The Project Gutenberg eBook of Remarks Upon the Solar and the Lunar Years, by Parker George

REMARKS
UPON
THE SOLAR AND THE LUNAR YEARS,
The Cycle of 19 Years, commonly called
THE GOLDEN NUMBER,
THE EPACT,

And a Method of finding the Time of
Easter, as it is now observed in most
Parts of Europe.

Being Part of a LETTER from

The Right Honourable

GEORGE EARL OF MACCLESFIELD

to

Martin Folkes Esq; President of the Royal Society,

and by him communicated to the same,
May 10. 1750.

A TABLE, shewing, by means of the Golden Numbers, the several Days on which the Paschal Limits or Full Moons, according to the Gregorian Account, have already happened, or will hereafter happen; from the Reformation of the Calendar in the Year of our Lord 1582, to the Year 4199 inclusive.

To find the Day on which the Paschal Limit or Full Moon falls in any given Year; Look, in the Column of Golden Numbers belonging to that Period of Time wherein the given Year is contained, for the Golden Number of that Year; over-against which, in the same Line continued to the Column intitled Paschal Full Moons, you will find the Day of the Month, on which the Paschal Limit or Full Moon happens in that Year. And the Sunday next after that Day is Easter Day in that Year, according to the Gregorian Account.

Remarks upon the Solar and the Lunar Years, the Cycle of 19 Years, commonly called the Golden Number, the Epact, and a Method of finding the Time of Easter, as it is now observed in most Parts of Europe. Being Part of a Letter from the Right Honourable George Earl of Macclesfield to Martin Folkes Esq; President of the Royal Society.

Of the Solar Year.

The mean Tropical Solar Year, or that mean Space of Time wherein the Sun, or Earth, after departing from any Point of the Ecliptic returns to the same again, consists, according to Dr. Halley's Tables, of 365^d, 5^h, 48^´, 55^´´: Which is less by 11^´, 5^´´, than the mean Julian Year, consisting of 365^d, 6^h, 0^´, 0^´´.

Hence the Equinoxes and Solstices anticipate, or come earlier than the Julian Account supposes them to do by 11´, 5´´, in each mean Julian Year; or 44^´, 20^´´ in every four; or 3^d, 1^h, 53^´, 20^´´, in every four hundred Julian Years.

In order to correct this Error in the Julian Year, the Authors of the Gregorian Method of regulating the Year, when they reformed the Calendar in the Beginning of October 1582, directed that three intercalary Days should be omitted or dropped in every four hundred Years; by reckoning all those Years, [2] whose Date consists of a Number of entire Hundreds not divisible by 4, such as 1700, 1800, 1900, 2100, &c. to be only Common, and not Bissextile or Leap Years, as they would otherwise have been; and consequently omitting the intercalary Days, which, according to the Julian Account, should have been inserted in the Month of February in those Years. But at the same time they order'd that every fourth hundredth Year, consisting of a Number of entire Hundreds, divisible by 4, such as 1600, 2000, 2400, 2800, &c. should still be consider'd as Bissextile or Leap Years, and, of consequence, that one Day should be intercalated as usual in those Years.

This Correction, however, did not entirely remove the Error: For the Equinoxes and Solstices still anticipate 1^h, 53^´, 20^´´ in every four hundred Gregorian Years.

But that Difference is so inconsiderable as not to amount to twenty-four Hours, or to one whole Day, in less than 5082 Gregorian Years.

Of the Lunar Year, Cycle of 19 Years, and the Epact.

The Space of Time betwixt one mean Conjunction of the Moon with the Sun and the next following, or a mean Synodical Month, is equal to 29^d, 12^h, 44^´, 3^´´, 2^´´´, 56^IV according to Mr. Pound's Tables of mean Conjunctions.

In each Cycle of 19 Lunar Years, there are 12 Common, and 7 Intercalary or Embolimæan Years, making together 235 Synodical Months.

It was thought, at the time of the General Council of Nice, which was holden in the Year of our Lord 325, that 19 Julian Solar Years were exactly equal to such a Cycle of 19 Lunar Years, or to 235 Synodical Months; and therefore, that, at the End of 19 Years, the New Moons or Conjunctions would happen exactly at the same Times, as they did 19 Years before: And upon this Supposition it was, that, some time afterwards, the several Numbers of that Cycle, commonly called the Golden Numbers, were prefixed to all those Days in the Calendar, on which the New Moons then happened in the respective Years corresponding to those Numbers; it being imagined, that whensoever any of those Numbers should for the future be the Golden Number of the Year, the New Moons would invariably happen on those Days in the several Months, to which that Number was prefixed.

