THE CALCULATION OF Solar Eclipses WITHOUT PARALLAXES.
A Calculation of the Great Eclipse of the Sun,
April 22d . 1715 in the Morning,
from Mr . Flamsteed's Tables; as corrected according to Sr . Isaac Newtons Theory of the Moon in the Astronomical Lectures, with its Construction for
London Rome and stockholme.
By W: Whiston
MA.
NB.
The Inquisitive are desir'd nicely to Observe whether in such Places where the Eclipse is plainly Total, there be not streaks of Red Light just before & after that Total Darkness; and how long it is visible; For if there be, it will imply that 'tis an Atmosphere about the Moon that is the occasion of it, & by its duration the height of the same Atmosphere - may in some measure be determined also.
The Sun's True Place and Anomaly
 
s
°
′
″
s
°
′
″
Anno Dom. 1701
9
20
43
50
3
7
40
10
Years—14
11
29
36
46
 
 
11
40
April, days—21
3
19
24
24
 
 
 
15
hours—21
 
 
51
45
3
7
52
5
minutes—36
 
 
1
29
Place of the Perihel.
Sun's Mean Motion
1
10
38
14
1
10
38
14
Equation Added
 
1
36
27
☉s Mean Motion
Sun's True Place
1
12
14
4
10
2
46
9
 
 
☉s Mean Anomaly
The General Eclipse by the Calculation
From Dr . Halley
 
h
′
h
′
Beginning—
7.
30
7
21
Middle—
9.
51
9
42
End—
12.
12
12
3
The Eclipse at London from the Calculation.
From Dr . Halley
 
h
′
h
′
Beginning—
8.
18
8
7
Middle—
9.
24
9
13
End—
10.
35
10
24
The Construction Explaind
This Scheme represents one half of the Inlightened Disk of the Earth as seen from its Center Projected at the distance of the Moon. The Elliptick Parallels, with their Hours, represent the Cities of
London, Rome,
and
Stockholme,
as plac'd at those Hours at different Times. The Principal strait Line divided by dotts represents the Path of the Moons Center ever the Disk of the Earth: And by the Hours in Larger and those above and below in smaller Characters, the Position of the Center is determind at those Times for those Places respectively. So that if with a pair of Compasse
we take from the proper Scale the Semediameter of the Penumbra, and carry it along the Path till it first reaches to, and then leaves the same minute on any Parallel that is the very time of its Begin̄ ing and Ending there. And if at any intermediate time in both you make Circles, one with the Moons Semidiameter on its Path; the other with the Suns on any of the 3 Parallels▪ the Intercepted part will shew the quantity of the Eclipse at that time in the Place to
which the Parallel you we does belong. And if you carry a Square along the Path, till the Perpendicular side cuts the same Hour and Minute there and in any Parallel, that is the Middle of the Eclipse there. Of all which you have examples in the Scheme. Only Note that the Center of the Penumbra at 21 after▪ 7 and at 3 after 12▪ which are the beginning and ending of the General Eclipse, extends beyond the▪ Copper Plate, and is to be supply'd by the Ren at the intersection of the proper Lines there to directed.
The Breadth of the intire Penumbra or paitial Eclipse upon this Perpendicular Plain, appears by the Construction to be no less than 1965 minutes or Geographical Mibes on each side of the Moons Path, or 3930 Miles in all; wch . correspond to many more on the Spherical Surface of the Earth: Nor is it all confind, as you may see here, to that Surface, but reaches▪ off a greatway into the empty Space beyond it Northward. The Lines which distinguish that breadth on each side into 12 parts denote so many Digits of the Suns Eclipses besides ⅓ for the Total shade) & the places both as to Long▪ 8▪ Lat: where the sun will at any Time be so much Eclipsed: And indeed I would willingly have procured a general Map here to have▪ shewd over what Countries and Places the intire Shadow would pass, as Doctor Halley has given us a particular Map of England for the Passage of the Total Shadow over it. But the nature of the Construction does not admit of that Projection (Such a Thing cannot be truly represented any other way than by the Copernicus; where there is a real Globe of the Earth, capable of a Diurnal motion, during the time of the Eclipse) the imposibility of which in all Perspective Projections of the Sphere renders that designs otherwise impracticable: Nor can I determin by this Construction whether the▪ Eclipse will be Total at London or not, because the Circles of the Sun and Moon at the Southern Limit seem here exactly coincident. But if we go by a Construction according to our Calculation the Digits Eclipsed at London will be hardly more than 11 ⅘ and the Shadow▪ will go full 30 Miles more Northward than in Dr . Halleys Map.
So that ye Middle of the General Eclipse in common or apparent Time▪ will be 50. 56. after Nine in the Morning▪ differing from Dr . Halley's Computation near 9 min. But Note that the Construction is accommodated to the Drs . Calculation.
Note also that hence the breadth of the Shadow of Total Darkness will be 98 Geographical Miles; and that its length on the Oblique Horizon of London will be near 150 Miles, as Dr . Halley's Description asserts.
But it must be here Observed that if in this Calculation the that and 6th New Equations of the Moon, taken from S. Isaac Newton's Theory, were neglected, this Calculation would be much nearer to Dr . Halley's, as it is now nearer to Mr . Flamsteed's. This Eclipse, if the Air prove clear for exact Observations, will go agreat
ay to determin how far those Equations are just; and how far they are necessary in the Calculation of the New and Full Moons, and of Eclipses, that happen only at those Times. St . Isaac Newton's third Equation, wch is no more than 13 to be Substr
ted, is here omitted, as very inconsiderable.
 
Moon's mean Motion
Motion of the Apogee
Motion of ye Nodr Retr.
 
s
°
′
″
s
°
′
″
s
°
′
″
Anno Dom.1701
10
15
19
50
11
8
18
20
4
27
24
20
Years—14
1
20
54
34
6
29
37
50
9
0
45
35
April▪ days—21
 
22
34
47
 
12
21
59
 
5
52
41
hours—21
 
11
31
46
 
 
5
51
 
 
2
47
minuted—36
 
 
1
46
 
 
 
10
 
 
 
5
(Mean Time)
 
 
 
 
 
 
 
 
9
6
41
8
Moon's mean Motion
1
10
40
43
6
20
24
10
7
20
43
12
Suns mean Anomaly
10
2
40
9
An: Eq: Add.
16
8
An Eq: subst.
7▪
41
Physical parts Substracted
 
 
9
50
6
20
40
18
7
20
35
31
6th Equation Substra
 
 
3
45
Mean Place Corrected
Mean Place Corrected
 
Sum
13
35
1
12
14
41
1
12
14
41
Moon's mean Place correcte
1
10
27
8
Suns True Place
Suns True Place
True Place of the Apogee
ulot
6
27
5
5
6
21
34
23
5
21
39
10
Moon's mean Anomaly
6
12
31
3
Annual Argument
☉
Distance from the Node
Equation Added
 
1
42
35
 
7
15
47
Equat: Subst.
25
19
Moon's Equat Place in its Orb
1
12
9
43
Equation Added
7
20
10
12
Sun's True Place
1
12
14
41
6
27
56
5
True Place of the Node
Moon's Distance from ye . sun
11
29
55
2
True Place of the Apogee
 
5
16
56
Variation substracted
 
 
 
2
Eccentricity
63
68
Inclination of ye Limit
Moon's True Place in it
Orbit.
1
12
9
41
☽ horary Mot.
38
0
Semidiam. ☉.
15
58
Node's True Place substract
7
20
10
12
☉
.
2
25
Semidiam. ☽ .
16
47
Argument of Latitude
5
21
59
29
☽ from ☉
35
35
Semid. Penum.
32
45
Reduction Added
 
 
2
0
35⌊ 6′ 60′115:
8
22
Semid. Disk
61
30
Moon's True Place in the Eclip
1
12
11
41
Eq. Time Add.
3
25
Ang
of ye ☽ way with ye Ecliptic
5°
35′
Moon's True North Latitude
 
 
44
10
Reduct Add.
3
9
Diff
of that ☉ & ☽ s Diam 98″or 98 Miles.
 
 
 
 
 
Sum of 3 Add.
14
56
Engrav'd and Sold by Iohn Senex at ye Globe in Salisbury Court near Fleet street. And Will: Taylor at the Ship in Paternoster Row. Where are sold Mr Whiston's Astronomical Lectures, his Taquet's Euclid, and ye Scheme of ye Solar System. Also ye newest Globes and Maps.
The Copernicus, or Universal Astronomical Instrument, being now finish'd, is sold by ye Author Mr . Whiston, at his House in Cross street Hatton Garden
THE CALCULATION OF
Solar Eclipses
WITHOUT PARALLAXES. WITH A SPECIMEN of the same in the Total Eclipse of the Sun,
May
11. 1724. Now first made Publick. To which is added, A PROPOSAL how, with the Latitude given, the Geographical Longitude of all the Parts of the Earth may be settled by the bare Knowledge of the Duration of Solar Eclipses, and especially of Total Darkness. WITH An ACCOUNT of some late Observations made with Dipping Needles, in order to discover the LONGITUDE and LATITUDE at Sea. By
WILL. WHISTON,
M. A. Sometime Professor of the Mathematicks in the University of
Cambridge.
LONDON:
Printed for J. SENEX in
Fleetstreet;
and W. TAYLOR in
Pater-Noster-Row.
1724.
LEMMATA:
OR, Preparatory Propositions.
I.
HE most useful and most remarkable
Cycle
or
Period
for the Revolution of Eclipses, both Solar and Lunar, mentioned by
Pliny, (Nat. Hist.
II.) and by him only of all the Ancients, is the Interval of 223° Synodical Months = 6585 Days: or = 18
Julian
Years: with 10 Days, when the Cycle
or Period contains 5 Leap-Years: and with 11 Days, when with 7 Hours 43′ ¼. In which Time the direct mean Motion of the
Moon
and her
Apogee
in the Ecliptick, is nearly so much
more;
and the Retrograde mean Motion of the
Nodes
nearly so much
less
than entire Revolutions, as the mean Motion of the
Moon from the Sun,
upon which all mean Conjunctions, Oppositions, and Eclipses properly depend, exceeds the like entire Revolutions Which Coincidences do therefore nearly restore the mean State of the Moon it self, its Apogee▪ Nodes, and Lunations: And produce an eminent Revolution of correspondent New Moons,
ull Moons, and Eclipses, after that Interval perpe
lly.
This appears by the following Calculation of all these mean Motions from the Astronomical Tables.
Mean Motion of the Sun or Earth.
 
s
°
′
″
Years 18
11
29
28
32
Days 11
0
10
50
32
Hours 7
0
0
17
15
′ 43
0
0
1
46
″15
0
0
0
1
Sum
0
10
48
6
Mean Motion of the Moon in the Ecliptick.
 
s
°
′
″
Years 18
7
11
37
22
Days 11
4
24
56
25
Hours 7
0
03
50
35
′ 43
0
00
23
36
″ 15
00
00
00
08
Sum
00
10
48
06
Mean Motion of the Apogee.
 
s
°
′
″
Years 18
0
12
23
53
Days 11
0
1
13
32
Hours 7
0
0
1
57
′ 43¼
0
0
0
12
Sum
0
13
39
34
Mean Motion of the Node backward.
 
s
°
′
″
Years 18
11
18
7
39
Days 11
0
0
34
57
Hours 7
0
0
0
56
′ 43 ¼
0
0
0
6
Sum
11
18
43
38
Mean Motion of the Moon from the Sun.
 
s
°
′
″
Years 18
7
11
58
50
Days 11
4
14
5
53
Hours 7
0
3
33
20
′ 43
0
0
21
50
″ 15
0
0
0
7
Sum
0
0
0
0
 
°
′
″
From the mean Motion of the Apogee
13
39
34
Substract that of the Moon in the Eclipt.
10
48
6
Remains—(2°⌊ 9)
2
51
28
To the mean Motion of the Moon
0
10
48
6
Add that of the Node
11
18
43
38
Sum
11
29
31
44
Difference from 12 Signs (0°⌊ 47)
0
0
28
16
Whence it appears that the Difference of the mean Motions of the Apogee and of the Nodes from that of the Moon her self in the Ecliptick, in such a Period, is but small: Not more in the former Case than 2° 51′28″ = 2°⌊ 9, nor in the latter than 28′ 16″ = 0°⌊ 47. Whence also it appears that the Lunar Apogee does in every such Cycle differ but 1/
2 of the entire Difference 2⌊ 9) 180 (62. That the Lunar Node only differs in that Cycle 1/
of the entire Difference 0°⌊ 47) 180 (383. And that the Anomaly of the Sun it self differs only 1/1
of the entite Anomaly: 10 8) 180 (16⌊ 6. Which Quantities being generally small, cannot occasion any great Inequality in the Times and Circumstances of New and Full Moons, or of Eclipses▪ nor by consequence greatly disturb the regular Succession of the same in any single Period; nor indeed very greatly in several successive Periods. For since the mean Motion of the
Moon from the Sun
is within a very small Matter ever certain and invariable, that Revolution is always just; and always determines the mean Time of all Conjunctions, Oppositions, and Eclipses rightly; and since the other Anomalies are but small, and always come right again in Length of Time, they cannot ever produce any very great Anomalies in our Calculations from them. As will farther appear under the following
Scholia.
N. B.
This Period for Eclipses, has of late been called, both by Mr.
Flamsteed,
and Dr.
Halley,
the
Saros,
or the
Chaldean Saros:
As if it were known and us'd by the old
Chaldeans,
and thence called by that Name. For which I know no sufficient Foundation. There is indeed a gross Mistake of
Pliny
's Number in
Suidas,
(who thus applies this term) 222 for 223 Months, as almost all the Editions of
Pliny
still have it; and He calls that Period by this
Chaldean
Name
Saros.
Yet the
Chaldeans
never, that we find, apply'd it to any other Period than that of 3600
Years
or
Days;
by which Period alone all the Antediluvian Reigns are determined both in
Abydenus
and
Berosus
themselves, from the anc
entest Records of that Kingdom. See my
New Theory,
the later Editions,
Hypoth.
X.
Lem.
to the third Argument; and
Appendix
to the
Essay
towards restoring the true Text of the Old Testament,
p.
203,—213.
SCHOLIA.
(1.) We may here Observe, that since the Limit for Eclipses of the Moon is about 11° 40′ = 700′ on each Side of the Node; as is the Limit for Eclipses of the Sun, about 16° 40′ = 1000′. If we divide 700, the Limit of the Moon's Eclipses, by 28′ 16″ = 28′⌊ 3, which is the Difference between the Revolution of the Moon to the Sun and of the Node above given, we shall have nearly Twenty Five for the Number of Cycles,, after a Central Lunar Eclipse in one of the Nodes, before the Moon goes off the Shadow of the Earth entirely at the same Node, and 450 Years (25
multiplier
18 = 450,) or double that Number 900 Years for the Time that the Moon begins to enter the Ecliptick Limit on one Side, till it goes out of it on the other. During which long Interval there will still be Eclipses of the Moon each Period. And if we divide 1000′ the Limit of the Sun's Eclipses, by the same Number 28′⌊ 3, we shall have nearly 35 for the Number of Periods after a Solar Central Eclipse at the Middle of the Earth, in one of the Nodes, before the Penumbra goes off the
North
or
South
Parts of the Disk of the Earth entirely at the same Time;
i. e.
630 Years. (35
multiplier
18 = 630,) or double that Number 1260 Years, from the Time that the Moon in any such Period begins to enter the Ecliptick Limit on one Side, till it goes out of on the other: During which longer Interval there will still be somewhere Eclipses of the Sun each Period. After which respective long Intervals of Time there will be no such Eclipses for much longer Intervals.
(2.) Since the utmost Latitude of the Moon that can permit any Lunar Eclipse, is about 62′, and the same utmost Latitude that can permit a Solar Eclipse is about 92′: If we divide the first Number by 25, or the last by 35, the Numbers of Revolutions for the Ecliptick Limits, we shall have about 2′⌊ 6 = 2′:36″ for the mean Alteration of the Moon's Latitude in each single Period all along; and this both for Solar and Lunar Eclipses. Which Latitude will be
South
during the one half of the long Period of the Ecliptick Limits before-mentioned; and
North
during the other half: Gradually increasing, and as gradually decreasing perpetually: As in the following Table.
A Table of the mean Latitudes of the Moon each single Cycle, either
North
or
South;
beginning at an Eclipse in one of the Nodes, without any Latitude at all.
Latitudes for
25
Cycles in Lunar, and
35
in Solar Eclipses.
Cycles
′
″
1
2
26
2
4
52
3
7
19
4
9
46
5
12
14
6
14
43
7
17
12
8
19
42
9
22
12
10
24
43
11
27
14
12
29
46
13
32
19
14
34
52
15
37
26
16
40
1
17
42
36
18
45
12
19
47
48
20
50
25
21
53
3
22
55
41
23
59
20
24
60
59
25
63
39
26
66
0
27
68
41
28
71
23
29
74
6
30
76
49
31
79
33
32
82
17
33
85
2
34
87
48
35
90
34
(3.) Since the principal Alteration in the Quantity and Duration of total Eclipses of the Sun, arises from the Difference there is at any Time between the real Distances, and apparent Diameters of the Sun and Moon, at the Time of such Eclipses, that Quantity and Duration must depend on the Difference of their mean Anomalies, which gives us that Difference of Distances and Diameters; and must therefore answer in each Cycle one with another, to the Differences of those mean Anomalies during that Interval; which in the Sun comes to 10°⌊ 8/180 or 1/16⌊ 6 of its entire Anomaly. And in the Moon to 2/
▪ 9/
or 1/
of its entire Anomaly. And since the whole mean Excentricity of the Moon is somewhat above three Times as great as that of the Sun, or as
/1000 to 17/100. The Differences of the Sun's Distances and Diameters will be but a little greater in each Period one with another, than those of the Moon.
(4.) When therefore the Anomalies of the Sun and Moon are of the same Species; I mean both ascending, or both descending; their Distances and Diameters will, one with another, increase or decrease nearly in the same Proportion; and the Quantity and Duration of total Darkness will alter but little in such a Period. But when those Anomalies are of the contrary Species; that is, the one ascending while the other descends; they will alter considerably. So that if the Sun be descending, and its apparent Diameter Increasing; while the Moon is ascending, and its apparent Diameter Decreasing, the Eclipse of the Sun will, each succeeding Cycle, afford a smaller total Shadow; till at last it afford no total Shadow at all; but the Eclipses become
Annular.
And if the Sun be Ascending while the Moon is Descending, the contrary will happen; and the total Shadow grow greater perpetually. From which Circumstances of the Sun and Moon in each Revolution of the Cycle duly considered, we may nearly determine whether any succeeding correspondent Eclipse will afford us a greater or less total Shadow, or whether the Eclipses will be only Annular.
(5.) From the like Circumstances we may also nearly determine whether such Eclipses will come somewhat sooner or later, than that of the mean Revolution of the Period before us. For if the Earth be much nearer its Aphelion, than the Moon its Apogaeon, at the end of any Cycle; and by consequence if the Earth then revolve comparatively slower, and the Moon swifter than ordinary; the meeting of the Luminaries will be accelerated. And if the Earth be much remoter from its Aphelion than the Moon from its Apogaeon, the contrary will happen; and the Moon will be later than ordinary e'er it overtake the Sun. So that in the former Case the Eclipse will come a little
before,
and in the latter a little
after
the proper Conclusion of that Period.
(6.) Since the Motion of the Node backward in one of these Periods does not quite reach to the Conjunction or Opposition, that Node must every Cycle go forward, with respect to the Lunations and Eclipses; and at the ascending Node the Moon will pass more
Southward,
and at the descending Node more
Northward
successively. Thus at the Solar Eclipse
May
1. 1706. the Moon, near its ascending Node, had greater
Northern
Latitude than it will have at the next corresponding Solar Eclipse,
May
11. 1724. And thus at the total Solar Eclipse,
April
22. 1715. the Moon near its descending Node, had less
Northern
Latitude than it will have at its corresponding great Eclipse,
May
2. 1733.
(7.) Since the Motion of the Moon's Apogee forward is greater in one of these Periods than that of the Lunations, that Apogee must also go forward every Cycle: And if at any one Solar Eclipse that Apogee be in quadrature with the Sun, after it had been in Conjunction, the Moon will the next Period descend by going backward in its Eclipses, towards the Perigee. And if at any one such Apogee it be in the quadrature, after it had been in Opposition, it will the next Cycle ascend: The Reverse of all which is true in Lunar Eclipses.
(8.) The Place and Motion of the Sun in its Ellipsis is so easily known, and that for many Ages, by bare Memory and Reflection, that a few Words will suffice. The Sun is now farthest from the Earth Eight Days after the longest Day; and nearest to it Eight Days after the shortest: And its Motion about 1 Degree in 72 Years. Whence it is evident, that it has, for all the past Ages of Astronomy, been about the Summer Solstice in our Apogee, and about the Winter Solstice in our Perigee; if I may use the T
rms of the
Ptolemaick
System. Nor is it therefore any Wonder that the greatest total Eclipses of the Sun have happened still in the Summer, and the g
eatest Annular ones in the Winter half Year: Since the farther the Sun is off in the Winter, the less must be its apparent Diameter, and by consequence the greater the Excess of the Moon's Diameter above it. On which Excess alone the Greatness of such Eclipses depends. And the Reverse is equally evident in the Case of Annular Eclipses in Winter: Nor indeed is it very strange, that Annular Eclipses are in these Parts of the World so rarely observed; since they most usually happen in Winter Days; which being short, must afford us a proportionably small Number of them. To the Inhabitants of the other Side of the Equator the Reverse must happen. But those Eclipses very rarely come to our Notice.
(9.) Since this Period reaches only from the middle of one general Eclipse to another, without regard to the Position of any particular Place on the Earth's Surface, arising from the diurnal Motion, we must remember that if an Eclipse of the Sun happens at any particular Place considerably before Noon, it will come sooner, and after Noon later than the proper Conclusion of this Period. Though it must be noted, that Eclipses of the Moon being absolute in their own Nature, are here wholly unconcerned; and no way subject to any Acceleration, Retardation, or Alteration on account of the diurnal Motion of the Earth.
(10.) The principal Alteration of the Time of the Day in all Eclipses depends on the Excess of this Period above an even Number of Days; which is 7 Hours and 43′¼. So that the Cycle does naturally put every correspondent Eclipse later than the foregoing, almost 8 Hours, or one third part of a Day; which thing, by reason of the intervening diurnal Motion, greatly alters all Eclipses, especially Solar, not only as to the bare Time of the Day when, but also as to the Places on the Earth where such correspondent Eclipse will be visible.
(11.) If therefore we join three of these Cycles together, those odd Hours and Minutes will amount nearly to a whole Day; and will therefore nearly bring the middle Point of the correspondent Eclipses to the same Time, in the same Place, and, in part, with the same Circumstances as before: Which a single Cycle cannot possibly do. Only with the Anticipation of 50′ in Time. Which three single Cycles therefore of 19,756 Days, or of 54 Years; with 32 or 33 Days; I call the
Grand Cycle.
And this will be, I think, of the greatest and readiest use in remote Eclipses of any other Period whatsoever.
Thus, for Example, there was a total Eclipse of the Sun on
Black Monday;
as it has thence been called ever since,
March
29. 1652. about Ten a Clock in the Morning.
Total,
I say it was in the
North
of
Ireland,
and the
Northwest
of
Scotland,
tho' not so at
London,
ot the remoter Parts of
England
and
Scotland.
To this if we add one Cycle, the Time of the next correspondent Eclipse will thus be discovered:
 
