CHAPTER I._Our Calendar The Julian calendar and its errors. How corrected by the Gregorian. Rules for finding the dominical letter, and the day of the week of any event from the days of Julius Caesar 46 B.C. to the year of our Lord four thousand; a new and easy method of fixing the date of Easter. Hebrew calendar; showing the correspondence in the date of events recorded in the Bible with our present Gregorian calendar. Illustrated by valuable tables and charts.

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OUR CALENDAR.

The Julian Calendar and Its Errors.

HOW CORRECTED BY THE GREGORIAN.

Rules For Finding the Dominical Letter,

AND THE DAY OF THE WEEK OF ANY EVENT FROM THE

DAYS OF JULIUS CAESAR 46 B. C. TO THE YEAR OF

OUR LORD FOUR THOUSAND--A NEW AND EASY

METHOD OF FIXING THE DATE OF EASTER.

HEBREW CALENDAR;

SHOWING THE CORRESPONDENCE IN THE DATE OF

EVENTS RECORDED IN THE BIBLE WITH OUR

PRESENT GREGORIAN CALENDAR.

ILLUSTRATED BY VALUABLE TABLES AND CHARTS.

BY REV. GEORGE NICHOLS PACKER,

CORNING, N. Y.

Entered according to Act of Congress, in 1890, 1892 and 1893,

BY REV. GEORGE NICHOLS PACKER,

In the Office of the Librarian of Congress, Washington, D. C.

WILLIAMSPORT, PA.:

FRED R. MILLER BLANK BOOK CO.

1893.

TO HON. HENRY W. WILLIAMS, JUSTICE OF THE SUPREME COURT OF PENNSYLVANIA

WHOM I HAVE FOUND A TRUE FRIEND IN POVERTY AND IN SICKNESS, AND FROM WHOM

I HAVE RECEIVED WORDS OF ENCOURAGEMENT AND COMFORT DURING MANY YEARS OF

ADVERSITY, AND AT WHOSE SUGGESTION THIS LITTLE VOLUME HAS BEEN WRITTEN,

AND BY WHOSE ASSISTANCE IT IS NOW PUBLISHED, THIS HUMBLE VOLUME IS

DEDICATED AS A TRIBUTE OF RESPECT BY THE AUTHOR.

PREFACE.

Many years ago, while engaged in teaching, the writer of this little

volume was in the habit of bringing to the attention of his pupils a few

simple rules for finding the dominical letter and the day of the week of

any given event within the past and the present centuries; further than

this he gave the subject no special attention.

A few years ago, having occasion to learn the day of the week of certain

events that were transpiring at regular intervals on the same day of the

same month, but in different years, he was led to investigate the subject

more thoroughly, so that he is now able to give rules for finding the

dominical letter and the day of the week of any event that has transpired

or will transpire, from the commencement of the Christian era to the year

of our Lord 4,000, and to explain the principles on which these rules

rest. When the investigations were entered upon he had no thought of

writing a book; but having been laid aside from active labor by ill

health, he found relief from the despondency in which sickness and poverty

plunged him by pursuing the study of the calendar, its history, and the

method of disposing of the fraction of a day found in the time required

for the revolution of the Earth in its orbit about the Sun.

He became so much interested in the study of this subject that he

frequently spoke of it to friends and acquaintances whom he met. On one

occasion, while speaking to Hon. H. W. Williams about some of the curious

results of the process by which the coincidence of the solar and the civil

year is preserved, it was suggested to him that he should put the story of

the calendar, its correction by Gregory, and the theory and results of

intercalation, in writing. It was urged that this would give increased

interest to the study, help the writer to forget his pains, and probably

enable him to realize a little money from the sale of his work to meet

pressing wants. Acting upon this suggestion, an effort has been made to

put into this little volume some of the most interesting facts relating to

the origin, condition, and practical operation of the calendar now in use;

together with rules for finding the day of the week on which any given day

of any month has fallen or will fall during four thousand years from the

beginning of our era.

The writer does not claim absolute originality for all that appears in the

following pages; on the contrary, he has made free use of all the

materials that came within his reach relating to the history of the

calendar and the work of its correction by Gregory. These materials,

together with his own calculations, he has arranged in accordance with a

plan of his own devising, so that the outline and the execution of the

work may be truly said to be original. Of its value the world must judge.

It has been prepared in weakness of body and in suffering, which have been

to some extent relieved by the mental occupation thus afforded, but which

may have nevertheless left their impress on the work. But let it be read

before pronouncing judgment upon it. Cicero could infer the littleness of

the Hebrew God from the smallness of the territory he had given his

people. To whom Kitto replies: "The interest and importance of a country

arise, not from its territorial extent, but from the men who form its

living soul; from its institutions bearing the impress of mind and spirit,

and from the events which grow out of the character and condition of its

inhabitants." So the value of a book does not consist in the size and

number of its pages, but from the knowledge that may be gained by its

perusal.

THE AUTHOR.

PREFACE

TO THE REVISED EDITION.

Soon after the publication of the former edition of this work, it was

suggested that a chapter be added on Easter; rules for fixing its date,

and also church festivals that depended upon the date of Easter. It was

suggested that this would add very much to the value of the work, if so

presented as to be brought within the comprehension of ordinary minds.

Knowing that the determination of Easter was an affair of considerable

nicety and complication, and had had the attention of our best minds, and

they had failed so to present it, that even among scholarly men, probably

not one in a hundred was able to determine its date without referring to

tables prepared for that purpose the author of this work felt as though he

was hardly competent for the task. Nevertheless it was undertaken, and the

work has been revised and enlarged by a Chapter on the Peculiarities of

the Roman Calendar, another on fixing the date of events prior to the

Christian era, and a third part on Easter, church festivals, and the

Hebrew Calendar. In the opinion of the author, the rules for determining

the date of Easter are so simplified by his new method that any person of

ordinary intelligence may understand them. How well he has succeeded the

public will decide.

G. N. P.

CONTENTS.

PART FIRST.

DEFINITIONS--HISTORY.

_Pages._

CHAPTER I.--Definitions 11

CHAPTER II.--History of the divisions of time, and the old

Roman Calendar 12

CHAPTER III--History of the reformation of the Calendar by

Julius Caesar 18

CHAPTER IV.--History of the reformation of the Julian Calendar

by Pope Gregory XIII 20

CHAPTER V.--Peculiarities of the Roman Calendar 26

PART SECOND.

MATHEMATICAL.

CHAPTER I.--Errors of the Julian Calendar 36

CHAPTER II.--Errors of the Gregorian Calendar 38

CHAPTER III.--Dominical Letter 39

CHAPTER IV.--Rule for finding the Dominical Letter 44

CHAPTER V.--Rule for finding the day of the week of any given

date, for both Old and New Styles 50

CHAPTER VI.--A simple method of finding the day of the week of

events, which occur quadrennially; the inaugural of

the Presidents, the day of the week on which they

have occurred and on which they will occur for the

next one hundred years 61

Some peculiarities concerning events which fall on

the 29th of February 64

CHAPTER VII.--Rule for finding the day of the week of events

prior to the Christian era 68

PART THIRD.

CYCLES--JULIAN PERIOD--EASTER.

CHAPTER I.--The Solar Cycle 73

CHAPTER II.--The Lunar Cycle 75

CHAPTER III.--The Lunar Cycle and Golden Number 77

CHAPTER IV.--Cycle of Indiction, and the Julian Period 79

CHAPTER V.--Easter 82

CHAPTER VI.--A new and easy method of fixing the date of Easter 88

CHAPTER VII.--Church feasts and fasts whose date depend on the

date of Easter 108

CHAPTER VIII.--Hebrew Calendar 124

APPENDIX 134

ERRATA 149

OUR CALENDAR.

PART FIRST.

DEFINITIONS. HISTORY.

CHAPTER I.

DEFINITIONS.

_a_--A Calendar is a method of distributing time into certain periods

adapted to the purposes of civil life, as hours, days, weeks, months,

years, etc.

_b_--The only natural divisions of time are the solar day, the solar year,

and the lunar month.

_c_--An hour is one of the subdivisions of the day into twenty-four equal

parts.

_d_--The true solar day is the interval of time which elapses between two

consecutive returns of the same terrestrial meridian to the Sun, the mean

length of which is twenty-four hours.

_e_--The week is a period of seven days, having no reference whatever to

the celestial motions, a circumstance to which it owes its unalterable

uniformity.

_f_--The lunar month is the time which elapses between two consecutive new

or full moons, and was used in the Roman calendar until the time of Julius

Caesar, and consists of 29d, 12h, 44m, 2.87s.

_g_--The calendar month is usually employed to denote an arbitrary number

of days approaching a twelfth part of a year, and has now its place in the

calendar of nearly all nations.

_h_--The year is either astronomical or civil. The solar astronomical year

is the period of time in which the Earth performs a revolution in its

orbit about the sun or passes from any point of the ecliptic to the same

point again, and consists of 365 days, 5 hours, 48 minutes and 49.62

seconds of mean solar time. Appendix A.

_i_--The civil year is that which is employed in chronology, and varies

among different nations, both in respect of the seasons at which it

commences and of its subdivisions.

CHAPTER II.

HISTORY OF THE DIVISIONS OF TIME AND THE OLD ROMAN CALENDAR.

_Day_--The subdivision of the day into twenty-four parts or hours has

prevailed since the remotest ages, though different nations have not

agreed either with respect to the epoch of its commencement or the manner

of distributing the hours. Europeans in general, like the ancient

Egyptians, place the commencement of the civil day at midnight; and reckon

twelve morning hours from midnight to midday and twelve evening hours from

midday to midnight. Astronomers, after the example of Ptolemy, regarded

the day as commencing with the Sun's culmination, or noon, and find it

most convenient for the purpose of computation to reckon through the whole

twenty-four hours. Hipparchus reckoned the twenty-four hours from midnight

to midnight.

