Cosmographia, or, A view of the terrestrial and cœlestial globes in a brief explanation of the principles of plain and solid geometry applied to surveying and gauging of cask : the doctrine of primum mobile : with an account of the Juilan & Gregorian calendars, and the computation of the places of the sun, moon, and fixed stars ... : to which is added an introduction unto geography / by John Newton ...

AN INTRODUCTION TO Astronomy. (Book 2)

The Second Book. (Book 2)

CHAP. I.

Of the Year Civil and Astronomical.

HAving shewed the Motion of the Pri∣mum Mobile, or Doctrine of the Sphere, which I call the Absolute Part of Astronomy; I come now un∣to the Comparative, that is, to shew the Motion of the Stars in reference to some cer∣tain Distinction of Time.

2. And the Distinction of Time is to be con∣sidered either according to Nature, or according to Institution.

3. The Distinction of Time according to Na∣ture, is that space of Time, in which the Planets do finish their Periodical Revolutions from one certain Point in the Zodiack, to the same again, and this in reference to the Sun is called a Year, in reference to the Moon a Month.

4. The Sun doth pass through the Zodiack in 365 Days, 5 Hours, and 49 Minutes. And the Moon doth finish her course in the Zodiack, and return into Conjunction with the Sun, in 29 Days, 12 hours, 44 Minutes, and 4 Seconds. And from the Motion of these two Planets, the Civil Year in every Nation doth receive its Institu∣tion.

5. Twelve Moons or Moneths is the measure of the Common Year, in Turkey in every Moneth they have 29 or 30 Days, in the whole Year 354 Days, and in every third Year 355 Days.

6. The Persians and Egyptians do also account 12 Moneths to their Year; but their moneths are proportioned to the Time of the Suns con∣tinuance in every of the Twelve Signs; in their Year therefore which is Solar, there are always 365 Days, that is eleven Days more than the Lu∣nar Year.

7. And the Iulian Year which is the Account of all Christendom, doth differ from the other in this; that by reason of the Sun's Excess in Moti∣on above 365 Days, which is 5 Hours, 49 Minutes, it hath a Day intercalated once in 4 Years, and by this intercalation, it is more agreeable to the Motion of the Sun, than the former, and yet there is a considerable difference between them, which hath occasioned the Church of Rome to make some further amendment of the Solar

Year, but hath not brought it to that exactness, which might be wished.

8. This intercalation of one Day once in 4 Years, doth occasion the Sunday Letter still to al∣ter till 28 Years be gone about; The Days of the Week which use to be signed by the seven first Letters in the Alphabet, do not fall alike in eve∣ry Common Year, but because the Year consisteth of 52 Weeks and one Day, Sunday this Year will fall out upon the next Year's Monday, and so for∣ward for seven years, but every fourth year con∣sisting of 52 weeks and two days, doth occasion the Sunday Letter to alter, till four times seven years, that is till 28 years be gone about. This Revolution is called the Cycle of the Sun, taking its name from the Sunday Letter, of which it sheweth all the Changes that it can have by rea∣son of the Bissextile or Leap-year. To find which of the 28 the present is, add nine to the year of our Lord, (because this Circle was so far gone about, at the time of Christs Birth) and di∣vide the whole by 28, what remaineth is the present year, if nothing remain the Cycle is out, and that you must call the last year of the Cycle, or 28.

9. This Intercalation of one day in four years, doth occasion the Letter F to be twice repeated in February, in which Moneth the day is added, that is, the Letter F is set to the 24 and 25 days of that Moneth, and in such a year S. Matthias day is to be observed upon the 25 day, and the next Sunday doth change or alter his Letter, from which leaping or changing, such a year is called Leap-year, aud the number of days in each Moneth is well expressed by these old Verses.

Thirty days hath September, April, June and No∣vember.

February hath 28 alone, All the rest have thirty and one.

But when of Leap-year cometh the Time, Then days hath February twenty and nine.

That this year is somewhat too long, is ac∣knowledged by the most skilful Astronomers, as for the number of days in a year the Emperours Mathematicians were in the right, for it is cer∣tain, that no year can consist of more than 365 days, but for the odd hours it is as certain that they cannot be fewer than five, nor yet so many as six; so then the doubt is upon the minutes, 60 whereof do make an hour, a small matter one would think, but how great in the consequence we shall see. The Emperours year being more than 10 minutes greater than the Suns, will in 134 years rise to one whole day, and by this means the Vernal or Spring Equinox, which in Iulius Caesar's time was upon the 24 of March, is now in our time upon the 10 of March, 13 days backward, and somewhat more, and so if it be let alone will go back to the first of March, and first of February, and by degrees more and more backward still.

10. To reform this difference, some of the late Roman Bishops have earnestly endeavoured. And the thing was brought to that perfection it now standeth, by Gregory the Thirteenth, in the year 1582. His Mathematicians, whereof Lili∣us was the Chief, advised him thus: That con∣sidering there had been an Agitation in the

Council of Nice somewhat concerned in this matter upon the motion of that Question, about the Celebration of Easter. And that the Fa∣thers of the Assembly, after due deliberation with the Astronomers of that time, had fixed the Vernal Equinox at the 21 of March, and con∣sidering also that since that time a difference of ten whole days had past over in the Calendar, that is, that the Vernal Equinox, which began upon the 21 of March, had prevented so much, as to begin in Gregorie's days at the 10 of the same, they advised, that 10 days should be cut off from the Calendar, which was done, and the 10 days taken out of October in the year 1582. as being the moneth of that year in which that Pope was born; so that when they came to the fifth of the moneth they reckoned the 15, and so the Equinox was come up to its place a∣gain, and happened upon the 21 of March, as at the Council of Nice.

But that Lilius should bring back the begin∣ning of the year to the time of the Nicene Council and no further, is to be marvelled at, he should have brought it back to the Emperours own time, where the mistake was first entered, and instead of 10, cut off 13 days; however this is the reason why these two Calendars differ the space of 10 days from one another. And thus I have given you an account of the year as it now stands with us in England, and with the rest of the Christian World in respect of the Sun, some o∣ther particulars there are between us and them which do depend upon the motion of the Moon, as well as of the Sun, and for the better under∣derstanding of them, I will also give you a brief

account of her revolution. But first I will shew you, how the day of the moneth in any year pro∣pounded in one Couutry, may be reduced to its correspondent time in another.

11. Taking therefore the length of the year, to be in several Nations as hath been before de∣clared, if we would find what day of the moneth in one Conntry is correspondent to the day of that moneth given in another, there must be some beginning to every one of these Accounts, and that beginning must be referred to some one, as to the common measure of the rest.

12. The most natural beginning of All Ac∣counts, is the time of the Worlds Creation, but they who could not attain to the Worlds Begin∣ning, have reckoned from their own, as the Ro∣mans from the building of Rome, the Greeks from their Olympicks, the Assyrians from Nabonassar, and all Christians from the Birth of Christ: the beginning of which and all other the most nota∣ble Epochaes, we have ascertained to their corre∣spondent times in the Julian Period, which Sca∣liger contrived by the continual Multiplication of those Circles, all in former time of good use, and two of them do yet remain; the Circles yet in use are those of the Sun and Moon, the one, to wit, the Sun, is a Circle of 28 years, and the Cir∣cle of the Moon is 19, as shall be shewed here∣after. The third Circle which now serves for no other use than the constituting of the Julian Period, is the Roman Indiction, or a Circle of 15 years; if you multiply 28 the Circle of the Sun, by 19 the Circle of the Moon, the Product is 532, which being multiplied by 15, the Circle of the Roman Indiction, the Product is 7980, the

Number of years in the Julian Period: whose admirable condition is to distinguish every year within the whole Circle by a several certain Cha∣racter, the year of the Sun, Moon, and Indiction being never the same again until the revolution of 7980 years be gone about, the beginning of this Period was 764 Julian years before the most reputed time of the Worlds Creation; which being premised, we will now by Example shew you how to reduce the years of Forreigners to our Julian years, and the contrary.

1. Example.

I desire to know at what time in the Turkish Account, the fifth of Iune in the year of our Lord 1640. doth fall.

The Julian years complete are 1648, and are thus turned into days, by the Table of days in Julian years.

1000 Julian years give days

365250

600 Julian years give days▪

219150

40 Julian years give days

14610

8 Years give days

2922

May complete

151

Days

5

The Sum is

602088

Now because the Turkish Account began Iuly 16. Anno Christi. 622. you must convert these years into days also.

600 Julian years give days

219150

20 Years give days

7305

1 Year giveth days

365

Iune complete

181

Days

15

The Sum is

227016

Which being substracted from

602088

There resteth days

375072

900 Turkish years give days

318930

There resteth

56142

02987

152

4

150 Turkish years give days

53155

There resteth

56142

02987

152

4

8 Turkish years give days

2835

There resteth

56142

02987

152

4

Giumadi. 4.

148

There resteth

56142

02987

152

4

Therefore the fifth of Iune 1649. in our En∣glish Account doth fall in the year 1058. of Ma∣homet, or the Turkish Hegira, the fourth day of the moneth Giumadi. 11

2. Example.

I desire to know upon what day of our Julian year the 17 day of the moneth in the 1069 year complete of the Persian Account from Ieshagile doth fall.

The beginning of this Epocha is from the Epo∣cha of Christ in complete days

230639

1000 Persian years give

365000

60 Years give

21900

9 Years give

3285

Chortal complete

90

Days complete

16

The Sum

620930

1000 Julian years Substracted

365250

There rests

255680

5

700 Julian years

255675

There rests

255680

5

Therefore it falls out in the Julian year from Christ 1700. the fifth day of Ianuary.

He that understands this may by the like me∣thod convert the years of other Epochas, into our Julian years and the contrary.

The Anticipation of the Gregorian Calendar is more easily obtained, for if you enter the Ta∣ble with the years of Christ complete, you have the days to be added to the time in the Julian Account, to make it answer to the Gregorian, which will be but ten days difference till the year 1700. and then the difference will be a day more, until the year 1800. and so forward three days difference more in every 400 years to come, un∣less our year shall be reformed as well as theirs.

CHAP. II.

Of the Cycle of the Moon, what it is, how placed in the Calendar, and to what pur∣pose.

THat the Civil Year in use with us and all Christians, doth consist of 365 days, and every fourth year of 366, hath been already shew∣ed, with the return of the Sunday Letter in 28 years. In which time the Moon doth finish her course in the Zodiack no less than twelve times, which twelve Moons, or 354 days, do fall short of the Sun's year, eleven days in every common year, and twelve in the Bissextile or Leap-year.

And by Observation of Meton an Athenian, it was found out about 432 years before Christ, that the Moon in nineteen years did return to be in Conjunction with the Sun on the self same day, and this Circle of nineteen years is called the Cy∣cle of the Moon, which being written in the Ca∣lendar against the day in every Moneth, in which the Moon did change, in Letters of Gold, was also called the Golden Number, or from the excellent use thereof, which was at first, only to find the New Moons in every Moneth for ever, but a∣mongst Christians it serveth for another purpose also, even the finding of the time when the Feast of Easter is to be observed. The New Moons by this Number are thus found. In the first year of the Circle, or when the Golden Number is 1, where the Number 1 was set in the Ca∣lendar in any Moneth, that day is New Moon, in

the second Year where you find the golden Num∣ber 2, in the third Year where you find the gol∣den Number 3, and so forward till the whole Circle be expired; then you must begin with one again, and run through the whole Circle as be∣fore.

2. And the reason why the Calendar begins with the golden Number 3, not 1, is this. The Christians in Alexandria had used this Circle of the Moon two Years before the Nicene Council. And in the first of these Years the new Moon next to the Vernal Equinox was upon the 27th Day of the Egyptian month Phamenoth answering to the 23d of our March, against that Day therefore they placed the golden Number 1. And because there are 29 Days and a half from one new Moon to another, they made the distance be∣tween the new Moons to be interchangeably 29 and 30 Days, and so they placed the same gol∣den Number against the 26 Day of Phurmuthi the Month following, and against the 26 Day of the Month Pachon and so forward, and upon this ground by the like progression was the golden Number set in the Roman Calendar; and so the golden Number 1 by their example was set a∣gainst March 23. April 21. Iune 19. Iuly 19. August 17. September 16. October 15. Novem∣ber 14. December. 13. But then because in the following Year the golden Number was 2. reckon∣ing 30 Days from the 13th of December, the golden Number 2 was set to Innuary 12. Febru∣ary 10. March 12. April 10. May 10. Iune 8. Iuly 8. August 6. September 5. October 4. November 3. December. 2. From whence reckon∣ing 13 Days as before, the golden Number 3

comes in course for the third Year to be set against the first of Ianuary.

But that you may know how the golden Num∣ber comes to be distributed in the Calendar ac∣cording to the form in which it now is, you must consider that in 19 Solar Years there are not only 228 Lunar Months or 12 times 19 Lunar Months but 235 for the 11 Days which the com∣mon Solar Year doth exceed the Lunar, do in 19 Years arise to 209 Days, out of which there may be appointed 7 Months, 6 whereof will con∣tain 30 Days apiece, and one Month 29 days; and these 7 Months are called Embolismical Months, because by a kind of injection or interposition they are reckoned in some of the 19 Years. And those Years in which they are reckoned are called Embolismical Years, to distinguish them from the common Years which always contain 354 Days, whereas 6 of these Embolismical Years do each of them contain 384 Days, and the seventh Embo∣lismical Year in which the Month of 29 Days is reckoned, doth contain 383 Days.

3. The Embolismical Years in the Cycle of the Moon are properly these Seven. 3, 6, 9, 11, 14, 17, 19. because in the third Year 11 Days being thrice reckoned do amount to 33 Days, that is one Month of 30 Days and 3 Days over. A∣gain in the sixth Year the 11 Days which the So∣lar exceed the Lunar, being thrice numbred, do amount to 33 Days. which with the 3 Days for∣merly reserved do make 36 Days, that is one Month of 30 Days and 6 Days over. Again in the Ninth Year there are also 33 Days, to which the 6 Days reserved being added, there will a∣rise one Month more and 9 Days over. But in

the Eleventh Year twice 11 Days being added to the 9 Days reserved, do make 31 Days, that is, one Month of 30 days and one day over, which be∣ing added to the supernumerary days in the four∣teenth Year do make another Month of 30 Days and 4 Days over, and these being added to the supernumerary Days in the sevententh Year do make another Month of 30 and 7 Days over, and these 7 Days being added to the 22 supernu∣merary Days in the Ninteenth Year of the Moons Cycle do make another Month of 29 Days.

4. But because there are 6939 Days and 18 Hours in 19 Solar Years, that is, 4 Days 18 Hours more then in the common and Embolismical Lu∣nar Years, in which the excess between the Lu∣nar and the Solar Year is supposed to be no more then 11 Days in each Year, whereas in every fourth Year the excess is one Day more, that is, 12 Days, that is, in 16 Years 4 Days, and in the remaining 3 Years three fourths of a day more. And that the new Moons after 19 Lunar Years or 235 Lunations do not return to the same days again, but want almost 5 days, it is evident that the civil Lunations do not agree with the Astro∣nomical and that there must be yet some kind of intercalation used.

