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THE CALCULATION OF Solar Eclipſes WITHOUT PARALLAXES.

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

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

The Sun's True Place and Anomaly
	s	°	′	″	s	°	′	″
Anno Dom. 1701	9	20	43	50	3	7	40	10
Years—14	11	29	36	46			11	40
April, days—21	3	19	24	24				15
hours—21			51	45	3	7	52	5
minutes—36			1	29	Place of the Perihel.
Sun's Mean Motion	1	10	38	14	1	10	38	14
Equation Added		1	36	27	☉s Mean Motion
Sun's True Place	1	12	14	4 [...]	10	2	46	9
					☉s Mean Anomaly

The General Eclipſe by the Calculation			From D^r. Halley
	h	′	h	′
Beginning—	7.	30	7	21
Middle—	9.	51	9	42
End—	12.	12	12	3

The Eclipſe at London from the Calculation.			From D^r. Halley
	h	′	h	′
Beginning—	8.	18	8	7
Middle—	9.	24	9	13
End—	10.	35	10	24

The Conſtruction Explaind

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

The Breadth of the intire Penumbra or paitial Eclipse upon this Perpendicular Plain, appears by the Conſtruction to be no leſs than 1965 minutes or Geographical Mibes on each side of the Moons Path, or 3930 Miles in all; w^ch. correſpond to many more on the Spherical Surface of the Earth: Nor is it all confind, as you may see here, to that Surface, but reaches▪ off a greatway into the empty Space beyond it Northward. The Lines which diſtinguiſh that breadth on each side into 12 parts denote so many Digits of the Suns Eclipseſ besides ⅓ for the Total shade) & the places both as to Long▪ 8▪ Lat: where the ſun will at any Time be so much Eclipsed: And indeed I would willingly have procured a general Map here to have▪ shewd over what Countries and Places the intire Shadow would paſs, as Doctor Halley has given us a particular Map of England for the Paſsage of the Total Shadow over it. But the nature of the Conſtruction does not admit of that Projec⯑tion (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 Projecti⯑ons 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 ex⯑actly coincident. But if we go by a Construction according to our Calculation the Digits Eclipsed at London will be hardly more than 11 ⅘ and the Shadow▪ will go full 30 Miles more Northward than in D^r. Halleys Map.

So that y^e Middle of the General Eclipse in com⯑mon or apparent Time▪ will be 50. 56. after Nine in the Morning▪ differing from D^r. Hal⯑ley's Computation near 9 min. But Note that the Conſtruction is ac⯑commodated to the D^rs. Calculation.

Note also that hence the breadth of the Shad⯑ow of Total Darkneſs will be 98 Geographi⯑cal Miles; and that its length on the Oblique Horizon of London will be near 150 Miles, as D^r. Halley's Deſcription aſserts.

But it must be here Observed that if in this Calculation the that and 6^th New Equations of the Moon, taken from S. Isaac New⯑ton's Theory, were neglected, this Calculation would be much near⯑er to D^r. Halley's, as it is now nearer to M^r. Flamſteed's. This Eclipſe, if the Air prove clear for exact Observations, will go agreat [...]ay to determin how far those Equations are juſt; and how far they are neceſsary in the Calculation of the New and Full Moons, and of Eclipses, that happen only at those Times. S^t. Isaac Newton's third Equation, w^ch is no more than 13 to be Substr [...]ted, is here omitted, as very inconsiderable.

	Moon's mean Motion				Motion of the Apogee				Motion of y^e Nodr Retr.
	s	°	′	″	s	°	′	″	s	°	′	″
Anno Dom.1701	10	15	19	50	11	8	18	20	4	27	24	20
Years—14	1	20	54	34	6	29	37	50	9	0	45	35
April▪ days—21		22	34	47		12	21	59		5	52	41
hours—21		11	31	46			5	51			2	47
minuted—36			1 [...]	46				10				5
(Mean Time)									9	6	41	8
Moon's mean Motion	1	10	40	43	6	20	24	10	7	20	43	12
Suns mean Anomaly	10	2	40	9	An: Eq: Add.		16	8	An Eq: subst.		7▪	41
Phyſical parts Substracted			9	50	6	20	40	18	7	20	35	31
[...] 6^th Equation Substra [...]			3	45	Mean Place Corrected				Mean Place Corrected
	Sum		13	35	1	12	14	41	1	12	14	41
Moon's mean Place correcte [...]	1	10	27	8	Suns True Place				Suns True Place
True Place of the Apogee [...]ulot	6	27	5 [...]	5	6	21	34	23	5	21	39	10
Moon's mean Anomaly	6	12	31	3	Annual Argument				☉ [...] Diſtance from the Node
Equation Added		1	42	35		7	15	47	Equat: Subst.		25	19
Moon's Equat Place in its Orb	1	12	9	43	Equation Added				7	20	10	12
Sun's True Place	1	12	14	41	6	27	56	5	True Place of the Node
Moon's Diſtance from y^e. ſun	11	29	55	2	True Place of the Apogee					5	16	56
Variation ſubſtracted				2	Eccentricity		63	68	Inclination of y^e Limit
Moon's True Place in it [...] Orbit.	1	12	9	41	☽ horary Mot.		38	0	Semidiam. ☉.		15	58
Node's True Place ſubſtract	7	20	10	12	☉ [...].		2	25	Semidiam. ☽.		16	47
Argument of Latitude	5	21	59	29	☽ from ☉		35	35	Semid. Penum.		32	45
Reduction Added			2	0	35⌊6′ 60′115:		8	22	Semid. Diſk		61	30
Moon's True Place in the Eclip [...]	1	12	11	41	Eq. Time Add.		3	25	Ang [...] of y^e ☽ way with y^e Ecliptic		5°	35′
Moon's True North Latitude			44	10	Reduct Add.		3	9	Diff [...] of that ☉ & ☽ ^s Diam 98″or 98 Miles.
					Sum of 3 Add.		14	56	Diff [...] of that ☉ & ☽ ^s Diam 98″or 98 Miles.

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

The Copernicus, or Universal Aſtronomical Inſtrument, being now finiſh'd, is sold by y^e Author M^r. Whiſton, at his Houſe in Croſs street Hatton Garden

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THE CALCULATION OF Solar Eclipſes WITHOUT PARALLAXES. WITH A SPECIMEN of the ſame in the Total Eclipſe of the Sun, May 11. 1724. 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.

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I.

[...]HE moſt uſeful and moſt remarkable Cycle or Period for the Revolution of Eclipſes, both Solar and Lunar, men⯑tioned 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 Moti⯑on 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 [2] Conjunctions, Oppoſitions, and Eclipſes properly depend, exceeds the like entire Revolutions Which Coincidences do therefore nearly reſtore the mean State of the Moon it ſelf, its Apogee▪ Nodes, and Lunations: And produce an eminent Revolution of correſpondent New Moons, [...]ull Moons, and Eclipſes, after that Interval perpe⯑ [...]lly.

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

Mean Motion of the Sun or Earth.
	s	°	′	″
Years 18	11	29	28	32
Days 11	0	10	50	32
Hours 7	0	0	17	15
′ 43	0	0	1	46
″15	0	0	0	1
Sum	0	10	48	6

Mean Motion of the Moon in the Ecliptick.
	s	°	′	″
Years 18	7	11	37	22
Days 11	4	24	56	25
Hours 7	0	03	50	35
′ 43	0	00	23	36
″ 15	00	00	00	08
Sum	00	10	48	06

[3]

Mean Motion of the Apogee.
	s	°	′	″
Years 18	0	12	23	53
Days 11	0	1	13	32
Hours 7	0	0	1	57
′ 43¼	0	0	0	12
Sum	0	13	39	34

Mean Motion of the Node backward.
	s	°	′	″
Years 18	11	18	7	39
Days 11	0	0	34	57
Hours 7	0	0	0	56
′ 43 ¼	0	0	0	6
Sum	11	18	43	38

Mean Motion of the Moon from the Sun.
	s	°	′	″
Years 18	7	11	58	50
Days 11	4	14	5	53
Hours 7	0	3	33	20
′ 43	0	0	21	50
″ 15	0	0	0	7
Sum	0	0	0	0

[4]

	°	′	″
From the mean Motion of the Apogee	13	39	34
Subſtract that of the Moon in the Eclipt.	10	48	6
Remains—(2°⌊9)	2	51	28

To the mean Motion of the Moon	0	10	48	6
Add that of the Node	11	18	43	38
Sum	11	29	31	44
Difference from 12 Signs (0°⌊47)	0	0	28	16

Whence it appears that the Difference of the mean Motions of the Apogee and of the Nodes from that of the Moon her ſelf in the Ecliptick, in ſuch a Period, is but ſmall: Not more in the former Caſe than 2° 51′28″ = 2°⌊9, nor in the latter than 28′ 16″ = 0°⌊47. Whence alſo it ap⯑pears 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 Pe⯑riod; 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 E⯑clipſes rightly; and ſince the other Anomalies are but ſmall, and always come right again in Length of Time, they cannot ever produce any [5] very great Anomalies in our Calculations from them. As will farther appear under the follow⯑ing Scholia.

