery of the LONGITUDE of the several Places of the Earth, by Total Eclipses of the Sun.

IT is humbly proposed, That Observations be made in all Places where Solar Eclipses are seen, of the exact Duration of the same; by either viewing the Beginning and Ending thereof through a Telescope, with a Glass smoaked in the Flame of a Candle, for saving the Eye of the Observer; or else by casting the Sun's Image through such a Telescope upon white Paper, and viewing the first and last Impression of the Moon's Shadow upon it. And that the Hour, Minute, and Second of some Pendulum Clock be carefully noted at the same Time: And that when the same Observations are transmitted for the Uses of Geography, the Latitudes of the Places be also set down and transmitted at the same Time: That the like Observations be also made in all Places where such Eclipses are total and visible, of the exact Beginning and Ending, with the Duration of Total Darkness; by the like Comparison of a Pendulum Clock, or other pendulous Body that vibrates Seconds, or half Seconds; and, with the Latitude be transmitted in the same Manner, and for the same Uses. How by the Help of these

〈4 pages missing〉

A Table of the Latitude of the Center of the total Shadow at the Eclipse, May 11. 1724. to every 10 Degrees of the Angle about the Pole, and every 7°⌊845. Longitude from London.

Angles at the Pole.

Longitude from London.

Latitde.

°

′

° East.

°

′

70

19

8⌊577

45

9

69

23

7⌊845

45

34

West.

59

23

0000

49

17

49

23

7⌊845

52

19

39

23

15⌊690

54

42

29

23

23⌊535

56

29

19

23

31⌊380

57

43

09

23

39⌊225

58

13

00

00

46⌊600

58

38

00

37

47⌊070

58

38

10

37

54⌊915

58

21

20

37

62⌊760

57

36

30

37

70⌊605

56

18

40

37

78⌊450

54

27

50

37

86⌊295

51

59

56

22

Midle.

90⌊021

50

16

60

37

94⌊140

48

51

70

37

101⌊985

45

5

80

37

109⌊830

40

40

90

37

117⌊675

☉

35

56

100

37

125⌊520

30

31

110

37

133⌊365

25

15

120

37

141⌊210

20

11

130

37

149⌊055

15

37

136

59

154⌊052

13

3

N. B. Here, as well as in the next Table, The Angles at the Pole differ by Ten Degrees; and the Longitude from London is found by the Proportion of 4⌊64 to 3⌊64. to be 7°⌊845 of Longitude from London for every such Ten Degrees. But if, as is generally most convenient, we would have the Differences of Longitude from London be even 10 Degrees; We must find the correspondent Angles at the Pole before we begin our Calculations by the inverse Proportion of 3⌊64: 4⌊64∷ 10°: 12′¾ the perpetual Addition of which Number will give us a Series for such a Calculation.

A Table of the Latitude of the Center of the total Shadow at the Eclipse, Sept. 4. 1717. to every 10 Degrees of the Angle at the Pole; and every 7°⌊167 Longitude from London.

Angles at the Pole.

Longitude from London.

Latitde.

°

′

° West.

°

North

11

33

12⌊248

30

18

4

28

7⌊167

30

20

East.

00

00

3⌊963

30

14

5

32

0⌊000

29

28

15

32

7⌊167

27

56

25

32

14⌊334

25

40

35

32

21⌊501

22

38

45

32

28⌊668

18

27

55

32

35⌊835

14

24

65

32

43⌊002

9

21

75

32

50⌊169

5

4

77

58

Midle.

51⌊903

Midle.

3

57

85

32

57⌊336

0

16

South.

95

32

64⌊503

4

32

105

32

71⌊670

9

55

115

32

78⌊837

14

54

125

32

86⌊004

19

16

135

32

193⌊171

23

00

145

32

100⌊338

25

57

155

32

107⌊505

28

8

165

32

114⌊672

29

34

168

39

116⌊909

30

14

3⌊53: 2⌊53 ∷ 10: 7⌊167.

2⌊53: 3⌊53 ∷ 10: 13⌊952.

A Calculation of the Total Eclipse of the Sun, A. D. 1724. May 11. Post Meridiem.

M.'s mean Mo.

Motin of Apogee.