This Difference amounts to a whole Day very nearly in 310.7 Years, the New Moons anticipating,[4] or falling earlier, by 24 Hours in that Space of Time, than they did before: And therefore now in the Year 1750, the New Moons happen above four Days and a half sooner, than the Times pointed out by the Golden Numbers in the Calendar.

In order therefore to preserve a sort of regular Correspondence betwixt the Solar and the Lunar Years, and to make the Golden Numbers, prefixed to the Days of the Month, useful for determining the Times of the New Moons, it would be necessary, when once those Golden Numbers should have been prefixed to the proper Days, to make them anticipate a Day at the End of every 310.7 Years, as the Moons will actually have done; that is to set them back one Day, by prefixing each of them to the Day preceding that, against which they before stood.

But as such a Rule would neither be so easily comprehended or retained in Memory, as if the Alteration was to be made at the End or at the Beginning of complete Centuries of Years; the Rule would be much more fit for Practice, and keep sufficiently near to the Truth, if those Numbers should be set back nine Days in the Space of 2800 Years; by setting them back one Day, first at the End of 400 Years, and then at the End of every 300 Years for eight times successively: whereby they would be set back, in the whole, nine Days in 2800 Years. After which they must again be set one Day back at the End of 400 Years, and so on, as in the preceding 2800 Years. By which means the Golden Numbers would always point out the mean Times of the New Moons, within a Day of the Truth.

[5] It is plain however that the Lunar Year will have lost one Day more than ordinary, with respect to the Solar Year, whenever the New Moons shall have anticipated a whole Day; as they will have done at those times, when it is necessary that the Golden Numbers should, by the Rule just now given, be set back one Day: and consequently the Epact, for that and the succeeding Years, must exceed by an Unit the several corresponding Epacts of the preceding 19 Years.

For the Epact is the Difference, in whole Days, betwixt the common Julian Solar and the Lunar Year; the former being reckoned to consist of 365, and the latter of only 354 Days. If therefore the Solar and the Lunar Year at any time should commence on the same Day, the Solar would, at the End of the Year, have exceeded the Lunar by 11 Days; which Number 11 would be the Epact of the next Year: 22 would be the Epact of the Year following, and 33 the Epact of the Year after that, the Epacts increasing yearly by 11. But as often as this yearly Addition makes the Epact exceed 30, those 30 are rejected as making an intercalary Month, and only the Excess of the Epact above 30 is accounted the true Epact for that Year. Thus when the Epact would amount to 31, 32, 33, 34, &c. the 30 is rejected, and the Epact becomes 1, 2, 3, 4, &c.

Since therefore the Lunar Year will have lost a Day more than ordinary, in respect of the Solar Year, whenever it is necessary to set the Golden Numbers one Day back, as was before observed; it follows, that the Epact must at the same time be increased by an Unit more than usual: the Difference betwixt the Solar and the Lunar Year having been just so[6] much greater than usual. That is, 12 must be added, instead of 11, to the Epact of the preceding, in order to form what will be the Epact of the then present Year. Which Addition of an Unit extraordinary to one Epact will occasion all the subsequent Epacts (which will follow each other in the usual manner, each exceeding the foregoing by 11) to be greater by an Unit than their respectively corresponding Epacts of the preceding 19 Years.

If therefore, instead of the Golden Numbers, the Epacts of the several Years were prefixed, in the manner the Gregorians have done, to the Days of the Calendar, in order to denote the Days on which the New Moons fall in those Years whereof those Numbers are the Epacts; there would never be Occasion to shift the Places of those Epacts in the Calendar; since the Augmentation by an Unit extraordinary of the Epacts themselves would answer the Purpose, and keep all tolerably right.

Thus in a very easy Method may the Course of the New Moons be pointed out, either by the Golden Numbers, or by the Epacts, according to the Julian Account or Manner of adjusting the Year, which goes on regular and uniform without any Variation.

But the regulating these things for those who use the Gregorian Account, is an Affair of more Intricacy; and for them it will require more Consideration to determine, when the Epacts are to be more than ordinarily augmented, and at what Times they are to continue in their usual Course; nay, to know when they are not only not to be extraordinarily augmented, but also when they are to be diminished by an Unit, by increasing one of them by 10 only instead of 11 as usual: and this happens much oftener with the[7] Gregorians, than the increasing one of them by 12 instead of 11. For, in every Gregorian Solar Year, whose Date consists of any Number of entire Hundreds not divisible by 4, it is supposed that the Equinox has anticipated one whole Day; and therefore one Day, that which ought to be the intercalary one, is omitted; and consequently the preceding Solar Year, where one Day was lost, exceeded the Lunar Year by 10 Days only instead of 11.