The Roman day, from sunrise to sunset, and the night, from sunset to

sunrise, were each divided at all seasons of the year into twelve hours,

the hour being uniformly one-twelfth of the day or the night, of course,

varied in length with the length of the day or night at different seasons

of the year.

_Week_--Although the week did not enter into the calendar of the Greeks,

and was not introduced at Rome till after the reign of Theodosius, A. D.

292, it has been employed from time immemorial in almost all Eastern

countries; and as it forms neither an aliquot part of a year nor of the

lunar months, those who reject the Mosaic recital will be at a loss to

assign to it an origin having much semblance of probability. In the

Egyptian astronomy the order of the planets, beginning with the most

remote, is Saturn, Jupiter, Mars, the Sun, Venus, Mercury, the Moon. Now,

the day being divided into twenty-four hours, each hour was consecrated to

a particular planet, namely: One to Saturn, the following to Jupiter,

third to Mars, and so on according to the above order; and the day

received the name of the planet which presided over its first hour. If,

then, the first hour of a day was consecrated to Saturn, that planet would

also have the 8th, the 5th and the 22d hours; the 23d would fall to

Jupiter, the 24th to Mars, and the 25th or the first hour of the second

day would belong to the Sun. In like manner the first hour of the third

day would fall to the Moon, the first hour of the fourth to Mars, of the

fifth to Mercury, of the sixth to Jupiter and the seventh to Venus. The

cycle being completed, the first hour of the eighth day would again return

to Saturn and all the others succeed in the same order. See table on the

17th page.

It will be seen by the table, and it is also recorded by Dio Cassius, of

the second Century, that the Egyptian week commenced with Saturday. On

their flight from Egypt the Jews, from hatred to their ancient oppressors,

made Saturday the last day of the week. It is stated that the ancient

Saxons borrowed the week from some Eastern nation, and substituted the

names of their own divinities for those of the gods of Greece. The names

of the days are here given in Latin, Saxon and English. It will be seen

that the English names of the days are derived from the Saxon.

LATIN. SAXON. ENGLISH.

_Dies Solis._ Sun's Day. SUNDAY.

_Dies Lunae._ Moon's Day. MONDAY.

_Dies Martis._ Tiw's Day. TUESDAY.

_Dies Mercurii._ Woden's Day. WEDNESDAY.

_Dies Jovis._ Thor's Day. THURSDAY.

_Dies Veneris._ Friga's Day. FRIDAY.

_Dies Saturni._ Seterne's Day. SATURDAY.

_Month_--The ancient Roman year contained but ten months and is indicated

by the names of the last four. September from Septem, seven; October from

Octo, eight; November from Novem, nine, and December from Decem, ten; July

and August were also denominated Quintilis and Sextilis, from Quintus

five, and Sex, six.

Quintilis was changed to July in honor of Julius Caesar, who was born on

the 12th of that month 98 B. C. Sextilis was changed to August by the

Roman Senate to flatter Augustus on his victories about 8 B. C. In the

reign of Numa Pompilius, about 700 B. C., two months were added to the

year, January at the beginning, and February at the end of the year. This

arrangement continued till 450 B. C., when the Decemvirs (ten magistrates)

changed the order, placing February after January, making March the third

instead of the first month of the Roman year.

_Year_--If the civil year correspond with the solar the seasons of the

year will always come at the same period. But if the civil year is

supposed to be too long (as is the case in the Julian year) the seasons

will go back proportionately; but if too short they will advance in the

same proportion. Now, as the ancient Egyptians reckoned thirty days to the

month invariably, and to complete the year, added five days, called

supplementary days, their year consisted of 365 days.

They made use of no intercalation, and by losing one-fourth of a day every

year, the commencement of the year went back one day in every period of

four years, and consequently made a revolution of the seasons in 1460

years. Hence the Egyptian year was called a vague or erratic year because

the first day of the year in the course of 1460 years wandered, as it

were, over all the seasons. Therefore 1460 Julian years of 365-1/4 days

each are equal to 1461 Egyptian years of 365 days each.

The ancient Roman year consisted of twelve lunar months, of twenty-nine

and thirty days alternately, which equals 354 days; but a day was added

to make the number odd, which was considered more fortunate, so that the

year consisted of 355 days.

This differed from the solar year by ten whole days and a fraction; but to

restore the coincidence, Numa ordered an additional or intercalary month

to be inserted every second year between the 23d and 24th of February,

consisting of twenty-two and twenty-three days alternately, so that four

years contained 1465 days, and the mean length of the year was

consequently 366-1/4 days, so that the year was then too long by one day.

As the error amounted to twenty-four days in as many years, it was ordered

that every third period of eight years, instead of containing four

intercalary months, two of twenty-two and two of twenty-three days,

amounting in all to ninety days, should contain only three of those months

of twenty-two days each, amounting to sixty-six days, thereby suppressing

twenty-four days in as many years, reducing the mean length of the year to

365-1/4 days.

Had the intercalations been regularly made the concurrence of the solar

and the civil year would have been preserved very nearly. But its

regulation was left to the pontiffs, who, to prolong the term of a

magistracy or hasten an annual election, would give to the intercalary

month a greater or less number of days, and consequently the calendar was

thrown into confusion, so that in the time of Julius Caesar there was a

discrepancy between the solar and the civil year of about three months;

the winter months being carried back into autumn and the autumnal into

summer.

A table of the order and the names of the planets in the Egyptian

astronomy illustrating the origin of the names of the days of the week:

---------+---------+--------+-------+-------+----------+-------

---------+---------+--------+-------+-------+----------+-------

1 | 2 | 3 | 4 | 5 | 6 | 7

8 | 9 | 10 | 11 | 12 | 13 | 14

15 | 16 | 17 | 18 | 19 | 20 | 21

22 | 23 | 24 | 1 | 2 | 3 | 4

5 | 6 | 7 | 8 | 9 | 10 | 11

12 | 13 | 14 | 15 | 16 | 17 | 18

19 | 20 | 21 | 22 | 23 | 24 | 1

2 | 3 | 4 | 5 | 6 | 7 | 8

9 | 10 | 11 | 12 | 13 | 14 | 15

16 | 17 | 18 | 19 | 20 | 21 | 22

23 | 24 | 1 | 2 | 3 | 4 | 5

6 | 7 | 8 | 9 | 10 | 11 | 12

13 | 14 | 15 | 16 | 17 | 18 | 19

20 | 21 | 22 | 23 | 24 | 1 | 2

3 | 4 | 5 | 6 | 7 | 8 | 9

10 | 11 | 12 | 13 | 14 | 15 | 16

17 | 18 | 19 | 20 | 21 | 22 | 23

24 | 1 | 2 | 3 | 4 | 5 | 6

7 | 8 | 9 | 10 | 11 | 12 | 13

14 | 15 | 16 | 17 | 18 | 19 | 20

21 | 22 | 23 | 24 | 1 | 2 | 3

4 | 5 | 6 | 7 | 8 | 9 | 10

11 | 12 | 13 | 14 | 15 | 16 | 17

18 | 19 | 20 | 21 | 22 | 23 | 24

---------+---------+--------+-------+-------+----------+-------

CHAPTER III.

HISTORY OF THE REFORMATION OF THE CALENDAR BY JULIUS CAESAR.

In order to put an end to the disorders arising from the negligence or

ignorance of the pontiffs, Julius Caesar, 46 B. C., abolished the use of

the lunar year and the intercalary month, and regulated the civil year

entirely by the Sun. With the advice and assistance of the astronomers,

especially Sosigenes of Alexandria, he fixed the mean length of the year

at 365-1/4 days, and decided that there should be three consecutive years

of 365 days, and a fourth of 366.

In order to restore the vernal equinox to the 24th of March, the place it

occupied in the time of Numa, two months, together consisting of 67 days,

were inserted between the last day of November and the first day of

December of that year. An intercalary month of 23 days had already been

added to February of the same year according to the old method, so that

the first Julian year commenced with the first day of January, 45 years

before Christ, and 709 from the foundation of Rome, making the year A. U.

C. 708 to consist of the prodigious number of 445 days, (i. e. 355 + 23 +

67 = 445). Hence it was called by some the year of confusion; Macrobius

said it should be named the last year of confusion.

There was also adopted at the same time a more commodious arrangement in

the distribution of the days throughout the several months. It was decided

to give to January, March, May, July, September and November each

thirty-one days; and the other months thirty, excepting February, which

in common years should have but twenty-nine days, but every fourth year

thirty; so that the average length of the Julian year was 365-1/4 days.

Augustus Caesar interrupted this order by taking one day from February,

reducing it to twenty-eight and giving it to August, that the month

bearing his name should have as many days as July, which was named in

honor of his great-uncle, Julius. In order that three months of thirty-one

days might not come together, September and November were reduced to

thirty days, and thirty-one given to October and December.

In the Julian calendar a day was added to February every fourth year, it

being the shortest month, which was called the additional or intercalary

day, and was inserted in the calendar between the 23d and 24th of that

month. In the ancient Roman calendar the first day of every month was

invariably called the calends. The 24th of February then was the 6th of

the calends of March--Sexto calendas; the preceding, which was the

additional or intercalary day, was called bis-sexto calendas (from _bis_,

twice, and _sextus_, six), twice the sixth day. Hence the term bis-sextile

as applied to every fourth year, commonly called leap-year. Appendix B.

CHAPTER IV.

HISTORY OF THE REFORMATION OF THE JULIAN CALENDAR BY POPE GREGORY XIII.

True enough, the year in which Julius Caesar reformed the ancient Roman

calendar was the last year of confusion, and the method adopted by him a

commodious one, and answered a very good purpose for a short time; but as

the years rolled on and century after century had passed away, astronomers

began to discover the discrepancy between the solar and the civil year;

that the vernal equinox did not occupy the place it occupied in the time

of Caesar, namely, the 24th of March, but was gradually retrograding

towards the beginning of the year, so that at the meeting of the Council

of Nice in 325 it fell on the 21st. Appendix C.