N. B. This Period for Eclipſes, has of late been called, both by Mr. Flamſteed, and Dr. Hal⯑ley, 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 Teſta⯑ment, p. 203,—213.

SCHOLIA.

(1.) We may here Obſerve, that ſince the Li⯑mit 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 E⯑clipſes, by 28′ 16″ = 28′⌊3, which is the Dif⯑ference 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 Cy⯑cles,, after a Central Lunar Eclipſe in one [6] of the Nodes, before the Moon goes off the Shadow of the Earth entirely at the ſame Node, and 450 Years (25 (multiplier) 18 = 450,) or double that Number 900 Years for the Time that the Moon begins to enter the Ecliptick Limit on one Side, till it goes out of it on the other. During which long Interval there will ſtill be Eclipſes of the Moon each Period. And if we divide 1000′ the Limit of the Sun's Eclipſes, by the ſame Number 28′⌊3, we ſhall have nearly 35 for the Number of Periods after a Solar Central Eclipſe at the Middle of the Earth, in one of the Nodes, before the Penumbra goes off the North or South Parts of the Diſk of the Earth entirely at the ſame Time; i. e. 630 Years. (35 (multiplier) 18 = 630,) or double that Number 1260 Years, from the Time that the Moon in any ſuch Period begins to enter the Ecliptick Limit on one Side, till it goes out of on the other: Du⯑ring which longer Interval there will ſtill be ſomewhere Eclipſes of the Sun each Period. Af⯑ter which reſpective long Intervals of Time there will be no ſuch Eclipſes for much longer Inter⯑vals.

(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, [7] and as gradually decreaſing perpetually: As in the following Table.

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

Latitudes for 25 Cycles in Lunar, and 35 in Solar Eclipſes.
Cycles	′	″
1	2	26
2	4	52
3	7	19
4	9	46
5	12	14
6	14	43
7	17	12
8	19	42
9	22	12
10	24	43
11	27	14
12	29	46
13	32	19
14	34	52
15	37	26
16	40	1
17	42	36
18	45	12
19	47	48
20	50	25
21	53	3
22	55	41
23	59	20
24	60	59
25	63	39
26	66	0
27	68	41
28	71	23
29	74	6
30	76	49
31	79	33
32	82	17
33	85	2
34	87	48
35	90	34

[8] (3.) Since the principal Alteration in the Quan⯑tity and Duration of total Eclipſes of the Sun, ariſes from the Difference there is at any Time between the real Diſtances, and apparent Diame⯑ters of the Sun and Moon, at the Time of ſuch Eclipſes, that Quantity and Duration muſt de⯑pend 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 Ano⯑maly. And in the Moon to 2/ [...] ▪ 9/ [...] or 1/ [...] of its en⯑tire Anomaly. And ſince the whole mean Excen⯑tricity 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 Di⯑ſtances and Diameters will, one with another, in⯑creaſe 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 In⯑creaſing; 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 Sha⯑dow [9] at all; but the Eclipſes become Annular. And if the Sun be Aſcending while the Moon is Deſcending, the contrary will happen; and the total Shadow grow greater perpetually. From which Circumſtances of the Sun and Moon in each Re⯑volution of the Cycle duly conſidered, we may nearly determine whether any ſucceeding corre⯑ſpondent Eclipſe will afford us a greater or leſs total Shadow, or whether the Eclipſes will be on⯑ly 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 com⯑paratively ſlower, and the Moon ſwifter than or⯑dinary; the meeting of the Luminaries will be accelerated. And if the Earth be much remoter from its Aphelion than the Moon from its Apo⯑gaeon, 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 Lu⯑nations 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 correſpond⯑ing Solar Eclipſe, May 11. 1724. And thus at [10] the total Solar Eclipſe, April 22. 1715. the Moon near its deſcending Node, had leſs Northern La⯑titude 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 aſ⯑cend: 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 Annu⯑lar [11] Eclipſes in Winter: Nor indeed is it very ſtrange, that Annular Eclipſes are in theſe Parts of the World ſo rarely obſerved; ſince they moſt uſually happen in Winter Days; which being ſhort, muſt afford us a proportionably ſmall Number of them. To the Inhabitants of the other Side of the Equator the Reverſe muſt happen. But thoſe Eclipſes very rarely come to our No⯑tice.

(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 Mo⯑tion, 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 Na⯑ture, are here wholly unconcerned; and no way ſubject to any Acceleration, Retardation, or Al⯑teration 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.

[12] (11.) If therefore we join three of theſe Cy⯑cles together, thoſe odd Hours and Minutes will amount nearly to a whole Day; and will there⯑fore nearly bring the middle Point of the corre⯑ſpondent Eclipſes to the ſame Time, in the ſame Place, and, in part, with the ſame Circumſtan⯑ces as before: Which a ſingle Cycle cannot poſ⯑ſibly 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 re⯑mote 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 Scot⯑land, 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 E⯑clipſe 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 lit⯑tle anticipated on that Account. But then, be⯑cauſe this Eclipſe was towards Evening, it would be much more retarded on that Account, than anticipated on the other; and the main Part of [13] the Eclipſe would happen after Sun-ſet, and be here inviſible.

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

	y	d	h	′	″
To A. D. 1670. April	00	08	5	43	15
Add	18	11	7	43	15
Sum 1688. April	00	19	13	26	30

So that the next correſpondent total Eclipſe ought to have been April 20th, 26′ ½ after one in the Morning; and was therefore, to be ſure, utterly inviſible to us here. To this Time if we add ano⯑ther 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 be⯑fore 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 remarka⯑ble for the raiſing of the Siege of Barcelona du⯑ring that total Darkneſs: Though I have been in⯑form'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, [14] both as to Time and Place. For if inſtead of tracing this correſpondent Eclipſe through the diſtinct Cycles we had at once taken our Grand Cycle of 54 Years, and 33 Days; we had come immediately to this Eclipſe: And by allowing the Anticipation of 50′ had been within about 20′ of the Calculation at London.

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

	y	d	h	′	″
To A. D. 1706. May	00	00	21	09	45
Add	18	10	07	43	15
Sum 1724 May	00	11	04	53	00

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

(12.) If we would now trace a few Lunar E⯑clipſes 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 Green⯑wich. Proceed therefore as is already directed to find the next correſpondent Eclipſe of the Moon thus:

[15]	y	d	h	′	″
To A. D. 1681/2 February.	00	11	10	59	00
Add	18	11	07	43	15
Sum 1699/1700 Febr.	00	22	18	42	15

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

	y	d	h	′	″
To A. D. 1699/1700, Feb.	00	22	18	42	15
Add	18	10	07	43	15
Sum 1717/18 March	00	05	02	25	30

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

	y	d	h	′	″
To A. D. 1717/18, March	00	05	02	25	30
Add	18	10	07	43	15
Sum 1736 March	00	15	10	08	45

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

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

N. B. As to the proper Quantity of the ſe⯑veral Alterations ariſing in each Period, which ought to be allowed for, they are nearly theſe: The Moon and Sun being about 31′ ⅔ in Diame⯑ter, 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 Spe⯑ctators the ſame, holds conſtantly: And though the diurnal Motion of the Earth removes all par⯑ticular 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 Ine⯑quality 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 Aphe⯑lio and Perihelio, and is the Difference of the Equa⯑tion belonging every where to the Addition of 10°⌊8, and is here 48″, which Space the Earth [17] goes in about 20′ of Time. So that the Diffe⯑rence 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 Alterati⯑on at the extreme Eccentricities of its Orbit is the Difference of the Equations at 2° 51′ ½ in Peri⯑gee and Apogee, according to thoſe extreme Ec⯑centricities: = 9′ which the Moon uſually goes in ſomewhat leſs than 20′. So that the Diffe⯑rence of Time, on this Account, muſt each Cy⯑cle be ſome Quantity leſs than 20 Minutes alſo.