y
d
h
′
″
To
A. D.
1652.
March,
00
28
22
00
00
Add
18
10
7
43
15
Sum 1670
April
00
8
5
43
15
So that the correspondent total Eclipse ought to have been
A. D.
1670.
April
the 8th 43′ ¼ past 5 a-Clock in the Evening. And because the Earth was then a small Matter nearer its Aphelion, than the Moon its Apogaeon, the Time would be a little anticipated on that Account. But then, because this Eclipse was towards Evening, it would be much more retarded on that Account, than anticipated on the other; and the main Part of the Eclipse would happen after Sun-set, and be here invisible.
To this Time if we add another Cycle, the next correspondent Eclipse will in like Manner be discovered:
 
y
d
h
′
″
To
A. D.
1670.
April
00
08
5
43
15
Add
18
11
7
43
15
Sum 1688.
April
00
19
13
26
30
So that the next correspondent total Eclipse ought to have been
April
20th, 26′ ½ after one in the Morning; and was therefore, to be sure, utterly invisible to us here. To this Time if we add another Cycle, we have the next correspondent total Eclipse thus:
 
y
d
h
′
″
To
A. D.
1688
April
00
19
13
26
30
Add
18
10
07
43
15
Sum 1706
May
00
00
21
09
45
So that the next correspondent total Eclipse was to have been
A. D.
1706. 9′ ¾ past Nine a-Clock in the Morning; which is not much before the Time when it was observ'd here; and was no other than that famous Eclipse which was total at
Cadiz, Barcelona, Marseilles, Geneva, Bern,
and
Zurich;
and became very remarkable for the raising of the Siege of
Barcelona
during that total Darkness: Though I have been inform'd by several there present, that it came to both Armies wholly unexpected, till the great Darkness of the Sky forced them to attend to it. Now this affords us also a remarkable Instance of the near Approximation of our
Grand Cycle,
both as to Time and Place. For if instead of tracing this correspondent Eclipse through the distinct Cycles we had at once taken our
Grand Cycle
of 54 Years, and 33 Days; we had come immediately to this Eclipse: And by allowing the Anticipation of 50′ had been within about 20′ of the Calculation at
London.
To this Time if we add another single Cycle, we shall have the next correspondent Eclipse thus:
 
y
d
h
′
″
To
A. D.
1706.
May
00
00
21
09
45
Add
18
10
07
43
15
Sum 1724
May
00
11
04
53
00
So that we ought hence to expect the total Eclipse next
May
11, 53′ past 4 in the Afternoon. And because both the Position of the Sun and Moon in their Ellipses, and the more considerable Alteration from the Time of the Day, which is here much farther in the Evening than the last was in the Morning; oblige us to suppose about 1h ¾ Retardation, we hence justly expect that this Eclipse will be the nearest Total at
London
about 40′ past Six in the Evening; as the exactest Calculations do determine.
(12.) If we would now trace a few Lunar Eclipses by this Cycle, we may do it according to the following Examples:
A. D.
1681/2,
Feb.
11. about 59′ past 10 a-Clock at Night, Mr.
Flamsteed
observ'd the Middle of a great and total Eclipse of the Moon at
Greenwich.
Proceed therefore as is already directed to find the next correspondent Eclipse of the Moon thus:
y
d
h
′
″
To
A. D.
1681/2
February.
00
11
10
59
00
Add
18
11
07
43
15
Sum 1699/1700
Febr.
00
22
18
42
15
So that this first correspondent total Eclipse of the Moon ought to have been
Feb.
23. 1699/1700, 42′ ¼ past Six a-Clock in the Morning; or in the day-time, and so must needs have been in great Part to us invisible.
 
y
d
h
′
″
To
A. D.
1699/1700,
Feb.
00
22
18
42
15
Add
18
10
07
43
15
Sum 1717/18
March
00
05
02
25
30
So that the next correspondent total Eclipse of the Moon ought to have been
March
5, 1717/1
. 25′ ½ after Two a-Clock in the Afternoon; which was in the day-time also; and so must equally with the former have been here invisible.
 
y
d
h
′
″
To
A. D.
1717/18,
March
00
05
02
25
30
Add
18
10
07
43
15
Sum 1736
March
00
15
10
08
45
So that the next total correspondent Eclipse of the Moon is hence to be expected
A. D.
1736.
March
15. 8h ¾ past 10 a-Clock at Night: which is about an Hour and half sooner than the Calculation. Which difference we shall presently find to be near the greatest Difference that can happen.
This may also be equally obtain'd by one entire
Grand Cycle
of 54 Years, and 33 Days; with the fore-mentioned Anticipation of 50′ which from
Feb.
11. 1681/2 10h 59′, brings us directly to
March
15, 1736, 9′ past 10 a-Clock at Night; or to somewhat above an Hour and half before the Calculation.
N. B.
As to the proper Quantity of the several Alterations arising in each Period, which ought to be allowed for, they are nearly these: The Moon and Sun being about 31′ ⅔ in Diameter, and the Digits of their Observation being 12. while the Difference of the Moon's Latitude, as we have seen, is about 2′ 36″ or the Twelfth Part of those Diameters, it is plain that the mean natural Alteration of every Period in the same Circumstances is about one Digit; though less in the lesser, and greater in greater Latitudes: which in the Moon, whose Eclipse is to all Spectators the same, holds constantly: And though the diurnal Motion of the Earth removes all particular Places, so much each Period as to render this Rule less observable in Solar Eclipses, yet after each grand Period, which nearly restores their former Position, it will hold in a good Degree there also, I mean so as to alter about 3 Digits therein: But besides that of the Digits eclipsed, we ought also to see what Alteration in Time may happen to each Period. Now as to the Inequality of the Sun's Motion, it is as we have seen 10°⌊ 8, and its greatest Velocity is at the Earth's Perihelion, and its least at its Aphelion: Its greatest Alteration therefore must be in
Aphelio
and
Perihelio,
and is the Difference of the Equation belonging every where to the Addition of 10°⌊ 8, and is here 48″, which Space the Earth goes in about 20′ of Time. So that the Difference of Time on this Account, must each Period be some Quantity less than 20′. And as to the Inequality of the Moon's Motion, it is also greatest at the Perigee and Apogee, and its greatest Alteration at the extreme Eccentricities of its Orbit is the Difference of the Equations at 2° 51′ ½ in Perigee and Apogee, according to those extreme Eccentricities: = 9′ which the Moon usually goes in somewhat less than 20′. So that the Difference of Time, on this Account, must each Cycle be some Quantity less than 20 Minutes also.
And now we come to the principal Alteration in Time that can happen in Eclipses; though it belongs only to those of the Sun: And that is the Time of the Day when they happen in any particular Place. Now because the Center of the Moon usually goes over an entire Diameter of the Disk of the Earth, in about three Hours and an half, Part of which is almost always before, and Part after Noon: while the odd Hours of a small Cycle 7h 43′ ¼, may reach equally from a Forenoon to an Afternoon's successive Eclipse; 'tis possible, such an Eclipse may appear an Hour and three Quarters later than the Period it self would determine it. Tho' usually this Alteration will not be near so great; especially when the Latitude of the Moon is very considerable. But then it is so easy to allow very nearly for this Inequality, upon a little Consideration, that it ought not to be objected against the Accuracy of this Period. If for Six Hours from Noon we allow about an Hour and half; and for two Hours, three Quarters of an Hour; we shall not err very much from the true Time.
Corollary.
If we would know what is the greatest Inequality in Digits and Time in a grand Period, made up of Three common ones, or of 54 Years, 32 or 33 Days, besides the constant Anticipation of 50′, we must say it may possibly, though it will very rarely, be almost thrice the Quantities already stated for a single Cycle: excepting the last and principal Difference, peculiar to Solar Eclipses: which is never much greater than that already mentioned.
N. B.
If any are not contented to know these Matters by such Approximations, but desire the utmost Accuracy; they must either make use of Dr.
Halley
's Equations, fitted to this Cycle, when published; or rather make use of Mr.
Flamsteed
's or Dr.
Halley
's most accurate Astronomical Tables, when published; with that Trigonometrical Calculation afterward, which I publish and exemplify in this Paper. In the former Part of which Work, this Cycle, with its proper Equations, will, at least, save us the one half of our Calculation, if it will not bring us it self to that utmost Accuracy: which indeed is hardly to be expected from it.
So that, upon the whole, If we duly consider the particular Circumstances of the Sun and Moon, with those of the Aphelion, Apogee and Node, and with the Times of the Day or Night when the Cycle ends, and rightly apply them to this single Cycle and to this grand Cycle, we shall be able nearly to determine the correspondent Eclipses with very small Trouble or Calculation.
II.
The Plane in which the Center of the Moon moves in Eclipses, is not that of the Ecliptick, but of the Orbit of the Moon, consider'd with the Annual Motion: Or it is a Plane inclined to the Plane of the Ecliptick in an Angle of about 5° 36′. Which in the Calculation of Eclipses is usually stiled,
The Angle of the Moon's visible Way.
This principal Plane I call,
The Lunar Plane.
III.
This
Lunar Plane
cuts the Sphere of the Earth, considered without its diurnal Motion, in a Circle whose Pole or Vertex is distant from the Pole of the Ecliptick in the same Angle. This Circle I call
The Lunar Circle.
IV.
In Eclipses which happen at the Solstices, and in the Nodes of the Moon's Orbit, the Distance of these Poles is exactly
Eastward
or
Westward.
In those which happen at the Equinoxes and Nodes, the Distance is exactly
North
and
South.
But in all other Cases it is Oblique.
V.
The Angle of that Obliquity is always compos'd of the Distance of the Sun and Moon from the Solstitial Colure; with the Distance of the same from the Nodes: And is sometimes the
Sum,
and sometimes the
Difference
of those Quantities.
VI.
The Distance between this Vertex and the Pole of the Earth, when Eclipses happen at the Solstices, is the
Sum
or
Difference
of the two forementioned Angles of Inclination; the one, of the Pole of the Equator and of the Pole of the Ecliptick = 23° 29′; the other of the Pole of the Ecliptick, and of the Vertex of the Lunar Circle = 5° 36′ nearly. But in all other Cases a Spherical Triangle must be solv'd, in order to find that Distance: Of which hereafter.
VII.
Since the Solstitial Colure is a great Circle, that is also a Meridian, or passes through the Poles of the Earth and Ecliptick: And since besides the Distance between the Vertex of the Lunar Circle and the Pole of the Earth, we shall want the Angle included between the Colure and that Line; this also must be obtained by the like Solution of a Spherical Triangle: Of which hereafter.
VIII.
Since the great Circle that passes thro' the Poles of the Earth▪ and of the Lunar Circle, and that alone cuts both those Circles, and their Parallels at Right Angles, That Meridian, and that alone wherein that Distance lies, is perpendicular to the
Path of the Moon's Center
along the other; and will determine the Point in that Path wherein
Center of the Shadow cuts that Meridian at R ght▪ Angles and approaches nearest of all to the Pole of the Earth: And indeed lays the Found
o
of our future Calculations. This Meri
I call
The Primary Meridian.
IX.
The angular Distance about the Pole of the Earth, of the Meridian that is directed to the Sun at the middle Point of the whole Eclipse from the primary Meridian, is composed of the
Sum
or
Difference
of the Angle made by the Solstitial Colure and the primary Meridian; and of the Complement of the Sun's Right Ascension at the middle of the general Solar Eclipse. Which Angle is of the greatest Consequence in our future Calculations. This Angle I call
the Primary Angle.
X.
Since the Motion of the Center of the Shadow of the Moon, in Solar Eclipses, is nearly even, and nearly recti-linear; since it is also in the Plane of the Lunar Circle; and is all one as if it were along a Line that touched that Circle at the Middle of the general Eclipse, the Point of Contact; we must divide each Quadrant of 90 Degrees into 90 or 180 unequal Parts: but so that the
Difference
of the Sines of those unequal Angles may be equal, and 1/9
or 1/
80 of the entire Radius: That so the first Sine
/
0 may be 1111/100000; the second 2222/100000; the third
/100000, &c. and this from the Table of natural Sines, with their corresponding Arcs or Angles at the Vertex, as follows:
Arcs.
D ff. Sines Equal.
Arcs.
Diff. Sine
Equal.
Diff.
°
′
Diff.
°
′
19⌊ 1
 