The venerable Bede, in the 8th century, observed that these phenomena took

place three or four days earlier than at the meeting of that council.

Roger Bacon, in the 13th century, wrote a treatise on this subject and

sent it to the Pope, setting forth the errors of the Julian calendar. The

discrepancy at that time amounted to seven or eight days.

Thus the errors of the calendar continued to increase until 1582, when the

vernal equinox fell on the 11th instead of the 21st of March. Gregory,

perceiving that the measure (of reforming the calendar) was likely to

confer great eclat on his pontificate, undertook the long desired

reformation; and having found the governments of the principal Catholic

states ready to adopt his views, he issued a brief in the month of March,

1582, in which he abolished the use of the ancient calendar, and

substituted that which has since been received in almost all Christian

countries under the name of the Gregorian calendar or New Style.

The edict of the Pope took effect in October of that year, causing the 5th

to be called the 15th of that month, thus suppressing ten days and making

the year 1582 to consist of only 355 days. So we see that the ten days

that had been gained by incorrect computation during the past 1257 years,

were deducted from 1582, restoring the concurrence of the solar and the

civil year, and consequently the vernal equinox to the place it occupied

in 325, namely, the 21st of March.

The Pope was promptly obeyed in Spain, Portugal, and Italy. The change

took place the same year in France, by calling the 10th the 20th of

December. Many other Catholic countries made the change the same year, and

the Catholic states of Germany the year following; but most of the

Protestant countries adhered to the Old Style until after the year 1700.

Among the last was Great Britain; she, after having suffered a great deal

of inconvenience for nearly two hundred years by using a different date

from the most of Europe, at length, by an act of Parliament, fixed on

September, 1752, as the time for making the much desired change, which was

done by calling the 3d of that month the 14th (as the error now amounted

to eleven days), adopting at the same time the Gregorian rule of

intercalation.

Russia is the only Christian country that still adheres to the Old Style,

and by using a different date from the rest of Europe is now twelve days

behind the true time. The discrepancy between solar and civil time does

not effect the day, for, as has already been shown, the mean length of the

day is twenty-four hours, and is marked by one revolution of the earth

upon its axis.

Nor does it effect the week, for the week is uniformly seven of those

days. But it effects the year, the month and the day of the month.

Russia, by adhering to the Old Style, has reckoned as many days and as

many weeks, and events have transpired on the same day of the week as they

have with us who have adopted the New Style; as Christian nations we are

observing the same day as the Sabbath.

When it was Tuesday, the 20th day of December, 1888, in Russia, it was

Tuesday, the 1st day of January, 1889, in those countries which have

adopted the New Style. Columbus sailed from Palos, in Spain, on Friday,

August 3d, 1492, Old Style, which was Friday, August 12th, New Style.

Washington was born on Friday, February 11th, 1732, Old Style, which was

Friday, February 22d, New Style.

Now, the difference in styles during the 15th century is nine days; during

the 16th and 17th centuries, ten days; the 18th century, eleven days, and

the 19th, twelve days. In regard to the sailing of Columbus, the change is

made by suppressing nine days, calling the 3d the 12th of August. In

regard to the birth of Washington, the change is effected by suppressing

eleven days, calling the 11th of February the 22d. As regards Russia, she

could have made the change last year by calling the 20th of December,

1888, the 1st day of January, 1889, thereby suppressing twelve days, and

making the year 1888 to consist of only 354 days, and the month of

December twenty days. The methods of computation, both Old and New Styles,

will be explained in another chapter.

To persons unacquainted with astronomy, the difference between Old and New

Styles would probably be better understood by the diagram on the 25th

page. The figures represent the ecliptic, which is the apparent path of

the Sun, or the real path of the Earth as seen from the Sun, in her annual

or yearly revolution around the Sun in the order of the months, as marked

on the ecliptic.

Attention is called to four points on the ecliptic, namely, the vernal

equinox, the autumnal equinox, the winter solstice, and the summer

solstice. These occur, in the order given above, on the 21st of March, the

21st of September, the 21st of December and the 21st of June. It has

already been stated that if the civil year correspond with the solar, the

seasons of the year will always come at the same period. Julius Caesar

found the ancient Roman year in advance of the solar; Gregory found the

Julian behind the solar year; so one reforms the calendar by

intercalation, the other by suppression. Appendix D.

Caesar restored the coincidence of the solar and the civil year, but failed

to retain it by allowing what probably appeared to him at the time a

trifling error in his calendar. The error, which was 11 minutes and 10.38

seconds every year, was hardly perceptable for a short period, but still

amounted to three days every 400 years. Hence the necessity in 1582 of

reforming the reformed calendar of Julius Caesar to restore the

coincidence. Appendix E.

From the meeting of the Council of Nice, in 325, to 1582, a period of 1257

years, there was found to be an error in the Julian calendar of ten days.

Now, in 1257 years the Earth performs 1257 annual and 459,109 daily

revolutions, after which the vernal equinox was found to occur on the 21st

of March, true or solar time; thus concurring with the vernal equinox of

325. But the erroneous Julian calendar would make the Earth perform

459,119 daily revolutions to complete the 1257 years, a discrepancy of ten

days, making the vernal equinox to fall on the 11th instead of the 21st.

It will be seen by the diagram that the ten days were deducted from

October, in 1582, making it a short month, consisting of only twenty-one

days.

The discrepancy between the Julian and Gregorian calendar amounts to

thirty days in 4000 years; three months in 12,175 years. Hence, in 12,175

years the equinoxes would take the place of the solstices, and the

solstices the place of the equinoxes. In 24,350 years, the vernal equinox

would take the place of the autumnal equinox, and the winter solstice the

place of the summer solstice.

And in 48,700 years, according to the Julian rule of intercalation, there

would be gained nearly 365-1/4 days, or one entire revolution of the

Earth. So, to restore the concurrence of the Julian and Gregorian years,

there would have to be suppressed 365-1/4 days, calling the 1st day of

January, 48,699, the 1st day of January, 48,700.

Thus would disappear from the Julian calendar twelve months, or one whole

year, it having been divided among the thousands of the preceding years.

[Illustration: The Julian calendar, reformed by Gregory XIII, Oct. 5th,

1582, by suppressing ten days, calling the 5th of Oct. the 15th, making it

a short month of 21 days, and the year to consist of 355 days.

The Roman calendar, reformed by Julius Caesar, 46 B. C., by intercalating

90 days, making that year to consist of 445 days, that is, 355 + 23 + 67 =

445 days, and "the last year of confusion."]

To make this subject better understood, let us suppose the solar year to

consist in round numbers of 365 days, and the civil year 366. It is

evident that at the end of the year of 365 days, there would still be

wanting one day to complete the civil year of 366 days, so one day must be

added to that year, and to every succeeding year, to complete the years of

366 days each, which would be the loss of one year of 365 days in 365

years. Hence, 364 years of 366 days each are equal to 365 years of 365

days each, wanting one day.

Again, let us suppose the civil year to consist of 364 days. It is evident

that at the end of the supposed solar year of 365 days, there would be an

advance or gain of one day in that year and every succeeding year, so that

in 365 years there would be a gain of 365 days or one whole year. Hence,

366 years of 364 days each are equal to 365 years of 365 days each,

wanting one day. Appendix F.

CHAPTER V.

PECULIARITIES OF THE ROMAN CALENDAR.

The Romans, instead of distinguishing the days of the month by the ordinal

numbers, first, second, third, etc., counted backwards from three fixed

points, namely, the Calends, the Nones, and the Ides.

Calends (Latin _Calandae_, from _Calare_, to call,) was so denominated

because it had been an ancient custom of the pontiffs to call the people

together on that day to apprise them of the festivals, or days that were

to be kept sacred during the month.

Nones (Latin _nonae_, from _nonus_, the ninth,) the ninth day before the

Ides.

Ides (Latin _idus_, supposed to be derived from an obsolete verb _iduare_,

to divide,) was near the middle of the month, either the 13th or the 15th

day.

The first day of each month was invariably called the Calends. The Nones

were the fifth, and the Ides the thirteenth, except in March, May, July,

and October, in which the Nones occurred on the seventh day and the Ides

on the fifteenth.

From these three points the days of the month were numbered--not forward,

but backward--as so many days before the Nones, the Ides, or the Calends,

the point of departure being counted in the reckoning, so that the last

day of every month was the second of the Calends of the following month.

It will be seen by the Roman and English calendar found on the following

pages, that there are six days of Nones in March, May, July and October,

and four of all the other months; also that all the months have eight days

of Ides. The number of days of Calends depend upon the number of days in

the month, and the day of the month on which the Ides fall.

If the month has thirty-one days and the Ides fall on the thirteenth,

there are nineteen days of Calends; but if the Ides fall on the fifteenth,

there are only seventeen days of Calends. As the Ides fall on the

thirteenth of all the months of thirty days, they have eighteen days of

Calends. February, the month of twenty-eight days, has only sixteen,

except in leap-year, when the sixth of the Calends is reckoned twice.

It may also be seen from the calendar that the Romans, after the first

day of the month, began to reckon so many days before the Nones, as 4th,

3d, 2d, then Nones; after the Nones, so many days before the Ides, as 8th,

7th, 6th, etc., and after the Ides, so many before the Calends of the next

month, the highest numbers being reckoned first.

In reducing the Roman calendar to our own, it should be remembered that in

reckoning backward from a fixed point, that the point of departure is

counted; also, that the last day of the month is not the point from which

the Calends are reckoned, but the first day of the following month. We

have then this rule for finding the English expression for any Latin date:

RULE.

If the given date be Calends, add two to the number of days in the month,

from which subtract the given date; if the date be Nones, or Ides, add one

to that of the day on which the Nones or Ides fall, from which subtract

the given date, and you will have the day of the month in our calendar. To

find the Latin expression for any English date, the preceding method is to

be reversed, upon the principle that if 5 - 3 = 2, then 5 - 2 = 3.