In order therefore to adapt the before-mention'd Rule to the Gregorian Account, and to know in what Years the Epacts should either be extraordinarily augmented or diminished, and the Golden Numbers should either be set backwards or forwards in the Calendar; the following Rules and Directions must be observed.

First. That in the Years 1800, 2100, 2700, 3000, &c. where the Number of entire Hundreds is divisible by 3, but not by 4, the Gregorian Solar, as well as the Lunar Year, will have lost a Day; and consequently the Difference betwixt them will be the same as usual: Therefore in those Years there must be no Alteration, either in the Epacts or the Golden Numbers; but the former must go on in the same manner, and the latter stand prefixed to the same Days in the Calendar, for another, as they did for the last hundred Years.

2dly. The like will happen in the Years 2000, 2800, 3200, &c. where the Number of entire Hundreds is divisible by 4, but not by 3: For neither the Gregorian Solar nor the Lunar Year is to be altered; and therefore the Epacts must go on, and the Golden Numbers stand, as they did before.

[8] But, 3dly, In the Years 2400, and 3600, whose Number of entire Hundreds is divisible both by 3 and 4, the Gregorian Solar Year goes on as usual, and the Lunar Year has lost a Day. The Difference therefore betwixt them being 12, the Epact of the preceding Year must be augmented by that Number instead of 11, in order to form the Epact of the then present Year; whereby a new Set of Epacts will be introduced, exceeding their precedent corresponding Epacts by an Unit: And the Golden Numbers must be set one Day back in the Calendar.

4thly and lastly, In the Years 1900, 2200, 2300, 2500, &c. where the Number of Hundreds is divisible neither by 3 nor 4; the Gregorian Solar Year having lost one Day, and the Lunar none, the Difference betwixt them being only 10; that Number only, and not 11, is to be added to the Epact of the preceding, in order to form the Epact of that, the then present Year; whereby a new Set of Epacts will be introduced, all of them less by an Unit than their precedent corresponding Epacts: And the Golden Numbers must be set a Day forwarder in the Calendar; that is, be prefixed to the Day following that, against which they stood in the precedent hundred Years.

This Method would preserve a sort of Regularity betwixt the Solar and the Lunar Years; and, by means of the Rules and Directions before-mentioned, the Days of the New Moons might be pointed out, either by the Golden Numbers or by the Epacts, placed in the Calendar for that Purpose; according to the Julian Account for ever, and according to the Gregorian Account till the Year 4199 inclusive, after[9] which there must be some little Variation made in the four last Precepts or Rules, but it would be to little Purpose now, to attempt the framing of a new Set of Rules for so distant a Time.

The Gregorians have chosen to make use of the Epacts to determine the Days of the New Moons, and follow pretty nearly the Rules prescribed above; except that they order the Epacts to have an additional Augmentation of an Unit eight times in 2500 Years, beginning with the Year 1800, as at the End of 400 Years; to which 400 Years if there be added three times seven hundred, or 2100 Years, the Period of 2500 Years will be completed in the Year 3900. After which they do not make their extraordinary Augmentation of an Unit in the Epacts, till at the End of another Term of 400 Years; which defers that Augmentation from the Year 4200 to the Year 4300. And this is the Reason that the Rules above delivered will require a Variation in the Year 4200; whereas it is directed in this Paper that the Epacts should be augmented, or (which is the same thing) the Golden Numbers be set back in the Calendar nine times in 2800 Years. This arises from the Gregorians supposing, that the Difference betwixt 19 Solar and as many Lunar Years would not amount to a whole Day in less than 312 Years and a half; whereas it has appeared above, that it would amount to a whole Day in 310.7 Years. But although the Rule prescribed in this Paper comes much nearer to the Truth, yet the Error in either Case is very inconsiderable, being so small as not to amount to a whole Day in many thousand Years; and therefore is not worth regarding.

A Method of finding the Time of Easter, as it is observed in most Parts of Europe.

From what has been already said, a Method may be obtained, for fixing, with sufficient Exactness, the Time of the Celebration of the Feast of Easter, which is governed by the Vernal Equinox, and by the Age of the Moon nearest to it. The former whereof, when once rightly adjusted, may (by the Corrections mentioned in that Part of this Paper which relates to the Solar Year) be made to continue to fall at very near the same time with, or at most not to differ a whole Day from the true Equinox: and the same Rules and Directions, which, as was before shewn, would, without any great Error, point out the Times of the first Day of the Moon, would with equal Certainty point out the fourteenth, fifteenth, or any other: And thus the Times of the Oppositions or the Full Moons might be as well marked out thereby, as those of the Conjunctions or the New Moons.