 
 
19⌊ 4
 
 
 
 
0
18
½
 
7
1
11
19⌊ 1
 
 
 
19⌊ 4
 
 
 
 
0
38
1
 
7
20
½
19⌊ 1
 
 
 
19⌊ 5
 
 
 
 
0
57
½
 
7
40
12
19⌊ 1
 
 
 
19⌊ 5
 
 
 
 
1
16
2
 
7
59
½
19⌊ 1
 
 
 
19⌊ 5
 
 
 
 
1
35
½
 
8
18
13
19⌊ 1
 
 
 
19⌊ 5
 
 
 
 
1
54
3
 
8
37
½
19⌊ 1
 
 
 
19⌊ 6
 
 
 
 
2
13
½
 
8
57
14
19⌊ 2
 
 
 
19⌊ 6
 
 
 
 
2
33
4
 
9
16
½
19⌊ 2
 
 
 
19⌊ 6
 
 
 
 
2
52
½
 
9
36
15
19⌊ 2
 
 
 
19⌊ 6
 
 
 
 
3
11
5
 
9
55
½
19⌊ 2
 
 
 
19⌊ 7
 
 
 
 
3
30
½
 
10
14
16
19⌊ 2
 
 
 
19⌊ 7
 
 
 
 
3
49
6
 
10
33
½
19⌊ 2
 
 
 
19⌊ 7
 
 
 
 
4
8
½
 
10
53
17
19⌊ 2
 
 
 
19⌊ 7
 
 
 
 
4
28
7
 
11
12
½
19⌊ 3
 
 
 
19⌊ 7
 
 
 
 
4
47
½
 
11
32
18
19⌊ 3
 
 
 
19⌊ 7
 
 
 
 
5
6
8
 
11
51
½
19⌊ 3
 
 
 
19⌊ 7
 
 
 
 
5
25
½
 
12
11
19
19⌊ 3
 
 
 
19⌊ 8
 
 
 
 
5
44
9
 
12
30
½
19⌊ 3
 
 
 
19⌊ 8
 
 
 
 
6
3
½
 
12
50
20
19⌊ 4
 
 
 
19⌊ 8
 
 
 
 
6
23
10
 
13
10
½
19⌊ 4
 
 
 
19⌊ 8
 
 
 
 
6
42
½
 
13
30
21
19⌊ 8
 
 
 
20⌊ 5
 
 
 
 
13
49
½
 
20
50
32
19⌊ 9
 
 
 
20⌊ 5
 
 
 
 
14
9
22
 
21
10
½
20
 
 
 
20⌊ 5
 
 
 
 
14
28
½
 
21
30
33
20
 
 
 
20⌊ 6
 
 
 
 
14
48
23
 
21
50
½
20
 
 
 
20⌊ 6
 
 
 
 
15
8
½
 
22
11
34
20
 
 
 
20⌊ 7
 
 
 
 
15
28
24
 
22
32
½
20
 
 
 
20⌊ 7
 
 
 
 
15
48
½
 
22
53
 
20⌊ 1
 
 
 
20⌊ 8
 
 
35
 
16
7
25
 
23
13
½
20⌊ 1
 
 
 
20⌊ 9
 
 
 
 
16
7
½
 
23
34
36
20⌊ 1
 
 
 
21
 
 
 
 
16
47
26
 
23
55
½
20⌊ 2
 
 
 
21
 
 
 
 
17
7
½
 
24
17
37
20⌊ 2
 
 
 
21⌊ 1
 
 
 
 
17
27
27
 
24
38
½
20⌊ 2
 
 
 
21⌊ 1
 
 
 
 
17
47
½
 
24
59
38
20⌊ 2
 
 
 
21⌊ 2
 
 
 
 
18
8
28
 
25
20
½
20⌊ 3
 
 
 
21⌊ 2
 
 
 
 
18
28
½
 
25
41
39
20⌊ 3
 
 
 
21⌊ 2
 
 
 
 
18
48
29
 
26
2
½
20⌊ 3
 
 
 
21⌊ 3
 
 
 
 
19
8
½
 
26
23
40
20⌊ 3
 
 
 
21⌊ 3
 
 
 
 
19
28
30
 
26
44
½
20⌊ 4
 
 
 
21⌊ 4
 
 
 
 
19
48
½
 
27
6
41
20⌊ 4
 
 
 
21⌊ 4
 
 
 
 
20
9
31
 
27
27
½
20⌊ 4
 
 
 
21⌊ 6
 
 
 
 
20
29
½
 
27
49
42
21⌊ 7
 
 
 
24⌊ 2
 
 
 
 
28
11
½
 
36
4
53
21⌊ 8
 
 
 
24⌊ 3
 
 
 
 
28
32
43
 
36
27
½
21⌊ 9
 
 
 
24⌊ 4
 
 
 
 
28
54
½
 
36
52
54
22⌊ 0
 
 
 
24⌊ 6
 
 
 
 
29
16
44
 
37
16
½
22⌊ 1
 
 
 
24⌊ 7
 
 
 
 
29
38
½
 
37
40
55
22⌊ 2
 
 
 
24⌊ 8
 
 
 
 
30
0
45
 
38
4
½
22⌊ 3
 
 
 
24⌊ 9
 
 
 
 
30
22
½
 
38
29
56
22⌊ 4
 
 
 
25⌊ 1
 
 
 
 
30
44
46
 
38
54
½
22⌊ 6
 
 
 
25⌊ 2
 
 
 
 
31
6
 
39
18
57
22⌊ 8
 
 
 
25⌊ 4
 
 
 
 
31
51
47
 
39
43
½
23
1
 
 
 
25⌊ 5
 
 
 
 
31
29
½
 
40
7
58
23⌊ 2
 
 
 
25⌊ 7
 
 
 
 
32
14
48
 
40
32
½
23⌊ 4
 
 
 
25⌊ 8
 
 
 
 
32
36
½
 
40
57
59
23⌊ 5
 
 
 
25⌊ 9
 
 
 
 
32
59
49
 
41
22
½
23⌊ 6
 
 
 
26⌊ 1
 
 
 
 
33
22
½
 
41
48
60
23⌊ 6
 
 
 
26⌊ 3
 
 
 
 
33
45
50
 
42
14
½
23
7
 
 
 
26⌊ 4
 
 
 
 
34
9
½
 
42
40
61
23
8
 
 
 
26⌊ 5
 
 
 
 
34
31
51
 
43
6
½
23
9
 
 
 
26
7
 
 
 
 
34
54
½
 
43
32
62
23⌊ 9
 
 
 
26⌊ 9
 
 
 
 
35
18
52
 
43
58
½
24⌊ 1
 
 
 
27⌊ 3
 
 
 
 
35
41
½
 
44
25
63
27⌊ 6
 
 
 
34
 
 
 
 
44
52
½
 
55
19
74
27⌊ 8
 
 
 
34⌊ 4
 
 
 
 
45
19
64
 
55
52
½
28⌊ 2
 
 
 
34⌊ 8
 
 
 
 
45
46
½
 
56
26
75
28⌊ 4
 
 
 
35⌊ 0
 
 
 
 
46
14
65
 
57
1
½
28⌊ 5
 
 
 
35⌊ 4
 
 
 
 
46
42
½
 
57
37
76
28⌊ 6
 
 
 
35⌊ 8
 
 
 
 
47
10
66
 
58
13
½
28⌊ 7
 
 
 
36⌊ 3
 
 
 
 
47
38
½
 
58
49
77
28⌊ 8
 
 
 
37
 
 
 
 
48
6
67
 
59
16
½
28⌊ 9
 
 
 
37⌊ 8
 
 
 
 
48
36
½
 
60
4
78
29
 
 
 
38⌊ 6
 
 
 
 
49
4
68
 
60
42
½
30
 
 
 
39⌊ 8
 
 
 
 
49
33
½
 
61
22
79
30
 
 
 
40⌊ 9
 
 
 
 
50
3
69
 
62
2
½
30⌊ 2
 
 
 
41⌊ 8
 
 
 
 
50
33
½
 
62
44
80
30⌊ 4
 
 
 
42⌊ 5
 
 
 
 
51
3
70
 
63
26
½
30⌊ 6
 
 
 
43
 
 
 
 
51
33
½
 
64
9
81
31
 
 
 
44⌊ 5
 
 
 
 
52
4
71
 
64
54
½
31⌊ 5
 
 
 
46
 
 
 
 
52
35
½
 
65
40
82
32
 
 
 
48
 
 
 
 
53
7
72
 
66
26
½
32⌊ 5
 
 
 
50
 
 
 
 
53
39
½
 
67
15
83
33
 
 
 
52
 
 
 
 
54
12
73
 
68
5
½
33⌊ 5
 
 
 
53
 
 
 
 
54
45
½
 
68
58
84
54
 
 
 
(86)
 
 
 
 
69
52
½
 
77
54
88
56
 
 
 
(97)
 
 
 
 
70
49
85
 
79
31
½
59
 
 
 
(116)
 
 
 
 
71
48
½
 
81
27
89
63
 
 
 
(69)
 
 
 
 
72
51
86
 
82
36
¼
67
 
 
 
(81)
 
 
 
 
73
58
½
 
83
57
½
72
 
 
 
(105)
 
 
 
 
75
10
87
 
85
42
¾
78
 
 
 
(258)
 
 
 
 
76
28
½
 
90
00
90
Where the Angle made by the Lunar Circle, and the Paral
el of the Latitude is considerable, instead of the first Number 19′⌊ 1 you must take for 1/1
0 the Numbers following; being in a reciprocal Proportion to the Secants of those Angles.
Angl.
 
0
19⌊ 1
5
19⌊ 1
10
18⌊ 8
15
18⌊ 4
20
17⌊ 9
25
17⌊ 3
30
16⌊ 5
N. B.
Where the Parallel is different from that at the very Middle of the Eclipse; as it usually is; you must increase or decrease the same Numbers 19′⌊ 1, &c. in the Proportion of the Cosines of the Latitude thus:
°
 
00
000
10
174
20
342
30
500
40
643
50
766
60
867
70
940
80
98⌊ 5
90
100⌊ 0
E. G.
If the Co-latitude at the Middle of the Eclipse be 60°, and come to be 40°; say, As 867 to 643, So is 19⌊ 1 to 14⌊ 2, which is there to be taken in its stead.
N. B.
The Hint that I had several Years ago, that in the Determination of Solar Eclipses the
Equality
of the
Difference
of Sines was made us
of by Dr.
Halley,
was the Occasion of the Discoveries in these Papers.
XI.
The perpendicular Distance of every Point of the Penumbra; and the like Distance of every Point of the total Shadow from the Path of the Moon's Center, may be discovered by Tables made from the natural Sines; where those Sines themselves, as before, differ equally, or in arithmetical Progression; according to the Duration of the whole Eclipse, or of total Darkness: and their Co-sines correspond to the Distances from that Path. Both which Tables here follow:
For entire Eclipses: Which are here suppos'd
108′¾
long in the Middle, and the Semidiameter of the Penumbra
2000
Miles.
Duration in Minut.
Distance in Miles.
Digts
1
2000
 
 
2
1999
 
 
3
1999
 
 
4
1999
 
 
5
1998
 
 
6
1997
 
 
7
1996
 
 
8
1995
 
 
9
1993
 
 
10
1991
 
 
11
1990
 
 
12
1988
 
 
13
1986
 
 
14
1984
 
 
15
1981
 
 
16
1979
 
 
17
1976
 
 
18
1973
 
 
19
1970
 
 
20
1966
 
 
21
1963
 
 
22
1959
 
 
23
1955
 
 
24
1951
 
 
25
1946
 
 
26
1942
 
 
27
1937
 
 
28
1932
 
 
29
1927
 
 
30
1922
 
 
31
1917
 
 
32
1911
 
 
33
1905
 
 
34
1899
 
 
35
1893
 
 
36
1886
 
 
37
1879
 
 
38
1872
 
 
39
1866
 
 
40
1859
 
 
41
1852
 
 
42
1844
 
 
 
 
1838
1
43
1835
 
 
44
1828
 
 
55
1820
 
 
56
1812
 
 
47
1803
 
 
48
1795
 
 
49
1786
 
 
50
1776
 
 
½
1771
 
 
51
1766
 
 
½
1761
 
 
52
1756
 
 
2/1
1751
 
 
53
1746
 
 
½
1741
 
 
54
1736
 
 
½
1731
 
 
55
1725
 
 
1720
 
 
56
1714
 
 
1708
 
 
57
1703
 
 
½
1697
 
 
58
1691
 
 
½
1685
 
 
59
1679
 
 
 
 
1676
2
½
1672
 
 
60
1667
 
 
½
1662
 
 
61
1655
 
 
½
1648
 
 
62
1642
 
 
½
1635
 
 
63
1628
 
 
½
1621
 
 
64
1614
 
 
½
1609
 
 
65
1603
 
 
½
1596
 
 
66
1589
 
 
½
1582
 
 
67
1575
 
 
½
1568
 
 
68
1561
 
 
½
1554
 
 
69
1546
 
 
½
1538
 
 
70
1534
 
 
½
1528
 
 
71
1521
 
 
½
1514
 
 
 
 
1513
3
72
1506
 
 
½
1498
 
 
73
1490
 
 
½
1482
 
 
74
1474
 
 
½
1456
 
 
75
1448
 
 
1439
 
 
76
1430
 
 
1424
 
 
77
1412
 
 
1403
 
 
78
1394
 
 
½
1384
 
 
79
1374
 
 
½
1364
 
 
80
1354
 
 
 
 
1350
4
½
1344
 
 
81
1333
 
 
½
1322
 
 
82
1312
 
 
½
1302
 
 
83
1291
 
 
½
1280
 
 
84
1269
 
 
½
1258
 
 
85
1246
 
 
½
1234
 
 
86
1222
 
 
½
1110
 
 
87
1098
 
 
 
 
1188
5
½
1186
 
 
88
1173
 
 
½
1160
 
 
9
1147
 
 
½
1134
 
 
90
1121
 
 
½
1107
 
 
91
1093
 
 
½
1079
 
 
92
1065
 
 
½
1050
 
 
93
1034
 
 
 
 
1025
6
½
1017
 
 
94
1000
 
 
½
983
 
 
95
971
 
 
½
955
 
 
96
937
 
 
½
919
 
 
97
902
 
 
½
884
 
 
98
864
 
 
 
863
7
½
844
 
 
99
824
 
 
½
804
 
 
100
784
 
 
¼
773
 
 
½
762
 
 
¾
751
 
 
101
739
 
 
¼
728
 
 
½
716
 
 
¾
704
 
 
 
 
700
8
102
691
 
 
¼
679
 
 
½
666
 
 
¾
653
 
 
103
639
 
 
¼
625
 
 
½
611
 
 
¾
597
 
 
104
582
 
 
¼
566
 
 
½
550
 
 
 
 
537
9
¾
534
 
 
105
517
 
 
¼
499
 
 
½
482
 
 
¾
463
 
 
106
443
 
 
¼
421
 
 
½
399
 
 
¾
376
 
 
 
 
375
10
107
352
 
 
¼
320
 
 
½
288
 
 
¾
258
 
 
108
226
 
 
 
 
212
11
¼
180
 
 
½
119
 
 
 
 
 
 
 