Motn of Node r tr.

s

°

′

″

s

°

′

″

s

°

′

″

Anno Dom.

1701

0

15

20

00

11

8

20

00

4

27

24

20

Years

20

4

13

34

05

3

3

50

15

26

50

15

and

3

28

9

10

4

1

59

32

1

27

59

9

Leap-Y. May, Days

11

9

29

17

03

14

42

21

6

59

•4

Hours

5

2

44

42

1

24

40

Minutes

15

8

14

4

2

(Mean Time.)

3

1

49

30

Moon's mean Motion

1

29

13

14

6

28

53

36

1

25

34

50

Sun's mean Anomal.

10

22

19

57

An. Eq. add

12

2

An. Eq. subst.

5

43

Physic. Parts substract

7

7

9

38

6

29

5

38

•

25

29

7

2d Equat. substract

2

31

Mean Pla. correct

Mean Pla. correct

3d Equat. add

10

2

16

2

1

38

45

2

1

38

45

6th Equat. add

2

6

Sun's true Place

Sun's true Place

Diff. substr.

7

22

7

2

33

7

6

9

38

Moon's mean Pl correct

1

29

5▪

5•

Annual A•g•ment

S's dist. from Node

Tru• Pl. of Apog. substr.

7

9

14

38

10

9

0

Equat. add

18

56

Moon's mean Anomal.

6

19

51

14

Equat. add

1

25

48

3

Equat. add

2

31.

7

7

9

14

38

True Pl. of Node

Moon's true Pl. in Orbit

2

1

36

59

True Pl. of Apogee

5

17

9

Sun▪• true Pla. substract

2

1

38

45

Eccentricity 6000

Inclinat. Limit.

Moon's dist. from Sun

11

29

58

14

′

″

M's h mo•.

37

42

Sun's

2

24

M's fro. Sun

35

18

35′ 18″: 60′∷ 1′ 46″: 3′ 2″ ▪

Eq. time add

3

53

Reduct. subst.

2

33

Diff. add

4

22

Ergo, Ecl pse is 11d 5h 19 22″

Temp. appar. deduct the Error of the Tables about▪

2′

22″

true Tim.

5h

17 0

′

″

S mid. Sun

15

53

Sem Moon

16

42

Sem. Penbra

•32

35

Semi. Difc.

61

38

Angle of the M's Way with the Ecliptick

5

36

Diff. Diamet r of the Sun & Moon, or Br•adth of total Shad.

0

98

Nodes true Pl. substract

1

25

48

3

Argument of Latitude

5

48

56

Reduction substract

1

30

Moon's tr. Pl. in Eclip.

2

1

35

29

Mo n's true North Lat.

32

14

At the Eclipse

32

19

A Calculation of the Sun's Place for the same Time.

Sun's mean Motion.

Motion of Perihelion.

s

°

′

″

s

°

′

″

A. D.

1701

9

20

43

40

3

7

44

30

Years

20

9

4

21

0

And

3

11

29

17

0

3

9

Leap-Y. May, Days

11

4

10

6

19

23

Hours

5

12

19

Minutes

15

37

3

8

9

2

Place of Perihelion.

2

90

28

59

Moon's mean Motion.

10

22

19

57

Sun's mean An•m.

(mean Time)

Sun's mean Motion

2

0

28

59

•quation add

1

9

46

Sun's true Place

2

1

38

45

A Calculation of the Total Eclipse of the Sun, A. D. 1727. Sept. 4. Mane.

M's mean Mot.

Motion Apogee.

Motn of Node retr.

s

°

′

•

s

°

′

•

s

°

′

•

Anno Dom.

1701

10

15

20

0

11

8

20

0

4

27

24

20

Years

20

4

13

34

5

3

3

50

15

26

50

15

And

6

2

9

28

55

8

4

5

44

3

26

1

29

Sept. Days

3

1

23

35

27

24

23

13

1

37

Hours

20

10

58

49

5

34

2

39

Minutes

35

19

13

10

5

(Mean Time.)

5

5

56

5

Moon's mean Motion

5

21

4

37

11

13

46

6

11

21

28

15

Sun's mean An ma•y

2

15

32

16

An. Eq. sub.