But in reducing a Roman date to a date of February in leap-year, for the

first twenty-four days, proceed according to the preceding rule as if the

month had only twenty-eight days, and to obtain the proper expression for

the remaining five days, regard the month as having twenty-nine days,

taking the Roman date from 31 instead of 30. Thus 31 - 6 = 25, while 30 -6

= 24; the former corresponding with sexto calendas, the latter with

bis-sexto calendas of the Julian calendar. By referring to the table on

the 35th page, one may easily learn how to find the English expression for

any Latin date, or the Latin expression for any English date.

It has already been stated that in January the Nones fall on the 5th, and

the Ides on the 13th, January then having thirty one days, the number from

which to subtract the Roman date to obtain the corresponding day of the

month is, for Nones, 5 + 1 = 6; for Ides, 13 + 1 = 14; for Calends, 31 + 2

= 33. Hence the first column in the table under January are the numbers 6,

14 and 33. February is the same as January for Nones and Ides. For Calends

in leap-year, for the first 24 days, it is 28 + 2 = 30; for the remaining

5 days it is 29 + 2 = 31. Hence for the first column under February are

the numbers 6, 14, 30 and 31. March is the same as January, except that

the Nones fall on the 7th and the Ides on the 15th, consequently we have

for Nones, 7 + 1 = 8, and for Ides, 15 + 1 = 16; hence, for the first

column under March we have the numbers 8, 16, and 33.

In the table the three months are taken to illustrate how easily the

change may be made from Roman to English, or from English to Roman date. A

complete calendar for 1892, both in Roman and English, which will be very

convenient for reference, may be found on the four following pages.

The first seven letters of the alphabet, used to represent the days of the

week, are placed in the calendar beside the days of the week. The letter

that represents Sunday is called the dominical or Sunday letter. The

letter that represents the first Sunday in any given year represents all

the Sundays in that year, unless it be leap-year, when two Sunday letters

are used. The first represents all the Sundays in January and February,

while the letter that precedes it represents all the Sundays for the rest

of the year.

The reason of this is, the day intercalated, or thrust in, between the

28th day of February and the 1st day of March so interrupts the order of

the letters that D, which always represents the 1st day of March, now

represents the 29th day of February, so that in 1892 it represents Monday,

the 29th day of February, also Tuesday the 1st day of March. As it

represented all the Mondays in January and February, it will now represent

all the Tuesdays for the rest of the year, while C, the letter preceding,

represented all the Sundays, will now represent all the Mondays, and B all

the Sundays. For January and February we have then C, Sunday; D, Monday;

E, Tuesday; F, Wednesday; G, Thursday; A, Friday, and B, Saturday. For the

rest of the year we have B, Sunday; C, Monday; D, Tuesday; E, Wednesday;

F, Thursday; G, Friday, and A, Saturday. See Part Second, chapters IV and

------------------++------------------++------------------

JAN., 1892. || FEB., 1892. || MARCH, 1892.

------------------++------------------++------------------

1|_Cal._ |a|Fri. || 1|_Cal._ |d|Mon. || 1|_Cal._ |d|Tues.

2| 4 |b|Sat. || 2| 4 |e|Tues.|| 2| 6 |e|Wed.

3| 3 |c|Sun. || 3| 3 |f|Wed. || 3| 5 |f|Thur.

4| 2 |d|Mon. || 4| 2 |g|Thur.|| 4| 4 |g|Fri.

5|_Non._ |e|Tues.|| 5|_Non._ |a|Fri. || 5| 3 |a|Sat.

6| 8 |f|Wed. || 6| 8 |b|Sat. || 6| 2 |b|Sun.

7| 7 |g|Thur.|| 7| 7 |c|Sun. || 7|_Non._ |c|Mon.

8| 6 |a|Fri. || 8| 6 |d|Mon. || 8| 8 |d|Tues.

9| 5 |b|Sat. || 9| 5 |e|Tues.|| 9| 7 |e|Wed.

10| 4 |c|Sun. ||10| 4 |f|Wed. ||10| 6 |f|Thur.

11| 3 |d|Mon. ||11| 3 |g|Thur.||11| 5 |g|Fri.

12| 2 |e|Tues.||12| 2 |a|Fri. ||12| 4 |a|Sat.

13|_Ides._|f|Wed. ||13|_Ides._|b|Sat. ||13| 3 |b|Sun.

14| 19 |g|Thur.||14| 16 |c|Sun. ||14| 2 |c|Mon.

15| 18 |a|Fri. ||15| 15 |d|Mon. ||15|_Ides._|d|Tues.

16| 17 |b|Sat. ||16| 14 |e|Tues.||16| 17 |e|Wed.

17| 16 |c|Sun. ||17| 13 |f|Wed. ||17| 16 |f|Thur.

18| 15 |d|Mon. ||18| 12 |g|Thur.||18| 15 |g|Fri.

19| 14 |e|Tues.||19| 11 |a|Fri. ||19| 14 |a|Sat.

20| 13 |f|Wed. ||20| 10 |b|Sat. ||20| 13 |b|Sun.

21| 12 |g|Thur.||21| 9 |c|Sun. ||21| 12 |c|Mon.

22| 11 |a|Fri. ||22| 8 |d|Mon. ||22| 11 |d|Tues.

23| 10 |b|Sat. ||23| 7 |e|Tues.||23| 10 |e|Wed.

24| 9 |c|Sun. ||24| 6 |f|Wed. ||24| 9 |f|Thur.

25| 8 |d|Mon. ||25| 6 |g|Thur.||25| 8 |g|Fri.

26| 7 |e|Tues.||26| 5 |a|Fri. ||26| 7 |a|Sat.

27| 6 |f|Wed. ||27| 4 |b|Sat. ||27| 6 |b|Sun.

28| 5 |g|Thur.||28| 3 |c|Sun. ||28| 5 |c|Mon.

29| 4 |a|Fri. ||29| 2 |d|Mon. ||29| 4 |d|Tues.

30| 3 |b|Sat. || | | | ||30| 3 |e|Wed.

31| 2 |c|Sun. || | | | ||31| 2 |f|Thur.

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

------------------++------------------++------------------

APRIL, 1892. || MAY, 1892. || JUNE, 1892.

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

1|_Cal._ |g|Fri. || 1|_Cal._ |b|Sun. || 1|_Cal._ |e|Wed.

2| 4 |a|Sat. || 2| 6 |c|Mon. || 2| 4 |f|Thur.

3| 3 |b|Sun. || 3| 5 |d|Tues.|| 3| 3 |g|Fri.

4| 2 |c|Mon. || 4| 4 |e|Wed. || 4| 2 |a|Sat.

5|_Non._ |d|Tues.|| 5| 3 |f|Thur.|| 5|_Non._ |b|Sun.

6| 8 |e|Wed. || 6| 2 |g|Fri. || 6| 8 |c|Mon.

7| 7 |f|Thur.|| 7|_Non._ |a|Sat. || 7| 7 |d|Tues.

8| 6 |g|Fri. || 8| 8 |b|Sun. || 8| 6 |e|Wed.

9| 5 |a|Sat. || 9| 7 |c|Mon. || 9| 5 |f|Thur.

10| 4 |b|Sun. ||10| 6 |d|Tues.||10| 4 |g|Fri.

11| 3 |c|Mon. ||11| 5 |e|Wed. ||11| 3 |a|Sat.

12| 2 |d|Tues.||12| 4 |f|Thur.||12| 2 |b|Sun.

13|_Ides._|e|Wed. ||13| 3 |g|Fri. ||13|_Ides._|c|Mon.

14| 18 |f|Thur.||14| 2 |a|Sat. ||14| 18 |d|Tues.

15| 17 |g|Fri. ||15|_Ides._|b|Sun. ||15| 17 |e|Wed.

16| 16 |a|Sat. ||16| 17 |c|Mon. ||16| 16 |f|Thur.

17| 15 |b|Sun. ||17| 16 |d|Tues.||17| 15 |g|Fri.

18| 14 |c|Mon. ||18| 15 |e|Wed. ||18| 14 |a|Sat.

19| 13 |d|Tues.||19| 14 |f|Thur.||19| 13 |b|Sun.

20| 12 |e|Wed. ||20| 13 |g|Fri. ||20| 12 |c|Mon.

21| 11 |f|Thur.||21| 12 |a|Sat. ||21| 11 |d|Tues.

22| 10 |g|Fri. ||22| 11 |b|Sun. ||22| 10 |e|Wed.

23| 9 |a|Sat. ||23| 10 |c|Mon. ||23| 9 |f|Thur.

24| 8 |b|Sun. ||24| 9 |d|Tues.||24| 8 |g|Fri.

25| 7 |c|Mon. ||25| 8 |e|Wed. ||25| 7 |a|Sat.

26| 6 |d|Tues.||26| 7 |f|Thur.||26| 6 |b|Sun.

27| 5 |e|Wed. ||27| 6 |g|Fri. ||27| 5 |c|Mon.

28| 4 |f|Thur.||28| 5 |a|Sat. ||28| 4 |d|Tues.

29| 3 |g|Fri. ||29| 4 |b|Sun. ||29| 3 |e|Wed.

30| 2 |a|Sat. ||30| 3 |c|Mon. ||30| 2 |f|Thur.

| | | ||31| 2 |d|Tues.|| | | |

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

--+---------------++------------------++------------------

JULY, 1892. || AUG., 1892. || SEPT., 1892.

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

1|_Cal._ |g|Fri. || 1|_Cal._ |c|Mon. || 1|_Cal._ |f|Thur.

2| 6 |a|Sat. || 2| 4 |d|Tues.|| 2| 4 |g|Fri.

3| 5 |b|Sun. || 3| 3 |e|Wed. || 3| 3 |a|Sat.

4| 4 |c|Mon. || 4| 2 |f|Thur.|| 4| 2 |b|Sun.

5| 3 |d|Tues.|| 5|_Non._ |g|Fri. || 5|_Non._ |c|Mon.

6| 2 |e|Wed. || 6| 8 |a|Sat. || 6| 8 |d|Tues.