5. Now therefore in distributing the golden Number throughout the Calendar. If the new Moons should interchangeably consist of 30 and 29 days, and so but 228 Lunations in 19 Years; we might proceed in the same order in which we have begun, and by which as hath been shewed the third Year of the Golden Number falls upon the Calends of Ianuary. But for as much as there are first six Lunations of 30 days apiece and

one of 29 days to be interposed, therefore there must be 6 times 2 Lunations together consisting of 30 days and once three Lunations of 29 days. And that respect may be also had to the Bissextile days, although they are not exprest in the Calen∣dar, that Lunation which doth contain the Bis∣sertile day, if it should have been 29 days, it must be 30, if it should have consisted of 30 days it must consist of 31.

6. And because it was thought convenient, as hath been shewed, to begin with the third Year of the Cycle of the Moon, because the Golden Num∣ber 3 is set to the Calends of Ianuary, therefore in this Cycle the Embolismical Years are, 2, 5, 8, 11, 13, 16, 19. But yet that it may ap∣pear, that these Years are in effect the same, as if we had begun with the first Year of the Gol∣den Number, save only that the eighth Year in∣stead of the ninth is to be accounted Embolismi∣cal, I have added the Table follwing, in which it is apparent that the former Embolismical years do agree with these last mentioned.

7. But as I said before, it was thought more convenient to begin the account from the num∣ber 3 set to the Calends of Ianuary, because by so reckoning 30 and 29 days to each Lunation interchangeably, the same Number 3 falls upon Ianuary 31. March 1, and 31. April 29. May 29. Iune 27. Iuly 27. August 25. September 14. October 23. November 22. December 21. As if the Lunar years were compleated upon the 20 of December there remain just 11 Days, which the Solar years doth exceed the Lunar.

8. And by ranking on and accounting 4 for the Golden Number of the next year, you will

find it set on Ianuary 20, February 18, March 20, April 18, May 18, Iune 16, Iuly 16, August 14, September 13, Octob. 12, No∣vemb. 11, Decemb. 10.

Cycle of the Moon.	Cycle of the Moon.	Embolismical Years.	Number of Days.
1	3		354
2	4		354
3	5	Embol.	384
4	6		354
5	7		354
6	8	Embol.	384
7	9		354
8	10		354
9	11	Embol.	384
10	12		354
11	13	Embol.	384
12	14		354
13	15		354
14	16	Embol.	384
15	17		354
16	18		354
17	19	Embol.	384
18	1		354
19	2	Embol.	384

9. But in going on, and taking 5 for the Golden Number in the third year, we must remember that that is an Embolis∣mical Year, and therefore that some∣where there must be 2 Months together of 30 days. And for this reason the Gol∣den Number 5, is set to Ianuary 9, Febru∣ary 7, March 9, April 7, May 7, Iune 5, Iuly 5, August 3, Se∣ptember 2, as also up∣on the second day of October, and not up∣on the first, that so there may be 2 Luna∣tions together of 30, and the same Num∣ber 5 is also set to the thirty first of October, to make the Lunation to consist of 29 days, and to the thirtieth of November in∣stead of the twenty ninth, that so a Lunation of

30 may again succeed as it ought.

10. In like manner in the sixth Year, having gone through the fourth and fifth as common years, you may see the Golden Number 8 set to the fifth of April, which should have been upon the fourth, and in the ninth Year the Golden Number 11 is set to the second of February which should have been upon the first.

And there is a particular reason, for which these numbers are otherwise placed from the eighth of March to the fifth of April, namely, that all the paschal Lunations may consist of 29 days: For thus from the eighth of March to the sixth of April, to both which days the Golden Number is 16, there are but 29 days. And from the ninth of March to the seventh of April, to both which days the Golden Number is 5, there are also 29 days, and so of the rest till you come to the fifth of April, which is the last Paschal Lunation, as the eighth of March is the first, but at any other time of the Year, the length of the Month in the Embolismical Year, may be fixed as you please.

12. And in this manner in the 17 years, in which the lunations of the whole Circle are fi∣nished, and in which the Golden Number is 19, the Month of Iuly is taken at pleasure, to the thir∣tieth day whereof is set the Golden Number 19, which should have been upon the thirty first, and the same Number being notwithstanding placed upon the twenty eighth of August, that by the two Lunations of 29 days together, it might be understood, that the seventh Embolismical Month consisting of 29 days is there inserted, instead of a Month of 30 days. In which place the Em∣bolismical or leaping Year of the Moon may

plainly be observed for that year is one day less than the rest, which the Moon doth as it were pass over. The which one day is again added to the 29 days of the last Month, that we may by that means come, as in other Years, to the Gol∣den Number, which sheweth the New Moon in Ianuary following. And for this reason the E∣pact then doth not consist of 11 but of 12 days. And thus you see the reason, for which the Gol∣den Numbers are thus set in the Calendar as here you see. In which we may also observe, that every following Number is made by adding 8 to the Number preceding, and every preceding Num∣ber is also made by adding 11 to the Number next following, and casting away 19 when the addition shall exceed it.

For Example, if you add 8 to the Golden Number 3 set against the first of Ianuary, it ma∣keth 11, to which add 8 more and it maketh 19, to which adding 8 it maketh 27, from which substracting 19 the remainer is 8, to which again adding 8, the sum is 19, to which adding 8 the sum is 24, from which deducting 19 the remainer is 5, and so of the rest. In like manner receding backward, to the 5 add 11 they make 16, to the 16 add 11 they make 27, from which deducting 19 the remainer is 8, to which 11 being added the same is 19, to which 11 being added the sum is 30, from which deducting 19 the remainer is 11, to which 11 being added the sum is 22, from which deducting 19 the remainer is 3. And by this we may see that every following number will be in use 8 years after the preceding, and every prece∣ding Number will be in use 11 years after the fol∣lowing, that is, the same will return to be in use after

8 Years and 11, and the other after 11 Years and 8, or once in 19 years.

CHAP. III.

Of the Vse of the Golden Number in finding the Feast of Easter.

THe Cycle of the Moon or Golden Number is a circle of 19 years, as hath been said al∣ready, which being distributed in the Calendar as hath been shewn in the last Chapter, doth shew the day of the New Moon for ever; though not exactly: But the use for which it was chiefly in∣tended, was to find the Paschal New Moons, that is, those new Moons on which the Feast of Easter and other moveable Feasts depend. To this purpose we must remember,

1. That the vernal Equinox is supposed to be fixed to the twenty first day of March.

2. That the fourteenth day of the Moon on which the Feast of Easter doth depend, can ne∣ver happen before the Equinox; though it may fall upon it or upon the day following.

3. That the Feast of Easter is never observed upon the fourteenth day of the Moon, but upon the Sunday following; so that if the fourteenth day of the Moon be Sunday, the Sunday follow∣ing is Easter day.

4. That the Feast of Easter may fall upon the fifteenth day of the Moon, or upon any other day unto the twenty first, inclusively.

5. That the Paschal Sunday is discovered by

the proper and Dominical Letter for every Year The which may be found as hath been already de∣clared, or by the proper Table for that purpose. Hence it followeth,

1. That the New Moon immediately prece∣ding the Feast of Easter, cannot be before the eighth day of March, for if you suppose it to be upon March 6, the Moon will be 14 days old March 19, which is before the Equinox, contra∣ry to the second Rule before given, and upon the seventh day of March there is no Golden Number fixed; and therefore the Golden Number 16, which standeth against March 8, is the first by which the Paschal New Moon may be disco∣vered.

2. It followeth hence, That the last Paschal New Moon cannot happen beyond the fifth day of April, because all the 19 Golden Numbers are expressed from the eighth of March to that day. And if a New Moon should happen upon the sixth of April, there would be two Paschal New Moons that year, one upon the eighth of March and a∣nother upon the sixth of April, the same Golden Number 16 being proper to them both, but this is absurd because Easter cannot be observed twice in one year.

3. It followeth hence, That the Feast of Easter can never happen before the twenty second day of March, nor after the twenty fifth day of April: For if the first New Moon be upon the eighth of March, and that the Feast of Easter must be upon the Sunday following the fourteenth day of the Moon; it is plain that the fourteenth day of the Moon must be March 21 at the soonest: So that supposing the next day to be Sunday, Easter can∣not

not be before March the twenty second. And because the fourteenth day of the last Moon fal∣leth upon the eighteenth day of April, if that day be Saturday, and the Dominical Letter D, Easter shall be upon the nineteenth day, but if it be Sun∣day, Easter cannot be till the twenty fifth.

4. It followeth hence, That although there are but 19 days, on which the fourteenth day of the Moon can happen, as there are but 19 Golden Numbers, yet there are 35 days from the twenty second of March to the twenty fifth of April, on which the Feast of Easter may happen, because there is no day within those Limits, but may be the Sunday following the fourteenth day of the Moon. And although the Feast of Easter can ne∣ver happen upon March 22, but when the four∣teenth day of the Moon is upon the twenty first, and the Sunday Letter D, nor upon the twenty fifth of April, but when the fourteenth day of the Moon is upon April 18, and the Dominical Letter C. Yet Easter may fall upon March 23, not only when the fourteenth day of the Moon is upon the twenty second day which is Saturday, but also if it fall upon the twenty first which is Friday. In like manner Easter may fall upon April 24, not only when the fourteenth day of the Moon is upon the eighteenth day which is Monday, but also if it happen upon the seventeenth being Sunday. And for the same reason it may fall oftner upon other days that are further distant from the said twenty second of March and twenty fifth of April.

5. It followeth hence, That the Feast of Easter may be easily found in any Year propounded: For the Golden Number in any Year being given, if

you look the same between the eighth of March and fifth of April both inclusively, and reckon 14 days from that day, which answereth to the Golden Number given, where your account doth end is the fourteenth day of the Moon: Then consider which is the Dominical Letter for that Year, and that which followeth next after the four∣teenth day of the Moon is Easter day. Example, In the year 1674 the Golden Number is 3, and the Sunday Letter D, which being sought in the Calendar between the aforesaid limits, the four∣teenth day of the Moon is upon April the thir∣teenth, and the D next following is April 19. And therefore Easter day that Year is April 19. Otherwise thus.

In March after the first C, Look the Prime wherever it be, The third Sunday after Easter day shall be. And if the Prime on Sunday be, Reckon that for one of the Three.

6. Thus the Feast of Easter may be found in the Calendar, and from thence a brief Table shew∣ing the same, may be extracted in this manner. Write in one Column the several Golden Num∣bers in the Calendar from the eighth of March to the fifth of April, in the same order observing the same distance. In the second Column set the Dominical Letters in number 35 so disposed, as that no Dominical Letter may stand against the Golden Number 16, but setting the Letter D against the Golden Number 5, write the rest in

this order. E, F, G, A, B, &c. and when you come to the Golden Number 8, set the Letter C, and there continue the Letters till you come to C again, because when the Golden Number is 16, which in the Calendar is set to the eighth day of March, is new Moon, and the fourteenth day of that Moon doth fall upon the twenty first, to which the Dominical Letter is C, upon which the Feast of Easter cannot happen; and therefore in the third Column containing the day in which the Feast of Easter is to be observed, is also void. But in the next place immediately following, to wit, against the letter D is set March 22, because if the fourteenth day of the Moon shall fall upon the twenty first of March being Saturday, the next day being Sunday, shall be the Feast of Easter.

To the Letters following, E, F, G, A, B, &c. are set 23, 24, 25, and so orderly to the last of March, and so forward till you come to the twen∣ty fifth of April, by which Table thus made, the Feast of Easter may be found until the Calendar shall be reformed.

For having found the Golden Number in the first Column, the Dominical Letter for the Year next after it, doth shew the Feast of Easter, as in the former Example, the Golden Number is 3 and the Dominicall Letter D, therefore Easter day is upon April 19. The other moveable Feasts are thus found.

Advent Sunday is always the nearest Sunday to St. Andrews, whether before or after.

Septuagesima Sunday is Nine Weeks before Easter.

Sexagesima Sunday is Eight Weeks before Easter.

Qainquagesima Sunday is Seven Weeks be∣fore Easter.

Quadragesima Sunday is Six Weeks before Easter.

Rogation Sunday is five Weeks after Easter.

Ascension day is Forty Days after Easter.

Whitsunday is Seven Weeks after Easter.

Trinity Sunday is Eight Weeks after Easter.

G. N.	D. L.	Easter.
XVI
V	D	22 March
	E	23
XIII	F	24
II	G	25
	A	26
X	B	27
	C	28
XVIII	D	29
VII	E	30
	F	31
XV	G	1 April
IV	A	2
	B	3
XII	C	4
I	D	5
	E	6
IX	F	7
	G	8
XVII	A	9
VI	B	10
	C	11
XIV	D	12
III	E	13
	F	14
XI	G	15
	A	16
XIX	B	17
VIII	C	18
	D	19
	E	20
	F	21
	G	22
	A	23
	B	24
	C	25

CHAP. IV.

Of the Reformation of the Calendar by Pope Gregory the Thirteenth; and substituting a Cycle of Epacts in the room of the Golden Number.

HItherto we have spoken of the Calendar which is in use with us, we will now shew you for what reasons it is alter'd in the Church of Rome, and how the Feast of Easter is by them observed.

The Year by the appointment of Iulius Caesar consisting of 365 days 6 hours, whereas the Sun doth finish his course in the Zodiack, in 365 days 5 hours 49 minutes or thereabouts, it cometh to pass that in 134 Years or less, there is a whole day in the Calendar more than there ought; in 268 years 2 days more; in 4002 years 3 days: and so since Iulius Caesar's time the vernal Equinox hath gone backward 13 or 14 days, namely from the 24 of March to the tenth. Now because the Equi∣nox was at the time of the Nicene Council upon the twenty first of March, when the time for the ob∣serving of Easter was first universally established, they thought it sufficient to bring the Equinox back to that time, by cutting off 10 days in the Calendar as hath been declared, and to prevent any anticipation for the time to come, have ap∣pointed, that the Leap-year shall be thrice omit∣ted in every 400 Years to come, and for memory sake, appointed the first omission to be account∣ed from the Year 1600, not from 1582, in which the reformation was made, because it was not only near the time, in which the emendation was begun, but also because the Equinox has not fully made an anticipation of 10 days from the

place thereof, at the time of the Nicene Council, which was March 21.

The Years then 1700, 1800, 1900, which should have been Bissextile Years, are to he ac∣counted common years, but the Year 2000 must be a Bissextile: In like manner the Years 2100, 2200, 2300, shall be common years, and the Year 2400 Bissextile, and so forward.

2. Again, because it was supposed that the Cycle of the Moon, or Golden Number was so fixed, that the new and full Moons would in eve∣ry 19 years return to the same days again; where∣as their not returning the same hours, but making an anticipation of one hour 27 minutes or there∣abouts, it must needs be that in 17 Cycles or lit∣tle more than 300 Years, there would be an anti∣cipation of a whole day. And hence it is evi∣dent that in 1300 Years since the Nicene Council, the New and Full Moons do happen more than 4 days sooner than the Cycle of the Moon or Gol∣den Number doth demonstrate: Whence also it comes to pass, that the fourteenth day of the Moon by the Cycle is in truth the eighteenth day, and so the Feast of Easter should be observed not from the fifteenth day of the Moon to the twenty first, but from the nineteenth to the twenty fifth.

3. That the Moon therefore being once brought into order, might not make any antici∣pation for the time to come, it is appointed that a Cycle of 30 Epacts should be placed in the Ca∣lendar instead of the Golden Number, answering to every day in the Year; to shew the New Moons in these days, not only for 300 Years or there∣abouts, but that there might be new Epacts with∣out

altering the Calendar, to perform the same thing upon other days as need shall require.