And now we come to the principal Alteration in Time that can happen in Eclipſes; though it belongs only to thoſe of the Sun: And that is the Time of the Day when they happen in any particular Place. Now becauſe the Center of the Moon uſually goes over an entire Diameter of the Diſk of the Earth, in about three Hours and an half, Part of which is almoſt always before, and Part after Noon: while the odd Hours of a ſmall Cycle 7^h 43′ ¼, may reach equally from a Fore⯑noon to an Afternoon's ſucceſſive Eclipſe; 'tis poſſi⯑ble, ſuch an Eclipſe may appear an Hour and three Quarters later than the Period it ſelf would deter⯑mine 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 ob⯑jected 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.

[18] Corollary. If we would know what is the great⯑eſt Inequality in Digits and Time in a grand Pe⯑riod, made up of Three common ones, or of 54 Years, 32 or 33 Days, beſides the conſtant An⯑ticipation of 50′, we muſt ſay it may poſſibly, though it will very rarely, be almoſt thrice the Quantities already ſtated for a ſingle Cycle: ex⯑cepting the laſt and principal Difference, peculi⯑ar 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 Ta⯑bles, when publiſhed; with that Trigonometri⯑cal Calculation afterward, which I publiſh and exemplify in this Paper. In the former Part of which Work, this Cycle, with its proper Equa⯑tions, 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 E⯑clipſes with very ſmall Trouble or Calculati⯑on.

II.

[19]

The Plane in which the Center of the Moon moves in Eclipſes, is not that of the Eclip⯑tick, 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 com⯑pos'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 Quan⯑tities.

VI.

[20]

The Diſtance between this Vertex and the Pole of the Earth, when Eclipſes happen at the Sol⯑ſtices, is the Sum or Difference of the two fore⯑mentioned Angles of Inclination; the one, of the Pole of the Equator and of the Pole of the E⯑cliptick = 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 Spheri⯑cal 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 be⯑ſides the Diſtance between the Vertex of the Lu⯑nar 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 here⯑after.

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 where⯑in [...] 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 Foun⯑d [...] o [...] of our future Calculations. This Meri⯑ [...] I call The Primary Meridian.

IX.

[21]

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 Sol⯑ſtitial 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 fu⯑ture Calculations. This Angle I call the Primary Angle.

X.

Since the Motion of the Center of the Sha⯑dow 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 Ra⯑dius: 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 cor⯑reſponding Arcs or Angles at the Vertex, as fol⯑lows:

[22]Arcs.			D ff. Sines Equal.	Arcs.			Diff. Sine [...] Equal.
Diff.	°	′	D ff. Sines Equal.	Diff.	°	′	Diff. Sine [...] Equal.
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
[23]19⌊8				20⌊5
	13	49	½		20	50	32
19⌊9				20⌊5
	14	9	22		21	10	½
20				20⌊5
	14	28	½		21	30	33
20				20⌊6
	14	48	23		21	50	½
20				20⌊6
	15	8	½		22	11	34
20				20⌊7
	15	28	24		22	32	½
20				20⌊7
	15	48	½		22	53
20⌊1				20⌊8			35
	16	7	25		23	13	½
20⌊1				20⌊9
	16	[...]7	½		23	34	36
20⌊1				21
	16	47	26		23	55	½
20⌊2				21
	17	7	½		24	17	37
20⌊2				21⌊1
	17	27	27		24	38	½
20⌊2				21⌊1
	17	47	½		24	59	38
20⌊2				21⌊2
	18	8	28		25	20	½
20⌊3				21⌊2
	18	28	½		25	41	39
20⌊3				21⌊2
	18	48	29		26	2	½
20⌊3				21⌊3
	19	8	½		26	23	40
20⌊3				21⌊3
	19	28	30		26	44	½
20⌊4				21⌊4
	19	48	½		27	6	41
20⌊4				21⌊4
	20	9	31		27	27	½
20⌊4				21⌊6
	20	29	½		27	49	42
[24]21⌊7				24⌊2
	28	11	½		36	4	53
21⌊8				24⌊3
	28	32	43		36	27	½
21⌊9				24⌊4
	28	54	½		36	52	54
22⌊0				24⌊6
	29	16	44		37	16	½
22⌊1				24⌊7
	29	38	½		37	40	55
22⌊2				24⌊8
	30	0	45		38	4	½
22⌊3				24⌊9
	30	22	½		38	29	56
22⌊4				25⌊1
	30	44	46		38	54	½
22⌊6				25⌊2
	31	6	[...]		39	18	57
22⌊8				25⌊4
	31	51	47		39	43	½
23 [...]1				25⌊5
	31	29	½		40	7	58
23⌊2				25⌊7
	32	14	48		40	32	½
23⌊4				25⌊8
	32	36	½		40	57	59
23⌊5				25⌊9
	32	59	49		41	22	½
23⌊6				26⌊1
	33	22	½		41	48	60
23⌊6				26⌊3
	33	45	50		42	14	½
23 [...]7				26⌊4
	34	9	½		42	40	61
23 [...]8				26⌊5
	34	31	51		43	6	½
23 [...]9				26 [...]7
	34	54	½		43	32	62
23⌊9				26⌊9
	35	18	52		43	58	½
24⌊1				27⌊3
	35	41	½		44	25	63
[25]27⌊6				34
	44	52	½		55	19	74
27⌊8				34⌊4
	45	19	64		55	52	½
28⌊2				34⌊8
	45	46	½		56	26	75
28⌊4				35⌊0
	46	14	65		57	1	½
28⌊5				35⌊4
	46	42	½		57	37	76
28⌊6				35⌊8
	47	10	66		58	13	½
28⌊7				36⌊3
	47	38	½		58	49	77
28⌊8				37
	48	6	67		59	16	½
28⌊9				37⌊8
	48	36	½		60	4	78
29				38⌊6
	49	4	68		60	42	½
30				39⌊8
	49	33	½		61	22	79
30				40⌊9
	50	3	69		62	2	½
30⌊2				41⌊8
	50	33	½		62	44	80
30⌊4				42⌊5
	51	3	70		63	26	½
30⌊6				43
	51	33	½		64	9	81
31				44⌊5
	52	4	71		64	54	½
31⌊5				46
	52	35	½		65	40	82
32				48
	53	7	72		66	26	½
32⌊5				50
	53	39	½		67	15	83
33				52
	54	12	73		68	5	½
33⌊5				53
	54	45	½		68	58	84
[26]54				(86)
	69	52	½		77	54	88
56				(97)
	70	49	85		79	31	½
59				(116)
	71	48	½		81	27	89
63				(69)
	72	51	86		82	36	¼
67				(81)
	73	58	½		83	57	½
72				(105)
	75	10	87		85	42	¾
78				(258)
	76	28	½		90	00	90

Where the Angle made by the Lunar Circle, and the Paral [...]el of the Latitude is conſide⯑rable, inſtead of the firſt Number 19′⌊1 you muſt take for 1/1 [...]0 the Numbers following; being in a reciprocal Proportion to the Se⯑cants 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

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

°
00	000
10	174
20	342
30	500
40	643
50	766
60	867
70	940
80	98⌊5
90	100⌊0

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

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

XI.

[28]

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 arithme⯑tical 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 Semi⯑diameter of the Penumbra 2000 Miles.
[29]Duration in Minut.	Diſtance in Miles.		Dig^ts
1	2000
2	1999
3	1999
4	1999
5	1998
6	1997
7	1996
8	1995
9	1993
10	1991
11	1990
12	1988
13	1986
14	1984
15	1981
16	1979
17	1976
18	1973
19	1970
20	1966
21	1963
22	1959
23	1955
24	1951
25	1946
26	1942
27	1937
28	1932
29	1927
30	1922
31	1917
32	1911
33	1905
34	1899
35	1893
36	1886
37	1879
38	1872
39	1866
40	1859
41	1852
42	1844
		1838	1
43	1835
44	1828
55	1820
56	1812
47	1803
48	1795
49	1786
50	1776
½	1771
51	1766
½	1761
52	1756
2/1	1751
53	1746
½	1741
54	1736
½	1731
55	1725
[...]	1720
56	1714
[...]	1708
57	1703
½	1697
58	1691
½	1685
59	1679
		1676	2
½	1672
60	1667
[30]½	1662
61	1655
½	1648
62	1642
½	1635
63	1628
½	1621
64	1614
½	1609
65	1603
½	1596
66	1589
½	1582
67	1575
½	1568
68	1561
½	1554
69	1546
½	1538
70	1534
½	1528
71	1521
½	1514
		1513	3
72	1506
½	1498
73	1490
½	1482
74	1474
½	1456
75	1448
[...]	1439
76	1430
[...]	1424
77	1412
[...]	1403
78	1394
½	1384
79	1374
½	1364
80	1354
		1350	4
½	1344
81	1333
½	1322
82	1312
½	1302
83	1291
½	1280
84	1269
½	1258
85	1246
½	1234
86	1222
½	1110
87	1098
		1188	5
½	1186
88	1173
½	1160
[...]9	1147
½	1134
90	1121
½	1107
91	1093
½	1079
92	1065
½	1050
93	1034
		1025	6
½	1017
94	1000
½	983
95	971
½	955
96	937
½	919
97	902
½	884
98	864
[31]		863	7
½	844
99	824
½	804
100	784
¼	773
½	762
¾	751
101	739
¼	728
½	716
¾	704
		700	8
102	691
¼	679
½	666
¾	653
103	639
¼	625
½	611
¾	597
104	582
¼	566
½	550
		537	9
¾	534
105	517
¼	499
½	482
¾	463
106	443
¼	421
½	399
¾	376
		375	10
107	352
¼	320
½	288
¾	258
108	226
		212	11
¼	180
½	119

		50	12
¾	0 [...]