I shall not at present take notice of the Canon of the Council of Nice, in the Year of our Lord 325, which directs the Time of celebrating the Feast of Easter, or of the Reasons upon which that Canon was founded. Nor shall I endeavour to explain the Rule now in Use in the Church of England for finding Easter: For, besides that such an Explanation would extend this Paper to an improper Length, those Points have already been treated of by several much abler Hands, and particularly by our Countryman the learned Dr. Prideaux. Nor is it my Intention to enter far into[11] the Methods used by the Gregorians, or those of the Church of Rome, or by any other Nations or Countries, for finding the Time of that Feast. As to our own, I shall only observe, that the Method now used in England, for finding the fourteenth Day of the Moon, or the Ecclesiastical Full Moon, on which Easter dependeth, is, by Process of Time, become considerably erroneous: as the Golden Numbers, which were placed in the Calendar, to point out the Days on which the New Moons fall in those Years of which they are respectively the Golden Numbers, do now stand several Days later in the same than those New Moons do really happen. Which Error, as was before observed, arises from the Anticipation of the Moons since the Time of the Council of Nice: And as the Vernal Equinox has also anticipated eleven Days since that time; neither that Equinox, nor the New Moons, do now happen on those Days upon which the Church of England supposes them so to happen.

When Pope Gregory XIII. reformed the Julian Solar Year, he likewise made a Correction as to the Time of celebrating the Feast of Easter, by placing the Epacts (which he directed to be made use of for the future instead of the Golden Numbers) much nearer to the true Times of the New Moons than the Golden Numbers then stood in the old Calendar: I say, much nearer to the true Times; because in fact the Epacts, as placed by him, were not prefixed to the exact Days upon which the New Moons then truly fell. And this was done with Design, and for a Reason which it is not material to the Purpose of this Paper to mention.

[12] But the Church of England, and that of Rome or the Gregorians, do still agree in this; that both of them mark (the former by the Golden Numbers, and the latter by the Epacts corresponding to them) the Days on which their Ecclesiastical New Moons are supposed to happen: And that fourteenth Day of the Moon inclusive, or that Full Moon, which falls upon, or next after, the 21st Day of March, is the Paschal Limit or Full Moon to both: And the Sunday next following that Limit or Full Moon, is by both Churches celebrated as Easter Day. But the 21st of March being reckoned, according to the Gregorian Account or the New Style, eleven Days sooner than by the Julian Account or the Old Style, which is still in Use amongst us; and their Ecclesiastical New Moons being three Days earlier than those of the Church of England; it happens that although the Church of England and that of Rome often do, yet more frequently they do not, celebrate the Feast of Easter upon the same natural Day.

It might however be easier for both, and could occasion no Inconvenience, now that Almanacks, which tell the exact Times of the New Moons, are in most Peoples Hands; if all the Golden Numbers and Epacts now prefixed to those Days of the Calendar, in our Book of Common Prayer, and in the Roman Breviary, on which the respective Ecclesiastical New Moons happen, were omitted in the Places where they now stand; and were set only against those fourteenth Days of the Moon, or those Full Moons, which happen betwixt the 21st Day of March and the 18th of April, both inclusive. Since no fourteenth Day or Full Moon, which happens before the 21st[13] of March, or after the 18th Day of April, can have any Share in fixing the Time of Easter. By which means the Trouble of counting to the fourteenth Day, and the Mistakes which sometimes arise therefrom, would be avoided.

We do as yet in England follow the Julian Account or the Old Style in the Civil Year; as also the Old Method of finding those Moons upon which Easter depends: Both of which have been shewn to be very erroneous.

If therefore this Nation should ever judge it proper to correct the Civil Year, and to make it conformable to that of the Gregorians, it would surely be adviseable to correct the Time of the Celebration of the Feast of Easter likewise, and to bring it to the same Day upon which it is kept and solemnized by the Inhabitants of the greatest Part of Europe, that is, by those who follow the Gregorian Account. For tho' I am aware, that their Method of finding the Time of Easter is not quite exact, but is liable to some Errors; yet I apprehend, that all other practicable Methods of doing it would be so too: And if they were more free from Error, they would probably be more intricate, and harder to be understood by Numbers of People, than the Method of determining that Feast either by a Cycle of Epacts, as is practiced by the Gregorians, or by that of 19 Years or the Golden Numbers, in the manner proposed in the following Part of this Paper: And it is of no small Importance, that a Matter of so general a Concern, as the Method of finding Easter is, should be within the Reach of the Generality of Mankind, at least as far as the Nature of the thing will admit.