 
50
12
¾
0
 
 
For Total Darkness of
166″ ⅔.
Duration in Secds .
Distance in Miles.
″
 
1
50⌊ 0
2
50⌊ 0
3
50⌊ 0
4
50⌊ 0
5
50⌊ 0
6
50⌊ 0
7
50⌊ 0
8
50⌊ 9
9
50⌊ 9
10
50⌊ 9
11
49⌊ 9
12
49⌊ 8
13
49⌊ 8
14
49⌊ 8
15
49⌊ 7
16
49⌊ 7
17
49⌊ 7
18
49⌊ 6
19
49⌊ 6
20
49⌊ 6
21
49⌊ 6
22
49⌊ 5
23
49⌊ 5
24
49⌊ 5
25
49⌊ 4
26
49⌊ 4
27
49⌊ 4
28
49⌊ 3
29
49⌊ 3
30
49⌊ 3
31
49⌊ 2
32
49⌊ 2
33
49⌊ 2
34
49⌊ 1
35
49⌊ 1
36
49⌊ 1
37
49⌊ 0
38
49⌊ 0
39
49⌊ 0
40
48⌊ 9
41
48⌊ 9
42
48⌊ 8
43
48⌊ 7
44
48⌊ 6
45
48⌊ 5
46
48⌊ 4
47
48⌊ 2
48
48⌊ 1
49
47⌊ 9
50
47⌊ 7
51
47⌊ 6
52
47⌊ 5
53
47⌊ 4
54
47⌊ 3
55
47⌊ 2
56
47⌊ 1
57
47⌊ 0
58
46⌊ 9
59
46⌊ 8
60
46⌊ 6
61
46⌊ 5
62
46⌊ 4
63
46⌊ 3
64
46⌊ 2
65
46⌊ 0
66
45⌊ 9
67
45⌊ 8
68
45⌊ 6
69
45⌊ 5
70
45⌊ 3
71
45⌊ 2
72
45⌊ 1
73
45⌊ 0
74
44⌊ 8
75
44⌊ 6
76
44⌊ 5
77
44⌊ 3
78
44⌊ 1
79
44⌊ 0
80
43⌊ 8
81
43⌊ 7
82
43⌊ 5
83
43⌊ 3
84
43⌊ 2
85
43⌊ 0
86
42⌊ 9
87
42⌊ 7
88
42⌊ 5
89
42⌊ 3
90
42⌊ 1
91
41⌊ 9
92
41⌊ 7
93
41⌊ 5
94
41⌊ 3
95
41⌊ 1
96
40⌊ 9
97
40⌊ 7
98
40⌊ 5
99
40⌊ 3
100
40⌊ 0
101
39⌊ 8
102
39⌊ 6
103
39⌊ 4
104
39⌊ 1
105
38⌊ 8
106
38⌊ 6
107
38⌊ 4
108
38⌊ 1
109
37⌊ 9
110
37⌊ 6
111
37⌊ 4
112
37⌊ 1
113
36⌊ 8
114
36⌊ 5
115
36⌊ 2
116
35⌊ 9
117
35⌊ 6
118
35⌊ 3
119
35⌊ 0
120
34⌊ 7
121
34⌊ 4
122
34⌊
123
33⌊ 8
124
33⌊ 4
125
33⌊ 0
126
32⌊ 7
127
32⌊ 4
128
32⌊ 1
129
31⌊ 7
130
31⌊ 3
131
30⌊ 9
132
30⌊ 5
133
30⌊ 1
134
29⌊ 7
135
29⌊ 3
136
28⌊ 9
137
28⌊ 5
138
28⌊ 0
139
27⌊ 5
140
27⌊ 1
141
26⌊ 6
142
26⌊ 1
143
25⌊ 6
144
25⌊ 1
145
24⌊ 6
146
24⌊ 0
147
23⌊ 5
148
23⌊ 0
149
22⌊ 4
150
21⌊ 8
151
21⌊ 1
152
20⌊ 5
153
19⌊ 8
154
19⌊ 1
155
18⌊ 3
156
17⌊ 6
157
16⌊ 7
158
15⌊ 9
159
14⌊ 9
160
14⌊ 0
161
12⌊ 9
162
11⌊ 7
163
10⌊ 4
164
8⌊ 9
165
7⌊ 0
166
4⌊ 4
166 ⅔
0⌊ 0
Semidiameter of the Umbra or Total Darkness = 50 Miles.
XII.
The Angles at the Vertex of the Lunar Circle, on each Side of the Point of Contact, by Reason of the perpendicular Situation of that Axis to its own Circle; are always right Angles: Only diminish'd in the Proportion of the Minutes describ'd by the Annual Motion during the Continuance of the Eclipse. Thus in our present Eclipse, which retains the Center of the Shadow near three Hours upon the Earth's Disk, in which Time the annual Motion amounts to about 8′; each of those right Angles in Strictness are to be esteem'd only 89° 56′, and both together 179° 52′. Only because the Refraction of the Rays of the Sun through our Atmosphere, requires a somewhat greater Increase of this Angle, than the annual Motion requires its Diminution, I shall wholly omit it, in all my Calculations hereafter.
XIII.
The Angles made at the Poles of the Earth, which shew the Difference of the two extreme Meridians, and limit the Extent of the entire central Eclipse, by reason of the Obliquity of the Earth's Axis to that Lunar Circle, are usually unequal to one another; and more or fewer than twice 90°, as the Eclipse happens at different Latitudes of the Moon, and Times of the Year.
XIV.
The Meridian that passes through the Middle or Central Point of general Solar Eclipses, is the same with that which passes through the Center of the Sun at the same Time, when the Moon has no Latitude, and the Eclipses are Central in the Plane of the Ecliptick; as also when they happen in either Solstices. Otherwise the Moon's Latitude being taken perpendicular to the Plane of the Ecliptick; and the nearest Distance of the Moon's Motion being taken perpendicular to the Lunar Circle, while the Meridians always pass through the Poles of the Earth; these Two Meridians will, generally speaking, be different, and their included Angle no otherwise to be known than by Trigonometry, as will appear hereafter.
XV.
The Dimensions of the Penumbra, or entire Eclipse, and the Extent of the total Shadow on the Earth, are continually different, according to the different Elevations of the Sun and Moon above any particular Horizon. For as the Moon is about the same Distance from every Place, when it is in its Horizon, as it is from the Earth's Center it self; with regard to which Center alone our first Calculations are always made: So when it is in the Zenith of any Place, it is one Semidiameter of the Earth nearer it; which Semidiameter being usually 1/60, and at our next Eclipse 1/55⌊ 5 of its entire Distance, as will appear hereafter, will deserve an Allowance. Nor will any lesser Elevation of the Moon be wholly inconsiderable in Eclipses, but in all accurate Determinations thereof must be particularly computed, in order to the distinct Knowledge of the Extent of such Eclipses; especially of the Breadth of the Total Shadow therein. Accordingly we are to observe that this Breadth of the Total Shadow will certainly be at this Eclipse considerably greater over
North America;
where the Luminaries are greater elevated above the Horizon; than over
Europe,
where they are much nearer it; as this Calculation requires: Of which hereafter.
XVI.
The
Figure
of the entire
Penumbra,
or general Eclipse; and of the
Umbra,
or Total Darkness; as they appear upon every Country, is different, on account of the different Obliquity of every Horizon; and will make Ovals or Ellipses of different Species perpetually. This in the vast
Penumbra
is best understood by such an Instrument as my
Copernicus;
or by the Perusal of a very scarce Book written by
P. Coursier, (Philos. Transact.
No . 343. p. 259.) and cited by Dr.
Halley:
Which distinctly treats of the Intersection of a Conical and Spherical Surface. But in the smaller
Umbra,
or Total Darkness, which is confined to a much narrower Compass, it very nearly approaches to the Intersection of a conick Surface with a Plane, which is a true Ellipsis.
XVII.
The
Species
of that Ellipsis depends on the Sun's Altitude above the Horizon at the Time of Total Darkness; as does the
Position
of its longer Axis on the Azimuth of the Sun at the same Time. Nor is it at all necessary that the Direction of this or any other Ellipsis should be along either of the Axes, but may as well be along any other Diameter whatsoever.
XVIII.
The
Direction
of the Center of the Shadow is according to the Direction of the Moon's Motion, along the Plane of the Lunar Circle, as compounded with the diurnal Motion, or with the Direction and Velocity of those Parts of the Earth over which it passes; and will be hereafter brought to Calculation. And indeed this Angle may be had, either by finding the several Points of the Path of the Moon's Way upon the Earth, in as many Meridians as we please, and drawing a curve Line through those Points; or by solving a Spherical Triangle, whose Sides are the Complements of the Latitudes of two neighbouring Places equally distant,
East
and
West,
from the Place you work for; and whose included Angle is the Angle at the Pole suited to the Difference of their Meridians; and taking half the the two Angles at the Base, the one internal and the other external, for the Angle desir'd▪ Of which hereafter.
XIX.
Every Ellipsis, made by the oblique Section of a Cone, has the Intersection of the
(Fig.
1.) Axis of the Cone
C
at some Distance from the Center of the Ellipsis
D.
And the Proportion of those unequal Divisions
B C
and
C A
are the same with that of the Sides of the Cone
V B
and
V A:
As appears by the
(Elem.
III. 6.) Elements of
Euclid.
Whence it is evident, that the proper Center of the total Shadow in Eclipses of the Sun, or that made by the Axis of the Cone, is not the same with the Center of the Elliptick Shadow; and that the Proportion of its Distance from that Center may be easily determin'd by the Proposition here refer'd to: Of which more hereafter.
Scholium.
This Ellipsis, when the Sun is of a considerable Altitude, is almost an exact one; but when the Sun is near the Horizon, it will be very long, and so less exact; because the Spherical Surface of the Earth is at that Distance more remote from a Plane.
XX.
The perpendicular
Breadth
of the Shadow is neither that of the longer, nor that of the shorter Axis: But that of the two longest Perpendiculars
(Fig.
2.)
A B
and
C D
drawn from the Tangents parallel to the Diameter
D B,
along which the Direction of the Motion is: The length of which Perpendiculars will be hereafter determined.
XXI.
The
Velocity
of the Motion of the Center of the Shadow is unequal; not only on account of the Difference of the Moon's own Motion, at the beginning and ending of the entire Eclipse; which indeed is very inconsiderable; but chiefly by reason of the Difference of the Obliquity of the Horizon all the Way of its Passage. However, since the several Points may, in all Meridians be distinctly found by Trigonometry, as we shall shew presently, this Inequality need create us no new Difficulty in the Determination of Eclipses.
XXII.
The Number of
Digits
eclips'd, which are twelfth Parts of the Sun's Diameter, with sexagesimal Parts of the same Digits, are always to be estimated as distinct from the total Shadow; and may be discovered by the help of the foregoing Table, p. 29, 30, 31. Where the Digits are already noted at every proper Distance from the Path of the Moons Center; and where the intermediate Fraction 1/1
2⌊ 6 is more exact than 1/6
; but which, by dividing that Number 1/16
⌊ 6 by 2⌊ 7 will give those Sexagesimals, without any farther Trouble. The Application of that Table will be taught hereafter.
XXIII.
The
Distance
of the Vertex of the conical Shadow of the Moon, which sometimes just reaches the Surface of the Earth, as in total Eclipses
sine morâ,
Sometimes does not reach it; as in Annular Eclipses; and sometimes would overreach it, if it were not intercepted, as in total Eclipses
cum morâ;
may be easily discovered at any time by the Analogy following: As
P C
the
(Fig.
3.) Semidiameter of the Moon: = 941 geographical Miles is to
CV
the Distance of the Moon from that Vertex = 215000∷ So is
R s
= 45 the smallest Semidiameter of the total Shadow, which is the same as of the circular Shadow it self, to
s V,
the Distance of that Vertex therefrom = 10316, which is thus: 941: 215000∷ 45: 10316.
XXIV.
The Determination of the Circumstances of Solar Eclipses, for any given Distance from the Path of the Moon's Center, either way, has no new Difficulty in it; but is to be made just as is that for the Center of the Penumbra. Only the Quantity of the Distance of the Vertex of the Lunar Plane from that Circle will be different; as the Path of the Moon's Center it self might be at another Eclipse, of otherwise the same Circumstances.
XXV.
If Two Bodies
A
and
B
set out together, the one from
A,
the other from
B:
and move evenly forward in a known Proportion as to Velocity; the Point C will be determined where the swifter will overtake the slower, and they will be coincident. Thus if the Velocity of
A, (Fig.
4.) be to that of
B,
as 5 to 1, the Proportion of the Lines
A B
to
B C
will be as 4 to 1, and if we add 1 to 4 = 5 we have the place
C
where the swifter will overtake the slower. Thus if their Velocities be to each other as 5⌊ 48 to 1, the Lines of their Motion
A B
and
B C,
will be as 4⌊ 48 to 1. So that if we take in the former Case ¼ of
A B
and in the other 1/4⌊ 4
of
A B,
and add it to
A B,
we gain
AC
the Distance of the Point
C
from
A.
Corollary.
If therefore
A
represent the Center of the Shadow of a Solar Eclipse, as it is plac'd at the Middle of the general Eclipse; and B
Greenwich
at the same Moment of absolute Time; and at a known Distance from the middle Point; and if the Velocity of the Center of the Shadow along the Circle be to the Velocity of
Greenwich
in its diurnal Motion as 5
48 to 1, if we ad 4⌊ 48 of their Distance at their setting out to that known Distance, we obtain the Point or Place where the Center of the Eclipse will overtake
Greenwich,
or the Time when the Eclipse will be at the Meridian of
Greenwich.
And this whether the Center and
Greenwich
move along the same Line as
A B C,
or two different Lines, as
A B C
and
a b c.
XXVI.
The Duration of Solar Eclipses is different, according as their Middle happe
s about Six in the Morning or Evening; or a
out Noon; or about any intermediate Time. If that happens about Six a-Clock, Morning or Evening, the diurnal Motion then neither much conspires with, nor opposes the proper Motion of the Center of the Shadow; and the Duration is almost the same as it would be if the Earth had no diurnal Motion at all. If that happens about Noon, the diurnal Motion most of all conspires with that proper Motion of the Center, and makes the Duration of the Eclipse the longest possible. If it happens in the intermediate Times, the diurnal Motion, in a less Degree, conspires with the other Motion, and makes the Duration of a me
n Quantity, between that of the other Cases. But if it happens considerably be
ore Six a-Clock in the Morning, or after Six a-Clock in the Evening, the diurnal Motion is backward, and short
ns that Duration proportionably. Of the Quantity of which Duration we shall enquire more hereafter.
PROBLEMS.
I.
To find the nearest Distance of the Path of the Moon s Center, to the Center of the Disk of the Earth, as seen at the Distance of the Mo
n in the total Eclipse of the Sun
A. D.
1724.
May
11°.
P. M.
This is equal to the Moon's true Latitude at the Time of the Conjunction in her own Orbit; and is set down in the Calculation 32′ 19″.
II.
To find the Sun's
Declination
at the Middle of the Eclipse.
As the Radius of the Circle: is to the Sine of the Sun's Longitude at that Time=61′ 39″∷ So is the Sine of the Sun's greatest Declination, = 23° 29′: to the Sine of the Sun's Declination then.
 
Rad.
 
10.
000000
 
 
 
 
S.
61°
39′
9.
944514
 
 
 
 
S.
23°
29′
9.
600409
 
°
″
 
S.
 
 
8.
544923
=
20
32
 
 
 
 
 
 
=
Declination
☉.
III.
To find the Sun's
Right Ascension
for the same Time.
As the Radius: to the Cosine of the Sun's greatest Declination∷ So is the Tangent of the Sun's Latitude: to the Tangent of the Sun's right Ascension.
Rad.
 
 
10.
 
 
 
 
Sin.
66
31
9.
962453
 
 
 
Tang.
61
39
10.
267952
 
°
′
Tang. Right Ascen.
10.
230405
=
59
31
 
Compl.
30
29
IV.
To find the
Distance
between the Vertex of the Lunar Circle and the Pole of the Earth.
Let
(Fig.
5.)
E P
represent the Distance between the two Poles of the Earth, and of the Ecliptick: = 23° 29′.
EV
the Distance between the Pole of the Ecliptick, and the Vertex of the Lunar Circle: = 5° 36′.
P EV
the Angle made by the Solstitial Colure
P E:
(in which the two Poles of the Earth, and of the Ecliptick always are) and that Arc
EV.
And because the Sun is here 28° 21′ distant from that Colure, which is the Complement of its Longitude from
Aries;
and the Ascending Node, or Argument of Latitude is then 5° 49′ distant from the Sun backward; The Sum of these Numbers gives 34° 10′, whose Complement is the Angle
V E R
= 55° 50′. In order then to gain
V P
proceed thus:
R.
 
 
10.
 
 
 
 
CS
VER.
55
50.
9.
749429
 
 
 
T.
VE.
5
36
8.
991451
 
°
′
T.
RE
 
 
8.
740880
=
3.
10.
 
+
23.
29.
 
=
26.
39.
Then say,
 
 
°
′
 
 
 
 
 
CS.
R E
3
9
9.
992343
 
 
 
CS.
V E
5
36
9.
997922
 
 
 
CS.
P R
26
39
9.
951222
 
 
 
 
 
 
 
19.
949144
 
°
′
 
CS.
P V
=
9.
949801
=
27.
0
V.
In the same Triangle
V P E,
to find the Angle
V P E,
included between the Colure
E P,
and the Prime Meridian
P V.
 
°
′
 
 
 
 
 
S.
P V
27
0.
9.
657047
 
 
 
S.
V E R
55
50.
9.
917719
 
 
 
S.
V E
5
36.
8.
989374
 
 
 
 
18.
907098
 
°
′
S
V P E
=
9.
249961
=
10
15
Corollary.
The
Primary Angle,
composed of the Complement of the Right Ascension, and of this Angle
V P E,
is = 40° 44′.
Compl.Rad.
30°
29′
+
10
15
=
40
44
VI.
To find the
Distance
of the Pole or Vertex of the Lunar Circle from the Circle it self.
As the Semidiameter of the Earth's Disk = 61′ 28″: To the Latitude of the Moon, or nearest Distance to the Path of the Moon's Center from the Center of the Disk; 32° 19′∷ So is the Radius: to the Sine of the Complement of that Distance. In Decimals thus: 61′⌊ 63: 32′⌊ 32∷ 10000: 5244 = S. 31° 38′, whose Complement is 58° 22′: equal to the Distance of the Lunar Circle from its Vertex.
VII.
To find the
Angle
included between the Meridian that passes through the Center of the general Eclipse, and that passing through the Center of the Sun at the same Time.
In the Triangle
(Fig.
6.)
M S P
where the Side
S P
is already given, = Complement of the Sun's Declination; find the Angle
M S P
thus:
R. 10.
S. of the Sun's Distance from the Solstitial Colure: = 28° 21′ 9.676562
S. of the Sun's greatest Declination= 23° 29′ 9. 600409
 
°
′
S. of an Angle
9. 276971 =
10.
54.
To which add the Angle of the Moon's Way: +
5
36
The Sum is the Angle
M S P
=
16.
30
Then, in the same Triangle
P M S,
we have two Sides:
M S
equal to the Latitude of the Moon, or the length of the Perpendicular to the Moon's way = 31° 38′. and
P S
= 69° 28′. and the included Angle
M S P
= 16° 30′. to find
M P S
thus:
R.
 
 
10.
 