18

59

An. Eq. add

9

7

Physical Parts add

11

23

11

13

27

7

11

21

37

22

2d Equat. substract

0

38

Mean Pl. Apo. cor.

Mean Pl. of No.

01

6th Equat. add

1

57

5

21

52

53

5

21

52

53

Difference add

12

42

Sun's true Place

Sun's true Place

M•on's mean Pl. correct

5

21

17

19

6

8

25

4•

6

0

15

31

True Pl. of Apog. subst.

11

16

23

44

Annual Argum.

S.'s dist. •ro. Nod.

Moon's mean Anomaly

6

4

53

35

Eq•. Ad

2

56

37

Eq. Ad.

0

48

Equation add

42

26

11

16

23

44

11

21

38

10

M.'s Correct. Pl. in Orb.

5

21

59

45

True Pl. of Apo.

True Pl. of Node

Sun's true Pla. substract

5

21

52

53

Eccentr.

663

37

5

17

20

Moon's Dist. from Sun

6

52

Inc•in. Limit.

Variation add

9

S'.s hor. mot.

2

26

Moon's

38

29

M. fr. S.

36

2

36⌊04: 60 ∷ 6⌊9: 11 30.

Ergo, Eclipse will be 20h 23 34

Mean Time.

Eq. tim. 2d.

4

38

Reduct. substr.

9

Diff. add

4

29

20h

28

03

Appar. Time.

Semid. S.

16

1

Semid. M.

16

50

Diff. =

00

49

Sum =

32

52

Parallax M. Horizon

62

7

Sun

10

Semid. D••k

61

57

Angle of the M.'s visible Way with the Ecliptick

5 36

20

M.'s true Pl. in Orbit

5

21

59

54

Nodes tr. Pl. substract

11

21

38

10

Argum•nt of Latitude

6

0

21

44

Reduction substract

6

Moon's true Pl. in Eclip.

5

21

51

48

Moon's true South Latit.

1

54

At the Eclipse

1

23

A Calculation of the Sun's Place for the same Time.

Sun's mean Motion.

Motion of Perihel.

s

°

′

•

s

°

′

″

A. D.

1701

9

20

43

40

3

7

44

30

Years

20

9

4

21

0

And

6

11

21

33

8

6

18

Sept. Days

3

8

2

28

9

40

Hours

20

49

17

3

8

12

28

Place of Perihel.

5

23

44

44

Sun's mean Motion

2

15

32

16

Sun's mean Anomaly.

Minutes

35

1

26

(Mean Time.)

Sun's mean Motion

5

23

44

44

Equat. substract

1

51

51

Sun's true Place

5

21

52

53

Some Account of Observations lately made with Dipping-Needles, in Order to discover the LONGITUDE and LATITUDE at Sea.

UPON the Receipt of the liberal Assistance of His most Excellent Majesty, King GEORGE; their Royal Highnesses the Prince and Princess of Wales, and many other of the Nobility and Gentry, my kind Friends, I sent last Year Four several Dipping-Needles to Sea; with Frames hung near the Center of Motion in Gimbols, to avoid the Shaking of the Ship; and with proper Instructions to the Masters of the Vessels: And this, in order to discover the State of Magnetism in the several Parts of the Globe; and to find whether accurate Observations could be made at Sea, and to determine whether the fundamental Theory I laid down from former Observations would hold or not; viz. "That Magnetick Variation and Dip are all deriv'd from one Spherical Magnet in the Center of our Earth; with an irregular Alteration of the Variation, according to the different Degrees of Strength of the several Parts of the Loadstone, as compounded with a very slow Revolution from East to West: And with a regular Alteration of the Dip, nearly according to the Line of Sines, from the Magnetick Pole to the Magnetick Equator; the Axis of that Equator being sufficiently Oblique to its Plane: All which is the Case of Spherical Loadstones here." Now having already received Four Journals from Four several Masters employ'd, I take this Occasion of returning my Benefactors hearty Thanks for their Assistance, and of giving them and the Publick some Account of the Success of these Observations; and what Consequences are naturally to be drawn from them; with the Difficulty hitherto met with in the Practice at Sea, and the proper Remedy for the same in future Trials.

