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invented; and to find out fuch a Cycle of Years, wherein by the Intercalation or Addition of one or more Months this might be effected, was the great Study and Endeavour of the Aftronomers of thofe Times. The firft Attempt that was made for this Purpose was that of the Dieteris, a Cycle of two Years, wherein an Intercalation was made of one Month; but in two Years Time, the Excefs of the Solar Year above the Lunar being only 22 Days, and a Lunar Month making 29 Days and an half, this Intercalation, instead of bringing the Lunar Year to a Reconciliation with the Solar, overdid it by 7 Days and an half. This Fault being foon perceived, for the mending of it the Tetraeteris was introduced, which was a Cycle of 4 Years, wherein it was thought that an Intercalation of one Month would bring all that to rights, which was over-done by the like Intercalation of the Dieteris. And this was contrived chiefly with a Refpect to the Olympic Games. For they being the chief of their Solemnities, and celebrated once every four Years, Care was taken to bring them every fourth Year as near as poffible to the fame Time of the Solar Year, in which they had been performed the Olympiad before. Now this Solemnity, according to the original Inftitution, was always to begin on the firft Full Moon after the Summer Solftice; and it was thought that an Intercalation of one Month in four Years would always bring it to this Time. But four Solar Years exceeding four Lunar Years 44 Days, the adding one Lunar Month, or 29 Days and an half, fell fhort of curing this Defect, upwards of 14 Days. This Fault likewife foon discovering itself, they intercalated alternatively, one four Years with one Month, and the next four Years with two Months, which brought it to the Octoeteris, or Cycle of eight Years, wherein by intercalating three Months, they thought they brought all to rights, and indeed it came much nearer to it, than any of the former Cycles. For by this Intercalation the eight Lunar Years were brought so near to the eight Solar Years, that they differed from them only by an Excefs of one Day and 14 Hours. And therefore this Cycle continued much longer in use than any of the reft. But at length the Error, by increafing every Year, grew great enough to be alfo difcovered, which produced the Invention of feveral other Cycles, till at length the Metonic Cycle of 19 Years took place, fo called from Meto an Athenian, the Inventor of it. This great Aftronomer found by Calculation, that if the Sun and Moon were fuppofed to let out together from any Point of the Zodiac, after 19 Solar Revolutions, they would meet again in the felf-fame Point, and

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begin a new Period exactly agreeing with the former. These 19 Solar Revolutions he found to contain 235 Lunations, which make 19 Lunar Years, and 7 Lunar Months, to be added to them by 7 Intercalations. So that the whole Cycle confifted of 12 Lunar Years of 12 Months each, and feven intercalated Years of thirteen Months, which correfponding to 19 Solar or Julian Years, the New and Full Moons after that Space, not only return to the fame Days of the Julian Year, but nearly to the fame Hours of the Day. A Courfe of Obfervations therefore determining the Days on which the New and Full Moons happen during one Revolution of this Cycle, will also serve for the next Revolution of the fame, and fo on in Succeffion. The chief Ufe of this Cycle among the Greeks being to fettle the Times of celebrating their Solemnities, and that of the Olympiads being the chief of them, on the fixing of which the fixing of all the reft did depend, it was in the first place applied to this Purpose. And as the Olympic Games were always to be celebrated on the firft Full Moon after the Summer Solftice, in order to fettle the Time of their Celebration, it was neceffary in the first Place to settle the Time of the Summer Solftice. This Meto, the Year he introduced his Cycle, obferved to be on the 21st Day of the Egyptian Month Phamenoth, which reduced to the Julian Year falls on the 27th of June. And therefore the Greeks having received this Cycle, did from this Time forward celebrate their Olympiads on the firft Full Moon after the 27th Day of our June; and henceforth also began their Year from the New Moon preceding. The Year in the Beginning of which the Olympic Games were celebrated, was in their Computation of Time, called the first Year of that Olympiad ; and in the Beginning of the fifth Year after they celebrated the next Olympiad, which made the Time from one Olympiad to another to be juft four Years, according to the Measure of the Years then used.