 
 
 
 
 
°
′
 
 
 
 
 
 
CS.M S P.
16.
30.
9.
981774
 
 
 
 
T.
S M.
31.
38.
9.
789585
 
 
 
 
T.
R M.
=
9.
771359
=
30
34
M R
 
 
 
 
 
+
31
38
M S
 
 
 
 
 
=
62
12
R S
Then say,
S.S P.
69.
28.
9.
971493
 
 
 
 
 
S.S R.
62.
12.
9.
946738
 
 
 
 
 
T.M S P.
16.
30.
9.
471605
 
 
 
 
 
 
 
 
9.
418343
 
°
′
 
 
T.
M P S
=
9.
446850
=
15
38
 
 
 
+
40
44
 
 
 
=
56
22
=
Angle with the Primary Meridian.
VIII.
To find the
Longitude
and
Latitude
of the Center of the Shadow at the Middle of the general Eclipse; or to solve the primary Triangle.
Under
Problem
IV. we have sound the Distance of the Vertex of the Lunar Circle from the Pole of the Earth = 27°. 0′. Under
Problem
VI. we have found the Distance of that Vertex from its Circle = 58°. 22′; and under the last
Problem
we have found the Angle at the Pole of the Earth, between the Primary Meridian, and that Meridian which passes through the Center of the Eclipse, = 56° 22′. From which
data
the Primary Triangle
(Fig.
7.) is thus to be solv'd:
R.
 
 
10.
 
 
°
′
 
 
CS.
C P Q.
56.
22.
9.
743412
T.
V P.
27.
0.
9.
707166
T.
P R. =
15.
46.
9.
450578
Then say,
CS.V P
27.
0.
9.
949880
 
 
 
CS.P R
15.
46.
9.
983345
 
 
 
CS.V C.
58.
22.
9.
719730
 
 
 
 
 
 
19.
703075
 
°
′
CS.P C.
 
 
9.
753195
=
55.
30
 
 
 
Deduct
PR
=
15.
46
 
 
 
Remt .
= P C.
=
39.
44
 
 
 
Ergo
Lat.
=
50.
16
To find the Angle at the Vertex
CV P,
proceed in this Manner:
°
′
 
 
 
 
 
S.
CV.
58.
22.
9.
930145
 
 
 
S.
C P Q.
56.
22.
9.
920436
 
 
 
S.
PC.
39.
44.
9.
805647
 
 
 
 
 
 
19.
726083
 
°
′
S.CVP.=
9.
795938
=
38
41
The following Analogy will give
V C P
the Complement of the Angle which the Direction of the Center of the Eclipse makes with the Meridian, that Direction being perpendicular to
V C.
 
°
′
 
 
 
 
 
S.
CV.
58.
22.
9.
930145
 
 
 
S. CP
Q.
56.
22.
9.
920436
 
 
 
S.
V P.
27.
0.
9.
657047
 
 
 
 
 
 
19.
577483
 
°
′
S.V C P.=
9.
647338
=
26.
21
Compl.
=
63.
39
Corollary.
Hence we also learn the most
Northern Latitude,
where the Center of the Shadow will cross the Meridian at Noon, and at right Angles: And this without any particular distinct Calculation. For
V Q
= 58° 22′
V P
27° 0′ =
P Q
= 31° 22′. whose Complement = 58° 38′. is that very
Northern Latitude.
IX.
To find the
Longitude
and
Latitude
of the Center of the Shadow, when it crosses the Meridian that passes through the Center of the Sun at the Middle of the Eclipse; or to solve the second principal Triangle.
If in the foregoing Triangle we suppose the Angle at the Pole to be equal to the Primary Angle, or 40° 44′. we may thus solve this Triangle:
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS. CP
Q.
40.
44.
9.
879529
 
 
 
T.
V P.
27.
0.
9.
707166
 
°
′
T.P R.
=
9.
586695
=
21
7
Then say,
CS.V P.
27.
0.
9.
949880
 
 
 
CS.P R.
21.
7.
9.
969811
 
 
 
CS.V C.
58.
22.
9.
719730
 
 
 
 
 
 
19.
689541
 
°
′
 
CS. CR.
=
9.
739661
=
56
42
 
 
 
 
Deduct
21
7
 
 
 
 
Remt . C P.
35
35
 
 
 
 
Ergo,
Lat.
=
54
25
To find the Angle at the Center of the Eclipse
V C P,
proceed thus:
 
°
′
 
 
 
 
 
S.
V C.
58.
22.
9.
930145
 
 
 
S.
C P Q.
40.
44.
9.
814607
 
 
 
S.
V P.
27.
0.
9.
657047
 
 
 
 
 
 
19.
471654
 
°
′
S.CV P.
=
9.
541509
=
20
22
X.
To find the
Longitude
and
Latitude
of the Center of the Shadow at its Entrance on the Disk of the Earth: Or to solve the third Principal Triangle.
Add the vertical Angle already found = 38° 41′ to a right Angle at the Vertex; = 90 + 38°. 41′ = 128°. 41′ this is equal to the Angle at the Vertex
CV P.
Substract this Angle from two right Angles. 180°—128°. 141′ = 51°. 19′, in order to gain the Supplement, whose Sines, &c. are the same with the others.
(Fig.
8.) Then say,
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
CV P.
51.
19.
9.
795891
 
 
 
T.
V P.
27.
0.
9.
707166
 
°
′
T.V R.
=
19.
503057
=
17
40
=V R
 
 
 
 
 
+
58
22
=V C
 
 
 
 
 
=
76
02
=C R
Then say,
 
°
′
 
 
 
 
 
CS.
V R.
17.
40.
9.
979019
 
 
 
CS.
V P.
27.
00.
9.
949880
 
 
 
CS.
C R.
76.
02.
9.
382661
 
 
 
 
 
 
19.
332541
 
°
′
CS.
C P.
=
9.
353522
=
76:
57
 
 
 
 
Ergo,
Lat.
13.
03
In order to find the Angle at the Pole
V P C,
whose Supplement is the Longitude of that Point where the Center of the Shadow enters the Disk of the Earth from the Primary Meridian, proceed thus:
 
°
′
 
 
 
 
 
S.
C P.
76.
57.
9.
988636
 
 
 
S.
CV P.
51.
19.
9.
892435
 
 
 
S.
C V.
58.
22.
9.
930145
 
 
 
 
 
 
19.
822580
 
80
°
S.V P C.
=
9.
833944
=
43
01
 
 
 
 
 
Suppl.
136:
59
To find the Angle
VC P,
proceed thus:
°
′
 
 
 
 
 
S.
C P.
76.
57.
9.
988636
 
 
 
S.
CV P.
51.
19.
9.
892435
 
 
 
S.
V P.
27.
00.
9.
657047
 
 
 
 
 
 
19.
549482
 
°
′
S.VC P.
=
9.
560846
=
21.
20
XI.
To find the Longitude and Latitude of the Center of the Shadow at its Exit from the Disk of the Earth; or to solve the fourth Principal Triangle.
Substract the Angle already found 38° 41′ from a right Angle 90° − 38° 41′ = 51° 19 = Angle at the Vertex
P V C (Fig.
9.) Then say,
R.
 
 
10.
 
 
 
 
 
 
 
°
′
 
 
 
 
 
 
 
CS.
P V C.
51.
19.
9.
795891
 
 
 
 
 
T.
P V.
27.
00.
9.
707166
 
°
′
 
 
 
T.V R.
9.
503057
=
17
40
 
 
 
 
 
 
from
58
22
 
 
 
 
 
 
Remnt
40
42
=
R C.
Then say,
 
°
′
 
 
 
 
 
CS.
V R.
17.
40.
9.
979019
 
 
 
CS.
P V.
27.
00.
9.
949880
 
 
 
CS.
R C.
40.
42.
9.
879746
 
 
 
 
 
 
19.
829626
 
°
′
CS.
P C
=
9.
850607
=
44.
51.
 
 
 
 
Ergo,
Lat.
45.
9.
In order to find the Angle
Q P C,
or the Longitude of that Point where the Center of the Shadow departs out of the Disk of the Earth, from the Primary Meridian, proceed thus:
°
′
 
 
 
 
 
S.
CP.
44.
51.
9.
848345
 
 
 
S.
PVC.
51.
19.
9.
892435
 
 
 
S.
VC.
58.
22.
9.
930145
 
 
 
 
 
 
19.
822580
 
°
′
S.QPC
=
9.
974235
=
70.
27
 
 
 
 
 
+
136
59
 
 
 
 
 
=
207
26
And for the Angle
VCP
thus;
 
°
′
 
 
 
 
 
S.
CP.
44.
51.
9.
848345
 
 
 
S.
PVC.
51.
19.
9.
892435
 
 
 
S.
VP.
27.
00.
9.
657047
 
 
 
 
 
 
19.
549482
 
°
′
S.
VCP.
=
9.
701137
=
30.
10.
Corollary.
Hence the Angular Motion of the Center of the Eclipse about the Pole of the Earth, if there were no diurnal Motion, is 207°. 26′.
XII.
To find the Time in which the Center of the Shadow will go over the Diameter of the Lunar Circle.
Say, first, 35′⌊ 3: 60′∷ 123⌊ 26: 209⌊ 6;
i. e.
As the Number of Minutes of a Degree pass'd over in an Hour: to an Hour∷ So is the entire Diameter of the Disk from the Calculation: to the Number of Minutes for that Passage.
Then say, 10000: 8514∷ 209′⌊ 6: 178′⌊ 4
i. e.
As the Radius: to the Sine of 58°: 22′ = the Distance of the Lunar Circle from its Pole or Vertex∷ So are the Minutes of the Passage over the entire Diameter: to the Minutes of the Passage over this Chord = 178′ 24″.
XIII.
To find the Proportion of the Velocities of the Center of the Shadow and of the diurnal Motion of the corresponding Point of the Earth at the Time of the Eclipse:
Say thus; As 178′⌊ 4 to 829′⌊ 6 = 207° 26′, or as 1 to 4⌊ 64, so is the Time of the Center of the Eclipse's Motion over the Diameter of the Lunar Circle: to the Timeof the diurnal Motion's going from the entranceto the Exit of the Center.
Corollary.
Hence the real Angular Motion of the Center of the Eclipse about the Pole of the Earth, is no more than 162° 40′. For 4⌊ 64: 3⌊ 64∷ 207° 26′: 162° 40′.
XIV.
To find the
Latitude
of any Place, over or near which the Center of any Shadow passes, to any known
Longitude
or
Time
given. And,
vice versa,
To find the
Longitude
or
Time
of the nearest Approach to any such Place to any known
Latitude.
This is no more than proceeding in the Calculations as hitherto; by taking any known Meridian or Time; or else any known Latitude for our Examples.
I shall therefore give three several Examples in both Cases; because of the great Dignity and Usefulness of the
Problem: viz.
For
Greenwich
the Meridian of the Tables; for
Dublin
more
Westward;
and for
Paris
more
Eastward.
Now I here suppose, from the Calculation and Construction of Eclipses, that the Middle of this general Eclipse will happen
May
11. 1724. 17′ past 5 a-Clock in the Afternoon; and that its Center will cross the Meridian of
Greenwich
41′ past 6. Upon which Hypothesis I thus compute:
°
′
 
°
′
From the Angle
100
15
=
6h
41
Deduct the Primary Angle
40
44
There remains the Angle of the Pole
Q P C
=
59
31
Then proceed thus:
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
Q P C.
59
31
9.
705254
 
 
 
T.
P V.
27
00
9.
707166
 
°
′
T.P R.
=
9.
412420
=
14
29
Say then,
CS. P V.
27
00
9.
949880
 
 
 
CS. P R.
14
29
9.
985974
 
 
 
CS. V C.
58
22
9.
719730
 
 
 
 
 
 
19.
705704
 
°
′
CS. R C.
=
9.
755824
=
55
15
 
 
 
Deduct
R P
=
14
29
 
 
 
Remains
P C
=
40
46
 
 
 
Ergo,
Lat.
49
14
N. B.
If we take Dr.
Halley
's Time 6h 36′, and substract 40°▪ 44′ out of 99° there remains 58°▪ 16′; and the Calculation will stand thus:
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
QP C.
58.
16.
9.
720958
 
 
 
T.
P V.
27.
0.
9.
707166
 
°
′
T.P R.
=
9.
428124
=
15
0
Then say,
 
°
′
 
 
 
 
 
CS.P V.
27.
0.
9.
949880
 
 
 
CS.P R.
15.
0.
9.
984944
 
 
 
CS.V C.
58.
22.
9.
719730
 
 
 
 
 
 
19.
704674
 
°
′
CS.R C.
=
9.
754794
=
55
21
 
 
 
Deduct
R P.
15
0
 
 
 
Remt .
P C.
40
21
 
 
 
Ergo,
Lat.
=
49
39
 
 
 
 
Diff.
00
25
N. B.
My Calculation differs from Dr.
Halley
's Scheme no less than a full Degree of a great Circle, in the Meridian; if our Difference of Time, which is about 5′, be allowed. And though we take the Doctor's own Time, yet do we differ in
Latitude
25 Minutes or Miles; by which Quantity the Doctor's Scheme brings the Center of the Eclipse nearer to
London
and
Greenwich
than this Calculation. The reason of which Difference I by no means understand. Time will discover which Determination is most accurate.
Dublin
is about 6° 22′
Westward
in Longitude from
Greenwich.
Let us find the Latitude of the Center of the Shadow, when it crosses the Meridian of
Dublin.
We must proceed thus: As 3⌊ 64 to 4⌊ 64 so is 6′. 22″ to 8′. 7″ Deduct then from the Angle at the Pole used for
Greenwich
this Difference of these Angles 59° 31′ − 8°. 7′ = 51° 24′. which is our Angle at the Pole for
Dublin.
So that if we use the former Figure with that Angle, we compute as before;
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
Q P C.
51.
24.
9.
795101
 
 
 
T.
P V.
27.
0.
9.
707166
 
°
′
T.
R P.
 
 
9.
502267
=
17.
38
Then say,
 
°
′
 
 
 
 
 
CS.
P V.
27.
0.
9.
949880
 
 
 
CS.
V C.
58.
22.
9.
719730
 
 
 
 
 
 
19.
698829
 
°
′
CS.
R C.
=
9.
748949
=
55
53
 
 
 
Deduct
R P.
17
38
 
 
 
Remnt
P C.
38
15
 
 
 
Ergo,
Lat.
51
45
Paris
is 2° 19′ more
Easterly
than
Greenwich,
Say therefore 3⌊ 64: 4⌊ 64∷ 2°. 19′: 2°. 57′ Now 59°. 31′ + 2°. 57′ = 62°. 28′ Then,
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
Q P C.
62.
28.
9.
664891
 
 
 
T.
P V.
27.
0.
9.
707166
 
°
′
T. R P.
=
9.
372057
=
13
15
Say then,
CS.
P V
27.
0.
9.
949880
 
 
 
CS.
R P
13.
15.
9.
988282
 
 
 
CS.
V C.
58.
22.
9.
719730
 
 
 
 
 
 
19.
708012
 
°
′
CS.
R C.
 
 
9
758132
=
55.
03
 
 
 
Deduct
R P
=
13.
15
 
 
 
Remt . =
P C.
41.
48
 
 
 
Ergo,
Lat.
=
48.
12
N. B.
By such Calculations we may determine the Latitude of the Center of the Shadow's Way, from its entry upon, till its exit out of the Disk of the Earth, to every known Meridian. A Specimen of which I intend to give presently for the several
East
and
West Longitudes
from
London
in the Eclipse before us: And another Specimen in the Eclipse,
Sept.
4. 1727.
If the
Latitudes
be given, as for the Meridian of
Greenwich
49°. 14′; For
Dublin
51°. 45′; For
Paris
48°. 12′: The Case will be that of a Spherical Triangle, when all the Sides are given; and the
Longitude
or
Time
is an Angle sought. Thus in the foregoing Figure for
Greenwich, V C
= 58°. 22′ is the Side against the Angle sought.
V P
= 27°. 0′, and
P C
= 40°. 46′. From which
data
we thus discover the Angle
Q C P.
 
 
°
′
 
 
 
 
 
 
V C
=
58
22
R.
10.
 
 
 
 
V P
=
27
00
S.
9.
657047
 
 
 
P C
=
40
46
S.
9.
814900
 
 
 
Sum of 3
=
126
08
Sum
19.
471947
 
 
 
½ Sum
 
63
4
S.
9.
950138
 
 
 
Diff. of
V C
4
42
S.
8.
913488
 
 
 
Double Radius
 
 
 
20.
 
 
 
 
The Sum
 
 
 
38.
863626
 
 
 
The Remainder
 
 
 
19.
391679
 
 
 
 
 
 
 
 
 
 
 
°
′
½ Remainder
 
 
 
9.
695839
= CS.
60
14
double
120.
28
from
180.
00
as before remains
59.
32
Add the Primary Angle
40.
44
Sum
100.
16
Equal in Time to
6h .
41′.
The Time past Noon of the Center's Transit over the Meridian of
Greenwich.
For
Dublin
thus:
 
 
°
′
 
 
 
 
 
 
 
V C
=
58
22
R.
10.
 
 
 
 
 
VP
27
00
S.
9.
657047
 
 
 
 
PC
38
15
S.
9.
791756
 
 
 
 
Sum of
3. 123
37
Sum
19.
44880
 
 
 
 
Half Sum
61
48 ½
 
9.
945159
 
 
 
 
Diff. of
V C
3
26 ½
 
8.
778383
 
 
 
 
Double Radius
 
20.
 