7|_Non._ |f|Thur.|| 7| 7 |b|Sun. || 7| 7 |e|Wed.

8| 8 |g|Fri. || 8| 6 |c|Mon. || 8| 6 |f|Thur.

9| 7 |a|Sat. || 9| 5 |d|Tues.|| 9| 5 |g|Fri.

10| 6 |b|Sun. ||10| 4 |e|Wed. ||10| 4 |a|Sat.

11| 5 |c|Mon. ||11| 3 |f|Thur.||11| 3 |b|Sun.

12| 4 |d|Tues.||12| 2 |g|Fri. ||12| 2 |c|Mon.

13| 3 |e|Wed. ||13|_Ides._|a|Sat. ||13|_Ides._|d|Tues.

14| 2 |f|Thur.||14| 19 |b|Sun. ||14| 18 |e|Wed.

15|_Ides._|g|Fri. ||15| 18 |c|Mon. ||15| 17 |f|Thur.

16| 17 |a|Sat. ||16| 17 |d|Tues.||16| 16 |g|Fri.

17| 16 |b|Sun. ||17| 16 |e|Wed. ||17| 15 |a|Sat.

18| 15 |c|Mon. ||18| 15 |f|Thur.||18| 14 |b|Sun.

19| 14 |d|Tues.||19| 14 |g|Fri. ||19| 13 |c|Mon.

20| 13 |e|Wed. ||20| 13 |a|Sat. ||20| 12 |d|Tues.

21| 12 |f|Thur.||21| 12 |b|Sun. ||21| 11 |e|Wed.

22| 11 |g|Fri. ||22| 11 |c|Mon. ||22| 10 |f|Thur.

23| 10 |a|Sat. ||23| 10 |d|Tues.||23| 9 |g|Fri.

24| 9 |b|Sun. ||24| 9 |e|Wed. ||24| 8 |a|Sat.

25| 8 |c|Mon. ||25| 8 |f|Thur.||25| 7 |b|Sun.

26| 7 |d|Tues.||26| 7 |g|Fri. ||26| 6 |c|Mon.

27| 6 |e|Wed. ||27| 6 |a|Sat. ||27| 5 |d|Tues.

28| 5 |f|Thur.||28| 5 |b|Sun. ||28| 4 |e|Wed.

29| 4 |g|Fri. ||29| 4 |c|Mon. ||29| 3 |f|Thur.

30| 3 |a|Sat. ||30| 3 |d|Tues.||30| 2 |g|Fri.

31| 2 |b|Sun. ||31| 2 |e|Wed. || | | |

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

------------------++------------------++------------------

OCT., 1892. || NOV., 1892. || DEC., 1892.

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

1|_Cal._ |a|Sat. || 1|_Cal._ |d|Tues.|| 1|_Cal._ |f|Thur.

2| 6 |b|Sun. || 2| 4 |e|Wed. || 2| 4 |g|Fri.

3| 5 |c|Mon. || 3| 3 |f|Thur.|| 3| 3 |a|Sat.

4| 4 |d|Tues.|| 4| 2 |g|Fri. || 4| 2 |b|Sun.

5| 3 |e|Wed. || 5|_Non._ |a|Sat. || 5|_Non._ |c|Mon.

6| 2 |f|Thur.|| 6| 8 |b|Sun. || 6| 8 |d|Tues.

7|_Non._ |g|Fri. || 7| 7 |c|Mon. || 7| 7 |e|Wed.

8| 8 |a|Sat. || 8| 6 |d|Tues.|| 8| 6 |f|Thur.

9| 7 |b|Sun. || 9| 5 |e|Wed. || 9| 5 |g|Fri.

10| 6 |c|Mon. ||10| 4 |f|Thur.||10| 4 |a|Sat.

11| 5 |d|Tues.||11| 3 |g|Fri. ||11| 3 |b|Sun.

12| 4 |e|Wed. ||12| 2 |a|Sat. ||12| 2 |c|Mon.

13| 3 |f|Thur.||13|_Ides._|b|Sun. ||13|_Ides._|d|Tues.

14| 2 |g|Fri. ||14| 18 |c|Mon. ||14| 19 |e|Wed.

15|_Ides._|a|Sat. ||15| 17 |d|Tues.||15| 18 |f|Thur.

16| 17 |b|Sun. ||16| 16 |e|Wed. ||16| 17 |g|Fri.

17| 16 |c|Mon. ||17| 15 |f|Thur.||17| 16 |a|Sat.

18| 15 |d|Tues.||18| 14 |g|Fri. ||18| 15 |b|Sun.

19| 14 |e|Wed. ||19| 13 |a|Sat. ||19| 14 |c|Mon.

20| 13 |f|Thur.||20| 12 |b|Sun. ||20| 13 |d|Tues.

21| 12 |g|Fri. ||21| 11 |c|Mon. ||21| 12 |e|Wed.

22| 11 |a|Sat. ||22| 10 |d|Tues.||22| 11 |f|Thur.

23| 10 |b|Sun. ||23| 9 |e|Wed. ||23| 10 |g|Fri.

24| 9 |c|Mon. ||24| 8 |f|Thur.||24| 9 |a|Sat.

25| 8 |d|Tues.||25| 7 |g|Fri. ||25| 8 |b|Sun.

26| 7 |e|Wed. ||26| 6 |a|Sat. ||26| 7 |c|Mon.

27| 6 |f|Thur.||27| 5 |b|Sun. ||27| 6 |d|Tues.

28| 5 |g|Fri. ||28| 4 |c|Mon. ||28| 5 |e|Wed.

29| 4 |a|Sat. ||29| 3 |d|Tues.||29| 4 |f|Thur.

30| 3 |b|Sun. ||30| 2 |e|Wed. ||30| 3 |g|Fri.

31| 2 |c|Mon. || | | | ||31| 2 |a|Sat.

--+-------+-+-----++--+-------+-+-----++--+-------+-+-----

--------------++----------------++----------------

JANUARY. || FEBRUARY. || MARCH.

--------------++----------------++----------------

_Cal._ 1 || _Cal._ 1 || _Cal._ 1

6 - 4 = 2 || 6 - 4 = 2 || 8 - 6 = 2

6 - 3 = 3 || 6 - 3 = 3 || 8 - 5 = 3

6 - 2 = 4 || 6 - 2 = 4 || 8 - 4 = 4

_Nones_ 5 || _Nones_ 5 || 8 - 3 = 5

14 - 8 = 6 || 14 - 8 = 6 || 8 - 2 = 6

14 - 7 = 7 || 14 - 7 = 7 || _Nones_ 7

14 - 6 = 8 || 14 - 6 = 8 || 16 - 8 = 8

14 - 5 = 9 || 14 - 5 = 9 || 16 - 7 = 9

14 - 4 = 10 || 14 - 4 = 10 || 16 - 6 = 10

14 - 3 = 11 || 14 - 3 = 11 || 16 - 5 = 11

14 - 2 = 12 || 14 - 2 = 12 || 16 - 4 = 12

_Ides_ 13 || _Ides_ 13 || 16 - 3 = 13

33 - 19 = 14 || 30 - 16 = 14 || 16 - 2 = 14

33 - 18 = 15 || 30 - 15 = 15 || _Ides_ 15

33 - 17 = 16 || 30 - 14 = 16 || 33 - 17 = 16

33 - 16 = 17 || 30 - 13 = 17 || 33 - 16 = 17

33 - 15 = 18 || 30 - 12 = 18 || 33 - 15 = 18

33 - 14 = 19 || 30 - 11 = 19 || 33 - 14 = 19

33 - 13 = 20 || 30 - 10 = 20 || 33 - 13 = 20

33 - 12 = 21 || 30 - 9 = 21 || 33 - 12 = 21

33 - 11 = 22 || 30 - 8 = 22 || 33 - 11 = 22

33 - 10 = 23 || 30 - 7 = 23 || 33 - 10 = 23

33 - 9 = 24 || 30 - 6 = 24 || 33 - 9 = 24

33 - 8 = 25 || 31 - 6 = 25 || 33 - 8 = 25

33 - 7 = 26 || 31 - 5 = 26 || 33 - 7 = 26

33 - 6 = 27 || 31 - 4 = 27 || 33 - 6 = 27

33 - 5 = 28 || 31 - 3 = 28 || 33 - 5 = 28

33 - 4 = 29 || 31 - 2 = 29 || 33 - 4 = 29

33 - 3 = 30 || || 33 - 3 = 30

33 - 2 = 31 || || 33 - 2 = 31

--------------++----------------++----------------

PART SECOND.

MATHEMATICAL.

CHAPTER I.

ERRORS OF THE JULIAN CALENDAR.

It will be necessary in the first place to understand the difference

between the Julian and Gregorian rule of intercalation. If the number of

any year be exactly divisible by four it is leap year; if the remainder be

1, it is the first year after leap-year; if 2, the second; if 3, the

third; thus:

1888 / 4 = 472, no remainder.

1889 / 4 = 472, remainder, 1.

1890 / 4 = 472, remainder, 2.

1891 / 4 = 472, remainder, 3.

1892 / 4 = 473, no remainder.

And so on, every fourth year being leap-year of 366 days.

This is the Julian rule of intercalation, which is corrected by the

Gregorian by making every centurial year, or the year that completes the

century, a common year, if not exactly divisible by 400; so that only

every fourth centurial year is leap-year; thus, 1,700, 1,800, and 1,900

are common years, but 2,000, the fourth centurial year, is leap year, and

so on.

By the Julian rule three-fourths of a day is gained every century, which

in 400 years amounts to three days. This is corrected by the Gregorian, by

making three consecutive centurial years common years, thus suppressing

three days in 400 years.

RULE.

Multiply the difference between the Julian and the solar year by 100, and

we have the error in 100 years. Multiply this product by 4 and we have the

error in 400 years. Now, 400 is the tenth of 4,000; therefore, multiply

the last product by 10, and we have the error in 4,000 years. Now, as the

discrepancy between the Julian and Gregorian year is three days in 400

years, making 3-400 of a day every year, so by dividing 365-1/4, the

number of days in a year, by 3-400, we have the time it would take to make

a revolution of the seasons.