Captain James Jolly set out in July, 1722. for Archangel, with one of my Dipping-Needles on Board. He, for some time, met with such Difficulties in the Practice, as confin'd to the Frame I had given him, that he was not at first able to make any good Observations at all. But after some Time, he took the Needle into his own Cabin; and without any Approach to the Center of Motion, or any Contrivance. for avoiding the Shaking of the Ship at all, having a clear and full Gale all along, but without any stormy Weather, He made me 28 very good Horizontal Observations, from the Latitude of 65 quite to Archangel: I say, Horizontal Observations only, as I desired him; the Needle, by an Accident before he went, being rendred incapable of making any other with sufficient Accuracy. In this Space the Needle altered its Velocity very greatly, as I expected it would: And 5 Vibrations which at first were perform'd in about 280″; beyond the North-Cape came to 250″; till towards Archangel it gradually returned to about 177″.

Captain Othniel Beal set out about the same Time for Boston in New-England, with the same Instrument, and made Four Observations of the Dip, both by the Vertical and Horizontal Vibrations, and by the Dip it self; Three upon the open Sea, and One in the Haven of Boston: Which in some small Manner differed one from another, but in the main agreed, and kept the due Analogy I expected. He greatly complained of the Shaking of the Ship; till in Boston Haven he made a nice Observation both Ways, which did not greatly differ: Tho' the greatest Part of of his Observations by the Dip it self were somewhat more agreeable to Analogy than the other. The Reason was, I take it, that, as he assured me, he always took great Care to avoid the Shaking of my Frame; which Frame tho' it very much avoided the slower and greater Oscillation of the Ship, yet made a quicker but lesser Oscillation it self: Which Fault I was sufficiently sensible of just before the Ships were going away, but was not able then to obviate; as I am prepared to do hereafter. After Captain Beal had made and sent me these Observations, he pursued his Voyage to Barbados, and thence to Charles Town in South Carolina; at both which Places he made Observations; but the best at Barbados. For before he came to Carolina, he observed the Axis of the Needle to shake; which made him take the Dip there otherwise than he ought to have done; which is the natural Occasion that the Dip there did not so well agree to Analogy as the rest. However, upon my Receipt of his first Journal, with the Four first Observations, especially the exact one at Boston, I formed a more exact Theory of the Proportion of the Alteration of the Dip in the Spherical Magnet of the Earth; and found it at this Distance of the Earth's Surface, not far from that in my Spherical Loadstone, at the Distance of about 9/10 of an Inch from its Surface; viz. Not exactly as the Line of Sines, where at the Middle of the Line the Angles are 60 and 30; but rather as 66 to 24. Which Rule therefore is what I now propose as much nearer than the other. By which Proportion I determined long before-hand the Dip at Barbados of 43° or 44°, as many of my Friends can witness: And when Captain Beal delivered me the Paper of this Observation at Barbados, before I opened it, or in the least knew what Dip it contain'd, I foretold to him from that Theory the very same Dip, which both himself and his Paper immediately assur'd me to be true; and whose Truth, as he inform'd me afterwards, was confirm'd by another Observation, made a little before in the open Sea, of about 45°.

Captain Tempest also, about the same Time, set out for Antegoa and St. Christopher's, with the same Instrument and Frame. In his Letter, da •ed last January, he greatly complains of the Shaking of my Frame; and proposes an Hint how it might be avoided: Which Method of its Avoidance I had long before thought of, and provided for accordingly; and which has been a full Year ready for Practice. Those Observations of his, that I have yet received; for I have not heard from him since January, but hope soon to here farther; were but Three, and all at open Sea; and but one of them made both the Ways that I desired: And, indeed, seem the least agreeable to Analogy of any of the rest. Only since that single Observation, which was also made by the horizontal Vibrations and vertical Oscillations, agrees very well to that Analogy; since they all three are about the same Quantity of 8 or 9 Degrees exceed that Analogy; and since very near the same Place, where the third Observation was made, I have a double Observation of Captain Beal's to correct the same; I rather conclude, that Captain Tempest made a Mistake, and placed the wrong Edge of the Needle upward in all the Three Observations: Which would naturally occasion such a Difference. When I receive the rest of his Observations, or his Needle again, I shall be able to judge better of that Matter. However, even these Observations agree in gross with all the rest, to the gradual Decrease of the Dip as you go nearer to the Equator: Tho' as they stand at present, they do not determine the accurate Proportion of that Alteration so well as the others.