 
 
 
 
The Sum
 
38.
723542
 
 
 
 
The Remainder
 
19.
274739
 
 
 
 
 
 
 
 
 
 
 
 
 
°
′
Half the Remainder
 
9.
637369
C S.
=
64
18
double
128.
36
from
180.
00
remains as before
51.
24.
To
51.
24.
Add
40.
44.
Sum 92. 08
=
6h
8½
Which deducted from 6h . 41′, leaves 32′ ½ for the Difference of the Angle at the Pole in Time. Say then 46⌊ 4: 3⌊ 64∷ 32′⌊ 5: 25′⌊ 4 which is the Difference in time of the Meridians of
Greenwich
and
Dublin.
For
Paris
thus:
 
 
°
′
 
 
 
 
 
 
 
VC
=
58
22
R.
10.
 
 
 
 
 
V P
=
27
00
S.
9.
657047
 
 
 
 
P C
=
41
48
S.
9.
823821
 
 
 
 
Sum of 3
127
10
 
19.
480868
 
 
 
 
Half Sum
63
35
 
9.
952105
 
 
 
 
Diff. of
VC
5
13
 
8.
958670
 
 
 
 
Double Radius
 
20.
 
 
 
 
 
Sum
 
38.
910775
 
 
 
 
Remainder
 
19▪
429907
 
 
 
 
 
 
 
 
 
 
 
 
 
°
′
Half Remainder
 
9.
714953
=
C S.
58.
45
double
117
30
As before, Remainder
62
30
Add, Primary Angle
40
44
Sum
103
11
In Time
6h
52½
From which deduct 6°. 41′. the Remainder 11′ ½ is the Difference of the Angle at the Pole in Time. Say then, 4⌊ 64: 3⌊ 64∷ 11
/2: 9′. which is the Difference in Time, of the Meridians of
Greenwich
and
Paris.
Corol.
(1.) The latter Branch of the Problem determines the Hour and Minute when the Centre of the Eclipse crosses the Meridian at any assign'd
Latitude;
and by a very small Allowance when the very middle of the Eclipse, or of Total Darkness happens in any Place very near the same.
Corollary
(2.) The same Branch determines the true
Difference of Meridians
in all Places over or near which the Center of the Eclipse passes, and so the Distance
East
and
West
from any known Meridian whatsoever. Thus because the Meridians of
Paris, Greenwich,
and
Dublin
do hence appear to be different as to the Angles of the Pole from the Primary Meridian, in the foregoing Angles, 62°. 28′. 59°. 31′. and 51°. 24′. respectively; It is plain, that the respective
Longitude
of
Paris
and
Greenwich,
when reduc'd from Angles of the Pole to the Difference of Meridians, is 2° 19′. and that of
Greenwich
and
Dublin
6°. 22′. and by Consequence of
Paris
and
Dublin
8°. 41′. Which is no other than the Foundation of my grand
Problem
of
Discovering the Geographical
Longitude
of Places by Solar Eclipses, from the
Latitude
given:
Of which more hereafter.
XV.
To find the Duration of a Solar Eclipse, along or near the Path of the Moon's Center, in any Place whatsoever.
From the Motion of the Moon from the Sun gain the Duration of the entire Eclipse; or the Time of that Center's Passage over the Diameter of the Penumbra, if there were no diurnal Motion during that Time, thus:
As the horary Motion of the Moon from the Sun, which is in angular Measure 35°. 18′. and is given in the Calculation; to an Hour or 60′ in Time∷ So let the Diameter of the Penumbra there given, also in angular Measure = 65°. 10′. be to a fourth Number: which will be the Number of Minutes requir'd. In Decimals thus: 35′⌊ 3:60′∷65′⌊ 17:110′⌊ 8=1h :50 ⅘. But then, because the diurnal Motion of the Earth must be compounded with this rectilinear Motion of the Moon; and that as we have already stated it, this Eclipse will cross the Meridian of
Greenwich
41′ past 6 a-Clock in the Evening; the one Part of the Duration of the Eclipse being before, and the other Part after that Time, both Intervals must be unequally affected by the diurnal Motion, and we must then take the former half-Duration 55′. 24″ distinctly. And since the diurnal Motion of
Greenwich
is in a larger Parallel than that of the Center of the general Eclipse; while its Obliquity to the Path of the Shadow's Center increases; their respective Motions will nearly keep the same Proportion all along; and so we may safely omit the Consideration of them both. We have also already discovered, that the Velocity of this Center's Motion is here, to that of the Velocity of the diurnal Motion, As 829⌊ 6 to 178⌊ 4. or as 4⌊ 64 to 1. And because the Eclipse begins about 14′ before 6, 14′ after 6 balances the same; and these 28 Minutes are almost all one, as if there were no diurnal Motion at all. So that we have only 27′ 24″ capable of Retardation in the first half: The middle Time of which will be about 30′ after 6. We must therefore look into the Table,
p.
22,—26 for the
Arc
97½, o
its Supplement 82 1/
, corresponding to 89¼ of the Sines. Where the Difference of half a Degree is instead of 19⌊ 1, as about Noon, no less than 160, which multiplied by 4⌊ 64 comes to 7424. So that the Motion is here retarded a 39th Part. Say then, As 39 to 38, so is 27⌊ 4 to 26⌊ 7 which 26⌊ 7 or 26′ 42″ is to be added to the 28 before excepted, for the Duration of the former Part of the Eclipse = 54′ 42″. The Middle of the second Part will be about 73 = 18°¼ backward beyond 90°. that is, 71° ¾ of the Arc, which corresponds to 85° ½ of the Sines, where the Difference of half a Degree is 61° 1/
. This multiplied by 4⌊ 64 gives 285. So that the Retardation is as 19⌊ 1 to 285, or 1 to 14⌊ 9. Say then, As 14⌊ 9 to 13⌊ 9: so is 55⌊ 4 to 51⌊ 7 = 51′ 42″, which is the Duration of the latter Part of the Eclipse, and added to the former Part of the Duration, gives us the whole Duration = 106′ 24″ = 1h 46′ 24″, without the Consideration of the Elevation of the Luminaries above the Horizon, which a small Matter enlarges that Duration: Of which hereafter.
N. B.
If we would be still more precisely nice, we may distinctly allow for the Difference of the Parallels, and the different Obliquity of the Direction; of which
p
26, 27. before: Which yet are here omitted, as very inconsiderable.
XVI.
To find the Duration of the Total Darkness along the Path of the Moon's Center; if the Luminaries were in the Horizon.
Say, As the Moon's Motion from the Sun in an Hour, or 60′; in the Calculation = 35′ 18″ is to those 60′: So is the Difference of the Diameters of the Sun and Moon in the same Calculation, = 1′ 38″ to a fourth Number: Which will be the entire Duration of the Total Darkness if the Luminaries were not at all elevated above the Horizon. In Decimals thus: 35′⌊ 3:60′∷1′⌊ 62:2′⌊ 73=2′46″⅔.
XVII.
To find the Altitude of the Sun above the Horizon, when the Center of the Shadow crosses the Meridian at
Greenwich.
In the Triangle
(Fig.
10.)
Z P S, Z P
is = Distance between the Zenith and Pole of the Earth: 38° 30′.
P S
is = Distance between the Pole of the Earth and Center of the Sun: = Complement of the Sun's Declination, or to 69° 28′. And the Angle
Z P S
= interval of Time, from the Meridian = 6h 41′ = 100° 15′, whose Supplement is 79° 45′.
Say then,
R.
 
 
10.
 
 
 
 
 
 
°
′
 
 
 
 
 
 
CS.Z P S
79
45
9.
250280
 
 
 
 
T.Z P
38
30
9.
900605
 
°
′
 
 
T. P R
=
9.
150885
=
8
3 =
P R
 
 
 
 
 
+
69
28 =
P S
 
 
 
 
 
=
77
31 =
R S
And,
 
°
′
 
 
 
 
 
 
 
CS. P R
8
3
9.
995699
 
 
 
 
 
CS. Z P
38
30
9.
893544
 
 
 
 
 
CS R S
77
31
9.
334766
 
 
 
 
 
 
 
 
19.
228310
 
°
′
 
 
 
CS. Z S
=
9.
232611
=
80
10
=
Z S
 
 
 
 
Ergo,
Altitude
=
9
50
 
 
XVIII.
To find the Azimuth of the Sun at the same Time and Place.
In the former Triangle the Complement of the Altitude being now found =
Z S
= 80° 10′. Proceed thus:
 
°
′
 
 
 
 
 
 
 
 
S. P S
80
10
 
9.
993572
 
 
 
 
 
S. Z PS
79
45
 
9.
993013
 
 
 
 
 
S. P S
69
28
 
9.
971493
 
 
 
 
 
 
 
 
 
19.
964506
 
°
′
 
 
S. PZS=
Suppl.
=
9.
970934
=
69
16
=
P Z
 
 
 
 
From the
West
20
44
 
 
XIX.
To find the Number of Minutes that any Solar Eclipse extends to, in a plain Perpendicular to the Axis of the Penumbra.
When the Parallax of the Moon is 57 ′ 17″, which is near its mean Quantity; the Moon's Distance is 60 Semidiameters of the Earth: And one Second on the Disk of the Earth as viewed at the Distance of the Moon; or on the Disk of the Moon viewed at the Distance of the Earth, is exactly one geographical Mile. And 60″ or one Minute, is one Degree upon the Disk of the Earth or Moon. But when the Moon's Parallax, as in the Calculation of this Eclipse, is 61′. 48″ whose Sine is 17976/1000000 or 1/55⌊ 5 of the Radius nearly: One Second is less than a Mile; and that in the Proportion of 55⌊ 5 to 60. So that about 65 geographical Miles correspond to 60″, or one Minute: And accordingly, 1781 such Miles will correspond to 32′ 35″, which in that Calculation, is the Semidiameter of the Penumbra, or the utmost perpendicular Extent of the Eclipse.
N. B.
We may always make use of the foregoing Tables,
pag.
29,—34 for the perpendicular Distances from the Path of the Moon's Centre, at all Durations of the Eclipses, and of Total Darkness, by still calling the Radius of every
Penumbra
2000 equal Parts or Miles, and the Radius of every
Umbra
or Total Shadow, 50 such Parts or Miles; and using them accordingly. And this without any other Inconvenience than the Supposition of other than geographical Miles: Which Inconvenience, by the Reduction of them to geographical Miles afterwards, will always, as here, come to nothing.
XX.
To find the Alteration there is in the Extent and Duration of Eclipses, and of total Darkness, on Account of the Elevation of the Luminaries at that Time above the Horizon.
This is ever, as the Radius: to the Sine of the Sun's Altitude: As compared with the particular Distance of the Moon at that Time. Thus at the Middle of this Eclipse at
Greenwich,
the Altitude of the Sun has been found to be 9°. 50′, whose Sine is 1708/10000, which divided by 55′⌊ 5 is equal to ½ of the whole: And in the Semidiameter of the
Penumbra,
as well as
Umbra,
amounts to about 6 Miles every Way. Thus also at the Middle of the Eclipse in
North-America,
where 'tis Central at Noon, the Sun will be about 52°▪ above the Horizon: whose Sine is 788/1000, which divided by 55⌊ 5 is 1/
0 nearly; which in the same Semidiameters, amounts to about 29 Miles every way. And for the Duration at
Greenwich
add to the common Duration already found, or to 1h 46′ 24″ the 1/324 Part of the same, or about ⅓ of a Minute both ways: which is about 2/
in the whole; which will bring the entire Duration from 1h 46′ 24″to full 1h 47′ and for Total Darkness will add about 6/50 of the whole total Darkness = 20″, and increase the Duration from 2′ 46⅔ to 3′ 6″
/3. But for the Alteration in
North America,
which is about 1/7
both ways, or 1/35 in the whole; this will there increase the entire Duration 4′ 12″, and from 1h 46′ 24″ bring it to 1h 50′ 36″, and for the total Darkness will add about 29/50 of the whole, or bring the 2′ 46″ ⅔ to 4′ 47″.
XXI.
To find the Species and Dimensions of the Ellipsis made by the Total Shadow at the same Places and Altitudes.
The shorter Axis of the Ellipsis, (which is the same with the Diameter of the Conick Shadow it self, in a Plane Perpendicular to the Axis;) is to the Longer, as the Sines before▪ mentioned. Or at
Greenwich,
as 1708 to 10000; and at
North America,
as 788 to 1000: And since that
horter Axis is in the Calculation 98 Minutes or Miles, of 65 to a Degree; they must be equal to 90 Geographical Miles; which by the last Problem must be increas'd to 96 and to 119 respectively. The golden Rule will therefore soon shew the Length of the longer Axis of the Shadow in both Places.
 
sine
rad.
miles
miles
For
Greenwich
1708:
10000∷
96:
562.
For
North-America
788:
1000∷
119:
151.
XXII.
To determine the central Point in the Elliptick Shadow for
Greenwich,
at the Middle of the same Eclipse.
B
Lemma
XIX. and in its Figure; As
V B:
to
V A∷
So is the Sine of
B A V
= 9° 34′ = 166: to the Sine of
A B V
= 10° 6′ = 175, and so is
B C:
to
A C.
And
componend
,
As 166 + 175 = 341 to 175∷ So is
B C + C A
= 562 to
C A,
which therefore is equal to 289. Also As 166 + 175 = 341 to 166∷ So is
B C + C A
= 562 to
B C;
which therefore is equal to 274.
For 341 : 175 ∷ 562 : 289.
And 341 : 166 ∷ 562 : 274.
Whence the Parts of the longer Axis being found, the larger
C A
= 289 Miles: the lesser
B C
= 274 Miles. The half of which,
D C
is the Distance between the central Point of the Shadow
D,
and the Center of the Ellipsis
C,
= 7½ Miles.
XXIII.
To determine the Angle of Direction of the Total Shadow over the Meridian of
Greenwich.
This to be done either by Construction, upon drawing the Curve Line of this Motion, through the several Points where, by Calculation, the Center of the Shadow crosses the Meridians of
Paris, Greenwich▪
and
Dublin;
and measuring the Angle it makes with the several Meridians; or more exactly for
Greenwich
by solving the Triangle
(Fig.
11.)
P V O,
where the Arc
V O
is equally distant
West,
as
Paris
is
East
from
Greenwich,
is to be found by the Method already made use of; thus:
Paris
Angle at the Pole = 62° 28′. Angle for
O V
at the Pole is less by twice the Difference of the Angles at the Pole for
Paris
and
Greenwich
= 5° 54′, And 62° 28′ − 5° 54′ = 56°: 34. First then find the Arc
V O
= Compl. Latitude for O, as in
Probl.
XIV. by this Analogy.
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
Q P C.
56.
34.
9.
741125
 
 
 
T.
P V.
27.
00.
9.
707166
 
°
′
 
T.
P R.
9.
448291
=
15
41
Then say,
°
′
 
 
 
 
 
CS.
P V.
27.
00.
9.
949880
 
 
 
CS.
P R.
15.
41.
9.
983523
 
 
 
CS.
V C.
58.
22.
9.
719730
 
 
 
 
 
 
19.
703253
 
°
′
CS.
P C
9.
753373
=
45.
29.
 
 
 
 
Deduct
15.
14.
 
 
 
 
Remt .
V O.
39.
48.
 
 
 
 
V G.
40.
46.
 
 
 
 
V P.
41.
48.
Then in the Triangle
V O
P whose included Angle is 4° 38′, and whose two Sides are 39° 48′ and 41° 48′. Find the Arc
P R
as usual thus:
R.
 
 
10.
 