SOLUTION.

(365 d, 6 h.) - (365 d, 5 h, 48 m, 49.62 s.) = (11 m, 10.38 s.) Now, (11

m, 10.38 s.) x 100 = 18 h, 37.3 m, the gain in 100 years. This is,

reckoned in round numbers, 18 hours, or three-fourths of a day. Now, (3/4

x 4) = (1 x 3) = 3: the Julian rule gaining three days, the Gregorian

suppressing three days in 400 years. (3 x 10) = 30, the number of days

gained by the Julian rule in 4,000 years. 365-1/4 / 3 400 = 48,700, so

that in this long period of time, this falling back 3/4 of a day every

century would amount to 365-1/4 days; therefore, 48,699 Julian years are

equal to 48,700 Gregorian years.

CHAPTER II.

ERRORS OF THE GREGORIAN CALENDAR.

By reference to the preceding chapter it will be seen that there is an

error of 37.3 minutes in every 100 years not corrected by the Gregorian

calendar; this amounts to only .373 of a minute a year, or one day in

3,861 years, and one day and fifty-two minutes in 4,000 years.

RULE.

To find how long it would take to gain one day: Divide the number of

minutes in a day by the decimal .373, that being the fraction of a minute

gained every year. To find how much time would be gained in 4,000 years,

multiply the decimal .373 by 4,000, and you will have the answer in

minutes, which must be reduced to hours.

SOLUTION.

(24 x 60) / .373 = 3,861, nearly; hence the error would amount to only one

day in 3,861 years.

(.373 x 4,000) / 60 = (24 h, 52 m,) = (1 d, 0 h, 52 m), the error in 4,000

years.

This trifling error in the Gregorian calendar may be corrected by

suppressing the intercalations in the year 4,000, and its multiples,

8,000, 12,000, 16,000, etc., so that it will not amount to a day in

100,000 years.

RULE.

Divide 100,000 by 4,000 and you will have the number of intercalations

suppressed in 100,000 years. Multiply 1 d, 52 m, (that being the error in

4,000 years) by this quotient, and you will have the discrepancy between

the Gregorian and solar year for 100,000 years. By this improved method we

suppress 25 days, so that the error will only amount to 25 times 52

minutes.

SOLUTION.

100,000 / 4,000 x (1 d, 52 m,) = (25d, 21 h, 40 m.) Now, (25d, 21 h, 40

m,) - 25 d = (21 h, 40 m,) the error in 100,000.

CHAPTER III.

DOMINICAL LETTER.

Dominical (from the Latin _Dominus_, Lord,) indicating the Lord's day or

Sunday. Dominical letter, one of the first seven letters of the alphabet

used to denote the Sabbath or Lord's day.

For the sake of greater generality, the days of the week are denoted by

the first seven letters of the alphabet, A, B, C, D, E, F, G, which are

placed in the calendar beside the days of the year, so that A stands

opposite the first day of January, B opposite the second, C opposite the

third, and so on to G, which stands opposite the seventh; after which A

returns to the eighth, and so on through the 365 days of the year.

Now, if one of the days of the week, Sunday for example, is represented by

F, Monday will be represented by G, Tuesday by A, Wednesday by B, Thursday

by C, Friday by D, and Saturday by E; and every Sunday throughout the year

will have the same character, F, every Monday G, every Tuesday A, and so

with regard to the rest.

The letter which denotes Sunday is called the Dominical or Sunday letter

for that year; and when the dominical letter of the year is known, the

letters which respectively correspond to the other days of the week become

known also. Did the year consist of 364 days, or 52 weeks invariably, the

first day of the year and the first day of the month, and in fact any day

of any year, or any month, would always commence on the same day of the

week. But every common year consists of 365 days, or 52 weeks and 1 day,

so that the following year will begin one day later in the week than the

year preceding. Thus the year 1837 commenced on Sunday, the following

year, 1838, on Monday, 1839 on Tuesday, and so on.

As the year consists of 52 weeks and 1 day, it is evident that the day

which begins and ends the year must occur 53 times; thus the year 1837

begins on Sunday and ends on Sunday; so the following year, 1838, must

begin on Monday. As A represented all the Sundays in 1837 and as A always

stands for the first day of January, so in 1838 it will represent all the

Mondays, and the dominical letter goes back from A to G; so that G

represents all the Sundays in 1838, A all the Mondays, B all the Tuesdays,

and so on, the dominical letter going back one place in every year of 365

days.

While the following year commences one day later in the week than the year

preceding, the dominical letter goes back one place from the preceding

year; thus while the year 1865 commenced on Sunday, 1866 on Monday, 1867

on Tuesday, the dominical letters are A, G and F, respectively. Therefore,

if every year consisted of 365 days, the dominical cycle would be

completed in seven years, so that after seven years the first day of the

year would again occur on the same day of the week.

But this order is interrupted every four years by giving February 29 days,

thereby making the year to consist of 366 days, which is 52 weeks and two

days, so that the following year would commence two days later in the week

than the year preceding, thus the year 1888 being leap-year, had two

dominical letters, A and G; A for January and February, and G for the rest

of the year. The year commenced on Sunday and ended on Monday, making 53

Sundays and 53 Mondays, and the following year, 1889, to commence on

Tuesday. It now becomes evident that if the years all consisted of 364

days, or 52 weeks, they would all commence on the same day of the week; if

they all consisted of 365 days, or 52 weeks and one day, they would all

commence one day later in the week than the year preceding; if they all

consisted of 366 days, or 52 weeks and two days, they would commence two

days later in the week; if 367 days or 52 weeks and three days, then three

days later, and so on, one day later for every additional day. It is also

evident that every additional day causes the dominical letter to go back

one place. Now in leap-year the 29th day of February is the additional or

intercalary day. So one letter for January and February, and another for

the rest of the year. If the number of years in the intercalary period

were two, and seven being the number of days in the week, their product

would be 2 x 7 = 14; fourteen, then, would be the number of years in the

cycle. Again, if the number of years in the intercalary period were three,

and the number of days in the week being seven, their product would be 3 x

7 = 21; twenty-one would then be the number of years in the cycle. But the

number of years in the intercalary period is four, and the number of days

in the week is seven, therefore their product is 4 x 7 = 28; twenty-eight

is then the number of years in the cycle.

This period is called the dominical or solar cycle, and restores the first

day of the year to the same day of the week. At the end of the cycle the

dominical letters return again in the same order, on the same days of the

month. Thus, for the year 1801, the dominical letter is D; 1802, C; 1803,

B; 1804, A and G; and so on, going back five places every four years for

twenty-eight years, when the cycle, being ended, D is again dominical

letter for 1829, C for 1830, and so on every 28 years forever, according

to the Julian rule of intercalation.

But this order is interrupted in the Gregorian calendar at the end of the

century by the secular suppression of the leap-year. It is not

interrupted, however, at the end of every century, for the leap-year is

not suppressed in every fourth centurial year; consequently the cycle will

then be continued for two hundred years. It should be here stated that

this order continued without interruption from the commencement of the era

until the reformation of the calendar in 1582, during which time the

Julian calendar, or Old Style was used.

It has already been shown that if the number of years in the intercalary

period be multiplied by seven, the number of days in the week, their

product will be the number of years in the cycle. Now, in the Gregorian

calendar, the intercalary period is 400 years; this number being

multiplied by seven, their product would be 2,800 years, as the interval

in which the coincidence is restored between the days of the year and the

days of the week.

This long period, however, may be reduced to 400 years; for since the

dominical letter goes back five places every four years, in 400 years it

will go back 500 places in the Julian and 497 in the Gregorian calendar,

three intercalations being suppressed in the Gregorian every 400 years.

Now 497 is exactly divisible by seven, the number of days in the week,

therefore, after 400 years the cycle will be completed, and the dominical

letters will return again in the same order, on the same days of the

month.

In answer to the question, "Why two dominical letters for leap-year?" we

reply, because of the additional or intercalary day after the 28th of

February. It has already been shown that every additional day causes the

dominical letter to go back one place. As there are 366 days in leap-year,

the letter must go back two places, one being used for January and

February, and the other for the rest of the year. Did we continue one

letter through the year and then go back two places, it would cause

confusion in computation, unless the intercalation be made at the end of

the year. Whenever the intercalation is made there must necessarily be a

change in the dominical letter. Had it been so arranged that the

additional day was placed after the 30th of June or September, then the

first letter would be used until the intercalation is made in June or

September, and the second to the end of the year. Or suppose that the end

of the year had been fixed as the time and place for the intercalation,

(which would have been much more convenient for computation,) then there

would have been no use whatever for the second dominical letter, but at

the end of the year we would go back two places; thus, in the year 1888,

instead of A being dominical letter for two months merely, it would be

continued through the year, and then passing back to F, no use whatever

being made of G, and so on at the end of every leap-year. Hence it is

evident that this arrangement would have been much more convenient, but we

have this order of the months, and the number of days in the months as

Augustus Caesar left them eight years before Christ. The dominical letter

probably was not known until the Council of Nice, in the year of our Lord

325, where, in all probability, it had its origin.

CHAPTER IV.

RULE FOR FINDING THE DOMINICAL LETTER.

Divide the number of the given year by 4, neglecting the remainders, and

add the quotient to the given number. Divide this amount by 7, and if the

remainder be less than three, take it from 3; but if it be 3 or more than

3, take it from 10 and the remainder will be the number of the letter

calling A, 1; B, 2; C, 3, etc.

By this rule the dominical letter is found from the commencement of the

era to October 5th, 1582. O. S. From October 15th, 1582, till the year

1700, take the remainder as found by the rule from 6, if it be less than

6, but if the remainder be 6, take it from 13, and so on according to

instructions given in the table on 49th page. It should be understood

here, that in leap-years the letter found by the preceding rule will be

the dominical letter for that part of the year that follows the 29th of

February, while the letter which follows it will be the one for January

and February.