[14] For which Reason, in case the Legislature of this Country should before the Year 1900, think fit to make our Civil Year correspond with that of the Gregorians, and also to celebrate all the future Feasts of Easter upon the same Days upon which they celebrate them; this last Particular might be easily effected, without altering the Rule of the Church of England for the finding of that Feast: And this only by advancing the Golden Numbers, prefixed to certain Days in the Calendar, 8 Days forwarder for the New Moons, or 21 Days forwarder for the fourteenth Days or Full Moons, than they now stand in our Calendar.

In order to explain this, it must be observed, that the Gregorian Account or the New Style is eleven Days forwarder than the Julian Account or the Old Style, which we still make use of; that is, the last Day of any of our Months is the eleventh Day of their next succeeding Month. If therefore their Ecclesiastical New Moons fell on the same Days with those of the Church of England, the Golden Number 14, which now stands against the last Day of February in our that is the Julian Calendar, should, when we should have adopted the Gregorian Calendar, be prefixed to the 11th Day of March. But since their Ecclesiastical New Moons happen 3 Days earlier than our Ecclesiastical New Moons at present do; so much should be deducted from those 11 Days, by which the Golden Numbers ought otherwise to be advanced; and the Golden Number 14 should not be placed against the 11th, but the 8th Day of March: Which being reckoned the first Day of the Moon, if we count on to the fourteenth Day of the same inclusive, that would be found to fall on the a 21st Day of March; on which[15] Day the Gregorian Paschal Limit or Full Moon will happen when the Golden Number is 14. And the like Course should be taken with the rest of the 19 Golden Numbers; which ought to be placed 8 Days forwarder than they now stand, if they are to point out the New Moon; or 21 Days forwarder than they are at present, if they are to mark the fourteenth Day of the Moon or the Full Moon: The latter of which, as has been shewn, would be more eligible, than to prefix those Numbers to the Days on which the New Moons happen.

Thus may the Rule and Method now used in the Church of England, be most easily adapted to shew the Time of Easter, as it is observed by the Gregorians, till the Year 1900; at which Time, and at the other proper succeeding Times, if the Golden Numbers in the Calendar shall either be advanced or set backward a Day, according the foregoing Rules and Directions for that Purpose, they will continue to shew us the New or the Full Moons, of the Church of Rome or the Gregorian Calendar with great Exactness, till the Year 4199: when, as has been already mentioned, there must be a little Variation made in those Rules and Directions.

There is however one Exception to those General Rules and Directions, which will be taken notice of in the next Paragraph.

Upon these Principles I framed the Table accompanying this Paper, and shewing, by means of the Golden Numbers, all the Gregorian Paschal Limits or Full Moons, from the Reformation of the Calendar, &c. by Pope Gregory to the Year 4199 inclusive. Which Space of Time is therein divided into sixteen [16] unequal Portions or Periods; at the Beginning of each of which, all the Golden Numbers, when once they shall have been properly placed in the Calendar, must either be advanced or set back one Day, with respect to the Place where they stood in the preceding Period, agreeably to the foregoing Rules: Except those Numbers which shall happen to stand against the 4th and 5th of April to shew the Paschal New Moons, or against the 17th and 18th of the same Month to mark out the Paschal Full Moons; both which Numbers at some Times, and only one of them at others, must keep the same Place for that, which was allotted to them in the immediately preceding Period.

In order to determine at what Times, and on what Occasions, this Exception is to take Place; let it be observed, that, in the Months of January, March, May, and some others in our present Calendar, as well as in the Table above mentioned, some of the Golden Numbers stand double or in Pairs, and follow one the other immediately; whilst others, on the contrary, generally stand single and by themselves.

Now, when any of those Pairs, or two Numbers which usually accompany each other, happen, in pursuance of the foregoing Rules, to be prefixed the one to the 4th and the other to the 5th of April for the New Moons, or the one to the 17th and the other to the 18th of April for the Paschal Limits or Full Moons: And when any of those Numbers, which generally stand single, are prefixed, according to the said Rules, to the 5th of April for the New Moons, or to the 18th for the Full Moons: In these Cases those Pairs or single Numbers that are so situated; must not be set forward or advanced at the Begin[17]ning of the next Period, but must keep their Places during another Period, if the foregoing Rules direct all the Golden Numbers to be advanced a Day; which must be complied with in respect to all the other Golden Numbers, except those so situated as above. Instances whereof may be seen in the Table, under the respective Periods beginning with the Years 1900, 2600, 3100, and 3800.