4. For the better understanding whereof, to the Calendar in use with us, we have annexed the Gregorian Calendar also: In the first Column whereof you have 30 numbers from 1 to 30, save only that in the place of 30 you have this Asterisk *, But they begin with the Calends of Ianuary, and we continued and repeated af∣ter a Retrograde order in this manner, *29, 28, 27, &c. and that for this cause especially, that the number being given which sheweth the New Moons in every Month for one Year, you might by numbring 11 upwards exclusively find the number which will shew the New Moons the Year following, to wit, the Number which fal∣leth in the eleventh place.

5. And these Numbers are called Epacts, be∣cause they do in order shew those 11 days, which are yearly to be added to the Lunar Year consist∣ing of 354 days, that it may be in conformity with the Solar Year consisting of 365 days. To this purpose, as hath been said concerning the Golden Number, these Epacts being repeated 12 times, and ending upon the twentieth day of De∣cember, the same Numbers must be added to the 11 remaining days, which were added to the first 11 days in the Month of Ianuary.

6. And because 12 times 30 do make 360, whereas from the first of Ianuary to the twentieth of December inclusively, there are but 354 days, you must know that to gain the other six days, the numbers 25 and 24 are in every other Month both placed against one day, namely, to February 5, April 5, Iune 3, August 1, September 29, and

November 27. But why these two Numbers are chosen rather then any other, and why in these 6 Months the number 25 is sometimes writ to XVI, sometimes to XXV in a common character, and why the number 19 is set to the last day of De∣cember in a common Character, shall be declared hereafter.

7. Here only note that this Asterisk * is set in∣stead of the Epact 30, because the Epact shew∣ing the Number of days which do remain after the Lunation in the Month of December, it may some∣times fall out that 2 Lunations may so end, that the one may require 30 for the Epact, and the other 0, which would, if both were written, cause some inconveniences, and therefore this * Aste∣risk is there set, that it might indifferently serve to both. And the Epact 29 is therefore set to the second day of Ianuary, because after the com∣pleat Lunation in the second of December there are 29 days, and for the like reason the Epact 28 is set against the third of Ianuary, because after the compleat Lunation in the third of December there are then 28 days over, and so the rest in order till you come to the thirtieth of Ianuary, where you find the Epact 1. because after the compleat Lunation on the thirtieth day there is only one day over.

8. And besides the shewing of the New Moons in every Month, which is and may be done by the Golden Number, the Epacts have this advan∣tage, that they may be perpetual and keep the same place in the Calendar in all future ages, which can hardly be effected with the Golden Number, for in little more then 700 years, the New Moons do make an anticipation of one day, and then it

will be necessary to set the Golden Number one degree backward, and so the Golden Number which at the time of the Nicene Council was set to the first of Ianuary, should in 300 years be set to the last of December, and so of the rest, but the Epacts being once fixed shall not need any such retraction or commutation. For as often as the New Moons do change their day either by Anticipation or by Suppression of the Bissextile year, you shall not need to do any more than to take another rank of 19 Epacts, insteed of those which were before in use. For instance, the Epacts which are and have been in use in the Church of Rome since the year of reformation 1582, and will continue till the year 1700, are these 10 fol∣lowing 1. 12. 23. 4. 15. 26. 7. 18. 29. 10. 21. 2. 13. 24. 5. 16. 27. 8. 19. And from the year 1700 the Epacts which will be in use are these. * 11. 22. 3. 14. 25. 6. 17. 28. 9. 20. 1. 12. 23. 4. 15. 26. 7. 18. and shall con∣tinue not only to the year 1800, but from thence until the year 1900 also; and although in the year 1800 the Bissextile is to be suppressed, yet is there a compensation for that Suppression, by the Moons Anticipation. To make this a little more plain, the motion of the Moon, which doth occasion the change of the Epact, must be more fully considered.

CHAP. V.

Of the Moons mean Motion, and how the Antici∣pation of the New Moons may be discovered by the Epacts.

THe Moon according to her middle motion doth finish her course in the Zodiack in 29 days, 12 hours 44 minutes, three seconds or there∣about, and therefore a common Lunar year doth consist of 354 days, 8 hours, 48 minutes, 38 seconds and some few thirds, but an Embolismi∣cal year doth consist of 383 days, 21 hours, 32 minutes, 41 seconds and somewhat more; and therefore in 19 years it doth exceed the motion of the Sun 1 hour, 27 minutes, 33 seconds feré.

2. Hence it cometh to pass, that although the New Moons do after 19 years return to the same days; yet is there an Anticipation of 1 hour, 27 minutes, 33 seconds. And in twice 19 years, that is, in 38 years, there is an Anticipation of 2 hours, 55 minutes, 6 seconds, and after 312 years and a half, there is an Anticipation of one whole day and some few Minutes. And therefore after 312 years no new Moon can happen upon the same day it did 19 years before, but a day sooner. Hence it comes to pass that in the Julian Calen∣dar, in which no regard is had to this Anticipa∣tion, the New Moons found out by the Golden Number must needs be erroneous, and from the time of the Nicene Council 4 days after the New Moons by a regular Computation.

3. And hence it follows also, that if the Gol∣den Number, after 312 were upon due conside∣ration

removed a day forwarder or nearer the be∣ginning of the Months, they would shew the New Moons for 312 years to come. And being again removed after those years, a day more would by the like reason do the same again. But it was thought more convenient so to dispose 30 Epacts, that they keeping their constant places, 19 of them should perform the work of the Gol∣den Number, until by this means there should be an Anticipation of one day. And when such an Anticipation should happen, those 19 Epacts be∣ing let alone, other 19 should be used, which do belong to the preceding day, without making any alteration in the Calendar.

4. And if this Anticipation would do the whole work, nothing were more plain, then to make that commutation of the 19 Epact once in 312 years: but because the detraction of the Bissex∣tile days doth variously interpose and cause the 19 Epacts sometimes to be changed into these that do precede, sometimes into these that follow, sometimes into neither, but to continue still the same; therefore some Tables are to be made, by which we may know, when the commutation was to be made and into what Epacts.

4. First therefore there was made a Table cal∣led Tabula Epactarum Expansa, in this manner.

First on the top were placed the 19 Golden Numbers in order, beginning with the Number 3, which in the old Calendar is placed against the Calends of Ianuary, and under every one of these Golden Numbers there are placed 30 Epacts all constituted from the lowest number in the first rank in which the Epact is 1, and in that first rank the Golden Number is 3, the rest from

thence towards the right Hand are made by the constant addition of it, and the casting away of 30, as often as they shall exceed that number, only when you come to the 27, the Epact under the Golden Number 19, there must be added 12 instead of 11, that so the Epact following may be 9 not 8, for the Reasons already given in this Discourse concerning the Golden Number and Embolismical years. And this rank being thus made, the other Epacts are disposed in their na∣tural order ascending upwards, and the number once again resumed after the Epact 30 or rather this Asterisk * set in the place thereof: only ob∣serve that under the Golden Number 12. 13. 14. 15. 16. 17. 18. 19. in the place of XX there is yet 25 in the common Character. And to the Epacts under the Golden Number 19, 12 must still be added to make that Epact under the Golden Number 1. As was said before concern∣ing the lowest Rank.

5. And on the left hand of these Epacts before those under the Golden Number 3. are set 30 Letters of the Alphabet, 19 in a small Character, and 11 in a great, in which some are passed by, for no other reason save only this, that their simili∣tude with some of the small Letters, should not occasion any mistake in their use, which shall be shewed in its place.

6. Besides this Table there was another Table made which is called Tabula AEquationis Epacta∣rum, in which there is a series of years, in which the Moon, by reason of her mentioned anticipa∣tion doth need AEquation, and in which the num∣ber of Epacts signed with the letters of the Al∣phabet, are to be changed; being otherwise AE∣quated

where it needeth, by the suppression of the Bissextile days.

7. But it supposeth, that it was convenient to suppress the Bissextiles once only in 100 years; and the Moon to be aequated, or as far as concerns her self, the rank of Epacts to be changed, once only in 300 years, and the 12 years and a half more, to be referred till after the years 2400, they do amount unto 100 years, and then an ae∣quation to be made: but then it must be made by reason of the interposing this hundred not in the three hundredth but the hundredth year. Moreover this aequation is to be made as in refe∣rece to the Moon only, because as the suppression of the Bissextiles intervene, the order of chang∣ing the ranks of Epacts is varied, as shall be shewed hereafter.

8. Again this Table supposeth, that seeing the New Moon at the time of the Nicene Council was upon the Calends of Ianuary, the golden Number 3 being there placed, that it would have been the same if the Epact * had been set to the same Calends, that is if the Epacts had been then in use. And therefore at that time the highest or last rank of Epacts was to be used, whose Index is P, and then after 300 years, the lowest or first rank should succeed, whose Index is a, (for the letters return in a Circle) and af∣ter 300 years more, the following rank whose Index is b and so forward; but that it is concei∣ved, that the New Moon in the Calends of Ia∣nuary, is more agreeable to the year of Christ 500, than the time of the Nicene Councel; and therefore as if the rank of Epacts under the let∣ter l were sutable to the year 500, it seemed

good to make use of that rank under the letter a in the year of Christ 800, and those under the let∣ter b, in the year 1100, and those under the let∣ter e in the year 1400.

9. Which being granted, because in the year 1582, ten days were cut off from the Calendar, we must run backward, or in an inverted order count 10 series, designed, suppose, by the letters b. a. P. N. M. H. G. F. E. D. so that from the year 1582 the series of Epacts whose literal Index is D, is to be used, and this is that rank of Epacts which is now used in the Church of Rome.

10. And therefore as if this Table had its be∣ginning from that year; the first number in the second column is 1582, and then in order un∣der it. 1600. 1700. 1800. 1900. 2000. &c. And in the third Column every fourth hundred year is marked for a Bissextile, that is, 1600. 2400. 2800, &c. and in the fourth Column to eve∣ry three hundreth▪ Year is set this Character C, to shew in what year the Moon by her Anticipati∣on of one day, doth need aequation; but in the year 1800 the double character is set CC, to sig∣nify that then another hundred years are got∣ten by the 12 years and a half reserved, besides and above the other 300 years; and this charact∣er is also set to the years 4300. 6800, and for the same reason.

But in the first Column, or on the left hand of these years are placed the Letters or Indices of those ranks of Epacts in the former Table, which are to be used in those years and when the Let∣ters are charged. Thus against the year 1600 the Letter D is continued, to shew that from that

year, to the year 1700 the rank of Epacts is still to be used, which do belong to that Letter. And for as much as the Letter C is set to the year 1700, it sheweth that that rank of Epacts is then to be used, which do belong thereto, and so of the rest.

11. The reason why these Letters in the first Column are sometimes changed in 100 years, sometimes in 200, sometimes not in less then 300 Years, and that they are some∣times taken forward, sometimes backward, according to the order of the Alphabet, is because the suppression of the Bissextiles do intervene with the lunar aequation: for if the Bissextile were only to be suppressed, in these 300 or sometimes 400 years, in which the Moon needeth aequation, the rank of Epacts in that case would need no commutation, but would continue the same for ever; and the gol∣den Number would have been sufficient, if the suppression of the Bissextile, and anticipation of the Moon, did by a perpetual compensati∣on cause the new Moons still to return to the same days: but because the Bissextile is ofttimes suppressed, when the Moon hath no aequation, the Moon hath sometimes an aequation when the Bis∣sextile is not suppressed, sometimes also both are to be done and sometimes neither; all which varieties may yet be reduced to these three Rules.

1. As often as the Bissextile is suppressed without any aequation of the Moon, then the let∣ter which served to that time shall be changed to the next below it contrary to the order of the Alphabet. And the new Moons shall be removed

one day towards the end of the Year.

2. As often as the Moon needeth aequation, without suppression of the Bissextile, then the Letter which was in use to that time shall be chan∣ged to the next above it according to the order of the Alphabet, that the New Moons may a∣gain return one day towards the beginning of the year.

3. As often as there is a Suppression and an aequation both, or when there is neither, the Letter is not changed at all but that which ser∣ved for the former Centenary, shall also conti∣nue in the succeeding; because the compensation so made, the New Moons do neither go for∣ward nor backward, but happen in the compass of the same days.

1. And this is enough to shew for what rea∣son the letters are so placed in the Table, as there you see them: for in the year 1600 the Bissextile being neither suppressed, nor the Moon aequated, the letter D used in the former Cen∣tenary or in the latter part thereof from the year 1582, is still the same.

In the year 1700, because there is a suppressi∣on, but no aequation, the commutation is made to the Letter C descending.

In the Year 1800, because there is both a sup∣pression and an aequation, the same letter C doth still continue.

In the Year 2400, because there is an aequation and no suppression, there is an ascension to the Letter A.

And thus you see not only the construction of this Table, but how it may be continued to any other Year, as long as the World shall last.

12. And by these two Tables we may easily know which rank of the 30 Epacts doth belong to, or is proper for any particular age: for as in our age, that is, from the Year 1600 to the Year 1700 exclusively, that series is proper whose Index is D. Namely, 23, 4, 15, 26, &c. so in the two Ages following, that is, from the Year 1700 to the Year 1900 exclusively, that series is proper whose In∣dex is C, namely these, 22, 3, 14, 25. and in the three ages following thence, that is from the Year 1900 to the Year 2100 exclusively, that series is proper whose Index is B, namely these, 21, 2, 13, 24, &c. And so for any other.

Hence also it may be known, which of the 19 doth belong to any particular Year, for which no more is necessary, than only to know the Gol∣den Number for the year given, which being sought in the head of the Table, and the Index of that Age in the side, the common Angle, or meeting of these two, will shew you the Epact desired: As in the year 1674 the Golden Number is 3 and the Index D; therefore in the common Angle I find 23 for the Epact that year, and shew∣eth the New Moons in every Month thereof.

And here it will not be unseasonable to give the reason, for which the Epact 25 not XXV is writ∣ten under the Golden Numbers 12, 13, 14, 15, 16, 17, 18, 19. namely, because the ranks of E∣pacts, which under these greater Numbers hath this Epact 25, hath also XXIV, it would follow that in these Ages in which any of these Ranks were in use, the New Moon in 19 years will hap∣pen twice upon the same days; in those six Months in which the Epacts XXV and XXIV are set to the same day: Whereas the New Moons do not hap∣pen

on the same day till 19 years be gone about. To avoid this inconvenience, the Epact 25 not XXV is set under these great numbers, and the Epact 25 is in the Calendar, in these Months set with the Epact XXVI, but in the other Months with the Epact XXV.

14. Hence it cometh to pass, 1. That in these Years the Epacts 25 and XXIV do never meet on the same day. 2. That there is no danger that the Epacts 25 and XXVI should in these 6 Months cause the same inconvenience, seeing that the E∣pacts 25 and XXVI are never both found in the same Rank. 3. That the Epact 25 may in other Months without inconvenience be set to the same day with the Epact XXVI, because in these there is no danger of their meeting with the Epact XXIV on the same days. 4. That there is no fear that the Epacts XXV and XXIV being set on the same days, should in future Ages cause the same inconvenience, because the Epacts XXV and XXIV are not found together in any of the other Ranks. But that either one or both of them are wanting. Besides, when one of these Epacts is in use, the o∣ther is not, and that only which is in use is pro∣per to the day. As in this our Age until the Year 1700 the Epacts in use are those in the rank whose Index is D. In which these two XXIV and XXV are not both found. And in the two following Ages, because the rank of Epacts in use is that whose Index is C, in which there is the Epact XXV, not XXIV, the New Moons are shewed by the Epact XXV not by XXIV. But because in three following Ages, the rank of Epacts in use is that whose Index is B, in which 25 and XXIV are both found, the New

Moons are shewed by the Epact XXIV when the golden Number is 6. And by the Epact 25 when the golden Number is 17, and not by the Epact XXV.