EXAMPLES.

To find the dominical letter for 1365, we have 1365 / 4 = 341 +; 1365 +

341 = 1706; 1706 / 7 = 243, remainder 5. Then 10 - 5 = 5; therefore E

being the fifth letter is the dominical letter for 1365.

To find the dominical letter for 1620, we have 1620 / 4 = 405; 1620 + 405

= 2025; 2025 / 7 = 289, remainder 2. Then 6 - 2 = 4; therefore, D and E

are the dominical letters for 1620; E for January and February, and D for

the rest of the year. The process of finding the dominical letter is very

simple and easily understood, if we observe the following order:

1st. Divide by 4.

2d. Add to the given number.

3d. Divide by 7.

4th. Take the remainder from 3 or 10, from the commencement of the era to

October 5th, 1582. From October 15th, 1582 to 1700, from 6 or 13. From

1700 to 1800, from 7, and so on. See table on 49th page.

We divide by 4 because the intercalary period is four years; and as every

fourth year contains the divisor 4 once more than any of the three

preceding years, so there is one more added to the fourth year than there

is to any of the three preceding years; and as every year consists of 52

weeks and one day, this additional year gives an additional day to the

remainder after dividing by 7. For example, the year

1 of the era consists of 52 w. 1 d.

2 years consist of 104 w. 2 d.

3 years consist of 156 w. 3 d.

(4 / 4) + 4 = 5 years consist of 260 w. 5 d.

Hence the numbers thus formed will be 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13,

15, and so on.

We divide by 7, because there are seven days in the week, and the

remainders show how many days more than an even number of weeks there are

in the given year. Take, for example, the first twelve years of the era

after being increased by one-fourth, and we have

1 / 7 = 0 remainder 1 Then 3 - 1 = 2 = B

2 / 7 = 0 " 2 " 3 - 2 = 1 = A

3 / 7 = 0 " 3 " 10 - 3 = 7 = G

5 / 7 = 0 " 5 " 10 - 5 = 5 = F E

6 / 7 = 0 " 6 " 10 - 6 = 4 = D

7 / 7 = 1 " 0 " 3 - 0 = 3 = C

8 / 7 = 1 " 1 " 3 - 1 = 2 = B

10 / 7 = 1 " 3 " 10 - 3 = 7 = A G

11 / 7 = 1 " 4 " 10 - 4 = 6 = F

12 / 7 = 1 " 5 " 10 - 5 = 5 = E

13 / 7 = 1 " 6 " 10 - 6 = 4 = D

15 / 7 = 2 " 1 " 3 - 1 = 2 = C B

From this table it may be seen that it is these remainders representing

the number of days more than an even number of weeks in the given year,

that we have to deal with in finding the dominical letter.

Did the year consist of 364 days, or 52 weeks, invariably, there would be

no change in the dominical letter from year to year, but the letter that

represents Sunday in any given year would represent Sunday in every year.

Did the year consist of only 363 days, thus wanting one day of an even

number of weeks, then these remainders, instead of being taken from a

given remainder, would be added to that number, thus removing the

dominical letter forward one place, and the beginning of the year, instead

of being one day later, would be one day earlier in the week than in the

preceding year.

Thus, if the year 1 of the era be taken from 3, we would have 3 - 1 = 2;

therefore, B being the second letter, is dominical letter for the year 1.

But if the year consist of only 363 days, then the 1 instead of being

taken from 3 would be added to 3; then we would have 3 + 1 = 4; therefore,

D being the fourth letter would be dominical letter for the year 1. The

former going back from C to B, the latter forward from C to D; or which

amounts to the same thing, make the year to consist of 51 weeks and 6

days; then 10 - 6 = 4, making D the dominical letter as before.

As seven is the number of days in the week, and the object of these

subtractions is to remove the dominical letter back one place every common

year, and two in leap-year, why not take these remainders from 7? We

answer, all depends upon the day of the week on which the era commenced.

Had G, the seventh letter been dominical letter for the year preceding the

era, then these remainders would be taken from 7; and 7 would be used

until change of style in 1582. But we know from computation that C, the

third letter, is dominical letter for the year preceding the era; so we

commence with three, and take the smaller remainders, 1 and 2 from 3; that

brings us to A. We take the larger remainders, from 3 to 6, from 3 + 7 =

10. We add the 7 because there are seven days in the week. We use the

number 10 until we get back to C, the third letter, the place from whence

we started. For example, we have

3 - 1 = 2 = B

3 - 2 = 1 = A

10 - 3 = 7 = G

10 - 4 = 6 = F

10 - 5 = 5 = E

10 - 6 = 4 = D

3 - 0 = 3 = C

The cycle of seven days being completed, we commence with the number three

again, and so on until 1582, when on account of the errors of the Julian

calendar, ten days were suppressed to restore the coincidence of the solar

and civil year. Now every day suppressed removes the dominical letter

forward one place; so counting from C to C again is seven, D is eight, E

is nine, and F is ten. As F is the sixth letter, we take the remainders

from 1 to 5, from 6; if the remainder be 6, take it from 6 + 7 = 13. Then

6 or 13 is used till 1700, when, another day being suppressed, the number

is increased to 7. And again in 1800, for the same reason, a change is

made to 1 or 8; in 1900 to 2 or 9, and so on. It will be seen by the table

on the 49th page that the smaller numbers run from 1 to 7; the larger ones

from 8 to 13.

From the commencement of the Christian era to October 5th, 1582, take the

remainders, after dividing by 7, from 3 or 10; from October 15th,

1582 to 1700 from 6 or 13

1700 to 1800 " 7

1800 to 1900 " 1 or 8

1900 to 2100 " 2 or 9

2100 to 2200 " 3 or 10

2200 to 2300 " 4 or 11

2300 to 2500 " 5 or 12

2500 to 2600 " 6 or 13

2600 to 2700 " 7

2700 to 2900 " 1 or 8

2900 to 3000 " 2 or 9

3000 to 3100 " 3 or 10

3100 to 3300 " 4 or 11

3300 to 3400 " 5 or 12

3400 to 3500 " 6 or 13

3500 to 3700 " 7

3700 to 3800 " 1 or 8

3800 to 3900 " 2 or 9

3900 to 4000 " 3 or 10

4000 to 4100 " 4 or 11

4100 to 4200 " 5 or 12

4200 to 4300 " 6 or 13

4300 to 4500 " 7

4500 to 4600 " 1 or 8

4600 to 4700 " 2 or 9

4700 to 4900 " 3 or 10

4900 to 5000 " 4 or 11

5000 to 5100 " 5 or 12

CHAPTER V.

RULE FOR FINDING THE DAY OF THE WEEK OF ANY GIVEN DATE, FOR BOTH OLD AND

NEW STYLES.

By arranging the dominical letters in the order in which the different

months commence, the day of the week on which any month of any year, or

day of the month has fallen or will fall, from the commencement of the

Christian era to the year of our Lord 4000, may be calculated. (Appendix

G.) They have been arranged thus in the following couplet, in which At

stands for January, Dover for February, Dwells for March, etc.

At Dover Dwells George Brown, Esquire,

Good Carlos Finch, and David Fryer.

Now if A be dominical or Sunday letter for a given year, then January and

October being represented by the same letter, begin on Sunday; February,

March and November, for the same reason, begin on Wednesday; April and

July on Saturday; May on Monday, June on Thursday, August on Tuesday,

September and December on Friday. It is evident that every month in the

year must commence on some one day of the week represented by one of the

first seven letters of the alphabet. Now let

January 1st be represented by A, Sun.

Feb. 1st (4 w. 3 d. from the preceding date) by D, Wed.

Mar. 1st 4 w. 0 d. " " " by D, Wed.

Apr. 1st 4 w. 3 d. " " " by G, Sat.

May 1st 4 w. 2 d. " " " by B, Mon.

June 1st 4 w. 3 d. " " " by E, Thur.

July 1st 4 w. 2 d. " " " by G, Sat.

Aug. 1st 4 w. 3 d. " " " by C, Tues.

Sept. 1st 4 w. 3 d. " " " by F, Fri.

Oct. 1st 4 w. 2 d. " " " by A, Sun.

Nov. 1st 4 w. 3 d. " " " by D, Wed.

Dec. 1st 4 w. 2 d. " " " by F, Fri.

Now each of these letters placed opposite the months respectively

represents the day of the week on which the month commences, and they are

the first letters of each word in the preceding couplet.

To find the day of the week on which a given day of any year will occur,

we have the following

RULE.

Find the dominical letter for the year. Read from this to the letter which

begins the given month, always reading from A toward G, calling the

dominical letter Sunday, the next Monday, etc. This will show on what day

of the week the month commenced; then reckoning the number of days from

this will give the day required.

EXAMPLES.

History records the fall of Constantinople on May 29th, 1453. On what day

of the week did it occur? We have then 1453 / 4 = 363 +; 1453 + 363 =

1816; 1816 / 7 = 259, remainder 3. Then 10 - 3 = 7; therefore, G being the

seventh letter is dominical letter for 1453. Now reading from G to B, the

letter for May, we have G Sunday, A Monday, and B Tuesday; hence May

commenced on Tuesday and the 29th was Tuesday.

The change from Old to New Style was made by Pope Gregory XIII, October

5th, 1582. On what day of the week did it occur? We have then 1582 / 4 =

395+; 1582 + 395 = 1977; 1977 / 7 = 282, remainder 3. Then 10 - 3 = 7;

therefore, G being the seventh letter, is dominical letter for 1582. Now

reading from G to A, the letter for October, we have G Sunday, A Monday,

etc. Hence October commenced on Monday, and the 5th was Friday.

On what day of the week did the 15th of the same month fall in 1582? We

have then 1582 / 4 = 395+; 1582 + 395 = 1977; 1977 / 7 = 282, remainder 3.