Captain Michel also, long after the rest, set out for Hamburgh with the same Instrument; though now without the Frame, which he was not willing to incumber himself with: and I suspected that in its present Contrivance it did more hurt than help the Nicety of the Experiments. I also by him, sent a Letter to the Reverend Mr. Eberhard, who was the Occasion of my studying this Matter, and was then Pastor of Altena, close by Hamburgh; desiring that he would there make the Experiment very exactly, and give me a particular Account of it. But I have not yet received his Answer.

Now the Observations here mentioned, as well as those many others I had by me before, do seem to me in general evidently to afford us the following Inferences:

(1.) That there is one Spherical Loadstone, and but one in the Center of our Earth; and that this Loadstone, like other Spherical Loadstones, has but one Northern Pole: Contrary to Dr. Halley's Hypothesis.

(2) That this Northern Pole is situated, contrary to the same Hypothesis also, a great Way to the East of our Meridian: And indeed, as I before had determined, about the Middle of the Distance between the North Cape and Nova Zembla. Captain Jolly's numerous Observations prove this most fully: While in Sailing towards that Point his horizontal Vibrations greatly increas'd in Number: And when he turned almost at right Angles, as he went down to Archangel, they soon diminished; and yet so little, after some time that it was evident he then sailed not far from a Parallel to that Northern Pole; and not very many Degrees from it neither; exactly according to my Expectations.

(3.) That the absolute Power of the internal Magnet is considerably different in different Places; and that without any certain Rule; as it is upon the Surface of our Terrellae or Spherical Loadstones here. This the various Number of Seconds to a vertical Oscillations, and all the Accounts in the other Observations fully prove; and by consequence this must cause different Variations in different Places, as is the Case of our Terrellae.

(4.) That there no where appears in open Seas any such Irregularity in the Dip, as we sometimes meet with near Shores, or at Land; and by consequence that Dr. Halley's grand Objection against the Discovery of the Longitude by the Dipping-Needle, taken from an Observation of his own, concerning such an Irregularity near the Shore at Cape Verd; and from his own Hypothesis of the four Magnetick Poles is utterly groundless. Nor indeed shall I •e at Rest, till▪I have sent a Dipping-Needle to Hudson's Bay, on purpose to determine this Dispute about the four Poles: For that Voyage being almost directly towards his second Northern Pole all the way, and about the same Distance all the way from mine; if this Voyage afford much the same Dip, it will demonstrate that there is but One Northern Pole; and that it is nearl• where I place it: But if that Dip greatly increase, it will demonstrate a s••ond Pole somewhere in those Parts of America, where Dr. Halley places it. And to this Decretory Experiment do I appeal for a final Determination of this Question. The Doctor seems to me to draw his Inferences from the Variation, which no Way proves any such double Poles; as being full as sensible on our Terrelle, which have no more than single ones; while he avoids all Observations from the Dip, which are still against him; and which are alone capable of discovering the exact Place of such Poles, either upon the Surface of the Earth, or of Terrellae. However, when one Set of Experiments with a Dipping-Needle, sent to Hudson's Bay, will certainly determine this Matter, 'tis a vain Thing to go on in the Way of Controversy about it.

In short, The Observations hitherto made, shew that the Foundations I go upon in this Discovery of the Longitude and the Latitude at Sea, are true and right: That the Terrestrial Magnetism is very regular and uniform, in the open Seas; that the Latitude in the Northern Parts may even, without any Avoidance of the Shaking of the Ship, in ordinary calm Weather, be in good Degree thereby discovered already; and that if I can sufficiently avoid the Shaking of the Ship, which I am now endeavouring, and have great Hopes of performing, both Latitude and Longitude may by this Method be discovered in the greatest Part of the sailing World. I say nothing here of another Method of Trial, which I am also pursuing, and which depends, like this, on the avoiding the main Part of the Ship's Agitation; and if effected will be more easy and universal than this. But as to giving any farther Account of that to the Publick, unless it succeed, I have no Intention at all.