15. And if it be asked why the Epact 19 in the common Character is set with the Epact XX against the last day of December; know that for the reasons before declared, the last Embo∣lismical Month within the space of 19 years, ought to be but 29 days and not 30, as the rest are; and therefore when the Epact 19 doth concur with the golden Number 19, the last Month or last Lunation beginning the second of December, shall end upon the 30 and not up∣on the 31 of that Month, and the New Moon should be supposed to happen upon the 31 un∣der the same Epact 19, that 12 being added to 19 and not 11, you may have one for the Epact of the year following, which may be found up∣on the 30 of Ianuary, as if the Lunation of 30 days had been accomplished the Day be∣fore.

CHAP. VI.

How to find the Dominical Letter and Feast of Easter according to the Gregorian account.

HAving shewed for what reason, and in what manner the Epacts are substituted in the place of the golden Number, and how the New Moons may be by them found in the Calendar for ever; I shall now shew you how to find the Feast of Easter and the other moveable

Feasts according to the Gregorian or new ac∣count; and to this purpose I must first shew you how to find the Dominical Letter, for that the Cycle of 28 years will not serve the turn, be∣cause of the suppression of the Bissextile once in a hundred years, but doth require 7 Cycles of 28 years apeice. The first whereof begins with CB, and endeth in D. The second begins with DC, and endeth in E. The third begins with ED, and endeth in F &c. The first of these Cycles began to be in use 1582, in which year the dominical Letter according to the Julian ac∣count was G, but upon the fifteenth day of October, that Year was changed to C: for the fifth of October being Friday and then called the fifteenth, the Letter A became Friday, B Satur∣day, and C Sunday, the remaining part of the year, in which the Cycle of the Sun was 23, and the second after the Bissextile or leap Year, and so making C, which answereth to the fifteenth year of that Circle, to be 23, the Circle will end at D; and consequently CB, which in the old account doth belong to the 21 year of the Circle, hath ever since been called the first, and so shall continue until the year 1700, in which the Bissextile being suppressed, the next Cycle will begin with DC as hath been said already. Under the first rank or order of Dominical Letters are written the years 1582 and 1600, under the se∣cond 1700, under the third 1800, under the fourth 1900 and 2000, under the fifth 2100, un∣der the sixth 2200 and under the seventh 2300 and 2400. And again under the first Order, 2500, under the second 2600, under the third 2700 and 2800, and so forward as far as you

please, always observing the same order, that the 100 Bissextile years may still be joyned with the not Bissextile immediately preceding.

1. And hence it appears, that the seven or∣ders of Dominical Letters, are so many Tables, successively serving all future Generations. For as the first Order serveth from the year 1582 and 1600 to the year 1700 exclusively, and the second Order from thence to the year 1800 ex∣clusively, so shall all the rest in like manner which here are set down, and to be set down at plea∣sure. And hence the Dominical Letter or Let∣ters may be found for any year propounded, as if it were required to find the dominical Letter for the year 1674, because the year given is contained in the centenary 1600. I find the Cycle of the Sun by the Rule already given to be 3. In the first order against the number 3, I find G for the Sunday Letter of that year, in like manner because the year 1750 is contained under the Centenary 1700, the Cycle of the Sun being 27, I find in the second rank the Let∣ter D answering to that Number, and that is the Dominical Letter for that year, and so of the rest.

3. Again for as much as the fifth Order is the same with that Table, which serves for the old account, therefore that order will serve the turn for ever where that Calendar is in use, and so this last will be of perpetual use to both the Calendars.

4. Now then to find the time in which the Feast of Easter is to be observed, there is but little to be added to that which hath been already said concerning the Julian Calendar. For the Pas∣chal

Limits are the same in both, the difference is only in the Epacts, which here are used in∣stead of the golden Number.

5. For the terms of the Paschal New Moons are always the eighth of March and the fifth of April: but whereas there are 11 days within these Limits to which no golden Number is affixed, there is now one day to which an Epact is not appointed, because there is no day within those Limits, on which in process of time a New Moon may not happen. And the reason for which the two Epacts XXV and XXIV are both set to the fifth of April, is first general, which was shew∣ed before, namly that by doing the same in 5 other Months, the 12 time 30 Epacts might be contracted to the Limits of the lunar Year which consists of 354 days: but there is a particular rea∣son also for it, that the Antients having appointed that all the Paschal lunations should consist of 29 days, it was necessary that some two of the E∣pacts should be set to one of these days in which the Paschal lunation might happen, the Epacts being 30 in number. And it was thought con∣venient to choose the last day, to which the E∣pact XXV belonging, the Epact XXIV should also be set; and hence by imitation it comes to pass, that these and not other Epacts are set to that day in other Months, in which two Epacts are to be set to the same days.

6. The use of these Epacts in finding the Feast of Easter, is the same with that which hath been shewed concerning the golden Numbers. For the Epact and the Sunday Letter for that year propounded being given, the Feast of Easter may be found in the Calendar after the same

manner. Thus in the year 1674, the Epact is 23 and the Sunday Letter G, and therefore reck∣oning fourteen days from the eighth of March to which the Epact is set, the Sunday following is March 25, which is the day on which the Feast of Easter is observed.

7. And hence as hath been shewed in the third Chapter concerning the Julian Calendar, a brief table may be made to shew the feast of Easter and the other moveable Feasts for ever, in which there is no other difference, save only that the Epacts as they are in this new Calendar, are to be used as the golden Numbers are, which stand in the old Calendar. And a Table having the golden Numbers of the old Calendar set in one Column, and the Epacts as they are in the new Calendar set in another, will indifferently shew the movable Feasts in both accounts, as in the Year 1674, the golden Number is 3 and the Sunday Letter according to the Julian ac∣count is D, according to the Gregorian G, and the Epact 23, and therefore according to this Table our Easter is April 19, and the other, to wit, the Gregorian, is March 25. The like may be done for any other year past or to come.

CHAP. VII.

How to reduce Sexagenary numbers into Decimal, and the Contrary.

EVery Circle hath antiently, and is yet ge∣nerally supposed to be divided into 300 de∣grees, each degree into 60 Minutes, each Mi∣nute into 60 Seconds, and so forward as far as need shall require. But this partition is some∣what troublesom in Addition and Subtraction, much more in Multiplication and Division; and the Tables hitherto contrived to ease that man∣ner of computation, do scarce sufficiently per∣form the work, for which they are intended. And although the Canon published by the lear∣ned H. Gellibrand, in which the Division of the Circle into 360 degrees is retained, but every degree is divided into 100 parts, is much bet∣ter than the old Sexagenary Canon, yet some are of opinion, that if the Antients had divided the whole circle into 100 or 1000 parts, it would have proved much better then either; only they think Custome such a Tyrant, that the alteration of it now will not be perhaps so advantagious; leaving them therefore to injoy their own opinions, they will not I hope be of∣fended if others be of another mind: for their sakes therefore, that do rather like the Deci∣mal way of calculation▪ Having made a Canon of artificial Signs and Tangents for the degrees and parts of a Circle divided into 100 parts, I shall here also shew you, how to reduce sexa∣genary Numbers into Decimal, and the contra∣ry,

as well in time as motion.

2. The parts of a Circle consisting of 360 degrees, may be reduced into the parts of a cir∣cle divided into 100 degrees or parts, by the rule of Three in this manner.

As 360 is to 100, so is any other Number of degrees, in the one, to the correspondent de∣grees and parts in the other.

But if the sexagenary degrees have Minutes and Seconds joyned with them, you must reduce the whole Circle as well as the parts propoun∣ded into the least Denomination, and so proceed according to the rule given.

Example. Let it be required to convert 125 degrees of the Sexagenary Circle, into their correspondent parts in the Decimal. I say, as 360 is to 100, so is 125 to 34, 722222, &c. that is, 34 degrees and 722222 Parts.

2. Example. Let the Decimal of 238 de∣grees 47 Minutes be required. In a whole Cir∣cle there are 21600 Minutes, and in 238 de∣grees, there are 14280 Minutes, to which 47 being added the sum is 14327. Now then I say if 21600 give 100, what shall 14327. The Answ. is 66, 3287 &c. In like manner if it were required to convert the Hours and Minutes of a Day into decimal Parts, say thus, if 24 Hours give 100, what shall any other number of Hours give. Thus if the Decimal of 18 hours were required, the answer would be 75, and the De∣cimal answering to 16 Hours 30 Minutes is 68, 75.

But if it be required to convert the Decimal Parts of a Circle into its correspondent Parts in Sexagenary. The proportion is; as 100 is

to the Decimal given, so is 360 to the Sexage∣nary degrees and parts required.

Example. Let the Decimal given be 349 722222, if you multiply this Number given by 360, the Product will be 1249999992, that is cutting off 7 Figures, 124 degrees and 9999992 parts of a degree. If Minutes be required, mul∣tiply the Decimal parts by 60, and from the product cut off as many Figures, as were in the Decimal parts given, the rest shall be the Mi∣nutes desired.

But to avoid this trouble, I have here exhi∣bited two Tables, the one for converting sex∣agenary degrees and Minutes into Decimals, and the contrary. The other for converting Hours and Minutes into Decimals, and the con∣trary. The use of which Tables I will explain by example.

Let it be required to convert 258 degrees 34′. 47″, into the parts of a Circle decimally divi∣ded.

The Table for this purpose doth consist of two Leaves, the first Leaf is divided into 21 Columns, of which the 1. 3. 5. 7. 9. 11. 13. 15. 17. 19 doth contain the degrees in a sexagenary Circle, the 2. 4. 6. 8. 10. 12. 14. 16. 18 and 20 doth contain the degrees of a Circle Decimally divided, answering to the for∣mer, and the last Column doth contain the De∣cimal parts, to be annexed to the Decimal de∣grees. Thus the Decimal degrees answering to 26 Sexagenary are 7, and the parts in the last Column are 22222222 and therefore the de∣grees and parts answering to 26 Sexagenary de∣grees are 7. 22222222.

In like manner the Decimal of 62 degrees, 17. 22222222. And the Decimal of 258 de∣grees, 34′. 47″, is thus found.

The Decimal of 258 degrees is

71.66666666

The Decimal of 34 Minutes is

.15747040

The Decimal of 47 seconds is

.00362652

Their Sum

71.82776358

is the Decimal of 258 degrees, 34′. 47″ as was required.

In like mauner the Decimal of any Hours and Minutes may be found by the Table for that purpose.

Example. Let the Decimal of 7 Hours 28′ be required.

The Decimal answering to 7h. is

29.16666667

The Decimal of 28 Minutes is

1.94444444

The Sum

31.11111111

is the Decimal Sought.

To find the degrees and Minutes in a sexage∣nary Circle, answering to the degrees and parts of a Circle Decimally divided, is but the contra∣ry work.

As if it were required to find the Degrees and minutes answering to this decimal 71. 02776359, the Degrees or Integers being sought in the 2. 4. 6 or 8 Columns &c. of the first Leaf of that Table, right against 71. I find 256 and in the last Column these parts 11111111, which being less than the Decimal given, I proceed

till I come to 6666667, which being the nea∣rest to my number given, I find against these parts under 71. Degrees 258, so then 258 are the de∣grees answering to the Decimal given and,

To find the Minutes and Seconds from

71.82776359

I Substract the num∣ber in the Table

71.66666667

The remainer is

16109692

which being Sought in the next Leaf under the title Minutes, the next leaf is

11747640

And the Minutes 34, and this number being Subtracted the remainer is

00362652

Which is the Decimal of 47 seconds, and so the degrees and Minutes answering to the De∣cimal given are 258 degrees 34′ and 47″, the like may be done for any other.

CHAP. VIII.

Of the difference of Meridiens.

HAving in the first part shewed how the pla∣ces of the Planets in the Zodiack may be found by observation, and how to reduce the time of an observation made in one Country, to the correspondent time in another, as to the day of the Month, by considering the several

measures of the year in several Nations, there is yet onething wanting, which is, by an obser∣vation made of a Planets place in one Country to find when the Planet is in that place in refe∣rence to another; as suppose the ☉ by observa∣tion was found at Vraniburg to be in ♈. 3^d. 13′. 14″. March the fourteenth 1583 at what time was the Sun in the same place at London? To re∣solve this and the like questions, the Longitude of places from some certain Meridian must be known; to which purpose I have here exhibi∣ted a Table shewing the difference of Meridians in Hours and Minutes, of most of the eminent places in England from the City of London, and of some places beyond the Seas also. The use whereof is either to reduce the time given under the Meridian of London to some other Meridi∣an, or the time given in some other Meridian to the Meridian of London.

1. If it be required to reduce the time given under the Meridian of London to some other Meridian, seek the place desired in the Catalogue, and the difference of time there found, either add to or subtract from the times given at Lon∣don, according as the Titles of Addition or Sub∣traction shew, so will the time be reduced to the Meridian of the other place as was required. Example. The same place at London was in the first Point of ♉, 6 Hours P. M. and it is required to reduce the same to the Meridian of Vraniburg I therefore seek in Vraniburg in the Catalogue of places, against which I find 50′ with the Letter A annexed, therefore I conclude, that the Sun was that day at Vraniburg in the first point of ♉, 6 Hours 50′. P. M.

2. If the time given be under some other Me∣ridian, and it be required to reduce the same to the Meridian of London, you must seek the place given in the Catalogue, and the difference of time there found, contrary to the Title is to be added or subtracted from the time there gi∣ven.

Example. Suppose the place of the Sun had been at Vraniburg, at 6 Hours 50′. P. M. and I would reduce the same to the Meridian of Lon∣don; against Vraniburg as before I find 50′ A. therefore contrary to the Title I Subtract 50′ and the remainder 6 Hours is the time of the Suns place in the Meridian of London.

CHAP. IX.

Of the Theory of the Sun's or Earth's Motion.

IN the first part of this Treatise we have spok∣en of the primary Motion of the Planets and Stars, as they are wheeled about in their di∣urnal motion from East to West, but here we are to shew their own proper motions in their several Orbs from West to East, which we call their second motions.

1. And these Orbs are supposed to be Ellip∣tical, as the ingenious Repler, by the help of Tycho's accurate observations, hath demonstra∣ted in the Motions of Mars and Mercury, and may therefore be conceived to be the Figure in which the rest do move.

2. Here then we are to consider what an El∣lipsis is, how it may be drawn, and by what Me∣thod

the motions of the Planets according to that Figure may be computed.

3. What an Ellipsis is Apollonius Pergaeus in Conicis, Claudius Mydorgius and others have well defined and explained, but here I think it suffici∣ent to tell the Reader, that it is a long Circle, or a circular Line drawn within or without a long Square; or a circular Line drawn between two Circles of different Diameters.

4. The usual and Mechanical way of drawing this Ellipsis is thus; first draw a line to that length which you would have the greatest Diameter to be, as the Line AP in Figure 8, and from the middle of this Line at X, set off with your com∣passes the Equal distance XM and XH.