 
 
 
 
°
′
 
 
 
 
 
CS.
O V. R.
04.
38.
9.
998577
 
 
 
T.
V O.
39.
48.
9.
920733
 
°
′
 
T.
V R.
9.
919310
=
39
48
 
 
 
 
from
41
42
 
 
 
 
Remnt 2° 6′ = PR.
Then for the Angle
VPO,
say:
 
°
′
 
 
 
 
 
S.
P R.
02.
06.
8.
563999
 
 
 
S.
V R.
39.
42.
9.
805343
 
 
 
T.
O V R.
04.
38.
8.
908719
 
 
 
 
 
 
18.
714062
 
°
′
T.
V P O.
=
10.
150063
=
54.
42.
And for
V O R;
 
°
′
 
 
 
 
 
S.
V O.
39.
42.
9.
805343
 
 
 
S.
V P O.
54.
42.
9.
911763
 
 
 
S.
VP.
41.
48.
9.
823821
 
 
 
 
 
 
19.
735584
 
°
′
S.
V O P =
9.
930241
=
58.
23
 
 
 
 
 
+
113
05
 
 
 
 
 
Half
56
32½
 
 
 
 
 
Compl.
33
27½
So that the Acute Angle
V G O,
which the Course of the Center of the Shadow makes with the Meridian of
Greenwich,
is 56° 32′. ½; And by consequence the Angle it makes with the Parallel of Latitude there is 33° 27′ ½. 5 or 6 Degrees mo
e than in Dr.
Halley
's Scheme.
Corollary.
The longer Axis of the total Shadow at the Meridian of
Greenwich,
makes an Angle of 12° 43′ ½ with the Direction of the Shadow.
for if from the Angle that the Directi
n of the Center of the Shadow makes with the Parallel of
Greenwich,
33
27½
We deduct the Angle that the longer Axis makes with the same Parallel, which is the Azimuth formerly discovered,
20
44
There remains this Angle
12
43½
XXIV.
To D termine the perpendicular Breadth of the Total Shadow, when that Shadow is nearest to
London;
or when the central Point is about the Meridian of
Padsto
in
Cornwall.
This
Problem
is, In an Ellipsis of 96 Miles broad, and 562 Miles long; whose longer▪ Axis makes an Angle with the Direction of the Shadow of 12° 43½; To determine the Length of the greatest Perpendicular that falls on that Diameter, along which the Direction of the Motion is. For twice that Perpendicular is the Breadth required. Now I have actually drawn this Ellipsis, or total Shadow, for my Map of this Eclipse. And I find by that Construction, that the Breadth of the total Shadow over
England
is about 155 Miles; or 30 Miles broader than in Dr.
Halley
's Scheme. Time will probably discover which is nearest the Truth.
N. B.
This total Shadow increases so greatly in Length as it goes
Eastward,
that it will reach from
Paris
to the utmost Boundary of the Sunsetting at once. And though the Center of the Shadow, by my Calculation, will end 8° ⅔
East
of the Meridian of
London,
in the Latitude of 45° 9′, or in the
Alps,
not far from
Milan
and
Turi
;
Yet by reason of the Extension and Breadth of the total Shadow, and the Refraction of the Rays of Light near the Horizon, it seems probable to me, that all
Switzerland
and
Lombardy,
as far as
Trent, Mantua. Cremona,
and
Parma;
nay, perhaps, as far as
Venice, Padua, Bononia, Ferrara, Ravenna
and
Florence;
may at the same Time be invelop'd in the Total Shadow; and that the Sun may set eclipsed at all or most of those Places at once.
XXV.
To determine the Digits eclipsed, and the Duration of the Eclipse, or of total Darkness, at any given Distance from the Path of the Moon's Center; and
vice versâ.
These Digits and Durations, if we do not consider the diurnal Motion, are immediately found to any given Distance in the Tables,
P.
29,—34. Thus, if the Digits be 4, or Duration 80′ 24″, the first Table gives us the Distance of 1350 Miles. And,
vice versâ
if the Distance be known to be 1350 Miles; the same Table gives us 4 Digits and 80′ 24″ Duration. And the like is true of the total Shadow, and the Distance or Duration in the second Table. Thus also we learn from the Map of this Eclipse, that the nearest Approach of the total Shadow to
London
will be about 40 geographical Miles: Which 40 Miles, in this Obliquity of the Motion, is about 31 perpendicular Miles. So that 32/16
= 12/6
of a Digit, is the Quantity of the Rim of Light that will be seen at
London,
when the Eclipse is greatest: And by consequence the Digits there eclipsed will be 11 48/60. Nor does it appear to me, that the total Darkness will come nearer to
London,
than 40 Miles; although Dr.
Halley
's Scheme brings it within 7 or 8 Miles. As for the Duration of the entire Eclipse here, it will be much the same as if it were Central: And has been already determined to 1h 47′. And for the total Darkness along the Path of the Center between
Exeter
and
Plimouth,
its Duration will be, as before stated, 3′ 6″½.
N. B.
As to the Alterations which will arise in more considerable Distances from the Path of the Moon's Center, proceed thus: First, find the Minutes of Duration corresponding to any given Distance in Miles,
e. g.
1350, thus; 2000: 1350∷ 32′⌊ 63: 22′⌊ 02. Or, As 2000 Miles, the Radius: to 1350 the Sine∷ So are the Number of Minutes for the Semidiameter of the Penumbra: to the Number of Minutes for that Sine. Now 32′ 19″ the present Length of the greatest Distance − 22′ 01″ = 10′ 18″. Then compute every Thing as if the Center of the Shadow were 10′ 18″, Latitude or nearest Distance; and as if the Chord of that Arc over the Earth's Disk were the Path of the Moon's Center: and the Chord of the Distance from the Center of the Penumbra were the Diameter of the Penumbra: And all will be discovered by the Rules before going; without any new Directions whatsoever. Only if this Distance be the contrary way to the present Example, we must here make use of Addition instead of Substraction.
N. B.
Upon confulting the
French
Ephemerides of
Des Places,
recommended, in some Degree, by
Cassini
himself, I find that he determines the Eclipse at
Paris
thus.
 
h
′
″
Beginning,
May
22.
(N. S.)
06
03
00
Middle
06
58
38
End
07
54
00
Whence the Duration there is
01
51
00
Digits eclipss'd
10
17/60
 
Latitude of the Moon at the Middle
39′
½
 
Difference from me in Digits
01
31/
 
In Latitude
07
10
If these Digits and this Latitude be right, Dr.
Halley,
and I, and those
English
Astronomers that have computed and constructed this E
lipse, who all, I think, do agree, that it will be to
at
Paris,
and that for two or three Miuutes also, are prodigiously mistaken. For if
Des Places
be in the right,
Copenhagen
and
Stockholm
will stand much fairer for the Pretence to a total Eclipse, than either
Dublin, London,
or
Paris.
Time will certainly determine who are in the right.
N. B.
If we add the Duration of the Eclipse, consider'd without Regard to the diurnal Motion, to the Time of the Center of the Shadows passing over the entire Disk, we gain the entire Duration of the Eclipse in general, thus:
 
h
′
″
To
1
50
48
Add
2
58
24
Sum
4
49
12
Therefore the general Eclipse by the Meridian of
London,
 
h
′
″
Begins
II.
52
24
Middle
V.
17
00
End
VII.
41
36
Duration
4h
49
12
Eclipse at
London,
Begins V.
45
00
Middle VI.
41
00
End VII.
32½
00
Duration 1h
47½
00
Digits eclipsed
11
48/60
Duration of Total Darknes
between
Exeter
and
Plimouth,
03
6½
N. B.
Dr.
Halley
's Times are still about 5′ sooner than mine, and his general Duration about 7′, and his Duration at
London
about 1′ longer.
N. B.
I cannot tell the Reason why my Original Calculation of this general Eclipse, which has been carefully made according to Sir
Isaac Newton
's famous
Theory of the Moon,
does here so much differ from Dr.
Halley
's Determination, as 5′ in Time; especially since both those Methods did very well agree, in the last celebrated total Eclipse,
Apr.
22. 1715. 'Tis Time alone that can determine between these two Methods of Calculation.
N. B.
I have lately been shewed an exact Scheme of this next Eclipse, according to Mr.
Flamsteed
's own Tables and Determination, and made in his Life-time; wherein the Digits eclipsed are 11 18/60 exactly according to my Determination in this Paper.
A PROPOSAL For the Discovery of the LONGITUDE of the several Places of the Earth, by Total Eclipses of the Sun.
IT is humbly proposed, That Observations be made in all Places where Solar Eclipses are seen, of the exact
Duration
of the same; by either viewing the Beginning and Ending thereof through a Telescope, with a Glass smoaked in the Flame of a Candle, for saving the Eye of the Observer; or else by casting the Sun's Image through such a Telescope upon white Paper, and viewing the first and last Impression of the Moon's Shadow upon it. And that the Hour, Minute, and Second of some Pendulum Clock be carefully noted at the same Time: And that when the same Observations are transmitted for the Uses of Geography, the Latitudes of the Places be also set down and transmitted at the same Time: That the like Observations be also made in all Places where such Eclipses are total and visible, of the exact Beginning and Ending, with the Duration of Total Darkness; by the like Comparison of a Pendulum Clock, or other pendulous Body that vibrates Seconds, or half Seconds; and, with the Latitude be transmitted in the same Manner, and for the same Uses. How by the Help of these
A Table of the
Latitude
of the Center of the total Shadow at the Eclipse,
May 11. 1724.
to every
10
Degrees of the Angle about the Pole, and every
7°⌊ 845. Longitude
from
London.
Angles at the Pole.
 
Longitude from
London.
 
Latitde .
°
′
 
° East.
 
°
′
70
19
 
8⌊ 577
 
45
9
69
23
 
7⌊ 845
 
45
34
 
 
 
West.
 
 
 
59
23
 
0000
 
49
17
49
23
 
7⌊ 845
 
52
19
39
23
 
15⌊ 690
 
54
42
29
23
 
23⌊ 535
 
56
29
19
23
 
31⌊ 380
 
57
43
09
23
 
39⌊ 225
 
58
13
00
00
 
46⌊ 600
 
58
38
00
37
 
47⌊ 070
 
58
38
10
37
 
54⌊ 915
 
58
21
20
37
 
62⌊ 760
 
57
36
30
37
 
70⌊ 605
 
56
18
40
37
 
78⌊ 450
 
54
27
50
37
 
86⌊ 295
 
51
59
56
22
Midle .
90⌊ 021
 
50
16
60
37
 
94⌊ 140
 
48
51
70
37
 
101⌊ 985
 
45
5
80
37
 
109⌊ 830
 
40
40
90
37
 
117⌊ 675
☉
35
56
100
37
 
125⌊ 520
 
30
31
110
37
 
133⌊ 365
 
25
15
120
37
 
141⌊ 210
 
20
11
130
37
 
149⌊ 055
 
15
37
136
59
 
154⌊ 052
 
13
3
N. B.
Here, as well as in the next Table,
The Angles at the Pole
differ by Ten Degrees; and the
Longitude from London
is found by the Proportion of 4⌊ 64 to 3⌊ 64. to be 7°⌊ 845 of Longitude from
London
for every such Ten Degrees. But if, as is generally most convenient, we would have the Differences of
Longitude from London
be even 10 Degrees; We must find the correspondent
Angles at the Pole
before we begin our Calculations by the inverse Proportion of 3⌊ 64: 4⌊ 64∷ 10°: 12′¾ the perpetual Addition of which Number will give us a Series for such a Calculation.
A Table of the Latitude of the Center of the total Shadow at the Eclipse,
Sept. 4. 1717.
to every
10
Degrees of the Angle at the Pole; and every
7°⌊ 167
Longitude
from
London.
Angles at the Pole.
 
Longitude from
London.
 
Latitde .
°
′
 
° West.
 
°
North
11
33
 
12⌊ 248
 
30
18
4
28
 
7⌊ 167
 
30
20
 
 
 
East.
 
 
 
00
00
 
3⌊ 963
 
30
14
5
32
 
0⌊ 000
 
29
28
15
32
 
7⌊ 167
 
27
56
25
32
 
14⌊ 334
 
25
40
35
32
 
21⌊ 501
 
22
38
45
32
 
28⌊ 668
 
18
27
55
32
 
35⌊ 835
 
14
24
65
32
 
43⌊ 002
 
9
21
75
32
 
50⌊ 169
 
5
4
77
58
Midle .
51⌊ 903
Midle .
3
57
85
32
 
57⌊ 336
 
0
16
 
 
 
 
 
South.
95
32
 
64⌊ 503
 
4
32
105
32
 
71⌊ 670
 
9
55
115
32
 
78⌊ 837
 
14
54
125
32
 
86⌊ 004
 
19
16
135
32
 
193⌊ 171
 
23
00
145
32
 
100⌊ 338
 
25
57
155
32
 
107⌊ 505
 
28
8
165
32
 
114⌊ 672
 
29
34
168
39
 
116⌊ 909
 
30
14
3⌊ 53: 2⌊ 53 ∷ 10: 7⌊ 167.
2⌊ 53: 3⌊ 53 ∷ 10: 13⌊ 952.
A Calculation of the Total Eclipse of the Sun,
A. D. 1724. May 11.
Post Meridiem.
 
 
M.'s mean Mo.
Motin of Apogee.
Motn of Node r tr.
 
 
s
°
′
″
s
°
′
″
s
°
′
″
Anno Dom.
1701
0
15
20
00
11
8
20
00
4
27
24
20
Years
20
4
13
34
05
3
3
50
15
 
26
50
15
and
3
 
28
9
10
4
1
59
32
1
27
59
9
Leap-Y.
May,
Days
11
9
29
17
03
 
14
42
21
 
6
59
4
Hours
5
 
2
44
42
 
 
1
24
 
 
 
40
Minutes
15
 
 
8
14
 
 
 
4
 
 
 
2
(Mean Time.)
 
 
3
1
49
30
Moon's mean Motion
1
29
13
14
6
28
53
36
1
25
34
50
Sun's mean Anomal.
10
22
19
57
An. Eq. add
12
2
An. Eq. subst.
5
43
Physic. Parts substract
7
7
9
38
6
29
5
38
25
29
7
2d Equat. substract
2
31
Mean Pla. correct
Mean Pla. correct
3d Equat. add
 
10
2
16
2
1
38
45
2
1
38
45
6th Equat. add
2
6
Sun's true Place
Sun's true Place
Diff. substr.
 
 
7
22
7
2
33
7
 
6
9
38
Moon's mean Pl correct
1
29
5▪
5
Annual A
g
ment
S's dist. from Node
Tru
Pl. of Apog. substr.
7
9
14
38
 
10
9
0
Equat. add
18
56
Moon's mean Anomal.
6
19
51
14
Equat. add
1
25
48
3
Equat. add
 
2
31.
7
7
9
14
38
True Pl. of Node
Moon's true Pl. in Orbit
2
1
36
59
True Pl. of Apogee
 
5
17
9
Sun▪
true Pla. substract
2
1
38
45
Eccentricity 6000
Inclinat. Limit.
Moon's dist. from Sun
11
29
58
14
 
′
″
M's h mo
.
37
42
Sun's
2
24
M's fro. Sun
35
18
35′ 18″: 60′∷ 1′ 46″: 3′ 2″ ▪
Eq. time add
3
53
Reduct. subst.
2
33
Diff. add
4
22
Ergo,
Ecl pse is 11d 5h 19 22″
Temp. appar. deduct the Error of the Tables about▪
2′
22″
true Tim.
5h
17 0
 
′
″
S mid. Sun
15
53
Sem Moon
16
42
Sem. Penbra
• 32
35
Semi. Difc.
61
38
Angle of the M's Way with the Ecliptick
5
36
Diff. Diamet r of the Sun & Moon, or Br
adth of total Shad.
0
98
Nodes true Pl. substract
1
25
48
3
Argument of Latitude
 
5
48
56
Reduction substract
 
 
1
30
Moon's tr. Pl. in Eclip.
2
1
35
29
Mo n's true
North
Lat.
 
 
32
14
At the Eclipse
 
 
32
19
A Calculation of the Sun's Place for the same Time.
 
Sun's mean Motion.
Motion of Perihelion.
 
s
°
′
″
s
°
′
″
A. D.
1701
9
20
43
40
3
7
44
30
Years
20
 
 
9
4
 
 
21
0
And
3
11
29
17
0
 
 
3
9
Leap-Y.
May,
Days
11
4
10
6
19
 
 
 
23
Hours
5
 
 
12
19
 
Minutes
15
 
 
 
37
3
8
9
2
Place of Perihelion.
2
90
28
59
Moon's mean Motion.
10
22
19
57
Sun's mean An
m.
(mean Time)
 
Sun's mean Motion
2
0
28
59
quation add
 
1
9
46
Sun's true Place
2
1
38
45
A Calculation of the Total Eclipse of the Sun,
A. D. 1727. Sept. 4. Mane.
 
M's mean Mot.
Motion Apogee.
Motn of Node retr.
 
s
°
′
s
°
′
s
°
′
Anno Dom.
1701
10
15
20
0
11
8
20
0
4
27
24
20
Years
20
4
13
34
5
3
3
50
15
 
26
50
15
And
6
2
9
28
55
8
4
5
44
3
26
1
29
Sept.
Days
3
 
1
23
35
 
27
24
23
 
13
1
37
Hours
20
 
10
58
49
 
 
5
34
 
 
2
39
Minutes
35
 
19
13
 
 
 
10
 
 
 
5
(Mean Time.)
 
 
5
5
56
5
Moon's mean Motion
5
21
4
37
11
13
46
6
11
21
28
15
Sun's mean An ma
y
2
15
32
16
An. Eq. sub.
18
59
An. Eq. add
9
7
Physical Parts add
 
 
11
23
11
13
27
7
11
21
37
22
2d Equat. substract
 
 
0
38
Mean Pl. Apo. cor.
Mean Pl. of No.
01
6th Equat. add
 
 
1
57
5
21
52
53
5
21
52
53
Difference add
 
 
12
42
Sun's true Place
Sun's true Place
M
on's mean Pl. correct
5
21
17
19
6
8
25
4
6
0
15
31
True Pl. of Apog. subst.
11
16
23
44
Annual Argum.
S.'s dist.
ro. Nod.
Moon's mean Anomaly
6
4
53
35
Eq
. Ad
2
56
37
Eq. Ad.
0
48
Equation add
 
 
42
26
11
16
23
44
11
21
38
10
M.'s Correct. Pl. in Orb.
5
21
59
45
True Pl. of Apo.
True Pl. of Node
Sun's true Pla. substract
5
21
52
53
Eccentr.
663
37
 
5
17
20
Moon's Dist. from Sun
 
 
6
52
 
Inc
in. Limit.
Variation add
 
 
 
9
S'.s hor. mot.
2
26
Moon's
38
29
M. fr. S.
36
2
36⌊ 04: 60 ∷ 6⌊ 9: 11 30.
Ergo,
Eclipse will be 20h 23 34
Mean Time.
Eq. tim. 2d.
4
38
Reduct. substr.
9
Diff. add
4
29
20h
28
03
Appar. Time.
Semid. S.
16
1
Semid. M.
16
50
Diff. =
00
49
Sum =
32
52
Parallax M. Horizon
62
7
Sun
 
10
Semid. D
k
61
57
Angle of the M.'s visible Way with the Ecliptick
5 36
20
M.'s true Pl. in Orbit
5
21
59
54
Nodes tr. Pl. substract
11
21
38
10
Argum
nt of Latitude
6
0
21
44
Reduction substract
 
 
 
6
Moon's true Pl. in Eclip.
5
21
51
48
Moon's true
South
Latit.
 
 
1
54
At the Eclipse
 
 
1
23
A Calculation of the Sun's Place for the same Time.
 
Sun's mean Motion.
Motion of Perihel.
 
s
°
′
s
°
′
″
A. D.
1701
9
20
43
40
3
7
44
30
Years
20
 
 
9
4
 
 
21
0
And
6
11
21
33
8
 
 
6
18
Sept.
Days
3
8
2
28
9
 
 
 
40
Hours
20
 
 
49
17
3
8
12
28
Place of Perihel.
5
23
44
44
Sun's mean Motion
2
15
32
16
Sun's mean Anomaly.
Minutes
35
 
 
1
26
(Mean Time.)
 