For Total Darkneſs of 166″ ⅔.
Duration in Sec^ds.	Diſtance in Miles.
″
1	50⌊0
2	50⌊0
3	50⌊0
4	50⌊0
5	50⌊0
6	50⌊0
7	50⌊0
8	50⌊9
9	50⌊9
10	50⌊9
11	49⌊9
12	49⌊8
13	49⌊8
14	49⌊8
15	49⌊7
16	49⌊7
17	49⌊7
18	49⌊6
19	49⌊6
20	49⌊6
21	49⌊6
22	49⌊5
23	49⌊5
24	49⌊5
25	49⌊4
26	49⌊4
27	49⌊4
28	49⌊3
29	49⌊3
30	49⌊3
31	49⌊2
32	49⌊2
33	49⌊2
34	49⌊1
35	49⌊1
36	49⌊1
37	49⌊0
38	49⌊0
39	49⌊0
40	48⌊9
41	48⌊9
42	48⌊8
43	48⌊7
44	48⌊6
45	48⌊5
46	48⌊4
47	48⌊2
48	48⌊1
49	47⌊9
50	47⌊7
51	47⌊6
52	47⌊5
53	47⌊4
54	47⌊3
55	47⌊2
56	47⌊1
57	47⌊0
58	46⌊9
59	46⌊8
60	46⌊6
61	46⌊5
62	46⌊4
63	46⌊3
64	46⌊2
65	46⌊0
66	45⌊9
67	45⌊8
68	45⌊6
69	45⌊5
70	45⌊3
71	45⌊2
72	45⌊1
[33]73	45⌊0
74	44⌊8
75	44⌊6
76	44⌊5
77	44⌊3
78	44⌊1
79	44⌊0
80	43⌊8
81	43⌊7
82	43⌊5
83	43⌊3
84	43⌊2
85	43⌊0
86	42⌊9
87	42⌊7
88	42⌊5
89	42⌊3
90	42⌊1
91	41⌊9
92	41⌊7
93	41⌊5
94	41⌊3
95	41⌊1
96	40⌊9
97	40⌊7
98	40⌊5
99	40⌊3
100	40⌊0
101	39⌊8
102	39⌊6
103	39⌊4
104	39⌊1
105	38⌊8
106	38⌊6
107	38⌊4
108	38⌊1
109	37⌊9
110	37⌊6
111	37⌊4
112	37⌊1
113	36⌊8
114	36⌊5
115	36⌊2
116	35⌊9
117	35⌊6
118	35⌊3
119	35⌊0
120	34⌊7
121	34⌊4
122	34⌊ [...]
123	33⌊8
124	33⌊4
125	33⌊0
126	32⌊7
127	32⌊4
128	32⌊1
129	31⌊7
130	31⌊3
131	30⌊9
132	30⌊5
133	30⌊1
134	29⌊7
135	29⌊3
136	28⌊9
137	28⌊5
138	28⌊0
139	27⌊5
140	27⌊1
141	26⌊6
142	26⌊1
143	25⌊6
144	25⌊1
145	24⌊6
146	24⌊0
147	23⌊5
148	23⌊0
[34]149	22⌊4
150	21⌊8
151	21⌊1
152	20⌊5
153	19⌊8
154	19⌊1
155	18⌊3
156	17⌊6
157	16⌊7
158	15⌊9
159	14⌊9
160	14⌊0
161	12⌊9
162	11⌊7
163	10⌊4
164	8⌊9
165	7⌊0
166	4⌊4
166 ⅔	0⌊0

Semidiameter of the Umbra or Total Dark⯑neſs = 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 di⯑miniſh'd in the Proportion of the Minutes de⯑ſcrib'd by the Annual Motion during the Conti⯑nuance of the Eclipſe. Thus in our preſent E⯑clipſe, 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 hereaf⯑ter.

XIII.

[35]

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 cen⯑tral 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 La⯑titudes 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 Me⯑ridians will, generally ſpeaking, be different, and their included Angle no otherwiſe to be known than by Trigonometry, as will appear hereaf⯑ter.

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 Cen⯑ter [36] it ſelf; with regard to which Center alone our firſt Calculations are always made: So when it is in the Zenith of any Place, it is one Semi⯑diameter of the Earth nearer it; which Semi⯑diameter being uſually 1/60, and at our next Eclipſe 1/55⌊5 of its entire Diſtance, as will appear here⯑after, will deſerve an Allowance. Nor will any leſſer Elevation of the Moon be wholly inconſi⯑derable in Eclipſes, but in all accurate Determi⯑nations 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 Eu⯑rope, where they are much nearer it; as this Cal⯑culation requires: Of which hereafter.

XVI.

The Figure of the entire Penumbra, or gene⯑ral Eclipſe; and of the Umbra, or Total Dark⯑neſs; as they appear upon every Country, is dif⯑ferent, 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 In⯑ſtrument as my Copernicus; or by the Peruſal of a very ſcarce Book written by P. Courſier, (Philoſ. Tranſact. N^o. 343. p. 259.) and cited by Dr. Hal⯑ley: Which diſtinctly treats of the Interſection of a Conical and Spherical Surface. But in the ſmaller Umbra, or Total Darkneſs, which is con⯑fined to a much narrower Compaſs, it very nearly approaches to the Interſection of a conick Sur⯑face with a Plane, which is a true Ellipſis.

XVII.

[37]

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 lon⯑ger Axis on the Azimuth of the Sun at the ſame Time. Nor is it at all neceſſary that the Directi⯑on 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 Mo⯑tion, 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 hereaf⯑ter 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 neigh⯑bouring Places equally diſtant, Eaſt and Weſt, from the Place you work for; and whoſe inclu⯑ded 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 [38] the Ellipſis D. And the Proportion of thoſe une⯑qual 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 Propo⯑ſition here refer'd to: Of which more hereaf⯑ter.

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 Spheri⯑cal 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 [39] the Horizon all the Way of its Paſſage. How⯑ever, ſince the ſeveral Points may, in all Meri⯑dians 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 ſexa⯑geſimal 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 forego⯑ing Table, p. 29, 30, 31. Where the Digits are alrea⯑dy noted at every proper Diſtance from the Path of the Moons Center; and where the interme⯑diate 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 Sha⯑dow 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 over⯑reach 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 geo⯑graphical 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 there⯑from [40] = 10316, which is thus: 941: 215000∷ 45: 10316.

XXIV.

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

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 coin⯑cident. 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 Cen⯑ter 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 [41] Point; and if the Velocity of the Center of the Shadow along the Circle be to the Velocity of Greenwich in its diurnal Motion as 5 [...] 48 to 1, if we ad 4⌊48 of their Diſtance at their ſetting out to that known Diſtance, we obtain the Point or Place where the Center of the Eclipſe will overtake Greenwich, or the Time when the E⯑clipſe 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, ac⯑cording 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 hap⯑pens in the intermediate Times, the diurnal Mo⯑tion, 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 hereaf⯑ter.

PROBLEMS.

[42]

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 Lati⯑tude 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 Decli⯑nation, = 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.

[43]

Rad.			10.
Sin.	66	31	9.	962453
Tang.	61	39	10.	267952		°	′
Tang. Right Aſcen.			10.	230405	=	59	31
				Compl.		30	29

IV.

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

Let (Fig. 5.) E P repreſent the Diſtance be⯑tween the two Poles of the Earth, and of the E⯑cliptick: = 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 Lati⯑tude is then 5° 49′ diſtant from the Sun back⯑ward; 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.

[44]

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 Di⯑ſtance 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 Meridi⯑an that paſſes through the Center of the general E⯑clipſe, and that paſſing through the Center of the Sun at the ſame Time.