5. Then take a piece of thred of the same length with the Diameter AP and fasten one end thereof in the point M and the other in the point H, and with your Pen extend the thred thus fastened to the point A, and from thence towards P keeping the thread stiff upon your Pen, draw a line from A by B to P, the line so drawn shall be half an Ellipsis, and in like manner you may draw the other half from P by D to A. In which because the whole thred is equal to the Diameter AP. therefore the two Lines made by thred in drawing of the Ellipsis, must in every point of the said Ellipsis be also e∣qual to the same Diameter AP. They that de∣sire a demonstration thereof geometrically, may consult Apollonius Pergaeus, Claudius Mydorgius or others, in their treatises of Conical Sections, this is sufficient for our present purpose, and from the equality of these two Lines with the Diameter, a brief Method of calculation of the

Planets place in an Ellipsis, is thus Demonstra∣ted by Dr. Ward now Bishop of Salisbury.

6. In this Ellipsis H denotes the place of the Suns Center, to which the true motion of the Planet is referred, M the other Focus whereun∣to the equal or middle motion is numbred, A the Aphelion where the Planet is farthest distant from the Sun and slowest in motion, P the Pe∣rihelion where the Planet is nearest the Sun and slowest in motion. In the points A and P the Line of the mean and true motion do convene, and therefore in either of these places the Pla∣net is from P in aequality, but in all other points the mean and true motion differ, and in D and C is the greatest elliptick AEquation.

8. Now suppose the Planet in B, the line of the middle motion according to this Figure is MB, the line of the true motion HB. The mean Anomaly AMB. The Eliptick aequati∣on or Prosthaphaeresis MBH, which in this Exam∣ple subtracted from AMB, the remainer AHB is the true Anomaly. And here note that in the right lined Triangle MBH, the side MH is al∣ways the same, being the distance of the Foci, the other two sides MB and HB are together equal to AP. Now then if you continue the side MB till BE be equal to BH and draw the line HE, in the right lined Triangle MEH, we have given ME=AD and MH with the Angle EMH, to find the Angles MEH and MHE which in this case are equal, because EB=BH by Contraction, and therefore the double of BEH or BHE=MBH, which is the Angle required.

And that which yet remaineth to be done, is

the finding the place of the Aphelion, the true Excentricity or distance of the umbilique points, and the stating of the Planets middle motion.

CHAP. X.

Of the finding of the Suns Apogeon, quantity of Excentricity aend middle motion.

THe place of the Suns Apogaeon and quantity of Excentricity may from the observati∣ons of our countrey man Mr. Edward Wright be obtained in this manner, in the years 1596, and 1497, the Suns entrance into ♈ and ♎ and into the midst of ♉. ♌. ♍. and ♒ were as in the Table following expressed.

	1596	1597
	D. H. M.	D. H. M.
Ianuary.	25. 00.07	24. 05.54	♒. 15
March.	9. 18.43	10. 00.37	♈. 0
April.	24. 21.47	25. 03.54	♉. 15
Iuly.	28. 01.43	28. 09.56	♌. 15
September.	12. 13.48	12. 19.15	♎. 0
October.	27. 15.23	27. 21.50	♍. 15

And hence the Suns continuance in the Nor∣thern Semicircle from ♈ to ♎ in the year 1596 being Leap year, was thus found.

 

d. h.

From the 1. of Ianuary to ☉ Entrance ♎.

256. 13. 48.

From the 1. of Iun to ☉ Entrance ♈

69. 18.43

Their difference.

186. 19.05

In the year 1597 from the 1 of Ianuary to the time of the ☉ Entrance into ♎.

255. 19.15

To the ☉ entrance into ♈.

69. 09.37

Their difference is

186. 18.38

And the difference of the Suns continuance in these Arks in the year 1596 and 1597 is 27′. and therefore the mean time of his continuance in those Arks is days 186. hours 18. minutes 51. se∣conds 30. And by consequence his continuance in the Southern Semicircle that is from ♎ to ♈ is 178 days. 11 hours, 8 minutes and 30 se∣conds.

In like manner in the year 1596 between his entrance into ♉ 15. and ♍ 15, there are days

185. 17.36

And in the year 1597 there are days

185. 17.56

And to find the middle motion answering to days 186. hours 18. Minutes 51. seconds 30 I say.

As 365 days, 6 hours, the length of the Julian, year is to 360, the degrees in a Circle.

So is 186 days, 18 hours, 51′. 30″ to 184 degrees. 03′. 56″.

In like manner the mean motion answering

to 185 days, 17^h. 46′ is 183 degrees, 02′.09.

Apparent motion from ♈ to ♎

180. 00.00

Middle motion

184. 03.56

Their Sum

364. 03.56

Half Sum is the Arch. SME

182. 01.58

In 1596 from 15 ♒ to 15 ♌ there are days 185, hours 01, minutes 36. In 1597. days 135. hours 4. 02′.

And the mean motion answering thereunto is. 182^d. 30′. 36″.

Apparent motion from

15 ♉ to 15 ♍. 180.

Middle motion

185. 17. 56. 181. 04.53

Half Sum is

183. 32. 26

From 15 ♒ to 15 ♌ Days. 185. 04^h. 02′

Apparent motion

180.

Middle motion

182. 30. 36

Half Sum

181. 15. 18

Now then in Fig. from PGC. 181. 32. 26 deduct NKD 180, the Remainer is DC+NP. 1. 32. 26. Therefore DC or NP. 46. 13, whose Sine is HA.

And from XPG. 181. 15. 18 deduct TNK 180, the Remainer is KG+TX 1. 15. 18. Therefore KG or TX 37. 39, whose Sine is HR.

Now then to find the Apogaeon.

As HA 46′.13″

5.12851105

To Rad. So HR 37′.39″

15.03948202

To Tang. HAR. 39^d.10′.04″

9.91097097

GAM.

45

Apogaeon

95. 49. 56.

Hence to find the excentricity AR.

As the Sine HAR. 39. 10.04

9.80043756

To Rad. So HR. 37.39

15.03948202

To RA. 1733.99

5.23904446

Or thus,

In the Triangle 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 we have given 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

As 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 37.39

5.03948202

To Rad. So 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 46. 13.

15.12851105

To Tang. R 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 50. 49.56

10.08902903

PAS.

45.

Apogaeon 95 deg. 49′. 56″. as before.

Then for the Excentricity RA.

As the Sine of R 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 50. 49. 56

9.88945938

Is to 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. 46′.13″

5.12851105

So is Radius. To RA.1734.01

5.23905167

And this agreeth with the excentricity, used

by Mr. Street in his Astron. Carolina, Pag. 23. But Mr. Wing as well by observation in former ages, as our own, in his Astron. Instaur. Pag. 39. doth find it to be 1788 or 1791. The work by both observations as followeth.

2. And first in the time of Ptolemy, Anno Christi 139 by comparing many observations to∣gether, he sets down for the measure nearest truth, the interval between the vernal Equinox and the Tropick of Cancer to be days 93. hours 23. and minutes 03. And from the Vernal to the Autumnal Equinox, days 186. hours 13. and minutes 5.

 

D.

The apparent motion from ♈ to ♎

90. 36.00

Middle motion for 93^d. 23^h. 3′. is

92. 36.42

The half Sum is GP

91. 18.21

Apparent motion from ♈ to ♎

180. 00.00

Middle motion for 186^d. 13^h. 5′. is

183. 52.03

The half Sum is GEK

181. 56.02

The half of GEK is GE.

90. 58.01

And GP less GE is

00. 20.20

Whose Sum is AC 59146.

 

Again from GEK 181. 56. 02. deduct the Semicircle FED 180. the remainer is the summ DK and FG. 1. 56. 2. and therefore DK=FG. 58′. 01″. whose sign is BC. 168755. L is the place of the Aphelion, and AB the Excentricity.

Now then in the Triangle ABC. in the Fig. 6 we have given the two sides AC and BC. To find the Angle BAC and the Hypotenuse AB.

For which the proportions are.

As the side AC. 59146

4.77192538

Is to the Radius.

10.00000000

So is the side BC▪ 168755

5.22725665

To Tang. BAC. 70. 41. 10.

10.45533127

Secondly for AB.

As the Sine of BAC. 70. 41. 10.

9.97484352

Is to the side AB. 168755.

5.22725665

So is the Radius.

10.00000000

To the Hypot. AB. 1788. 10.

5.25241313

Therefore the Aphelion at that time was in II 10. 41. 10. And the excentricity. 1788.

3. Again Anno Christi 1652 the Suns place by observation was found to be as followeth.

April. 24.	hours. 10.	♉. 15
October. 27.	hours. 7. 10′	♍. 15
Ianuary. 24.	hours. 11.20′	♒. 15
Iuly. 27.	hours. 16.30,	♌. 15

Hence it appeareth that the Sun is running through one Semicircle of the Ecliptick, that is from ♉ 15 to ♍ 15. 185 days 21 hours and 10′. And through the other Semicircle from ♒ 15 to ♌ 15, days 184. hours 5. therefore the Suns mean motion, according to the practice in the last example, from ♉ 15 to ♍ 15 is 181. 30. 26. and from ♒ 15 to ♌ 15. 181. 16. 30.

Now then in Fig. 7. if we subtract the semi∣circle of the Orb KMH. 180. from WPV 181. 36. 26. the remainer is the sum of KW and HV

1. 36. 26. the Sine of half thereof 48′. 13″ is e∣qual to AC. 140252.

Again the mean motion of the Sun in his Orb from ♒ 15 to ♉ 15 is the Arch SKP. 181. 16. 30. whose excess above the Semicircle being bi∣sected is 38. 15. whose Sine CB. 111345. now then in the Triangle ABC to find the Angle BAC, the proportion is.

As the side AC. 140252

5.14690906

Is to the Radius.

10.00000000

So is the Side CB 111345

5.04667072

To Tang. BAC. 38. 36. 21,

9.89966166

Which being deducted out of the Angle. 69 A ♌. 45 it leaveth the Angle 69 AL 6. 33. 39. the place of the ☉ Aphelion sought, and this is the quantity which we retain.

And for the excentricity BC.

As the Sum of BAC. 38. 26. 21

9.79356702

Is to the Radius.

10.00000000

So is the side BC 111345

5.04667072

To the Hypot. AB. 179103

5.25310370

So then Anno Christi. 1652. Aphel.

96. 33.39

Anno Christi. 139. the Aphelion

70. 41.10

Their difference is

25. 52.29

And the difference of time is 1513 Julian years.

Hence to find the motion of the Aphelion for 2. years, say I, if 1513 years give 25. 52.29, what shall one year give, and the answer is 00^d. 01′

01″. 33‴. 56^iv. 44^v. that is in Decimal num∣bers. 0. 00475. 04447. 0555.

And the motion for. 1651 years. 7. 84298. 4208862, which being deducted from the place of the Aphelion Anno Christi. 1652—26. 82245. 3703703. The remainer, viz. 18. 97946. 9494841 is the place thereof in the beginning of the Christian AEra, which being reduced is, 68 deg. 19. min. 33. sec. 56. thirds.

4. The Earths middle motion, Aphelion and Excentricity being thus found, we will now shew how the same may be stated to any particular time desired, and this must be done by help of the Sun or Earths place taken by observation. In the 178 year then from the death of Alexan∣der, Mechir the 27 at 11 hours P. M. Hippar∣cus found in the Meridian of Alexand. that the Sun entered ♈ 0. the which Vernal Equinox happened in the Meridian of London according to Mr. Wings computation at 9 hours 14′, and the Suns Aphelion then may thus be found.

The motion of the Aphelion for one year, was before found to be. 0. 00475. 04447. 0555. there∣fore the motion thereof for one day is 0. 00001. 501491722. The Christian AEra began in the 4713 year compleat of the Julian Period, in which there are days 1721423. The AEra A∣lexandri began November the twelfth, in the year 4390 of the Julian Period, in which there are 1603397 days. And from the death of A∣lexander to the 27 of Mechir 178, there are days 64781, therefore from the beginning of the Julian Period, to the 178 year of the AEra A∣lexandri, there are days 1668178 which being deducted from the days in the Christian AEra,

1721423, the remainer is 53245, the number of days between the 178 year after the death of Alexander, Mechir 27, and the beginning of the Christian AEra.

Or thus. From the AEra Alexandri to the AEra Christi there are 323 Julian years, and 51 days, that is 118026 days. And from the AEra Alexandri to the time of the observation, there are 64781 days, which being deducted from the former, the remainer is 53245 as before. Now then if you multiply the motion of the A∣phelion for one day, viz. 0. 00001. 3014917 by 53245, the product is 0. 69297. 9255665, which being deducted from the place of the Aphelion in the beginning of the Christian AEra, before found. 18. 97946. 9494841. the remainer 18. 28649. 0239176 is the place of the Aphelion at the time of the observation, that is in Sexage∣nary numbers. deg. 65. 49′. 53″.

5. The place of the Aphelion at the time of the observation being thus found to be deg. 65. 49′. 53″. The Suns mean Longitude at that time, may be thus computed.

In Fig. 8. In the Triangle EMH we have gi∣ven the side ME 200000, the side MH 3576, the double excentricity before found, and the An∣gle EMH 114. 10′. 07″. the complement of the Aphelion to a Semicircle, to find the Angle MEH, for which the proportion is,

As the Summ of the sides, is to the difference of the sides, so is the Tangent of the half Summ of the opposite Angles, to the Tangent of half their difference.

The side ME. 200000.

 

The side MH 3576.

 

Z. Of the sides. 203576. Co. ar.

4.69127343

X. Of the sides. 196424.

5.29321855

Tang. ½ Z Angles. 32′. 54′. 56.

9.91111512

Tang. ½ X Angles. 31. 59. 21.

 

Angle MEH. 0. 55. 35.

9.79560710

The double whereof is the Angle MBH 1. 51. 10. which being Subtracted from 360 the re∣mainer 358. 08. 50. is the estimate middle mo∣tion of the Sun, from which subtracting the A∣phelion before found, 65. 49. 53. the remain∣er 292. 18. 57. is the mean Anomaly by which the absolute AEquation may be found according to the former operation.

Z. ME+MH. 203576. Co. ar.

4.69127343

X. ME-MH. 196424

5.29321855

Tang. ½ Anom. 56. 09. 28.

10.17359517

Tang. ½ X. 55. 12. 18.

10.15808715

Differ. 00. 57. 10.

 

Doubled 1. 54. 20, which added to the mid∣dle motion before found gives the ☉ true place ♈. 00. 3′. 10″, which exceeds the observation 3′. 10″. therefore I deduct the same from the middle motion before found, and the remainer 358. 05. 50. is the middle motion at the time of the observation of Hipparchus, to which if you add the middle motion of the Sun for 53245 days, or for 323 AEgyptian years 131 days, 280. 46. 08′ the Summ, rejecting the whole Circles, is 278. 51. 48 the Suns mean Longitude in the beginning of the Christian AEra.

6. But one observation is not sufficient, where∣by

to state the middle motion for any desired Epocha, we will therefore examine the same by another observation made by Albategnius at A∣racta in the year of Christ 882, March: 15. hours 22. 21. but in the Meridian of London at 18 hours. 58′.

The motion of the Aphelion for 881 years, 74 days is 3. 806068653737, which being ad∣ded to the place thereof in the beginning of the Christian AEra, the place at the time of the obser∣vation will be found to be 22. 785538148578, that is reduced, Deg. 82. 01′. 40″. And hence the AEquation according to the former operati∣ons is Deg. 2. 01′. 16″ which being deducted from a whole Circle, the remainer 357^d. 58′. 44″ is the estimate middle motion at that time, from which deducting the Aphelion deg. 82. 01. 40. the remainer 275. 57. 04 is the mean ano∣maly, and the AEquation answering thereto is deg. 2. 02′. 18″ which being added to the middle motion before found, gives the ☉ place ♈. 00. 01′. 02″ which exceeds the observation 01′. 02″. therefore deduct the same from the middle motion before found, the remainer 357. 57′. 22″ is the middle motion of the ☉ at the time of the observation, from which deducting the middle motion for 881 years, 74 days, 18 hours, 58 minutes, viz. 80^d. 06′. 10″. the re∣mainer 277 deg. 51′. 12″. is the ☉ mean Lon∣gitude in the beginning of the Christian AEra.