But if, in Conformity to the foregoing Rules, all the Golden Numbers are to be set one Day backward; those Pairs or single Numbers, tho' situated as is above-mentioned, must not keep their Places, but must move one Day backward like all the other Golden Numbers; as they may be seen to do in the Periods beginning with the Years 2400 and 3600.

To give a plain and intelligible Account of the Reason, on which the Directions now given with respect to this Exception are founded, would extend this Paper, already too long, far beyond its due and proper Bounds. I shall therefore content myself with observing, that it depends chiefly upon the Nature of the Menses Pleni and Menses Cavi, into which the Lunar Year is usually divided: and that, in order to make use of the Golden Numbers for finding the Time of the Gregorian Easter, it will be necessary not only to conform to the general Rules laid down in the former Part of this Paper; but also to follow the Directions just now given, with respect to the above-mentioned Exception to those general Rules.

But I should not do Justice to Peter Davall of the Middle Temple, Esq; Secretary of the Royal[18] Society, did I not here acknowledge, that, before I had so fully considered these Matters as I have since done, I had the first Hint of applying the Golden Numbers to find the Gregorian Paschal Limit or full Moon, from him; who has since that time composed and drawn up Tables, &c. which may possibly be of considerable and general Use in this Nation hereafter.

*** END OF THE PROJECT GUTENBERG EBOOK REMARKS UPON THE SOLAR AND THE LUNAR YEARS, THE CYCLE OF 19 YEARS, COMMONLY CALLED THE GOLDEN NUMBER, THE EPACT, AND A METHOD OF FINDING THE TIME OF EASTER, AS IT IS NOW OBSERVED IN MOST PARTS OF EUROPE ***

Updated editions will replace the previous one—the old editions will be renamed.

Creating the works from print editions not protected by U.S. copyright law means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for an eBook, except by following the terms of the trademark license, including paying royalties for use of the Project Gutenberg trademark. If you do not charge anything for copies of this eBook, complying with the trademark license is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. Project Gutenberg eBooks may be modified and printed and given away—you may do practically ANYTHING in the United States with eBooks not protected by U.S. copyright law. Redistribution is subject to the trademark license, especially commercial redistribution.

START: FULL LICENSE

PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free distribution of electronic works, by using or distributing this work (or any other work associated in any way with the phrase “Project Gutenberg”), you agree to comply with all the terms of the Full Project Gutenberg™ License available with this file or online at www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing Project Gutenberg™ electronic works

1.A. By reading or using any part of this Project Gutenberg™ electronic work, you indicate that you have read, understand, agree to and accept all the terms of this license and intellectual property (trademark/copyright) agreement. If you do not agree to abide by all the terms of this agreement, you must cease using and return or destroy all copies of Project Gutenberg™ electronic works in your possession. If you paid a fee for obtaining a copy of or access to a Project Gutenberg™ electronic work and you do not agree to be bound by the terms of this agreement, you may obtain a refund from the person or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be used on or associated in any way with an electronic work by people who agree to be bound by the terms of this agreement. There are a few things that you can do with most Project Gutenberg™ electronic works even without complying with the full terms of this agreement. See paragraph 1.C below. There are a lot of things you can do with Project Gutenberg™ electronic works if you follow the terms of this agreement and help preserve free future access to Project Gutenberg™ electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the Foundation” or PGLAF), owns a compilation copyright in the collection of Project Gutenberg™ electronic works. Nearly all the individual works in the collection are in the public domain in the United States. If an individual work is unprotected by copyright law in the United States and you are located in the United States, we do not claim a right to prevent you from copying, distributing, performing, displaying or creating derivative works based on the work as long as all references to Project Gutenberg are removed. Of course, we hope that you will support the Project Gutenberg™ mission of promoting free access to electronic works by freely sharing Project Gutenberg™ works in compliance with the terms of this agreement for keeping the Project Gutenberg™ name associated with the work. You can easily comply with the terms of this agreement by keeping this work in the same format with its attached full Project Gutenberg™ License when you share it without charge with others.

1.D. The copyright laws of the place where you are located also govern what you can do with this work. Copyright laws in most countries are in a constant state of change. If you are outside the United States, check the laws of your country in addition to the terms of this agreement before downloading, copying, displaying, performing, distributing or creating derivative works based on this work or any other Project Gutenberg™ work. The Foundation makes no representations concerning the copyright status of any work in any country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other immediate access to, the full Project Gutenberg™ License must appear prominently whenever any copy of a Project Gutenberg™ work (any work on which the phrase “Project Gutenberg” appears, or with which the phrase “Project Gutenberg” is associated) is accessed, displayed, performed, viewed, copied or distributed:

This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.