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

R. 10.
S. of the Sun's Diſtance from the Solſtitial Co⯑lure: = 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 in⯑cluded 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	=	An⯑gle with the Primary Meridian.

VIII.

[46]

To find the Longitude and Latitude of the Center of the Shadow at the Middle of the ge⯑neral 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, be⯑tween the Primary Meridian, and that Meridian which paſſes through the Center of the Eclipſe, = 56° 22′. From which data the Primary Tri⯑angle (Fig. 7.) is thus to be ſolv'd:

R.			10.
	°	′
CS. C P Q.	56.	22.	9.	743412
T. V P.	27.	0.	9.	707166
T. P R. =	15.	46.	[...]9.	450578

Then ſay,

CS.V P	27.	0.	9.	949880
CS.P R	15.	46.	9.	983345
CS.V C.	58.	22.	9.	719730
			19.	703075		°	′
CS.P C.			9.	753195	=	55.	30
			Deduct PR		=	15.	46
			Rem^t. = P C.		=	39.	44
			Ergo Lat.		=	50.	16

To find the Angle at the Vertex CV P, pro⯑ceed in this Manner:

[47]	°	′
S. CV.	58.	22.	9.	930145
S. C P Q.	56.	22.	9.	920436
S. PC.	39.	44.	9.	805647
			19.	726083		°	′
S.CVP.=			9.	795938	=	38	41

The following Analogy will give V C P the Complement of the Angle which the Direction of the Center of the Eclipſe makes with the Me⯑ridian, 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 Nor⯑thern 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 Me⯑ridian that paſſes through the Center of the Sun at the Middle of the Eclipſe; or to ſolve the ſecond principal Triangle.

[48] If in the foregoing Triangle we ſuppoſe the Angle at the Pole to be equal to the Primary An⯑gle, or 40° 44′. we may thus ſolve this Trian⯑gle:

R.			10.
	°	′
CS. CP Q.	40.	44.	9.	879529
T. V P.	27.	0.	9.	707166		°	′
T.P R.		=	[...]9.	586695	=	21	7

Then ſay,

CS.V P.	27.	0.	9.	949880
CS.P R.	21.	7.	9.	969811
CS.V C.	58.	22.	9.	719730
			19.	689541		°	′
	CS. CR.	=	9.	739661	=	56	42
				Deduct		21	7
				Rem^t. C P.		35	35
				Ergo, Lat.	=	54	25

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

	°	′
S. V C.	58.	22.	9.	930145
S. C P Q.	40.	44.	9.	814607
S. V P.	27.	0.	9.	657047
			19.	471654		°	′
S.CV P.		=	9.	541509	=	20	22

X.

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

Add the vertical Angle already found = 38° 41′ to a right Angle at the Vertex; = 90 + 38°. 41′ = 128°. 41′ this is equal to the Angle at the [49] Ver⯑tex CV P. Subſtract this Angle from two right Angles. 180°—128°. 141′ = 51°. 19′, in or⯑der 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:

[50]	°	′
S. C P.	76.	57.	9.	988636
S. CV P.	51.	19.	9.	892435
S. V P.	27.	00.	9.	657047
			19.	549482		°	′
S.VC P.		=	9.	560846	=	21.	20

XI.

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

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

R.			10.
	°	′
CS. P V C.	51.	19.	9.	795891
T. P V.	27.	00.	9.	707166		°	′
	T.V R.		[...]9.	503057	=	17	40
				from		58	22
				Remn^t		40	42	=	R C.

Then ſay,

	°	′
CS. V R.	17.	40.	9.	979019
CS. P V.	27.	00.	9.	949880
CS. R C.	40.	42.	9.	879746
			19.	829626		°	′
CS. P C		=	9.	850607	=	44.	51.
				Ergo, Lat.		45.	9.

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

[51]	°	′
S. CP.	44.	51.	9.	848345
S. PVC.	51.	19.	9.	892435
S. VC.	58.	22.	9.	930145
			19.	822580		°	′
S.QPC		=	9.	974235	=	70.	27
					+	136	59
					=	207	26

And for the Angle VCP thus;

	°	′
S. CP.	44.	51.	9.	848345
S. PVC.	51.	19.	9.	892435
S. VP.	27.	00.	9.	657047
			19.	549482		°	′
S. VCP.		=	9.	701137	=	30.	10.

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

XII.

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

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

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

XIII.

[52]

To find the Proportion of the Velocities of the Center of the Shadow and of the diurnal Moti⯑on 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:

[53]	°	′		°	′
From the Angle	100	15	=	6^h	41
Deduct the Primary Angle				40	44
There remains the Angle of the Pole Q P C			=	59	31

Then proceed thus:

R.			10.
	°	′
CS. Q P C.	59	31	9.	705254
T. P V.	27	00	9.	707166		°	′
T.P R.		=	[...]9.	412420	=	14	29

Say then,

CS. P V.	27	00	9.	949880
CS. P R.	14	29	9.	985974
CS. V C.	58	22	9.	719730
			19.	705704		°	′
CS. R C.		=	9.	755824	=	55	15
			Deduct R P		=	14	29
			Remains P C		=	40	46
			Ergo, Lat.			49	14

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

R.			10.
	°	′
CS. QP C.	58.	16.	9.	720958
T. P V.	27.	0.	9.	707166		°	′
T.P R.		=	[...]9.	428124	=	15	0

[54] Then ſay,

	°	′
CS.P V.	27.	0.	9.	949880
CS.P R.	15.	0.	9.	984944
CS.V C.	58.	22.	9.	719730
			19.	704674		°	′
CS.R C.		=	9.	754794	=	55	21
			Deduct R P.			15	0
			Rem^t. P C.			40	21
			Ergo, Lat.		=	49	39
				Diff.		00	25

N. B. My Calculation differs from Dr. Hal⯑ley'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 Green⯑wich 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 Meri⯑dian 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

[55] Then ſay,

	°	′
CS. P V.	27.	0.	9.	949880
CS. V C.	58.	22.	9.	719730
			19.	698829		°	′
CS. R C.		=	9.	748949	=	55	53
			Deduct R P.			17	38
			Remn^t P C.			38	15
			Ergo, Lat.			51	45

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

R.			10.
	°	′
CS. Q P C.	62.	28.	9.	664891
T. P V.	27.	0.	9.	707166		°	′
T. R P.		=	[...]9.	372057	=	13	15

Say then,

CS. P V	27.	0.	9.	949880
CS. R P	13.	15.	9.	988282
CS. V C.	58.	22.	9.	719730
			19.	708012		°	′
CS. R C.			9	758132	=	55.	03
			Deduct R P		=	13.	15
			Rem^t. = P C.			41.	48
			Ergo, Lat.		=	48.	12

N. B. By ſuch Calculations we may deter⯑mine the Latitude of the Center of the Sha⯑dow's Way, from its entry upon, till its exit out of the Diſk of the Earth, to every known Meri⯑dian. A Specimen of which I intend to give preſently for the ſeveral Eaſt and Weſt Longitudes [56] from London in the Eclipſe before us: And ano⯑ther Specimen in the Eclipſe, Sept. 4. 1727.

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

		°	′
V C	=	58	22	R.	10.
V P	=	27	00	S.	9.	657047
P C	=	40	46	S.	9.	814900
Sum of 3	=	126	08	Sum	19.	471947
½ Sum		63	4	S.	9.	950138
Diff. of V C		4	42	S.	8.	913488
Double Radius					20.
The Sum					38.	863626
The Remainder					19.	391679
								°	′
½ Remainder					9.	695839	= CS.	60	14
double								120.	28
from								180.	00
as before remains								59.	32
Add the Primary Angle								40.	44
Sum								100.	16
Equal in Time to								6^h.	41′.

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

[57] For Dublin thus:

		°	′
V C	=	58	22	R.	10.
VP		27	00	S.	9.	657047
PC		38	15	S.	9.	791756
Sum of		3. 123	37	Sum	19.	44880 [...]
Half Sum		61	48 ½		9.	945159
Diff. of V C		3	26 ½		8.	778383
Double Radius					20.
The Sum					38.	723542
The Remainder					19.	274739
									°	′
Half the Remainder					9.	637369	C S.	=	64	18
double									128.	36
from									180.	00
remains as before									51.	24.
To									51.	24.
Add									40.	44.
Sum 92. 08								=	6^h	8½

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

[58] For Paris thus:

		°	′
VC	=	58	22	R.	10.
V P	=	27	00	S.	9.	657047
P C	=	41	48	S.	9.	823821
Sum of 3		127	10		19.	480868
Half Sum		63	35		9.	952105
Diff. of VC		5	13		8.	958670
Double Radius					20.
Sum					38.	910775
Remainder					19▪	429907
									°	′
Half Remainder					9.	714953	=	C S.	58.	45
double									117	30
As before, Remainder									62	30
Add, Primary Angle									40	44
Sum									103	11
In Time									6^h	52½

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

Corol. (1.) The latter Branch of the Problem de⯑termines the Hour and Minute when the Centre of the Eclipſe croſſes the Meridian at any aſſign'd La⯑titude; 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.