By the first observation it is deg.

278. 51′. 48″

By the second

277. 51. 12

Their difference is

1. 00. 36

He that desires the same to this or any other Epocha, to more exactness, must take the pains to compare the Collection thereof from sun∣dry Observations, with one another, this is sufficient to shew how it is to be found. Here therefore I will only add the measures set down by some of our own Nation, and leave it to the Readers choice to make use of that which pleaseth him best.

The ☉ mean Longitude in the beginning of the Christian AEra according to.

Vincent Wing is

9. 8^d. 00′. 31″

Tho. Street is

9. 7. 55. 56

Iohn Flamsted is

9. 7. 54. 39

By our first Computation

9. 8. 51. 48

By our second

9. 7. 51. 12

In the Ensuing Tables of the ☉ mean Lon∣gitude, we have made use of that measure gi∣ven by Mr. Flamsted, a little pains will fit the Tables to any other measure.

CHAP. XI.

Of the quantity of the Tropical and Sydereal Year.

THe year Natural or Tropical (so called from the Greek word 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, (which sig∣nifies to turn) because the year doth still turn or return into it self) is that part of time in which the ☉ doth finish his course in the Zodiack

by coming to the same point from whence it began.

2. That we may determine the true quanti∣ty thereof, we must first find the time of the ☉ Ingress into the AEquinoctial Points, about which there is no small difference amongst Astrono∣mers, and therefore an absolute exactness is not to be expected, it is well that we are arrived so near the Truth as we are. Leaving it therefore to the scrutiny of after Ages, to make and com∣pare sundry Observations of the ☉ entrance in∣to the AEquinoctial Points, it shall suffice to shew here how the quantity of the Tropical year may be determined, from these following obser∣vations.

3. Albategnius, Anno Christi 882 observed the ☉ entrance into the Autumnal AEquinox at Aracta in Syria to be Sept. 19. 1 hour 15′ in the Morning. But according to Mr. Wings cor∣rection in his Astron. Instaur. Page 44, it was at 1 hour 43′ in the Morning, and therefore ac∣cording to the ☉ middle motion, the mean time of this Autumnal AEquinox was Sept. 16. 12^h. 14′. 25″. that is at London at 8^h. 54′. 25″.

4. Again by sundry observations made in the year 1650. the second from Bissextile as that of Albategnius was, the true time of the ☉ in∣gress into ♎ was found to be Sept. 12. 14^h. 40′. and therefore his ingress according to his mid∣dle motion was Sept. 10. 13^h. 02.

5. Now the interval of these two observa∣tions is the time of 768 years, in which space by subtracting the lesser from the greater, I find an anticipation of 5 days, 9 hours, 52′. 25″. which divided by 768 giveth in the quotient

10′. 55″. 39 which being subtracted for 365 days, 6 hours, the quantity of the Julian year, the true quantity of the Tropical year will be 365 days, 5 hours, 49′. 04″. 21‴. Others from other observations have found it some∣what less, our worthy countryman Mr. Edward Wright takes it to be 365^d. 5 hours. 48′.

Mr. Iohn Flamsted, 5^h. 29′. Mr. Tho. Street 5^h. 49′. 01″. taking therefore the Tropical year to consist of 365 days, 5 hours, 49 Minutes, the Suns mean motion for one day is 0 deg. 59′. 8″. 19‴. 43^iv. 47^v. 21^vi. 29^vii. 23^viii. or in de∣cimal Numbers, the whole Circle being divi∣ded into 100 degrees, the ☉ daily motion is 0. 27379. 08048. 11873.

6. The Sydereal or Starry year is found from the Solar by adding the Annual Motion of the eighth Orb or praecession of the AEquinoctial Points thereunto, that praecession being first converted into time.

7. Now the motion of the fixed Stars is found to be about 50″. in a years time, as Mr. Wing hath collected from the several observations of Timocharis, Hipparchus, Tycho and others; and to shew the manner of this Collection, I will mention onely two, one in the time of Timo∣charis, and another in the time of Tycho.

8. Timocharis then as Ptolemy hath it in his Almagist, sets down the Virgins Spike more northwardly than the AEquinoctial, 1 deg. 24′. the time of this observation is supposed to be about 291 years before Christ, the Latitude 1 deg. 59′ South, and therefore the place of the Star was in ♍. 21^d. 59′. And by the ob∣servation of Tycho 1601 current, it was in ♎

18. 16′. and therefore the motion in one year 50″, which being divided by 365 days, 6 hours, the quotient is the motion thereof in a days time. 00′. 8‴. 12^iv. 48^v. 47^vi. 18^vii. 30^viii. 13^ix. and in decimal Numbers, the motion for a year is 00385. 80246. 91358. The motion for a day. 00001. 05626. 95938.

9. Now the time in which the Sun moveth 50″, is 20′. 17″. 28‴, therefore the length of the sydereal year is 365 days, 6 hours, 9′. 17″. 28‴. And the Suns mean motion for a day 59′. 8″. 19‴. 43^iv. 47^v. 21^vi. 29^vii. 23^viii. converted into time is 00. 03′. 56″. 33‴. 18^iv. 55^v. 9^vi. 23^vii. 57^viii. which being added to the AEquinocti∣al day, 24 hours, giveth the mean solar day, 24 hours. 3.′ 56″. 33‴. 18^iv. 55. 9. 23. 57.

10. And the daily motion of the fixed Stars, being converted into time is 32^iv. 51^v. 15^vi. 9^vii. 14^viii. 24^ix. and therefore the AEquinoctial day being 24 hours, the sydereal day is 24 hours, 00′. 00″. 00‴. 32^iv. 51. 15. 9. 14 24.

11. Hence to find the praecession of the AEqui∣noctial Points, or Longitude of any fixed Star, you must add or subtract the motion thereof, from the time of the observation, to the time given, to or from the place given by observation, and you have your desire.

Example. The place of the first Star in A∣ries found by Tycho in the year 1601 current, was in ♈. 27^d. 37′. 00. and I would know the place thereof in the beginning of the Christian AEra.

The motion of the fixed Stars for 1600 years,

22^d. 13′. 20″

Which being deducted from the place found by observ.

27. 37. 00

The remainer.

5. 231. 40

is the place thereof in the beginning of the Christian AEra.

12. Having thus found the ☉ middle moti∣on, the motion of the Aphelion and fixed Stars, with their places, in the beginning of the Chri∣stian AEra; we will now set down the num∣bers here exhibited AEra Christi. Mr. Wing from the like observations, takes the ☉ motion to be as followeth.

The ☉ mean Longitude

9. 8. 00. 31

Place of Aphelion

2. 8. 20. 03

The Anomaly

06. 29. 40. 28

The which in decimal Numbers are

The ☉ mean Longitude

77. 22460. 86419

Place of the Aphelion

18. 98171. 29629

The Anomaly

58. 24289. 56790

The mean motions for one year.

The ☉ mean Longitude

99. 93364. 37563. 34

The Aphelion

00. 00475. 04447. 05

The ☉ mean Anomaly

99. 92889. 33116. 29

The ☉ mean motions for one day.

The ☉ mean Longitude

00. 27379. 08048. 11

The Aphelion

00. 00001. 30149. 17

The mean Anomaly

00. 27377. 77898. 94

And according to these measures are the Ta∣bles made shewing the ☉ mean Longitude and Anomaly, for Years, Months, Days and Hours.

CHAP. XII.

The Suns mean motions otherwise stated.

SOme there are in our present age, that will not allow the Aphelion to have any motion, or alteration, but what proceeds from the mo∣tion of the fixed Stars, the which as hath been shewed, do move 50 seconds in a year, and hence the place of the first Star in Aries, in the begin∣ning of the Christian AEra was found to be ♈. 5. 23^d. 40.

Now then, if from the place of the Aphelion Anno Christi. 1652 as was shewed in the tenth Chapter, deg. 96. 33′. 39. we deduct the mo∣tion of the fixed Stars for that time. 28. 19. 12. the remainer 68. 14. 27 is the constant place of the Aphelion; but Mr. Street in his Astrono∣mia Carolina Page 23, makes the constant place of the Aphelion to be 68^d. 20. 00, and the ☉ ex∣centricity 1732.

And from the observation of Tycho 1590 March the eleventh. in the Meridian of V∣raniburg, but reduced to the Meridian of London. March the tenth, hour 23. 2′. He

[illustration]

[illustration]

determines the Earths mean Anomaly thus.

The place of the Sun observed

♈. 0. 33. 19

The praecession of AEquinox

0. 27. 27. 22

The Earths Sydereal Longitude

5. 03. 05. 57

The place of the Aphelion Subtract

8. 08. 20. 00

The Earths true Anomaly

8. 24. 45. 57

AEquation Subtract

1. 58. 47

The remainer is the Estimate M. Anom.

8. 22. 47. 10

AEquation answering thereto add.

1. 58. 27

The Earths true Anomaly

8. 24. 45. 37

The place of the Aphelion

8. 08. 20. 00

Praecession of the AEquinox

0. 27. 27. 22

Place of the Sun

♈. 00. 32. 59

But the place by observation

♈. 00. 33. 19

The difference is

001. 001. 20

Which being added to the mean Anom.

8. 22. 47. 10

The mean Anomaly is

8. 22. 47. 30

The absolute AEquation

1. 58. 27

The true Anomaly

8. 24. 45. 57

Agreeing with observation.

 

And so the mean Anomaly AEra Christi is 6. 23. 19. 56. But Mr. Flamsted according to whose measure the ensuing Tables are compo∣sed, takes the mean Anomaly AEra Christi. to be 6. 24. 07. 091. The place of the Aphelion to be 8, 08. 23. 50. And so the Praecession of the AEquinox and Aphelion in the beginning of

the Christian AEra. 8, 13. 47. 30. in decimal Numbers.

AEra Christi.

The Suns mean Anomaly

56. 69976. 85185

The Suns Apogaeon and Praec. AEq.

20. 49768. 51851

The ☉ mean motions for one Year.

The ☉ mean Longitude

99. 93364. 37563. 34

The Praecession of AEquin.

00385. 80246. 91

The ☉ mean Anomal.

99. 92978. 57316. 43

The ☉ mean Motions for one Day.

The ☉ mean Longitude

00. 27379. 08048. 11

The Praecession of AEqui.

00. 00001. 05699. 30

The ☉ mean Anom.

00. 27378. 02348. 81

CHAP. XIII.

How to Calculate the Suns true place by either of the Tables of middle motion.

VVRite out the Epocha next before the given time, and severally under that set the motions belonging to the years, months and days compleat, to the hours, scruples, cur∣rent every one under his like (only remember that in the Bissextile years after the end of Fre∣bruary the days must be increased by an unite) then adding all together, the sum shall be the ☉ mean motion for the time given.

Example.

Let the given time be Anno Christi 1672. Fe∣bruary 23. hours 11. 34′. 54″. by the Tables of the ☉ mean Longitude and Anomaly, the num∣bers are as followeth.

		M. Longitud.	M. Anomal.
The Epocha	1660	80. 67440.	53.79815
Years	11	99. 81766.	99.76526
Ianuary		08. 48751.	08.48711
Day.	23	06. 29718.	06.29688
Hours	11	00. 12548.	00.12548
	34	00. 00646.	00.00646
	54	00. 00017.	00.00017
		95. 40886.	68.47951

By the Tables of the Suns mean Anomaly and praecession of the AEquinox, the numbers are these.

		Anomaly.	Praece. AEqui.
The Epocha	1660	53. 76721.	26.90200
Years	11	99. 77520.	00.04243
Ianuary		8. 48718.	00032
Days	23	6. 29694.	00024
Hours	11	0. 12548.	26.94499
	34′	.00646.	68.45882
	54″	.00035.	95.40381
☉ mean Anomaly		68.45882

There is no great difference between the ☉ mean Longitude and Anomaly found by the Ta∣bles of mean Longitude and Anomaly, and that found by the Tables of mean Anomaly and Pre∣cession of the AEquinox. The method of finding the Elliptical AEquation is the same in both, we will instance in the latter only, in which the ☉ mean Anomaly is Degrees 68. 45882. And the precession of AEquin. deg. 26. 94499.

But because there is no Canon of Sines and Tan∣gents as yet published, suitable to this division of the Circle into an 100 deg. or parts: We must first convert the ☉ mean Anomaly, and prec. of of the AEquin, given, into the degrees and parts of the common Circle: And this may be done either into degrees and decimal parts of a degree, or into deg. and minutes: if it were required to be done into degrees and minutes, the Table here exhibited for that purpose will serve the turn, but if it be required to be done into degrees and de∣cimal parts, I judge the following method to be more convenient.

Multiply the degrees and parts given by 36, the Product, if you cut off one figure more towards the right hand than there are parts in the num∣ber given, shall be the degrees and parts of the common Circle.

Anomaly. 68. 45882	Praec. AEquinox. 26. 94499
36		36
41075292		16166994
20537646		8983497
Anom. 246. 451752		Prae. AEq. 97. 001964

And if you multiply the parts of these Pro∣ducts, you will convert them into minutes.

Otherwise thus. Multiply the degrees and parts given by 6 continually, the second Product, if you cut off one figure more towards the right hand than are parts in the number given, shall be the degrees and parts of the common Circle. The third Product of the parts only shall give minutes, the fourth seconds, and so forward as far as you please. Example.

☉ Mean Anom. 68. 45882

Praec. AEq. 26. 94499

6

6

6

6

41075292

16166994

246.451752

97.001964

6

6

6

6

27.10512

0.11784

6

6

6

6

6.3072

7.0704

And thus the mean Anom. is deg. 246. 451742 or 27′. 06. The Prec. AEq. 97. 001964. or 00′. 07″.

Hence to find the Elliptical AEquation in degrees and decimal parts: In Fig. 8. we have given in the right lined plain Triangle EMH, the sides ME, and MH, and the Angle EMH, 66. 451742. the excess of the mean Anomaly above a Semicircle, to find the Angle MEH.

The side ME

200000

The side MH

3468

Zcru.	203468	Co. ar.	4.69150389
Xcru.	196532		5.29343327
t frac12 Zangle.	56.774129		10.18374097
t frac12 Xangle.	55.857087		10.16867813

MEH. 0. 917042 the double whereof is the Angle MBH. 1. 834084 or Elliptick AEqua∣tion sought, which being added to the mean A∣nomaly and praecession of the AEquinox, because the Anomaly is more than a Semicircle, the same is the Suns true place.

The ☉ mean Anomaly

246.451742

The Praecession of the AEquinox

97.001964

Elliptick AEquation

1.834084

The Suns true place.

345.287790

But because the Elliptick AEquation thus found doth not so exactly agree to observation as is de∣sired, Bullialdus in Chap. 3. of his Book entituled Astronomiae Philolaicae fundamenta clarius explicata, Printed at Paris, 1657. shews how to correct the same by an Angle applied to the Focus of middle motion, subtended by the part of the ordinate line, intercepted between the Ellipsis and the Cir∣cle circumscribing it. This Mr. Street maketh use of in his Astronomia Carolina, and this I thought not amiss to add here.