I have thus given you a full and I hope intelligible Account of the Metonic Cycle, fo famous in ancient Chronology; and ftill known among us under the Name of the Cycle of the Moon or Golden Number. Upon the ceafing of the Greek Solemnities, the Ufe of this Cycle alfo ceased, and fo continued for feveral Centuries, till at length after the Council of Nice, the Chriftians introduced it into their Calendar, and made ufe of it in fettling Eafter, and the other moveable Feafts. For by a Decree of that Council, Eafter-day was fixed to the Sunday after the firft Full Moon that followed next after the vernal Equinox: Thus it became neceffary in the Chriftian Church as well as among the Greeks, to calculate

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the New and Full Moons in the Heavens, and adjust them to the Solar Courfe. And as a better Cycle for this Purpose than the 19 Years Cycle was not to be found, because none other can bring the Course of the Sun and Moon to a nearer Agreement, the Chriftian Church accordingly pitched upon it, as the best Rule they could follow for the fixing of their Eafter. And fo great a Value did they fet upon it by reason of its great Ufefulness in Ecclefiaftical Computations, that the Numbers of it were written in the ancient Calendars in golden Letters; from whence in our present Almanacks, that Number of this Cycle, which accords with the Year for which the Almanack is made, is called the Golden Number. Now the Golden Number for any Year whatsoever of the Chriftian Era, may be eafily found by the following plain Rule. The first Year of Christ according to the Computation in Old Stile, fell in with the 2d Year of the Lunar Cycle, and therefore if to the given Year of the Chriftian Æra you add one, and divide the Sum by 19, the Quotient fhews the Number of Revolutions of the Cycle from the Beginning of the faid Æra, and the Remainder after Divifion is the Golden Number required; but if nothing remains, the Golden Number is 19. Suppofe for Example the Golden Number of the. Year 1746 were required: Then 1746 added to I makes 1747, and that divided by 19, gives 91 for the Quotient, with a Remainder of 18. And therefore 18 is the Golden Number for that Year; and 91 the Quotient expreffes the Number of Revolutions of this Cycle from the Beginning of the Year preceding the Birth of Chrift.

P. But does this Cycle of 19 Years bring the Solar and Lunar Revolutions to fo exact an Agreement, as to be always an invariable Rule in this Cafe?

G. Altho' the Metonic Cycle comes indeed very near the Truth, fo as to fhew the Lunations for the Space of three Centuries without the Error of a fingle Day; yet the Difference continually increafing, grows in Time to be confiderable. For 19 Lunar Years, and 7 intercalated Months, of which this Cycle confifts, falling fhort of 19 Julian Years almost an Hour and an half, hence it hath followed, that in every one of the Years of this Lunar Cycle, the New Moons and Full Moons have happened juft fo much fooner each Month, than in the fame Years of the Cycle immediately preceding. And hereby it hath come to pafs, that after the elapfing of fo many Rounds of this Cycle, as have revolved from the Times of the Nicene Council to the Year 1746, the New Moons and Full Moons in the Heavens, have anticipated the New

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and Full Moons in the Calendar of the Common Prayer Book, four Days and an half; because the New Moons and Full Moons are there ftated, not according to the present Times, but according to the Times of that Council. Thefe laft are called Ecclefiaftical New Moons, to diftinguish them from the true ones in the Heavens; and the general Table or Rule for finding Eafter for ever may ftill be applied, if we make the proper Allowance above defcribed. That is, in calculating the New Moons we must reckon four Days and an half before the Time affigned by the Calendar; or, which amounts to the fame, call the Day of the New Moon as you find it by the Calendar, the fifth Day of the Moon's Age. In the Gregorian Reformation of the Calendar, the Golden Number is thrown out, and the Epact introduced in the Place of it. But as it is not my Intention here to meddle with Ecclefiaftical Computations any farther than is neceffary to give a clear Idea of the Cycles that constitute the Julian Period, I fhall here conclude my Obfervations upon the Lunar Cycle, which I have endeavoured to explain in the moft full and diftinct Manner, not only because of its great Note in ancient Computation, but also for the diftinguished Place it ftill retains in our Civil Calendar.