[59] Corollary (2.) The ſame Branch determines the true Difference of Meridians in all Places over or near which the Center of the Eclipſe paſſes, and ſo the Diſtance Eaſt and Weſt from any known Meridian whatſoever. Thus becauſe the Meridi⯑ans 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 An⯑gles, 62°. 28′. 59°. 31′. and 51°. 24′. reſpective⯑ly; 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 Diame⯑ter 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 Num⯑ber of Minutes requir'd. In Decimals thus: 35′⌊3:60′∷65′⌊17:110′⌊8=1^h:50 ⅘. [60] But then, becauſe the diurnal Motion of the Earth muſt be compounded with this rectilinear Motion of the Moon; and that as we have alrea⯑dy 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 diur⯑nal Motion, and we muſt then take the former half-Duration 55′. 24″ diſtinctly. And ſince the di⯑urnal 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 E⯑clipſe begins about 14′ before 6, 14′ after 6 ba⯑lances 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 Retar⯑dation in the firſt half: The middle Time of which will be about 30′ after 6. We muſt there⯑fore 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 back⯑ward [61] beyond 90°. that is, 71° ¾ of the Arc, which correſponds to 85° ½ of the Sines, where the Dif⯑ference of half a Degree is 61° 1/ [...]. This multi⯑plied by 4⌊64 gives 285. So that the Retardati⯑on 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 Du⯑ration, gives us the whole Duration = 106′ 24″ = 1^h46′ 24″, without the Conſideration of the Elevation of the Luminaries above the Horizon, which a ſmall Matter enlarges that Duration: Of which hereafter.

N. B. If we would be ſtill more preciſely nice, we may diſtinctly allow for the Difference of the Parallels, and the different Obliquity of the Di⯑rection; 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 Lu⯑minaries 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 Dia⯑meters of the Sun and Moon in the ſame Calcu⯑lation, = 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 Ho⯑rizon, when the Center of the Shadow croſſes the Meridian at Greenwich.

[62] In the Triangle (Fig. 10.) Z P S, Z P is = Di⯑ſtance 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 Me⯑ridian = 6^h 41′ = 100° 15′, whoſe Supplement is 79° 45′.

Say then,

R.			10.
	°	′
CS.Z P S	79	45	9.	250280
T.Z P	38	30	9.	900605		°	′
	T. P R	=	[...] 9.	150885	=	8	3 =	P R
					+	69	28 =	P S
					=	77	31 =	R S

And,

	°	′
CS. P R	8	3	9.	995699
CS. Z P	38	30	9.	893544
CS R S	77	31	9.	334766
			19.	228310		°	′
	CS. Z S	=	9.	232611	=	80	10	=	Z S
				Ergo, Altitude	=	9	50

XVIII.

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

In the former Triangle the Complement of the Altitude being now found = Z S = 80° 10′. Pro⯑ceed 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.

[63]

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 Di⯑ſtance 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 ex⯑actly one geographical Mile. And 60″ or one Mi⯑nute, 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 Se⯑cond is leſs than a Mile; and that in the Propor⯑tion 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 Semi⯑diameter of the Penumbra, or the utmoſt perpen⯑dicular Extent of the Eclipſe.

N. B. We may always make uſe of the forego⯑ing 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 Dark⯑neſs, by ſtill calling the Radius of every Penum⯑bra 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 Suppoſi⯑tion of other than geographical Miles: Which In⯑convenience, by the Reduction of them to geogra⯑phical Miles afterwards, will always, as here, come to nothing.

XX.

[64]

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 Pe⯑numbra, 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 Hori⯑zon: whoſe Sine is 788/1000, which divided by 55⌊5 is 1/ [...]0 nearly; which in the ſame Semidiameters, a⯑mounts to about 29 Miles every way. And for the Duration at Greenwich add to the common Dura⯑tion already found, or to 1^h 46′ 24″ the 1/324 Part of the ſame, or about ⅓ of a Minute both ways: which is about 2/ [...] in the whole; which will bring the entire Duration from 1^h 46′ 24″to full 1^h 47′ and for Total Darkneſs will add about 6/50 of the whole total Darkneſs = 20″, and increaſe the Duration from 2′ 46⅔ to 3′ 6″ [...]/3. But for the Al⯑teration in North America, which is about 1/7 [...] both ways, or 1/35 in the whole; this will there increaſe the entire Duration 4′ 12″, and from 1^h 46′ 24″ bring it to 1^h 50′ 36″, and for the total Darkneſs will add about 29/50 of the whole, or bring the 2′ 46″ ⅔ to 4′ 47″.

XXI.

[65]

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 Geo⯑graphical 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.

[66] Whence the Parts of the longer Axis being found, the larger C A = 289 Miles: the leſſer B C = 274 Miles. The half of which, D C is the Di⯑ſtance 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 Cen⯑ter of the Shadow croſſes the Meridians of Paris, Greenwich▪ and Dublin; and meaſuring the An⯑gle 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,

[67]	°	′
CS. P V.	27.	00.	9.	949880
CS. P R.	15.	41.	9.	983523
CS. V C.	58.	22.	9.	719730
			19.	703253		°	′
CS. P C			9.	753373	=	45.	29.
				Deduct		15.	14.
				Rem^t. V O.		39.	48.
				V G.		40.	46.
				V P.		41.	48.

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

R.			10.
	°	′
CS. O V. R.	04.	38.	9.	998577
T. V O.	39.	48.	9.	920733		°	′
	T. V R.		[...]9.	919310	=	39	48
				from		41	42
				Remn^t 2° 6′ = PR.

Then for the Angle VPO, ſay:

	°	′
S. P R.	02.	06.	8.	563999
S. V R.	39.	42.	9.	805343
T. O V R.	04.	38.	8.	908719
			18.	714062		°	′
T. V P O.		=	10.	150063	=	54.	42.

[68] And for V O R;

	°	′
S. V O.	39.	42.	9.	805343
S. V P O.	54.	42.	9.	911763
S. VP.	41.	48.	9.	823821
			19.	735584		°	′
S. V O P =			9.	930241	=	58.	23
					+	113	05
					Half	56	32½
					Compl.	33	27½

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

Corollary. The longer Axis of the total Sha⯑dow 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.

[69] This Problem is, In an Ellipſis of 96 Miles broad, and 562 Miles long; whoſe longer▪ Axis makes an Angle with the Direction of the Sha⯑dow of 12° 43½; To determine the Length of the greateſt Perpendicular that falls on that Dia⯑meter, along which the Direction of the Motion is. For twice that Perpendicular is the Breadth required. Now I have actually drawn this Ellip⯑ſis, or total Shadow, for my Map of this E⯑clipſe. 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 diſco⯑ver 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 Sun⯑ſetting 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 proba⯑ble to me, that all Switzerland and Lombardy, as far as Trent, Mantua. Cremona, and Parma; nay, perhaps, as far as Venice, Padua, Bononia, Ferra⯑ra, 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 Du⯑ration of the Eclipſe, or of total Darkneſs, at [70] any given Diſtance from the Path of the Moon's Center; and vice verſâ. Theſe Digits and Du⯑rations, 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″ Dura⯑tion. And the like is true of the total Shadow, and the Diſtance or Duration in the ſecond Ta⯑ble. Thus alſo we learn from the Map of this Eclipſe, that the neareſt Approach of the total Shadow to London will be about 40 geographical Miles: Which 40 Miles, in this Obliquity of the Motion, is about 31 perpendicular Miles. So that 32/16 [...] = 12/6 [...] of a Digit, is the Quantity of the Rim of Light that will be ſeen at London, when the Eclipſe is greateſt: And by conſequence the Digits there eclipſed will be 11 48/60. Nor does it appear to me, that the total Darkneſs will come nearer to London, than 40 Miles; although Dr. Halley's Scheme brings it within 7 or 8 Miles. As for the Duration of the entire Eclipſe here, it will be much the ſame as if it were Central: And has been already determined to 1^h 47′. And for the total Darkneſs along the Path of the Cen⯑ter 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 Ra⯑dius: to 1350 the Sine∷ So are the Number of [71] Minutes for the Semidiameter of the Penumbra: to the Number of Minutes for that Sine. Now 32′ 19″ the preſent Length of the greateſt Di⯑ſtance − 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 go⯑ing; without any new Directions whatſoever. Only if this Diſtance be the contrary way to the preſent Example, we muſt here make uſe of Ad⯑dition 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 Pa⯑ris, 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, [72] than either Dublin, London, or Paris. Time will certainly determine who are in the right.