In Fig. 9. let ABCPDF be supposed an El∣lipsis, and the Circle AGPK described upon the extremes of the transverse Diameter, and the Or∣dinates KN and OB extended to G and M in

[illustration]

the Periphery of the Circle: then by the 21 of the first of Apollonius.

XN. GX∷OB tang. OEB. OM tang. OEM.

And the Angle OEM-OEB=BEM=ETY, the variation to be deducted from the Elliptick AEquation ETH, the Remainer is the absolute AEquation YTS in the first Quadrant.

In the second and third Quadrants, the variati∣on or difference between the mean and corrected Anomaly, must be added to the Elliptick AEqua∣tion, to find the true and absolute AEquation.

For XN. XG. QV. tang. QEV. the comp. m. Anom. QR. t. QER. and the Angle VER=ECO is the va∣riation, and ECO+ECH=OCH is the absolute AEquation sought in the second Quadrant.

Again, XN. SG∷a D, tang. a ED. a b, tang. aEB. And aEB—aED=DEf the variation=EFO and EfO+EfH=OfH the absolute AEquation sought in the third Quadrant.

Lastly, in the fourth Quadrant of mean Ano∣maly it is.

XN. XG∷ch. tang. eEH. eg. tang. eFg. and hEg is the variation: And EFH—〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉=〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 the absolute AEquation sought in the fourth Quadrant.

And to find XN the conjugate Semi-diameter, in the right angled Triangle ENX, we have gi∣ven, EN=AX and EX the semi-distance of the umbilick points. And Mr. Briggs in Chap. 19. of his Arithm. Logar. hath shewed, that the half Sum of the Logarith. of the sum and difference of the Hypotenuse, and the given leg. shall be the Loga∣rith. of the other leg.

Now then EN=AX. 100000
The Leg EX.	1734
Their Sum	101734	5.00745001
Their difference	98266	4.99240328
	The Z of the Logarithms,	9.99985329
	frac12; Z. Logarith. XN. 99983	4.99992664

Now then in the former Example the mean Anomaly is 246 deg. 451741. and the excess a∣bove a semicircle is the ang. aED. 66. 451742. Therefore.

As XN. 99983

4.99992664

Is to XG. 100000

5.00000000

So is the tang. aED 66.451742

10.36069857

To the tang. aEB 66.455296

10.36077193

aEB—aED=DEf .003544 the variation, which being added to the Elliptick AEquation be∣fore found, the absolute AEquation is 1. 837628. and therefore the ☉ true place 345. 291334. that is X. 15. 17. 28.

CHAP. XIV.

To find the place of the fixed Stars.

THe annual motion of the fixed Stars is, as hath been shewed, 50 Seconds, hence to find their places at any time assigned, we have exhibited a Table of the Longitude and Latitude of some of the most fixed Stars, from the Cata∣logue

[illustration]

of noble Tycho for the year of our Lord 1600 compleat. Now then the motion of the fixed Stars according to our Tables being com∣puted, for the difference of time between 1600 and the time propounded, and subtracted from the place in the Table, when the time given is before 1600, or added to it, when the time gi∣ven is after; the Summ or difference shall be the place desired. The Latitude and Magnitude are still the same.

Example. Let the given time be 1500, the difference of time is 100 years, and the moti∣on of the fixed Stars for 100 years is 0. 38580.

The place of the 1 * in ♈, 1600

7.67129

Motion for 100 years subtract

0.38580

Place required in the year 1500

7.28549

2. Example.

Let the time given be

1674.

The place of the first Star in ♈ 1600 was

7.67129

Motion for 60 years is

0.23148

Motion for 14 years is

0.05401

Place required in the year 1674 compl.

7.95678

CHAP. XV.

Of the Theory of the Moon, and the finding the place of her Apogaeon, quantity of excentricity and middle motion.

THe Moon is a secondary Planet, moving a∣bout the Earth, as the Earth and other

Planets do about the Sun, and so not only the Earth but the whole System of the Moon, is also carried about the Sun in a year. And hence, ac∣cording to Hipparchus, there arises a twofold, but according to Tycho a three-fold Inequality in the Moons Motion. The first is Periodical and is to be obtained after the same manner, as was the excentrick AEquation of the Sun or Earth: in order whereunto, we will first shew how the place of her Apogaeon and excentricity may be found.

At Bononia in Italy, whose Longitude is 13 degrees Eastward from the Meridian of London, Ricciolus and others observed the apparent times of the middle of three lunar Eclipses to be as fol∣loweth.

• The first 1642. April the 4. at 14 hours and 4 Minutes.
• The second 1642, September 27 at 16 hours and 46 minutes.
• The third 1643. September 17 at 7 hours and 31 Minutes.

The equal times reduced to the Meridian of London, with the places of the Sun in these three observations, according to Mr. Street in the 25 Page of his Astronomia Carolina, are thus.

Anno Mens. D. h.

d.

1642. April 4. 13. 37.

♈. 25. 6. 54

1642. Septemb. 27. 15. 57

♎. 14. 50. 09

1643. Sehtemb. 17. 6. 46

♎ 4. 20. 20

Hence the place of the Moon in the first obser∣vation

is in ♎ 25. 6′. 54. in the second ♈ 14. 50. 9. in the third ♓ 4. 20. 20. Now then in Fig. 10. let the Circle BHDGFE denote the Moons AEquant T the Center of the Earth, the Semidiameters TD, TE and TF the apparent places of the Moon, in the first, second and third observations, C the Center of the Excentrick, CD, CE and CF the Lines of middle motion.

From the first observation to the second there are

176^d. 2^h. 20′

The true motion of the Moon is deg.

169. 43. 15″

The motion of the Apogaeon subtract

19. 37. 07

The motion of the true Anomaly is the arch DE

150. 06. 08

The motion of the mean Anomaly DCE

140. 42. 28

From the first observation to the third, there are

530^d. 17^h. 9.

The true motion of the Moon is degrees

159. 13. 26

The motion of the Apogaeon subtract

159. 07. 32

The motion of the true Ano∣maly is the Arch DF

100. 05. 54

The motion of the mean Anomaly DCF

93. 46. 45

And deducting the Arch DGF from the Arch DFE, the re∣mainer is the Arch FE

50. 00. 14

And deducting the Angle DCF from the Angle DCE, the re∣mainer is the Angle FCE

46. 55. 43

Suppose 10.00000000 the Logarithm of DC, continue FC to H, and with the other right Lines compleat the Diagram.

1. In the Triangle DCH we have given the Angle DCH 86. 13. 15. the complement of DCF 93. 46. 45 to a Semicircle. The Angle DHC 50. 02. 57. The half of the Arch DF and the side CD 1000000. To find CH.

As the Sine of DHC 50. 02. 57

9.88456640

To the Side DC, so the Sine of HDC 43. 43. 48.

19.83964197

To the Side CA

9.95507557

2 In the Triangle HCE we have given CH as before, the Angle CHE 25. 00. 07. The half of the Arch FE, the Angle HCE 133. 04. 17 the complement of FCE, and by consequence the Angle CEH 21. 55. 36 To find the Side CE.

As the Sine of CEH 21. 55. 36

9.57219707

To the Side CH

19.95507557

So is the Sine of CHE 25. 00. 07

9.62597986

To the Sine CE

• 19.58105543
• 10.00885836

3. In the Triangle DCE, we have given DC. CE and the Angle DCE 140. 42. 28. whose complement 39. 17. 32 is the Summ of the An∣gles, to find the Angle CED and DE,

As the greater Side CE

10.00885836

Is to the lesser Side DC

10.00000000

So is the Radius

10.00000000

To the tang. of 44. 24. 54

19.99114164

Which subtracted from 45. 2

the remainer is the half.

Difference of the acute angles 35. 16.

 

As the Radius.

To the tang. of the com. 35. 16

8.01109962

Is to the tang. of the frac12; Z. 19. 38. 46

9.55265735

To the tang. of frac12; X. 00. 12. 35

7.56375697

Their Sum 19. 51. 21. is the angle—CDE.

 

Their difference 19. 26. 11. is the angle CED.

 

As the Sine of CED. 19. 26. 11.

9.52216126

Is to the Sine of DCE. 140. 42. 28.

9.80159290

So is the Side EC.

10.00000000

To the Side DE.

10.27943164

4. In the Isosceles Triangle DTE we have gi∣ven the Side DE, the angle DTE 150. 06. 08 whose complement 29. 53. 52 is the Summ of the other two angles, the half whereof is the angle TDE 14. 56. 56 which being subtracted from the angle CDE. 19. 51. 21 the remainer is the angle CDT. 4. 54. 25.

As the Sine of DTE 150. 06. 08 Co. ar.

0.30237482

Is to the Sine of DET. 14. 56. 56

9.41154778

So is the Side DE

10.27943164

To the Side DT

9.99335424

5. In the Triangle CDT we have given DC. DT and the angle CDT, to find CTD and CT.

As the Side DT

9.99335424

Is to the Side DC

10.00000000

So is the Rad.

10.00000000

To the tang. of 26. 18

10.00664576

• Deduct 45.
• As the Radius.

Is to the Sine of the remainer 0. 26. 18.

7.88368672

So is the tang. of the frac12; Z angle 87. 32. 57

11.36854996

To the tang. frac12; X angle 10. 08. 04

9.25223668

Their Summ 97. 41. 01

is the angle CTD

As the Sine of CTD. 97. 41. 01. Co. ar.

0.00391693

Is to the Side DC

10.00000000

So is the Sine of CDT 4. 54. 25

8.93215746

To the Side CT

8.93607439

 

s. d.

The place of the Moon in the first Observation

6. 25. 06. 54

The true Anomaly CTD sub.

3. 07. 41. 01

The place of the Apogaeon

3. 17. 25. 53

☽ place in the first Observation

6. 25. 06. 54

The AEquation CDT Add.

04. 54. 25

The ☽ mean Longitude

7. 00. 01. 19

From which subtract the place of the Apogeon

3. 27. 25. 53

There rests the mean Anomaly BCD

3. 12. 35. 26

And for the excentricity in such parts, as the Radius of the AEquant is 100000 the Proporti∣on is.

DT

9.99335424

CT

8.93607439

100000

5.00000000

8764

3.94272015

And this is the Method for finding the place of the Moons Apogaeon and excentricity. And from these and many other Eclipses as well Solar as Lunar, Mr. Street limits the place of the ☽ Apogaeon to be at the time of the first observation 21′. 04″ more, and the mean Anomaly 20. 41″ less, and the excentricity 8765 such parts as the Radius of the AEquant is 100000.

And by comparing sundry observations both antient and modern, he collects the middle mo∣tion of the Moon, from her Apogaeon, to be in the space of four Julian years or 146 days, 53 revolutions, 0 Signes, 7 degrees, 56 minutes, 45 Seconds. And the Apogaeon from the AEqui∣nox 5 Signes, 12 degrees, 46 minutes. And hence the daily motion of her mean Anomaly will be found to be 13^d. 03′. 53″. 57‴. 09^iv. 58^v. 46^vi. Of her Apogaeon 0. 06. 41. 04. 03. 25. 33.

And according to these Measures, if you de∣duct the motion of the ☽ mean Anomaly for 1641 years April

4. hours 13. 37′, viz.

8. 22. 02. 00.

from

3. 121. 35. 26

The remainer is

6. 201. 33. 26

from which abating 20′. 41″ the ☽ mean Anom. AEra Chr. 6. 20. 12. 45.

 

[illustration]

In like manner the motion of her Apogaeon for the same time is

6. 05. 311. 57

which being deducted from

3. 17. 25. 57

The remainer is

9. 11. 55. 56

To which if you add

21.04

The Sum

91. 121. 15200

is the place of the ☽ Apogaeon in the beginning of the Christian AEra.

 

CHAP. XVI.

Of the finding of the place and motion of the Moons Nodes.

ANno Christi 1652, March 28, hour. 22. 16′, the Sun and Moon being in conjunction, Mr. Street in Page 33, computes the ☽ true place in the Meridian of London to be in ♈. 19. 14. 18 with latitude North 46′. 15″.

And Anno Christi 1654 August 1. hour. 21. 19′. 30″ was the middle of a Solar Eclipse at London. at which time the Moons true place was found to be in ♌ 18. 58′. 12″ with North La∣titude 32′. 01″.

• 1654 August 1. 21. 19′. 30″ ☽ place ♌ 18. 58. 12
• 1652 March 28. 22. 16. 00 ☽ place ♈ 19. 14. 18

From the first observation to the second there are 27 years, 4 months, 5 days, 23 hours 03′. 30″.

Mean motion of the Nodes in that time, deg.

45. 19. 41

The true motion of the ☽

119. 43. 54

Their Summ is in Fig. 11. The angle DPB

165. 03. 35

Therefore in the oblique angled Spherical Triangle DPB we have given BP. 89. 13. 45 the complement of the Moons Latitude in the first Observation 2. PD 89. 27. 50 the comple∣ment of the Moons Latitude in the second obser∣vation, and the angle DPB 165. 03. 35, whose complement to a Semicircle is DPF 14. 56. 25. The angle PBD is required.

1. Proportion.

As the Cotangent of PD 89. 27. 50

9.97114485

Is to the Radius

10.00000000

So is the Cosine of DPF 14. 56. 25

9.98506483

To the tang. of PF 89. 26. 42

12.01191998

BP 89. 13. 45

 

Their Z is FPB 178. 40. 27. whose complement Is the Arch FG 1. 19. 33.

2. Proportion.

As the Sine of FP 89. 26. 42. Co. ar.

0.00002037

Is to the Cotang. of DPF 14. 56. 25

10.57376158

So is the Sine of FG 1. 19. 33

8.36418419

To the Cotang. of FGD 85. 02. 56

8.93796614

FGD=PBD inquired.

 

And in the right angled Spherical Triangle BA☊ right angled at A we have given AB 046′. 15″ the Latitude in the first observation, and the Angle AB☊=PBD 85. 02. 56. to find A☊ the Longitude of the Moon from the ascending Node.

As the Cot. of AB☊ 85. 02. 56

8.93796614

Is to the Radius

10.00000000

So is the Sine of AB 0.46′. 15″

8.12882290

To the tang. of A☊ 8. 49. 17

9.19085676

2. To find the Angle A☊B.

As the tang. of AB 0. 46. 15

8.12886212

Is to the Radius

10.00000000

So is the Sine of A☊ 8. 49. 17

9.18569718

To the Cotang. of A☊B 5. 0. 41

11.05682506

The angle of the ☽ orbite with the Ecliptick

 

The first observed place of the ☽

♈. 19. 14. 18

A☊ Subtract

8. 49. 17

There rests the true place of the ☊

♈. 10. 25. 01

The retrograde motion whereof in 4 Julian years or 2461 days, is by other observations found to be Sign 2. deg. 17. 22′. 06″. and there∣fore the daily motion deg. 0. 03′. 10″. 38‴. 11^iv. 35^v.

And the motion thereof for 1651 years, March 28. h. 22. 16′, viz. Sign 8. deg. 18. 26′. 58″ being added to the place of the Node before found Sig. 0. 10. 25: 01. Their Sum is the place thereof in the beginning of the Christian AEra Sign 8. deg. 28. 51′. 59″.

But the Rudolphin Tables as they are corrected by Mr. Horron and reduced to the Meridian of London, do differ a little from these measures, for according to these Tables, the Moons mean mo∣tions are.

AEra Christi.

