THE CALCULATION OF Solar Eclipses WITHOUT PARALLAXES.

A Calculation of the Great Eclipse of the Sun, April 22 d. 1715 in the Morning, from M r. Flamsteed's Tables; as corrected accord­ing to S r. 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 D r. 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 D r. 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; w ch. 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 D r. Halleys Map.

So that y e Middle of the General Eclipse in com­mon or apparent Time▪ will be 50. 56. after Nine in the Morning▪ differing from D r. Hal­ley's Computation near 9 min. But Note that the Construction is ac­commodated to the D rs. 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 D r. Halley's Description asserts.

But it must be here Observed that if in this Calculation the that and 6 th New Equations of the Moon, taken from S. Isaac New­ton's Theory, were neglected, this Calculation would be much near­er to D r. Halley's, as it is now nearer to M r. 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. S t. Isaac Newton's third Equation, w ch 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 y e 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
[...] 6 th 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 y e. sun 11 29 55 2 True Place of the Apogee   5 16 56
Variation substracted       2 Eccentricity 63 68 Inclination of y e 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 y e ☽ way with y e 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 y e Globe in Salisbury Court near Fleet street. And Will: Taylor at the Ship in Paternoster Row. Where are sold M r Whiston's Astronomical Lectures, his Taquet's Euclid, and y e Scheme of y e Solar System. Also y e newest Globes and Maps.

The Copernicus, or Universal Astronomical Instrument, being now finish'd, is sold by y e Author M r. 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 [Page 2] 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

[Page 3]

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

[Page 4]

  ° ′ ″
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 [Page 5] 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 [Page 6] 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, [Page 7] 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

[Page 8] (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 [Page 9] 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 [Page 10] 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 [Page 11] 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.

[Page 12] (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 [Page 13] 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, [Page 14] 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 1 h 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:

[Page 15] 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. 8 h 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.

[Page 16] 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 10 h 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 [Page 17] 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 7 h 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.

[Page 18] 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:

[Page 22]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
[Page 23]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
[Page 24]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
[Page 25]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
[Page 26]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

[Page 27] 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.
[Page 29]Duration in Minut. Distance in Miles. Dig ts
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    
[Page 30]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    
[Page 31]   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 Sec ds. 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
[Page 33]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
[Page 34]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 [Page 36] 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. N o. 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 [Page 38] 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 [Page 39] 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 [Page 40] = 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 [Page 41] 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.

[Page 43]

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.

[Page 45] 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
      Rem t. = P C. = 39. 44
      Ergo Lat. = 50. 16

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

[Page 47] ° ′          
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.

[Page 48] 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
        Rem t. 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 [Page 49] 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:

[Page 50] ° ′          
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    
        Remn t 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:

[Page 51] ° ′          
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:

[Page 53] ° ′   ° ′
From the Angle 100 15 = 6 h 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 6 h 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

[Page 54] 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
      Rem t. 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

[Page 55] 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
      Remn t 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
      Rem t. = 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 [Page 56] 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 6 h. 41′.

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

[Page 57] 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 = 6 h 8½

Which deducted from 6 h. 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.

[Page 58] 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 6 h 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.

[Page 59] 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=1 h:50 ⅘. [Page 60] 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 [Page 61] 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″ = 1 h46′ 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.

[Page 62] 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 = 6 h 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 1 h 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 1 h 46′ 24″to full 1 h 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 1 h 46′ 24″ bring it to 1 h 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.

[Page 66] 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,

[Page 67] ° ′          
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.
        Rem t. 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
        Remn t 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.

[Page 68] 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.

[Page 69] 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 [Page 70] 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 1 h 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 [Page 71] 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, [Page 72] 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 4 h 49 12

Eclipse at London,

Begins V. 45 00
Middle VI. 41 00
End VII. 32½ 00
Duration 1 h 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.

[Page 73] 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

[...]
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.   Latit de.
° ′   ° 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 Mid le. 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 [Page 80] 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.   Latit de.
° ′   ° 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 Mid le. 51⌊903 Mid le. 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.

[Page 81]

A Calculation of the Total Eclipse of the Sun, A. D. 1724. May 11. Post Meridiem.
    M.'s mean Mo. Moti n of Apogee. Mot n of N ode 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 11 d 5 h 19 22″
Temp. appar. deduct the Error of the Tables a­bout▪ 2′ 22″
true Tim. 5 h 17 0
  ′ ″
S mid. Sun 15 53
Sem Moon 16 42
Sem. Pen bra •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

[Page 82]

A Calculation of the Total Eclipse of the Sun, A. D. 1727. Sept. 4. Mane.
  M's mean Mot. Motion Apogee. Mot n of N ode 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 20 h 23 34
Mean Time.
Eq. tim. 2d. 4 38
Reduct. substr. 9
Diff. add 4 29
20 h 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 [Page 84] 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 [Page 85] 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 [Page 86] 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 [Page 87] 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 [Page 88] 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. [Page 89] 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 [Page 90] 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

[Page 91]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

[Page 92] 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 [Page 93] 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 [Page 94] 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 y e 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. 10 d. 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 1 h. 12′.

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