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

	h	′	″
To	1	50	48
Add	2	58	24
Sum	4	49	12

Therefore the general Eclipſe by the Meridian of London,

	h	′	″
Begins	II.	52	24
Middle	V.	17	00
End	VII.	41	36
Duration	4^h	49	12

Eclipſe at London,

Begins V.	45	00
Middle VI.	41	00
End VII.	32½	00
Duration 1^h	47½	00
Digits eclipſed	11	48/60
Duration of Total Darkneſ [...] between Exeter and Plimouth,	03	6½

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

[73] N. B. I cannot tell the Reaſon why my Ori⯑ginal 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 Me⯑thods 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 eclip⯑ſed are 11 18/60 exactly according to my Determina⯑tion in this Paper.

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

[74]

IT is humbly propoſed, That Obſervations be made in all Places where Solar Eclipſes are ſeen, of the exact Duration of the ſame; by ei⯑ther 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 Geo⯑graphy, 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

[...]

[79]

A Table of the Latitude of the Center of the total Shadow at the Eclipſe, May 11. 1724. to every 10 Degrees of the Angle about the Pole, and every 7°⌊845. Longitude from London.
Angles at the Pole.			Longitude from London.		Latit^de.
°	′		° Eaſt.		°	′
70	19		8⌊577		45	9
69	23		7⌊845		45	34
			Weſt.
59	23		0000		49	17
49	23		7⌊845		52	19
39	23		15⌊690		54	42
29	23		23⌊535		56	29
19	23		31⌊380		57	43
09	23		39⌊225		58	13
00	00		46⌊600		58	38
00	37		47⌊070		58	38
10	37		54⌊915		58	21
20	37		62⌊760		57	36
30	37		70⌊605		56	18
40	37		78⌊450		54	27
50	37		86⌊295		51	59
56	22	Mid^le.	90⌊021		50	16
60	37		94⌊140		48	51
70	37		101⌊985		45	5
80	37		109⌊830		40	40
90	37		117⌊675	☉	35	56
100	37		125⌊520		30	31
110	37		133⌊365		25	15
120	37		141⌊210		20	11
130	37		149⌊055		15	37
136	59		154⌊052		13	3

N. B. Here, as well as in the next Table, The Angles at the Pole differ by Ten Degrees; and the Longitude from London is found by the Proporti⯑on of 4⌊64 to 3⌊64. to be 7°⌊845 of Longitude [80] from London for every ſuch Ten Degrees. But if, as is generally moſt convenient, we would have the Differences of Longitude from London be even 10 Degrees; We muſt find the corre⯑ſpondent Angles at the Pole before we begin our Calculations by the inverſe Proportion of 3⌊64: 4⌊64∷ 10°: 12′¾ the perpetual Addition of which Number will give us a Series for ſuch a Calculation.

A Table of the Latitude of the Center of the total Shadow at the Eclipſe, Sept. 4. 1717. to every 10 Degrees of the Angle at the Pole; and every 7°⌊167 Longitude from London.
Angles at the Pole.			Longitude from London.		Latit^de.
°	′		° Weſt.		°	North
11	33		12⌊248		30	18
4	28		7⌊167		30	20
			Eaſt.
00	00		3⌊963		30	14
5	32		0⌊000		29	28
15	32		7⌊167		27	56
25	32		14⌊334		25	40
35	32		21⌊501		22	38
45	32		28⌊668		18	27
55	32		35⌊835		14	24
65	32		43⌊002		9	21
75	32		50⌊169		5	4
77	58	Mid^le.	51⌊903	Mid^le.	3	57
85	32		57⌊336		0	16
					South.
95	32		64⌊503		4	32
105	32		71⌊670		9	55
115	32		78⌊837		14	54
125	32		86⌊004		19	16
135	32		193⌊171		23	00
145	32		100⌊338		25	57
155	32		107⌊505		28	8
165	32		114⌊672		29	34
168	39		116⌊909		30	14

3⌊53: 2⌊53 ∷ 10: 7⌊167.

2⌊53: 3⌊53 ∷ 10: 13⌊952.

[81]

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 N^ode r tr.

′

″

′

″

′

″

Anno Dom.

1701

Years

and

Leap-Y. May, Days

[...]4

Hours

Minutes

(Mean Time.)

Moon's mean Motion

Sun's mean Anomal.

An. Eq. add

An. Eq. ſubſt.

Phyſic. Parts ſubſtract

[...]

2d Equat. ſubſtract

Mean Pla. correct

3d Equat. add

6th Equat. add

Sun's true Place

Diff. ſubſtr.

Moon's mean Pl correct

5▪

5 [...]

Annual A [...]g [...]ment

S's diſt. from Node

Tru [...] Pl. of Apog. ſubſtr.

Equat. add

Moon's mean Anomal.

Equat. add

31.

True Pl. of Node

Moon's true Pl. in Orbit

True Pl. of Apogee

Sun▪ [...] true Pla. ſubſtract

Eccentricity 6000

Inclinat. Limit.

Moon's diſt. from Sun

	′	″
M's h mo [...].	37	42
Sun's	2	24
M's fro. Sun	35	18
35′ 18″: 60′∷ 1′ 46″: 3′ 2″ ▪
Eq. time add	3	53
Reduct. ſubſt.	2	33
Diff. add	4	22
Ergo, Ecl pſe is 11^d 5^h 19 22″
Temp. appar. deduct the Error of the Tables a⯑bout▪	2′	22″
true Tim.	5^h	17 0

	′	″
S mid. Sun	15	53
Sem Moon	16	42
Sem. Pen^bra	^•32	35
Semi. Difc.	61	38
Angle of the M's Way with the Ecliptick	5	36
Diff. Diamet r of the Sun & Moon, or Br [...]adth of total Shad.	0	98

Nodes true Pl. ſubſtract

Argument of Latitude

Reduction ſubſtract

Moon's tr. Pl. in Eclip.

Mo n's true North Lat.

At the Eclipſe

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

Sun's mean Motion.

Motion of Perihelion.

′

″

′

″

A. D.

1701

Years

And

Leap-Y. May, Days

Hours

Minutes

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

[...]quation add

Sun's true Place

[82]

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

M's mean Mot.

Motion Apogee.

Motⁿ of N^ode retr.

′

[...]

′

[...]

′

[...]

Anno Dom.

1701

Years

And

Sept. Days

Hours

Minutes

(Mean Time.)

Moon's mean Motion

Sun's mean An ma [...]y

An. Eq. ſub.

An. Eq. add

Phyſical Parts add

2d Equat. ſubſtract

Mean Pl. Apo. cor.

Mean Pl. of No.

6th Equat. add

Difference add

Sun's true Place

M [...]on's mean Pl. correct

4 [...]

True Pl. of Apog. ſubſt.

Annual Argum.

S.'s diſt. [...]ro. Nod.

Moon's mean Anomaly

Eq [...]. Ad

Eq. Ad.

Equation add

M.'s Correct. Pl. in Orb.

True Pl. of Apo.

True Pl. of Node

Sun's true Pla. ſubſtract

Eccentr.

663

Moon's Diſt. from Sun

Inc [...]in. Limit.

Variation add

S'.s hor. mot.	2	26
Moon's	38	29
M. fr. S.	36	2
36⌊04: 60 ∷ 6⌊9: 11 30.
Ergo, Eclipſe will be 20^h 23 34
Mean Time.
Eq. tim. 2d.	4	38
Reduct. ſubſtr.		9
Diff. add	4	29
20^h	28	03
Appar. Time.

Semid. S.	16	1
Semid. M.	16	50
Diff. =	00	49
Sum =	32	52
Parallax M. Ho⯑rizon	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

Nodes tr. Pl. ſubſtract

Argum [...]nt of Latitude

Reduction ſubſtract

Moon's true Pl. in Eclip.

Moon's true South Latit.

At the Eclipſe

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

Sun's mean Motion.

Motion of Perihel.

′

[...]

′

″

A. D.

1701

Years

And

Sept. Days

Hours

3	8	12	28
Place of Perihel.
5	23	44	44
Sun's mean Motion
2	15	32	16
Sun's mean Anomaly.

Minutes

(Mean Time.)