P. What other Cycles befides this of the Moon, are made ufe of in the Compofition of the Julian Period?

G. The Julian Period, befides the Lunar Cycle, takes in alfo two others: That of the Sun as it is commonly called, and the Cycle of Indiction. The Solar Cycle is fo called, not from expreffing any Number or Series of Solar Revolutions, but because by its Help we know the Dominical Letter, or the Character of Sunday. But to enable you the better to comprehend this, I muft obferve, that as we divide Time into Weeks, and defcribe the Day of the Week by seven several Names; fo are those Days diftinguished in the Calendar by seven Letters fet in alphabetical Order before them, and repeated to them in a conftant Round throughout -the whole Year. Thefe Letters are the first seven of the Alphabet, A, B, C, D, E, F, G; and the Custom is, to affign the Letter A to the firft Day of the Year; which if it happens to be a Sunday, then A is the Letter for Sunday, or the Dominical Letter; and the reft are applied in Order to the other Days of the Week. Now as the Number of Days in a Week are feven, and the Number of Letters applied to them alfo feven, it is evident, that whatever Letter answers to the first Sunday of the Year, will ftand for Sunday all the Year round, the Revolution of Days and Letters being in

this Refpect the fame, and perfectly coinciding. It is manifeft Jikewife, that if the Year was made up of an exact Number of Weeks, the Dominical Letter would continue conftantly and invariably the fame, because the firft Day of the Year would always fall upon the fame Day of the Week, and of Confequence create no Interruption or Difturbance in the Order and Succeffion of Letters. But as this is not the Cafe, the odd Day or Days muft unavoidably break in upon the Series, and to take Account of these Alterations, is the Defign of the Solar Cycle. As the common Julian Year confifts of 52 Weeks and one Day; if the first Day of the Year falls upon a Sunday making A the Dominical Letter, then will the laft Day of the Year alfo fall upon a Sunday, and the firft Day of the next fucceeding Year will be Monday. But as the Letter A is always appropriated to the first Day of the Year, it now of courfe becomes the Characteristick of Monday, and the Letter that in due Order of Succeffion falls to Sunday, is G, which therefore becomes the Dominical Letter of the Year. A like Train of Things will also shift the Dominical Letter of the enfuing Year back by one Letter, and throw it upon F. And this Revolution, were it allowed to run on without Interruption, would be determined in feven Years.

But it fo happens in the Julian Computation, that every 4th Year is a Leap-Year, confifting of 366 Days, which make 52 Weeks and 2 Days, and in this Cafe the Dominical Letter will be fhifted back by two Letters, and fall the following Year upon the next Letter fave one in a retrograde Order. Thus if the Dominical Letter at the Beginning of a LeapYear be A, it will not the following Year fall upon G as in the first Cafe, but by a double Retrogreffion, because of the two odd Days, it is fhifted back to F. And it is farther to be obferved of thefe Leap-Years, that the fame Dominical Letter is not as in common Years, continued to the End of the Year, as might have been done, and the two odd Days fuffered then to effect the double Change; but it has been judged more convenient to change the Dominical Letter in the Month of February, when the intercalary Day is inferted, Whatever therefore is the Sunday Letter at the Beginning of a Leap-Year, fo continues till towards the End of February but then, by reafon of the Intercalation, the 23d and 24th Days are denoted by the fame Letter, in which Cafe it is evident that the Dominical Letter muft for the Remainder of that Year go one Place back. If therefore the Dominical Letter in the Beginning of the Year be A, after the 24th of February

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