 
Sun's mean Motion
 
5
23
44
44
Equat. substract
 
 
1
51
51
Sun's true Place
 
5
21
52
53
Some Account of Observations lately made with Dipping-Needles, in Order to discover the LONGITUDE and LATITUDE at Sea.
UPON the Receipt of the liberal Assistance of His most Excellent Majesty, King
GEORGE;
their Royal Highnesses the Prince and Princess of
Wales,
and many other of the Nobility and Gentry, my kind Friends, I sent last Year Four several Dipping-Needles to Sea; with Frames hung near the Center of Motion in Gimbols, to avoid the Shaking of the Ship; and with proper Instructions to the Masters of the Vessels: And this, in order to discover the State of Magnetism in the several Parts of the Globe; and to find whether accurate Observations could be made at Sea, and to determine whether the fundamental Theory I laid down from former Observations would hold or not;
viz.
"That Magnetick Variation and Dip are all deriv'd from one Spherical Magnet in the Center of our Earth; with an irregular Alteration of the Variation, according to the different Degrees of Strength of the several Parts of the Loadstone, as compounded with a very slow Revolution from
East
to
West:
And with a regular Alteration of the Dip, nearly according to the Line of Sines, from the Magnetick Pole to the Magnetick Equator; the Axis of that Equator being sufficiently Oblique to its Plane: All which is the Case of Spherical Loadstones here."
Now having already received Four Journals from Four several Masters employ'd, I take this Occasion of returning my Benefactors hearty Thanks for their Assistance, and of giving them and the Publick some Account of the Success of these Observations; and what Consequences are naturally to be drawn from them; with the Difficulty hitherto met with in the Practice at Sea, and the proper Remedy for the same in future Trials.
Captain
James Jolly
set out in
July,
1722. for
Archangel,
with one of my Dipping-Needles on Board. He, for some time, met with such Difficulties in the Practice, as confin'd to the Frame I had given him, that he was not at first able to make any good Observations at all. But after some Time, he took the Needle into his own Cabin; and without any Approach to the Center of Motion, or any Contrivance. for avoiding the Shaking of the Ship at all, having a clear and full Gale all along, but without any stormy Weather, He made me 28 very good Horizontal Observations, from the Latitude of 65 quite to
Archangel:
I say,
Horizontal Observations
only, as I desired him; the Needle, by an Accident before he went, being rendred incapable of making any other with sufficient Accuracy. In this Space the Needle altered its Velocity very greatly, as I expected it would: And 5 Vibrations which at first were perform'd in about 280″; beyond the
North-Cape
came to 250″; till towards
Archangel
it gradually returned to about 177″.
Captain
Othniel Beal
set out about the same Time for
Boston
in
New-England,
with the same Instrument, and made Four Observations of the Dip, both by the Vertical and Horizontal Vibrations, and by the Dip it self; Three upon the open Sea, and One in the Haven of
Boston:
Which in some small Manner differed one from another, but in the main agreed, and kept the due Analogy I expected. He greatly complained of the Shaking of the Ship; till in
Boston
Haven he made a nice Observation both Ways, which did not greatly differ: Tho' the greatest Part of of his Observations by the Dip it self were somewhat more agreeable to Analogy than the other. The Reason was, I take it, that, as he assured me, he always took great Care to avoid the Shaking of my Frame; which Frame tho' it very much avoided the slower and greater Oscillation of the Ship, yet made a quicker but lesser Oscillation it self: Which Fault I was sufficiently sensible of just before the Ships were going away, but was not able then to obviate; as I am prepared to do hereafter. After Captain
Beal
had made and sent me these Observations, he pursued his Voyage to
Barbados,
and thence to
Charles Town
in
South Carolina;
at both which Places he made Observations; but the best at
Barbados.
For before he came to
Carolina,
he observed the Axis of the Needle to shake; which made him take the Dip there otherwise than he ought to have done; which is the natural Occasion that the Dip there did not so well agree to Analogy as the rest. However, upon my Receipt of his first Journal, with the Four first Observations, especially the exact one at
Boston,
I formed a more exact Theory of the Proportion of the Alteration of the Dip in the Spherical Magnet of the Earth; and found it at this Distance of the Earth's Surface, not far from that in my Spherical Loadstone, at the Distance of about 9/10 of an Inch from its Surface;
viz.
Not exactly as the Line of Sines, where at the Middle of the Line the Angles are 60 and 30; but rather as 66 to 24. Which Rule therefore is what I now propose as much nearer than the other. By which Proportion I determined long before-hand the Dip at
Barbados
of 43° or 44°, as many of my Friends can witness: And when Captain
Beal
delivered me the Paper of this Observation at
Barbados,
before I opened it, or in the least knew what Dip it contain'd, I foretold to him from that Theory the very same Dip, which both himself and his Paper immediately assur'd me to be true; and whose Truth, as he inform'd me afterwards, was confirm'd by another Observation, made a little before in the open Sea, of about 45°.
Captain
Tempest
also, about the same Time, set out for
Antegoa
and
St. Christopher
's, with the same Instrument and Frame. In his Letter, da
ed last
January,
he greatly complains of the Shaking of my Frame; and proposes an Hint how it might be avoided: Which Method of its Avoidance I had long before thought of, and provided for accordingly; and which has been a full Year ready for Practice. Those Observations of his, that I have yet received; for I have not heard from him since
January,
but hope soon to here farther; were but Three, and all at open Sea; and but one of them made both the Ways that I desired: And, indeed, seem the least agreeable to Analogy of any of the rest. Only since that single Observation, which was also made by the horizontal Vibrations and vertical Oscillations, agrees very well to that Analogy; since they all three are about the same Quantity of 8 or 9 Degrees exceed that Analogy; and since very near the same Place, where the third Observation was made, I have a double Observation of Captain
Beal
's to correct the same; I rather conclude, that Captain
Tempest
made a Mistake, and placed the wrong Edge of the Needle upward in all the Three Observations: Which would naturally occasion such a Difference. When I receive the rest of his Observations, or his Needle again, I shall be able to judge better of that Matter. However, even these Observations agree in gross with all the rest, to the gradual Decrease of the Dip as you go nearer to the Equator: Tho' as they stand at present, they do not determine the accurate Proportion of that Alteration so well as the others.
Captain
Michel
also, long after the rest, set out for
Hamburgh
with the same Instrument; though now without the Frame, which he was not willing to incumber himself with: and I suspected that in its present Contrivance it did more hurt than help the Nicety of the Experiments. I also by him, sent a Letter to the Reverend Mr.
Eberhard,
who was the Occasion of my studying this Matter, and was then Pastor of
Altena,
close by
Hamburgh;
desiring that he would there make the Experiment very exactly, and give me a particular Account of it. But I have not yet received his Answer.
Now the Observations here mentioned, as well as those many others I had by me before, do seem to me in general evidently to afford us the following Inferences:
(1.) That there is one Spherical Loadstone, and but one in the Center of our Earth; and that this Loadstone, like other Spherical Loadstones, has but one
Northern
Pole: Contrary to Dr.
Halley
's Hypothesis.
(2) That this
Northern
Pole is situated, contrary to the same Hypothesis also, a great Way to the
East
of our Meridian: And indeed, as I before had determined, about the Middle of the Distance between the
North Cape
and
Nova Zembla.
Captain
Jolly
's numerous Observations prove this most fully: While in Sailing towards that Point his horizontal Vibrations greatly increas'd in Number: And when he turned almost at right Angles, as he went down to
Archangel,
they soon diminished; and yet so little, after some time that it was evident he then sailed not far from a Parallel to that
Northern
Pole; and not very many Degrees from it neither; exactly according to my Expectations.
(3.) That the absolute Power of the internal Magnet is considerably different in different Places; and that without any certain Rule; as it is upon the Surface of our
Terrellae
or Spherical Loadstones here. This the various Number of Seconds to a vertical Oscillations, and all the Accounts in the other Observations fully prove; and by consequence this must cause different Variations in different Places, as is the Case of our
Terrellae.
(4.) That there no where appears in open Seas any such Irregularity in the Dip, as we sometimes meet with near Shores, or at Land; and by consequence that Dr.
Halley
's grand Objection against the Discovery of the Longitude by the Dipping-Needle, taken from an Observation of his own, concerning such an Irregularity near the Shore at
Cape Verd;
and from his own Hypothesis of the four Magnetick Poles is utterly groundless. Nor indeed shall I
e at Rest, till▪ I have sent a Dipping-Needle to
Hudson's Bay,
on purpose to determine this Dispute about the four Poles: For that Voyage being almost directly towards his second
Northern
Pole all the way, and about the same Distance all the way from mine; if this Voyage afford much the same Dip, it will demonstrate that there is but One
Northern
Pole; and that it is nearl
where I place it: But if that Dip greatly
increase,
it will demonstrate a s
ond Pole somewhere in those Parts of
America,
where Dr.
Halley
places it. And to this Decretory Experiment do I appeal for a final Determination of this Question. The Doctor seems to me to draw his Inferences from the
Variation,
which no Way proves any such double Poles; as being full as sensible on our
Terrelle,
which have no more than single ones; while he avoids all Observations from the
Dip,
which are still against him; and which are alone capable of discovering the exact Place of such Poles, either upon the Surface of the Earth, or of
Terrellae.
However, when one Set of Experiments with a Dipping-Needle, sent to
Hudson's Bay,
will certainly determine this Matter, 'tis a vain Thing to go on in the Way of Controversy about it.
In short, The Observations hitherto made, shew that the Foundations I go upon in this Discovery of the
Longitude
and the
Latitude
at Sea, are true and right: That the Terrestrial Magnetism is very regular and uniform, in the open Seas; that the Latitude in the
Northern
Parts may even, without any Avoidance of the Shaking of the Ship, in ordinary calm Weather, be in good Degree thereby discovered already; and that if I can sufficiently avoid the Shaking of the Ship, which I am now endeavouring, and have great Hopes of performing, both
Latitude
and
Longitude
may by this Method be discovered in the greatest Part of the sailing World. I say nothing here of another Method of Trial, which I am also pursuing, and which depends, like this, on the avoiding the main Part of the Ship's Agitation; and if effected will be more easy and universal than this. But as to giving any farther Account of that to the Publick, unless it succeed, I have no Intention at all.
N. B.
The original Journals are all in the Hands of my great Friend and Patron
Samuel Molyneux,
Esq
Secretary to his Royal Highness, the Prince of
Wales,
and Fellow of the Royal Society: Which Journals, when I have compleated the rest of the Observations I hope to procure, I intend to publish entire, for the more full Satisfaction of the curious,
A Table of the Angle of Inclination below the Horizon, in
Dipping-Needles,
to every
1/9
Part of their respective equal Distances from the Magnetick Poles and Equator.
Dist. from the Pole.
Dip.
Dist. from the Equat
Dip.
°
°
°
°
°
°
1
89
30
1
08
41
2
89
00
2
12
23
3
88
27
3
15
14
4
87
59
4
17
41
5
87
29
5
19
51
6
86
59
6
21
50
7
86
38
7
23
41
8
85
58
8
25
24
9
85
27
9
27
2
10
84
57
10
28
36
11
84
27
11
30
6
12
83
56
12
31
32
13
83
26
13
32
55
Dist. from the Pole.
Dip.
Dist. from the Pole.
Dip.
°
°
°
°
°
°
14
82
55
14
34
15
15
82
24
15
35
33
16
81
54
16
36
50
17
81
23
17
38
4
18
80
52
18
39
16
19
80
21
19
40
28
20
79
49
20
41
37
21
79
18
21
42
45
22
78
47
22
43
54
23
78
16
23
44
58
24
77
44
24
46
2
25
77
12
25
47
6
26
76
41
26
48
9
27
76
8
27
49
10
28
75
36
28
50
12
29
75
4
29
51
12
30
74
32
30
52
12
31
73
59
31
53
11
32
73
26
32
54
9
33
72
54
33
55
6
34
72
20
34
56
1
35
71
47
35
57
0
36
71
14
36
57
56
37
70
39
37
58
52
38
70
5
38
59
47
39
69
31
39
60
41
40
68
57
40
61
35
41
68
22
41
62
49
42
67
47
42
63
22
43
67
12
43
64
15
44
66
36
44
65
8
45
66
00
45
66
0
N. B.
I take the
Northern
Pole of the Terrestrial Magnet to be about the Meridian of
Archangel,
in the Latitude of 75½. Its Equator to be nearly a great Circle, intersecting the Earth's Equator about 2½ Degrees
Eastward
of the Meridian of
London;
and in its opposite Point. And that its utmost Latitude
Northward
is in the Gulph of
Bengall
about 12½ Degrees; and as much
South
in the opposite Point, in the great
South Sea.
And that the
Souther
Pole is nearly circular; its Radius 40 Degrees of a great Circle, and its Center in a Meridian
Eastward
from
Ceilon
about 4½ Degrees, and about 68½ Latitude.
N. B. London
is nearly 1
⌊ 5/84 = 2⌊ 6/90 distance from the
North
Pole of the Magnet, whence its Dip will be at 74
/
, which is certainly so in Fact.
Boston
in
New-England
is 51/
=
⌊
/9
, whence its Dip will be about 68° 22′, which Captain
Beal
found to be so in Fact.
Barbados
is about 26⌊
/1
= 22/90⌊ 5 distant from the Equator of the Magnet, whence its Dip ought to be about 44° ½, as Captain
Beal
also found it to be in Fact.
St. Helena
is about 14/49 =
⌊
/9
, whence its Dip ought to be about 47° 50′ as Dr.
Halley
found it to be in Fact. And so every where in the main Ocean, at considerable Distances from the Shores.
N. B.
If the Dip of any Needles be somewhat different at
London,
add or substract a proportionable Part of the Dip elsewhere. And you will have nearly the true Dip at any other Place with that Needle Thus if your Needle differ from the other 2° or 120′, and shew the Dip at
London
72° 45′ instead of 74° 45′, which is its-proper Dip in this Table; and you require the true Dip by this Needle for
Boston
in
New-England, Southward;
which in the Table is 68° 22′, proceed thus. Because the equal Distance of
Boston
from the Magnetick Equator is 49 Parts of 60⌊ 4, the like Distance of
London
from that Equator; deduct
9/60⌊ 4 120′ = 97′ = 1° 37′ out of the Tabular Dip 68° 22′. The Remainder is 66° 45′, for the true Dip at
Boston
with that Needle. Thus if you want the true Dip, by the same Needle, at
Dronthem
in
Norway, Northward:
Because the equal Distance of
Dronthem
from the Magnetick Pole is 15⌊ 2 Parts of 29⌊ 6 the Distance of
London
from that Pole; deduct 15/29⌊ 2/6 12′ = 62′ = 1° 2′ out of the Tabular Dip 82° 30′, and the Remainder, 81° 28′ is the true Dip at
Dronthem,
with that Needle: And so in all other Cases whatsoever.
N. B.
The Table before set down, supposes that the true Dip differs according to such a Line of Sines, whose middle Point gives 66° on one Side, and 24 on the other; and is made by adding or substracting 8 to the Complement of the Dip found by the natural Sines for every 1/
0 of equal Distances from the Equator or Pole.
N. B.
If any desire to calculate by Trigonometry the Distances of all Places from the magnetick Equator of Poles, and the Distances of that Equator and those Poles in every particular Case, both made use of in the foregoing Calculations, it is thus to be done:
In the
(Fig.
12.) Triangle
B L A
we have
B L
the Co-Latitude of
London; B A
the Co-Latitude of the magnetick
North
Pole; and the included Angle,
A B L
= the Distance of the Meridian of that Pole from the Meridian of
London;
to find the Angle
Q A M
and the Side
A L.
Then in the Triangle
Q A M,
we have the Angles
Q A M
and
Q M A,
and the Side
A M,
= the Distance of the Magnetick Pole from the Magnetick Equator, to find
A Q.
So we have the Proportion of
A L
to
AQ, Q. E. I.
But since the
Data
are not yet sufficiently exact for the Calculation, measuring is sufficient.
FINIS.
to fold out at the end of ye Book
Fig.
1.
Fig.
2.
Fig.
3.
Fig.
4.
Fig.
5.
Fig.
7.
Fig.
6.
Fig.
9.
Fig.
8.
Fig.
10.
Fig.
11.
Fig.
12.
ERRATA.
PAge
1.
lines
12, 13.
read,
5 Leap. Days; and with 11 Days when 4 Leap Days.
P.
7.
over against
21, &c.
read,
53. 2. 55. 40. 58. 19. 60. 58. 63. 38. 66. 19. 69. 0. 71. 42. 74. 24. 77. 7. 79. 50. 82. 34. 85. 19. 88. 4. 90. 50.
P.
10.
l.
32.
read,
Summer.
P.
13.
l.
7.
r.
10d .
l.
8.
and
10,
and
16.
r.
18.
P.
13.
l.
24.
r. Bern, Zurich,
and
Pillaw
near
Koningsberg
in
Prussia. Dele p.
16.
l.
32.
to p.
17.
l.
10.
and instead of it read thus:
Its greatest Alteration therefore must be at the mean Distance; and is the Difference of the Equation belonging there to the Addition of 10°⌊ 8 = 17. which Space the Moon usually goes in about 36 of time. So that the Difference on this Account must, each Period, be usually less than 36′. And as to the Moon's own Motion, it has also its greatest Alteration at its mean Distance; and is the Difference of the Equation at 2°. 51. ½=17. which the Moon usually goes in about 36′. of time. So that the Difference on this Account must, each Period, be usually less than 36▪ . also, and on both Accounts less than 1h . 12′.