The Moons mean Lon∣gitude is

Sign. 04. deg. 02. 25. 55

The Moons Apogaeon

Sign. 09. deg. 13. 46. 59

The Moons mean Anomaly

Sign. 06. deg. 18. 38. 56

The Moons Node Re∣trograde

Sign. 08. deg. 28. 33. 16

And according to these measures, the Moons mean motions in decimal Numbers are.

AEra Christi.

The Moons mean Longi∣tude, deg.

34. 00887.345677

The Moons Apogaeon, deg.

78. 82862.654320

The Moons mean Anomaly, deg.

55. 18024.691357

The Moons Node Retro∣grade, deg.

74. 69845.679010

[illustration]

The ☽ mean motion for one year.

The Moons mean Longi∣tude, deg.

35. 94001. 44893. 1

The Moons Apogeaon, deg.

11. 29551. 126365

The Moons mean Anoma∣ly, deg.

24. 64450. 322566

The Moons Node Retro∣grade, deg.

05. 36900. 781604

The ☽ mean motion for one day.

The Moons mean Longi∣tude, deg

03. 66010. 962873

The Moons Apogaeon, deg.

00. 03094. 660620

The Moons mean Anoma∣ly, deg.

03. 62916. 302253

The Moons Node Retro∣grade, deg.

00. 01470. 961045

And according to these measures are the Ta∣bles made shewing the Moons mean Longitude, Apogaeon, Anomaly, and Node retrograde for Years, Months, Days and Hours.

And hence to compute the Moons true place in her Orbit, I shall make use of the Method, which Mr. Horron in his Posthumas works lastly pub∣lished by Mr. Flamsted, in which from the Ru∣dolphin Tables he sets down these Dimensi∣ons.

The Moons mean Semidiameter deg.

00. 15′. 30″

Her mean distance in Semid. of the Earth Deg.

11. 47. 22

The half whereof deg. 5. 53. 41. he adds 45 the whole is

deg. 50. 53. 41

Whose Artificial cotangent is

9.91000022

And the double thereof makes this standing Numb.

9.82000044

	Greatest 6685. 44
The Moons	Mean 5523. 69	Excentricity
	Least 4361. 94
And her greatest variation 00. 36′. 27″.

These things premised his directions for com∣puting the Moons place, are as followeth.

CHAP. XVII.

How to Calculate the Moons true place in her Orbit.

TO the given time find the true place of the Sun, or his Longitude from the Vernal AEquinox, as hath been already shewed.

2. From the Tables of the Moons mean moti∣ons, write out the Epocha next before the gi∣ven time, and severally under that set the moti∣ons, belonging to the years, months and days compleat, and to the hours and scruples cur∣rent, every one under his like (only remember that in the Bissextile years, after the end of Fe∣bruary, the days must be increased by one Unite) then adding them all together, the Summ shall be

the Moons mean motions for the time given: But in her Node Retragrade you must leave out the Radix or first number, and the Summ of the remainer being deducted from the Radix, shall be the mean place of her Node required.

3. Deduct the Moons Apogaeon from the ☉ true place, the rest is the annual Augment, the tan∣gent of whose Complement 180 or 360, being ad∣ded to the artificial Number given 9. 82000044. the Summ shall be the tangent of an Arch, which being deducted from the said Complement, gi∣veth the Apogaeon AEquation to be added to the mean Apogaeon, in the first and third quadrants of the annual Augment, and Subtracted in the second and fourth, their Summ or difference is the true Apogaeon.

4. The true Apogaeon being Deducted from the ☽ mean Longitude gives the Moons mean Anomaly.

5. Double the annual Augment, and to the Cosine thereof add the Logarithm of 1161. 75. the difference between the Moons mean and ex∣tream Excentricity, viz. 3. 06511268, the Summ shall be the Logarithm of a number which being added to the mean Excentricity, if the double annual Augment be in the first or fourth quadrants; or Subtracted from it, if in the second or third quadrants; the Summ or difference shall be the Moons true Excentricity.

6. The Moons true Excentricity being taken for a natural Sine, the Arch answering thereto shall be the ☽ greatest Physical AEquation.

7. To the half of the Moons greatest Physical AEquation add 45 deg. the cotagent of the Summ is the artificial Logarithm of the Excentrick.

To the double whereof if you add the tang. of half the mean Anomaly, the Summ shall be the tangent of an Arch, which being added to half the mean Anomaly, shall give the Excentrick A∣nomaly.

8. To the Logarithm of the Excentrick, add the tangent of half the Excentrick Anomaly, the Summ shall be the tangent of an Arch, whose double shall be the Coequated Anomaly, and the difference between this and the mean Anomaly is the terrestrial Equation, which being added to the Moons mean Longitude, if the mean Anoma∣ly be in the first Semicircle, or Subtracted from it, if in the latter, the Summ or difference shall be the place of the Moon first Equated.

9. From the place of the Moon first Equated, Deduct the true place of the Sun, and double the remainer, and to the Sine of the double add the Sine of the greatest variation 0. 36. 27, viz. 8. 02541571, the Summ shall be the Sine of the true variation, at that time, which being added to the Moons place first Equated, when her single distance from the Sun is in the first or third qua∣drants, or Subtracted when in the second or fourth, the Summ or difference shall be the Moons true place in her Orbit.

Example.

Let the given time be Anno Christi 1672. Feb. 23. h. 11. 34′. 54″ at which time the Suns true place is in ♓ 15. 29133 and the Moons middle motions are as followeth.

	☽ Longitude	☽ Apogaeon	☊ Retrograde
1660	13. 36650.	41. 78372.	55.85177
11.	02. 66032.	24. 31246.	59.08943
Ianuary.	13. 46339.	00.95934	.45599
D. 23	84. 18252.	.71177	.33832
H. 11	1. 67755.	.01418	.00674
34′	.08641.	.00072	.00054
54	.00228.	.00012	.00001
☽ Longitude	15. 43897.	67.78229	59.89082
				95.96094

These Numbers reduced to the Degrees and Parts of the common Circle are for the ☽ mean Longitude.

55.580292

The ☽ Apogaeon.

244.015956

The ☉ true place is

345.29133

The ☽ Apogaeon subtract.

244.01595

The Annual Augment.

101.27538

The Complement whereof is

78.72462

The Tang. of deg. 78. 72462

10.70033391

The standing Number.

9.82000044

The Tang. of deg. 73. 20288

10.52033435

Their difference. 5. 52174 the Apogaeon Equation

 

Mean Apogaeon 244. 01595

 

Their difference 238. 49421 is the true Apogaeon.

 

Secondly.

The ☽ mean Longitude.

55.58029

The true Apogaeon subtract.

238.49421

Rests the ☽ mean Anom. correct.

177.08608

Or thus.

The ☽ mean Anomaly in the Tables for the time propounded, will be found to be 67. 78221, which converted into the deg. and parts of the common Circle is

171.56434

To which the Apogaeon Equation being added

5.52174

Their Sum is the mean Anom. correct.

177.08608

And hence it appears that working by the mean Anomaly instead of the mean Longitude, the true Apogaeon Equation must be added to the mean Anomaly, in the second and fourth Qua∣drants of the ☽ Annual Augment, and sub∣tracted from it in the first and third.

Thirdly.

The Annual Augment. 101. 27538 being dou∣bled is deg. 202. 55076, the Cosine of whose excess above 180, that is the Cosine of 22. 54076 is

9.96545577

The Logarithm of 1161. 75

3.06511268

The Logarithm of 1072. 92

3.03056845

The ☽ mean Excentr. 5523. 69

 

Their difference 4450. 77 is the ☽ true Ex∣centricity.

Which taken as a natural Sine, the Arch answering thereunto Deg. 2. 55094 is the ☽ greatest Physical Equation.

 

Fourthly.

To the half of the Physical Equation. deg. 01. 27547 add 45 degrees, the Sum is deg. 46. 27547, the Cotangent whereof; viz. 9. 98080957 is the Logarithm of the Excen∣trick, the double of which Logarithm is

9.96161914

Tangent frac12 Anomaly corrected 88. 54304

11.59455229

Tang. of deg. 88. 40849

11.55620143

Their Sum deg. 176. 95153 is the excentrick A∣nomaly.

 

Fifthly.

The Logarithm of the Excen∣trick is

9.98080957

Tang. frac12 excent. Anom. 88. 475765

11.57505878

Tangent of deg. 88. 407268

11.55586835

The double whereof 176. 814536

is the coequa∣ted Anomaly.

M. Anomaly correct.

177.086080

Their difference 0. 271544

is the Equati∣on sought to

be subst. from ☽ mean Long.

55.580292

The Remainer 55. 308748

is the ☽ place first Equated.

Sixthly,

From the place of the ☽ first E∣quated.

55.308748

Deduct the true place of the Sun

345.291330

The Remainer is the Distance of the ☽ à ☉

70.017418

The double whereof is 140. 034836. The Sine of whose Complement to a Semi-circle, 39. 965164 is

9.80775260

The Sine of the greatest variation

8.02541571

The Sine of the true var. 0. 390206

7.83316831

The ☽ place first Equa. 55. 308748

 

The ☽ place in Orbit 55. 698954 that is in Sex∣agenary Numbers. 8. 25. 41. 54.

 

CHAP. XVIII.

To compute the true Latitude of the Moon, and to reduce her place, from her Orbit to the Ecliptick.

THe greatest Obliquity of the Moon's Orb with the Ecliptick or Angle A ☊ B Fig. 11. is by many Observations confirmed to be 5 De∣grees just, at the time of the Conjunction or Op∣position of the Sun and Moon, but in her Quar∣ters deg. 5. 18′. Now then then find her Latitude at all times, the said Mr. Horrox refers us to pag. 87. in the Rudolphin Tables, to find from thence the Equation of the Nodes, and Inclination limi∣tis menstrui, in this manner.

1. From the mean place of the Node, deduct

the ☉ true place, the Remainer is the distance of the ☉ from the ☊. with which entring the said Table, he finds the Equation of the Node and Inclination limitis menstrui, which being added to or subtracted from the Nodes mean place ac∣cording to the title, the Sum or difference is the true place of the Node, which being deducted from the place of the Moon in her Orb, the Re∣mainer shall be the Augment of Latitude or Di∣stance of the Moon from the Node, or Leg A ☊.

2. With the Augment of Latitude, enter the Table of the Moon's Latitude, and take thence her Simple and Latitude and Increase answering to it. Then say, as the whole excess of Latitude 18′, or in Decimals 30. is to the Inclination of the Monethly limit: So is the increase of Latitude to the Part Proportional; which being added to the sim∣ple Latitude, will give you the true Latitude of the Moon.

3. With the same Augment of Latitude, en∣ter the Table of Reduction, and take thence the Reduction and Inclination answering thereto: Then say again, as 18′. 00″. or 0. 30. is to the In∣clination of the Monethly limit: So is the increase of Reduction, to the Part Proportional; which being added to the simple Reduction, shall give the true, to be added to, or subtracted from the place of the Moon in the Ecliptick.

Example. By the former Chapter, we found the mean motion of the Node to be 95. 96094, which reduced to the Degrees and Parts of the common Circle is

345.459384

And the Suns true place to be

345.291334

Their difference is the distance ☉ à ☊

. 168050

with which entring the Table, Entituled Ta∣bula AEquationis Nodorum Lunae. I find the Node to need no Equation, and the Inclinati∣on limitis menstrui to be deg. 00. 30.

The place of the ☽ in her Orbit

55.698954

The Nodes true place, subtract.

345.459384

The Augment of Latitude

70.239570

2. With this Augment of Latitude I enter the Table shewing the Moons simple Latitude, and thereby find her simple Latitude to be De∣grees. 04. 70476. North; And the in∣crease

00.28234

And therefore the Moons true Lati∣tude is deg.

4.98610

3. With the same Augment of Latitude, I en∣ter the Table of Reduction, and thereby find the Reduction to be

00.06955

And the increase of Reduction to be deg.

00.00855

And therefore the whole Reduction to be sub.

00.07810

From the ☽ place in her Orbit

55.69895

The ☽ true place in the Ecliptick

55.62085

That is in Sexagenary Numbers.

8. 25. 37′. 15″.

CHAP. XIX.

To find the Mean Conjunction and Opposition of the Sun and Moon.

TO this purpose we have here exhibited a Ta∣ble shewing the Moons mean motion from the Sun, the construction whereof is this: By the Tables of the Moons mean motions, her mean

Longitude AEra Christi is

34.0088734567

The ☉ mean Anomaly.

56.6997085185

Praecession of the AEquinox.

20.4976851851

Their Sum is the ☉ mean lon∣git. AEra a Christi.

77.4973937036

Which being deducted from the ☽ mean longitude, the remainer is the Moons mean

56.8114797531

distance from the Sun, in the beginning of the Christian AEra.

 

In like manner the Moons mean distance from the Sun in a year or a day is thus found.

☉ Anomaly for a year.

99.9297857316

Praecession of the AEquinox.

0038580246

Their Sum subtract.

99.9336437562

From the ☽ mean Longitude.

35.9400144893

Moons distance from the ☉.

36.0063707331

Moons distance from the Sun in a days time.

☉ mean Anomaly.

27378.02348

Praecession of the AEquinox.

1.05699

Their Sum subtract.

27379.08047

From the ☽ mean Longitude.

03. 66010.96287

☽ Daily motion from the ☉.

03. 38631.88240

And according to these measures are the Ta∣bles made, shewing the Moons mean motion from the Sun, by which the mean conjunction of the ☽ and Moon may be thus computed.

To the given year and Month gather the mid∣dle motions of the Moon from the Sun, and take the complement thereof to a whole Circle, from which subtracting continually the nearest lesser middle motions, the day, hour, and minute en∣fuing thereto is the mean time of the Conjun∣ction.

Example, Anno Christi 1676. I would know the time of the mean Conjunction or New Moon in October.

Epocha 1660

32.697283

Years Compl. 15.

50.254463

Septemb. Compl.

24.465038

1. day for Leap-year.

03.386318

Their Sum is the Moons motion from the ☉.

10.803102

Complement to a whole Circle.

89.196898

Days 26 Subtract.

88.044289

Hours 8. substract.

• 1.152609
• 1.128772

Minutes 10 Subtract.

• 0.023837
• 0.023516

The Remainer giveth 8″.

.00321

Therefore the mean Conjunction in October, 1676. was the 26 day, 10 min. 8 seconds after 8 at night.

And to find the mean opposition. To the com∣plement of the middle motion, add a semicircle, and then subtract the nearest lesser middle moti∣ons as before, the day, hour, and minute ensuing thereto, shall be the mean opposition required.

Example, Anno Christi, 1676. I desire to know the mean opposition in November.

Epocha 1660

32.697283

Years Compl. 15

50.254463

October Compl.

29.440922

1 day for Leap-year.

03.386318

The ☽ mean motion from the ☉

15.778986

Complement to a whole Circle.

84.221014

To which add a Semicircle.

50.

The Sum is

34.221014

Day 10 subtract.

33.863188

Hours 2.

• .357826
• .282193

Minutes 32.

• .075633
• .075251

The Remainer giveth 9 seconds.

.000382

Therefore the Full Moon or mean Opposition of the Sun and Moon was November the 10th, Hours 2, 32′ 09″. The like may be done for a∣ny other.

And here I should proceed to shew the manner of finding the true Conjunction or Opposition of the Sun and Moon, but there being no decimal Canon yet extant, suitable to the Tables of mid∣dle motions here exhibited, I chuse rather to re∣fer my Reader to Mr. Street's Astronomia Carolina, for instructions in that particular, and what else shall be found wanting in this Subject.