N. B. The original Journals are all in the Hands of my great Friend and Patron Samuel Molyneux, Esq Secretary to his Royal Highness, the Prince of Wales, and Fellow of the Royal Society: Which Journals, when I have compleated the rest of the Observations I hope to procure, I intend to publish entire, for the more full Satisfaction of the curious,

A Table of the Angle of Inclination below the Horizon, in Dipping-Needles, to every 1/9• Part of their respective equal Distances from the Magnetick Poles and Equator.

Dist. from the Pole.

Dip.

Dist. from the Equat

Dip.

°

°

°

°

°

°

1

89

30

1

08

41

2

89

00

2

12

23

3

88

27

3

15

14

4

87

59

4

17

41

5

87

29

5

19

51

6

86

59

6

21

50

7

86

38

7

23

41

8

85

58

8

25

24

9

85

27

9

27

2

10

84

57

10

28

36

11

84

27

11

30

6

12

83

56

12

31

32

13

83

26

13

32

55

Dist. from the Pole.

Dip.

Dist. from the Pole.

Dip.

°

°

°

°

°

°

14

82

55

14

34

15

15

82

24

15

35

33

16

81

54

16

36

50

17

81

23

17

38

4

18

80

52

18

39

16

19

80

21

19

40

28

20

79

49

20

41

37

21

79

18

21

42

45

22

78

47

22

43

54

23

78

16

23

44

58

24

77

44

24

46

2

25

77

12

25

47

6

26

76

41

26

48

9

27

76

8

27

49

10

28

75

36

28

50

12

29

75

4

29

51

12

30

74

32

30

52

12

31

73

59

31

53

11

32

73

26

32

54

9

33

72

54

33

55

6

34

72

20

34

56

1

35

71

47

35

57

0

36

71

14

36

57

56

37

70

39

37

58

52

38

70

5

38

59

47

39

69

31

39

60

41

40

68

57

40

61

35

41

68

22

41

62

49

42

67

47

42

63

22

43

67

12

43

64

15

44

66

36

44

65

8

45

66

00

45

66

0

N. B. I take the Northern Pole of the Terrestrial Magnet to be about the Meridian of Archangel, in the Latitude of 75½. Its Equator to be nearly a great Circle, intersecting the Earth's Equator about 2½ Degrees Eastward of the Meridian of London; and in its opposite Point. And that its utmost Latitude Northward is in the Gulph of Bengall about 12½ Degrees; and as much South in the opposite Point, in the great South Sea. And that the Souther• Pole is nearly circular; its Radius 40 Degrees of a great Circle, and its Center in a Meridian Eastward from Ceilon about 4½ Degrees, and about 68½ Latitude.

N. B. London is nearly 1•⌊5/84 = 2⌊6/90 distance from the North Pole of the Magnet, whence its Dip will be at 74 •/•, which is certainly so in Fact. Boston in New-England is 51/••• = •⌊•/9•, whence its Dip will be about 68° 22′, which Captain Beal found to be so in Fact. Barbados is about 26⌊•/1•• = 22/90⌊5 distant from the Equator of the Magnet, whence its Dip ought to be about 44° ½, as Captain Beal also found it to be in Fact. St. Helenais about 14/49 = ••⌊•/9•, whence its Dip ought to be about 47° 50′ as Dr. Halley found it to be in Fact. And so every where in the main Ocean, at considerable Distances from the Shores.

N. B. If the Dip of any Needles be somewhat different at London, add or substract a proportionable Part of the Dip elsewhere. And you will have nearly the true Dip at any other Place with that Needle Thus if your Needle differ from the other 2° or 120′, and shew the Dip at London 72° 45′ instead of 74° 45′, which is its-proper Dip in this Table; and you require the true Dip by this Needle for Boston in New-England, Southward; which in the Table is 68° 22′, proceed thus. Because the equal Distance of Boston from the Magnetick Equator is 49 Parts of 60⌊4, the like Distance of London from that Equator; deduct •9/60⌊4 120′ = 97′ = 1° 37′ out of the Tabular Dip 68° 22′. The Remainder is 66° 45′, for the true Dip at Boston with that Needle. Thus if you want the true Dip, by the same Needle, at Dronthem in Norway, Northward: Because the equal Distance of Dronthem from the Magnetick Pole is 15⌊2 Parts of 29⌊6 the Distance of London from that Pole; deduct 15/29⌊ 2/6 12′ = 62′ = 1° 2′ out of the Tabular Dip 82° 30′, and the Remainder, 81° 28′ is the true Dip at Dronthem, with that Needle: And so in all other Cases whatsoever.