1.E.2. If an individual Project Gutenberg™ electronic work is derived from texts not protected by U.S. copyright law (does not contain a notice indicating that it is posted with permission of the copyright holder), the work can be copied and distributed to anyone in the United States without paying any fees or charges. If you are redistributing or providing access to a work with the phrase “Project Gutenberg” associated with or appearing on the work, you must comply either with the requirements of paragraphs 1.E.1 through 1.E.7 or obtain permission for the use of the work and the Project Gutenberg™ trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg™ electronic work is posted with the permission of the copyright holder, your use and distribution must comply with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed by the copyright holder. Additional terms will be linked to the Project Gutenberg™ License for all works posted with the permission of the copyright holder found at the beginning of this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg™ License terms from this work, or any files containing a part of this work or any other work associated with Project Gutenberg™.

1.E.5. Do not copy, display, perform, distribute or redistribute this electronic work, or any part of this electronic work, without prominently displaying the sentence set forth in paragraph 1.E.1 with active links or immediate access to the full terms of the Project Gutenberg™ License.

1.E.6. You may convert to and distribute this work in any binary, compressed, marked up, nonproprietary or proprietary form, including any word processing or hypertext form. However, if you provide access to or distribute copies of a Project Gutenberg™ work in a format other than “Plain Vanilla ASCII” or other format used in the official version posted on the official Project Gutenberg™ website (www.gutenberg.org), you must, at no additional cost, fee or expense to the user, provide a copy, a means of exporting a copy, or a means of obtaining a copy upon request, of the work in its original “Plain Vanilla ASCII” or other form. Any alternate format must include the full Project Gutenberg™ License as specified in paragraph 1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying, performing, copying or distributing any Project Gutenberg™ works unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing access to or distributing Project Gutenberg™ electronic works provided that:

• You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg™ works calculated using the method you already use to calculate your applicable taxes. The fee is owed to the owner of the Project Gutenberg™ trademark, but he has agreed to donate royalties under this paragraph to the Project Gutenberg Literary Archive Foundation. Royalty payments must be paid within 60 days following each date on which you prepare (or are legally required to prepare) your periodic tax returns. Royalty payments should be clearly marked as such and sent to the Project Gutenberg Literary Archive Foundation at the address specified in Section 4, “Information about donations to the Project Gutenberg Literary Archive Foundation.”
• You provide a full refund of any money paid by a user who notifies you in writing (or by e-mail) within 30 days of receipt that s/he does not agree to the terms of the full Project Gutenberg™ License. You must require such a user to return or destroy all copies of the works possessed in a physical medium and discontinue all use of and all access to other copies of Project Gutenberg™ works.
• You provide, in accordance with paragraph 1.F.3, a full refund of any money paid for a work or a replacement copy, if a defect in the electronic work is discovered and reported to you within 90 days of receipt of the work.
• You comply with all other terms of this agreement for free distribution of Project Gutenberg™ works.

1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™ electronic work or group of works on different terms than are set forth in this agreement, you must obtain permission in writing from the Project Gutenberg Literary Archive Foundation, the manager of the Project Gutenberg™ trademark. Contact the Foundation as set forth in Section 3 below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend considerable effort to identify, do copyright research on, transcribe and proofread works not protected by U.S. copyright law in creating the Project Gutenberg™ collection. Despite these efforts, Project Gutenberg™ electronic works, and the medium on which they may be stored, may contain “Defects,” such as, but not limited to, incomplete, inaccurate or corrupt data, transcription errors, a copyright or other intellectual property infringement, a defective or damaged disk or other medium, a computer virus, or computer codes that damage or cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right of Replacement or Refund” described in paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation, the owner of the Project Gutenberg™ trademark, and any other party distributing a Project Gutenberg™ electronic work under this agreement, disclaim all liability to you for damages, costs and expenses, including legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a defect in this electronic work within 90 days of receiving it, you can receive a refund of the money (if any) you paid for it by sending a written explanation to the person you received the work from. If you received the work on a physical medium, you must return the medium with your written explanation. The person or entity that provided you with the defective work may elect to provide a replacement copy in lieu of a refund. If you received the work electronically, the person or entity providing it to you may choose to give you a second opportunity to receive the work electronically in lieu of a refund. If the second copy is also defective, you may demand a refund in writing without further opportunities to fix the problem.

1.F.4. Except for the limited right of replacement or refund set forth in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg™ electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg™ electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg™ work, (b) alteration, modification, or additions or deletions to any Project Gutenberg™ work, and (c) any Defect you cause.

Section 2. Information about the Mission of Project Gutenberg™

Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life.

Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org.

Section 3. Information about the Project Gutenberg Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws.

The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form accessible by the widest array of equipment including outdated equipment. Many small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS.

The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate.

International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and credit card donations. To donate, please visit: www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg™ electronic works

Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.

Project Gutenberg™ eBooks are often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition.

Most people start at our website which has the main PG search facility: www.gutenberg.org.

This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.

Golden Numbers from the Year 1583 to 1699, and so on to 4199, all inclusive.																Paschal Full Moons.
1583 to 1699	1700 to 1699	1900 to 2199	2200 to 2299	2300 to 2399	2400 to 2499	2500 to 2599	2600 to 2899	2900 to 3099	3100 to 3399	3400 to 3499	3500 to 3599	3600 to 3699	3700 to 3799	3800 to 4099	4100 to 4199	Days of the Month, and Sunday Letters
3	14	..	6	17	6	17	..	9	..	1	12	1	12	..	4	March 21. C
..	3	14	..	6	..	6	17	..	9	..	1	..	1	12	..	22. D
11	..	3	14	..	14	..	6	17	..	9	..	9	..	1	12	23. E
..	11	..	3	14	3	14	..	6	17	..	9	..	9	..	1	24. F
19	..	11	..	3	..	3	14	..	6	17	..	17	..	9	..	25. G
8	19	..	11	..	11	..	3	14	..	6	17	6	17	..	9	26. A
..	8	19	..	11	..	11	..	3	14	..	6	..	6	17	..	27. B
16	..	8	19	..	19	..	11	..	3	14	..	14	..	6	17	28. C
5	16	..	8	19	8	19	..	11	..	3	14	3	14	..	6	29. D
..	5	16	..	8	..	8	19	..	11	..	3	..	3	14	..	30. E
13	..	5	16	..	16	..	8	19	..	11	..	11	..	3	14	31. F
2	13	..	5	16	5	16	..	8	19	..	11	..	11	..	3	April 1. G
..	2	13	..	5	..	5	16	..	8	19	..	19	..	11	..	2. A
10	..	2	13	..	13	..	5	16	..	8	19	8	19	..	11	3. B
..	10	..	2	13	2	13	..	5	16	..	8	..	8	19	..	4. C
18	..	10	..	2	..	2	13	..	5	16	..	16	..	8	19	5. D
7	18	..	10	..	10	..	2	13	..	5	16	5	16	..	8	6. E
..	7	18	..	10	..	10	..	2	13	..	5	..	5	16	..	7. F
15	..	7	18	..	18	..	10	..	2	13	..	13	..	5	16	8. G
4	15	..	7	18	7	18	..	10	..	2	13	2	13	..	5	9. A
..	4	15	..	7	..	7	18	..	10	..	2	..	2	13	..	10. B
12	..	4	15	..	15	..	7	18	..	10	..	10	..	2	13	11. C
1	12	..	4	15	4	15	..	7	18	..	10	..	10	..	2	12. D
..	1	12	..	4	..	4	15	..	7	18	..	18	..	10	..	13. E
9	..	1	12	..	12	..	4	15	..	7	18	7	18	..	10	14. F
..	9	..	1	12	1	12	..	4	15	..	7	..	7	18	..	15. G
17	..	9	..	1	..	1	12	..	4	15	..	15	..	7	18	16. A
6	17	17	9	..	9	..	1	12	12	4	15	4	15	15	7	17. B
14	6	6	17	9	17	9	9	1	1	12	4	12	4	4	15	18. C
																19. D
																20. E
																21. F
																22. G
																23. A
																24. B
																25. C

For 19 Julian Solar Years contain	6939^d,	18^h,	0^´,	0^´´,	0^´´´
Whereas 235 Synodical Months contain only	6939,	16,	31,	56,	30
And are therefore less than 19 Julian Solar Years by	0,	1,	28,	3,	30.

The Project Gutenberg eBook of Remarks upon the solar and the lunar years, the cycle of 19 years, commonly called the golden number, the epact, and a method of finding the time of Easter, as it is now observed in most parts of Europe

Transcriber's note

REMARKS
UPON
THE SOLAR AND THE LUNAR YEARS,
The Cycle of 19 Years, commonly called
THE GOLDEN NUMBER,
THE EPACT,
And a Method of finding the Time of
Easter, as it is now observed in most
Parts of Europe.

Of the Solar Year.

Of the Lunar Year, Cycle of 19 Years, and the Epact.

A Method of finding the Time of Easter, as it is observed in most Parts of Europe.

Transcriber's Note

THE FULL PROJECT GUTENBERG LICENSE