Sun's mean Motion

Equat. ſubſtract

Sun's true Place

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

[83]

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 Obſer⯑vations could be made at Sea, and to deter⯑mine 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 Mag⯑netick Pole to the Magnetick Equator; the Axis of that Equator being ſufficiently Oblique [84] to its Plane: All which is the Caſe of Sphe⯑rical Loadſtones here."’ Now having already received Four Journals from Four ſeveral Ma⯑ſters 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 Pra⯑ctice 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 Diffi⯑culties 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 Wea⯑ther, He made me 28 very good Horizontal Ob⯑ſervations, from the Latitude of 65 quite to Arch⯑angel: 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 ex⯑pected 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 [85] Inſtrument, and made Four Obſervations of the Dip, both by the Vertical and Horizontal Vibra⯑tions, 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 ſome⯑what 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 Sha⯑king of my Frame; which Frame tho' it very much avoided the ſlower and greater Oſcillation of the Ship, yet made a quicker but leſſer Oſ⯑cillation 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 pre⯑pared 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 Theo⯑ry 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 Di⯑ſtance [86] of about 9/10 of an Inch from its Surface; viz. Not exactly as the Line of Sines, where at the Middle of the Line the Angles are 60 and 30; but rather as 66 to 24. Which Rule therefore is what I now propoſe as much nearer than the other. By which Proportion I determi⯑ned 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 pro⯑vided for accordingly; and which has been a full Year ready for Practice. Thoſe Obſervations of his, that I have yet re⯑ceived; for I have not heard from him ſince Ja⯑nuary, 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 Ana⯑logy; ſince they all three are about the ſame [87] Quantity of 8 or 9 Degrees exceed that Analogy; and ſince very near the ſame Place, where the third Obſervation was made, I have a double Obſerva⯑tion of Captain Beal's to correct the ſame; I ra⯑ther conclude, that Captain Tempeſt made a Mi⯑ſtake, 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 De⯑creaſe of the Dip as you go nearer to the Equa⯑tor: Tho' as they ſtand at preſent, they do not determine the accurate Proportion of that Alte⯑ration 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. Eber⯑hard, 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 par⯑ticular Account of it. But I have not yet re⯑ceived 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 follow⯑ing Inferences:

(1.) That there is one Spherical Loadſtone, and but one in the Center of our Earth; and that this [88] Loadſtone, like other Spherical Loadſtones, has but one Northern Pole: Contrary to Dr. Halley's Hypotheſis.

(2) That this Northern Pole is ſituated, con⯑trary 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 Zem⯑bla. Captain Jolly's numerous Obſervations prove this moſt fully: While in Sailing towards that Point his horizontal Vibrations greatly in⯑creas'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 Pla⯑ces; and that without any certain Rule; as it is upon the Surface of our Terrellae or Spherical Loadſtones here. This the various Number of Se⯑conds 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 Ter⯑rellae.

(4.) That there no where appears in open Seas any ſuch Irregularity in the Dip, as we ſome⯑times 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 Hypothe⯑ſis of the four Magnetick Poles is utterly ground⯑leſs. [89] Nor indeed ſhall I [...]e at Reſt, till▪I have ſent a Dipping-Needle to Hudſon's Bay, on purpoſe to determine this Diſpute about the four Poles: For that Voyage being almoſt directly towards his ſecond Northern Pole all the way, and about the ſame Di⯑ſtance all the way from mine; if this Voyage af⯑ford 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 great⯑ly increaſe, it will demonſtrate a ſ [...]ond Pole ſomewhere in thoſe Parts of America, where Dr. Halley places it. And to this Decretory Experi⯑ment 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 Obſervati⯑ons from the Dip, which are ſtill againſt him; and which are alone capable of diſcovering the exact Place of ſuch Poles, either upon the Sur⯑face of the Earth, or of Terrellae. However, when one Set of Experiments with a Dipping-Needle, ſent to Hudſon's Bay, will certainly de⯑termine 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 Diſ⯑covery of the Longitude and the Latitude at Sea, are true and right: That the Terreſtrial Magne⯑tiſm 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 De⯑gree thereby diſcovered already; and that if I can ſufficiently avoid the Shaking of the Ship, which I am now endeavouring, and have great [90] Hopes of performing, both Latitude and Longi⯑tude 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 Agitati⯑on; and if effected will be more eaſy and univer⯑ſal than this. But as to giving any farther Ac⯑count 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 com⯑pleated 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 Ho⯑rizon, in Dipping-Needles, to every 1/9 [...] Part of their reſpective equal Diſtances from the Mag⯑netick 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

[91]Diſt. from the Pole.	Dip.		Diſt. from the Pole.	Dip.
°	°	°	°	°	°
14	82	55	14	34	15
15	82	24	15	35	33
16	81	54	16	36	50
17	81	23	17	38	4
18	80	52	18	39	16
19	80	21	19	40	28
20	79	49	20	41	37
21	79	18	21	42	45
22	78	47	22	43	54
23	78	16	23	44	58
24	77	44	24	46	2
25	77	12	25	47	6
26	76	41	26	48	9
27	76	8	27	49	10
28	75	36	28	50	12
29	75	4	29	51	12
30	74	32	30	52	12
31	73	59	31	53	11
32	73	26	32	54	9
33	72	54	33	55	6
34	72	20	34	56	1
35	71	47	35	57	0
36	71	14	36	57	56
37	70	39	37	58	52
38	70	5	38	59	47
39	69	31	39	60	41
40	68	57	40	61	35
41	68	22	41	62	49
42	67	47	42	63	22
43	67	12	43	64	15
44	66	36	44	65	8
45	66	00	45	66	0

[92] N. B. I take the Northern Pole of the Ter⯑reſtrial Magnet to be about the Meridian of Arch⯑angel, in the Latitude of 75½. Its Equator to be nearly a great Circle, interſecting the Earth's E⯑quator about 2½ Degrees Eaſtward of the Meridi⯑an 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 circu⯑lar; 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. Bo⯑ſton in New-England is 51/ [...] = [...] ⌊ [...]/9 [...], whence its Dip will be about 68° 22′, which Captain Beal found to be ſo in Fact. Barbados is about 26⌊ [...]/1 [...] = 22/90⌊5 diſtant from the Equator of the Magnet, whence its Dip ought to be about 44° ½, as Captain Beal alſo found it to be in Fact. St. Helenais about 14/49 = [...] ⌊ [...]/9 [...], whence its Dip ought to be about 47° 50′ as Dr. Halley found it to be in Fact. And ſo eve⯑ry where in the main Ocean, at conſiderable Di⯑ſtances from the Shores.

N. B. If the Dip of any Needles be ſomewhat different at London, add or ſubſtract a proportio⯑nable 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 [93] Boſton in New-England, Southward; which in the Table is 68° 22′, proceed thus. Becauſe the equal Diſtance of Boſton from the Magnetick Equator is 49 Parts of 60⌊4, the like Diſtance of London from that Equator; deduct [...]9/60⌊4 120′ = 97′ = 1° 37′ out of the Tabular Dip 68° 22′. The Re⯑mainder 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 Nor⯑way, 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 Tabu⯑lar 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 add⯑ing or ſubſtracting 8 to the Complement of the Dip found by the natural Sines for every 1/ [...]0 of e⯑qual Diſtances from the Equator or Pole.

N. B. If any deſire to calculate by Trigonome⯑try the Diſtances of all Places from the magnetick Equator of Poles, and the Diſtances of that E⯑quator 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 [94] in the Triangle Q A M, we have the Angles Q A M and Q M A, and the Side A M, = the Diſtance of the Magnetick Pole from the Magne⯑tick Equator, to find A Q. So we have the Pro⯑portion of A L to AQ, Q. E. I.

But ſince the Data are not yet ſufficiently ex⯑act for the Calculation, meaſuring is ſufficient.

FINIS.

Appendix A

[]

to fold out at the end of y^e Book

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 7.

Fig. 6.

Fig. 9.

Fig. 8.

Fig. 10.

Fig. 11.

Fig. 12.

Appendix B ERRATA.

[]

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

Distributed by the University of Oxford under a Creative Commons Attribution-ShareAlike 3.0 Unported License

Citation Suggestion for this Object: TextGrid Repository (2020). TEI. 4879 The calculation of solar eclipses without parallaxes With a specimen of the same in the total eclipse of the sun May 11 1724 By Will Whiston. University of Oxford Text Archive. . https://hdl.handle.net/21.T11991/0000-001A-5E38-D

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

The Conſtruction Explaind

LEMMATA: OR, Preparatory Propoſitions.

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SCHOLIA.

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PROBLEMS.

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

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

Appendix A

Appendix B ERRATA.

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