N. B. The Table before set down, supposes that the true Dip differs according to such a Line of Sines, whose middle Point gives 66° on one Side, and 24 on the other; and is made by adding or substracting 8 to the Complement of the Dip found by the natural Sines for every 1/•0 of equal Distances from the Equator or Pole.

N. B. If any desire to calculate by Trigonometry the Distances of all Places from the magnetick Equator of Poles, and the Distances of that Equator and those Poles in every particular Case, both made use of in the foregoing Calculations, it is thus to be done:

In the (Fig. 12.) Triangle B L A we have B L the Co-Latitude of London; B A the Co-Latitude of the magnetick North Pole; and the included Angle, A B L = the Distance of the Meridian of that Pole from the Meridian of London; to find the Angle Q A M and the Side A L. Then in the Triangle Q A M, we have the Angles Q A M and Q M A, and the Side A M, = the Distance of the Magnetick Pole from the Magnetick Equator, to find A Q. So we have the Proportion of A L to AQ, Q. E. I.

But since the Data are not yet sufficiently exact for the Calculation, measuring is sufficient.

FINIS.

to fold out at the end of ye Book
Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 7.

Fig. 6.

Fig. 9.

Fig. 8.

Fig. 10.

Fig. 11.

Fig. 12.

ERRATA.

PAge 1. lines 12, 13. read, 5 Leap. Days; and with 11 Days when 4 Leap Days. P. 7. over against 21, &c. read, 53. 2. 55. 40. 58. 19. 60. 58. 63. 38. 66. 19. 69. 0. 71. 42. 74. 24. 77. 7. 79. 50. 82. 34. 85. 19. 88. 4. 90. 50. P. 10. l. 32. read, Summer. P. 13. l. 7. r. 10d. l. 8. and 10, and 16. r. 18. P. 13. l. 24. r. Bern, Zurich, and Pillaw near Koningsberg in Prussia. Dele p. 16. l. 32. to p. 17. l. 10. and instead of it read thus: Its greatest Alteration therefore must be at the mean Distance; and is the Difference of the Equation belonging there to the Addition of 10°⌊8 = 17. which Space the Moon usually goes in about 36 of time. So that the Difference on this Account must, each Period, be usually less than 36′. And as to the Moon's own Motion, it has also its greatest Alteration at its mean Distance; and is the Difference of the Equation at 2°. 51. ½=17. which the Moon usually goes in about 36′. of time. So that the Difference on this Account must, each Period, be usually less than 36▪. also, and on both Accounts less than 1h. 12′.

Information about this book

Title statement

The calculation of solar eclipses without parallaxes: With a specimen of the same in the total eclipse of the sun, May 11. 1724. ... By Will. Whiston, ...

Whiston, William, 1667-1752.

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ID [ota]

http://ota.ox.ac.uk/id/4879

ID [isbn10]

1106018788

ID [isbn13]

9781106018786

ID [TCP]

K084724.000

ID [BIBNO]

CW3310808211

ID [ECMS]

0590900400

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— The calculation of solar eclipses without parallaxes: With a specimen of the same in the total eclipse of the sun, May 11. 1724. ... By Will. Whiston, ..., Whiston, William, 1667-1752.. [4],94,[2]p.,plate ; 8⁰. London :: printed for J. Senex; and W. Taylor,. 1724.. [With a half-title and a final errata leaf.] [Reproduction of original from the British Library.] [English Short Title Catalog, ESTCT105408.] [Electronic data. Farmington Hills, Mich. : Thomson Gale, 2003. Page image (PNG). Digitized image of the microfilm version produced in Woodbridge, CT by Research Publications, 1982-2002 (later known as Primary Source Microfilm, an imprint of the Gale Group).]

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