Then 6 - 3 = 3; therefore, C being the third letter, is the dominical

letter for 1582. Now reading from C to A, the letter for October, we have

C Sunday, D Monday, E Tuesday, etc. Hence October commenced on Friday, and

the 15th was Friday.

How is this, says one? You have just shown by computation that October,

1582, commenced on Monday, you now say that it occurred on Friday. You

also stated that the 5th was Friday; you now say that the 15th was Friday.

This is absurd; ten is not a multiple of seven. There is nothing absurd

about it. The former computation was Old Style, the latter New Style, the

Old being ten days behind the new.

As regards an interval of ten days between the two Fridays, there was

none; Friday, the 5th, and Friday, the 15th, was one and the same day;

there was no interval, nothing ever occurred, there was no time for

anything to occur; the edict of the Pope decided it; he said the 5th

should be called the 15th, and it was so.

Hence to October the 5th, 1582, the computation should be Old Style; from

the 15th to the end of the year New Style.

On what day of the week did the years 1, 2 and 3, of the era commence?

None of these numbers can be divided by 4; neither are they divisible by

7; but they may be treated as remainders after dividing by 7. Now each of

these numbers of years consists of an even number of weeks with remainders

of 1, 2 and 3 days respectively. Hence we have then for the year 1, 3 - 1

= 2; therefore, B being the second letter, is the dominical letter for the

year 1. Now reading from B to A, the letter for January, we have B Sunday,

C Monday, D Tuesday, etc. Hence January commenced on Saturday.

Then we have for the year 2, 3 - 2 = 1; therefore A being the first

letter, is dominical letter for the year 2; hence it is evident that

January commenced on Sunday. Again we have for the year 3, 10 - 3 = 7;

therefore, G being the seventh letter, is dominical letter for the year 3.

Now reading from G to A, the letter for January, we have G Sunday, A

Monday; hence January commenced on Monday.

On what day of the week did the year 4 commence? Now we have a number that

is divisible by 4, it being the first leap-year in the era, so we have 4 /

4 = 1; 4 + 1 = 5; 5 / 7 = 0, remainder 5. Then 10 - 5 = 5; therefore, E

being the 5th letter, is dominical letter for that part of the year which

follows the 29th of February, while F, the letter that follows it, is

dominical letter for January and February. Now reading from F to A, the

letter for January, we have F Sunday, G Monday, A Tuesday; hence January

commenced on Tuesday.

Now we have disposed of the first four years of the era; the dominical

letters being B, A, G, and F, E. Hence it is evident, while one year

consists of an even number of weeks and one day, two years of an even

number of weeks and two days, three years of an even number of weeks and

three days, that every fourth year, by intercalation, is made to consist

of 366 days; so that four years consist of an even number of weeks and

five days; for we have (4 / 4) + 4 = 5, the dominical letter going back

from G in the year 3, to F, for January and February in the year 4, and

from F to E for the rest of the year, causing the following year to

commence two days later in the week than the year preceding.

The year 1 had 53 Saturdays; the year 2, 53 Sundays; the year 3, 53

Mondays, and the year 4, 53 Tuesdays and 53 Wednesdays, causing the year 5

to commence on Thursday, two days later in the week than the preceding

year. Now what is true concerning the first four years of the era, is true

concerning all the future years, and the reason for the divisions,

additions and subtractions in finding the dominical letter is evident.

The Declaration of Independence was signed July 4, 1776. On what day of

the week did it occur? We have then 1776 / 4 = 444; 1776 + 444 = 2220;

2220 / 7 = 317, remainder 1. Then 7 - 1 = 6, therefore F and G are the

dominical letters for 1776, G for January and February, and F for the rest

of the year. Now reading from F to G, the letter for July, we have F

Sunday, G Monday; hence July commenced on Monday, and the fourth was

Thursday. On what day of the week did Lee surrender to Grant, which

occurred on April 9th, 1865? We have then 1865 / 4 = 466+; 1865 + 466 =

2331; 2331 / 7 = 333, remainder 0. Then 1 - 0 = 1; therefore, A being the

first letter, is dominical letter for 1865. Now reading from A to G, the

letter for April, we have A Sunday, B Monday, C Tuesday, etc. Hence April

commenced on Saturday, and the 9th was Sunday.

Benjamin Harrison was inaugurated President of the United States on

Monday, March 4, 1889. On what day of the week will the 4th of March fall

in 1989? We have then 1989 / 4 = 497+; 1989 + 497 = 2486; 2486 / 7 = 355,

remainder 1. Then 2 - 1 = 1; therefore, A being the first letter, is

dominical letter for 1989. Now, reading from A to D, the letter for March,

we have A Sunday, B Monday, C Tuesday, and D Wednesday; hence March will

commence on Wednesday, and the 4th will fall on Saturday. Columbus landed

on the island of San Salvador on Friday, October 12, 1492. On what day of

the month and on what day of the week will the four hundredth anniversary

fall in 1892?

The day of the month on which Columbus landed is, of course, the day to be

observed in commemoration of that event. The Julian calendar, which was

then in use throughout Europe, and the very best that had ever been given

to the world, made the year too long by more than eleven minutes. Those

eleven minutes a year had accumulated, from the council of Nice, in 325,

to the discovery of America, in 1492, to nine days, so that the civil year

was nine days behind the true or solar time; that is, when the Earth, in

her annual revolution, had arrived at that point of the ecliptic

coinciding with the 21st of October, the civil year, according to the

Julian calendar, was the 12th.

Now, to restore the coincidence, the nine days must be dropped, or

suppressed, calling what was erroneously called the 12th of October, the

21st. Since the Julian calendar was corrected by Gregory, in 1582, we have

so intercalated as to retain, very nearly, the coincidence of the solar

and the civil year. It has already been shown in Chapter III, (q. v.) that

in the Gregorian calendar, the cycle which restores the coincidence of the

day of the month and the day of the week, is completed in 400 years; so

that after 400 years, events will again transpire in the same order, on

the same day of the week. Now, as Columbus landed on Friday, October 21st,

1492, so Friday, October 21st, 1892, is the day of the month and also the

day of the week to be observed in commemoration of that event. We have

then 1892 / 4 = 473; 1892 + 473 = 2365; 2365 / 7 = 337, remainder 6. Then

8 - 6 = 2; therefore, B and C are dominical letters for 1892, C for

January and February, and B for the rest of the year. Now, reading from B

to A, the letter for October, we have B Sunday, C Monday, etc. Hence

October will commence on Saturday and the 21st will be Friday.

Although there was an error of thirteen days in the Julian calendar when

it was reformed by Gregory, in 1582, there was a correction made of only

ten days. There was still an error of three days from the time of Julius

Caesar to the Council of Nice, which remained uncorrected. Gregory restored

the vernal equinox to the 21st of March, its date at the meeting of that

council, not to the place it occupied in the time of Caesar, namely, the

24th of March. Had he done so it would now fall on the 24th, by adopting

the Gregorian rule of intercalation. Appendix H.

If desirable calculations may be made in both Old and New Styles from the

year of our Lord 300. There is no perceptible discrepancy in the

calendars, however, until the close of the 4th century, when it amounts to

nearly one day, reckoned in round numbers one day. Now in order to make

the calculation, proceed according to rule already given for finding the

dominical letter, and for New Style take the remainders after dividing by

seven from the numbers in the following table:

From 400 to 500 From 4 or 11

" 500 " 600 " 5 " 12

" 600 " 700 " 6 " 13

" 700 " 900 " 7

" 900 " 1000 " 1 " 8

" 1000 " 1100 " 2 " 9

" 1100 " 1300 " 3 " 10

" 1300 " 1400 " 4 " 11

" 1400 " 1500 " 5 " 12

" 1500 " 1700 " 6 " 13

It will be found by calculation that from the year

400 to 500 the discrepancy is 1 day

500 " 600 " " " 2 "

600 " 700 " " " 3 "

700 " 900 " " " 4 "

900 " 1000 " " " 5 "

1000 " 1100 " " " 6 "

1100 " 1300 " " " 7 "

1300 " 1400 " " " 8 "

1400 " 1500 " " " 9 "

1500 " 1700 " " " 10 "

Hence the necessity, in reforming the calendar in 1582, of suppressing ten

days. (See table on 59th page.) On what day of the week did January

commence in 450? We have then 450 / 4 = 112+; 450 + 112 = 562; 562 / 7 =

80, remainder 2. Then 3 - 2 = 1; therefore, A being the first letter, is

dominical letter for 450, Old Style, and January commenced on Sunday. For

New Style we have 4 - 2 = 2; therefore, B being the second letter, is

dominical letter for the year 450. Now reading from B to A, the letter for

January, we have B Sunday, C Monday, D Tuesday, etc.

Hence, January commenced on Saturday. Old Style makes Sunday the first

day; New Style makes Saturday the first and Sunday the second. On what day

of the week did January commence in the year 1250? We have then 1250 / 4 =

312+; 1250 + 312 = 1562; 1562 / 7 = 223, remainder 1. Then 3 - 1 = 2;

therefore, B being the second letter, is dominical letter for the year

1250, Old Style. Now, reading from B to A, the letter for January, we have

B Sunday, C Monday, etc. Hence January commenced on Saturday. B is also

dominical letter, New Style; for we take the remainder after dividing by

7, from the same number.

As both Old and New Styles have the same dominical letter, so both make

January to commence on the same day of the week; but Old Style, during

this century, is seven days behind the true time, so that when it is the

first day of January by the Old, it is the eighth by the New.

Vernal equinox in the time of Numa, about 700 B. C.

It is here seen by the errors of the Julian Calendar

the Vernal Equinox is made to occur

three days earlier every 400 years, so

that in 1582 it fell on the

11th instead of the

21st of March. 18

11 By suppressing 10 d

Hou

By the Gregorian rule of intercalation the coincidence of

years.

March 24, 46 B. C.

23 | | 1 A. D.

| Restored |

22 | | 100 " "

| by |

21 | | 300 " "

20 | | | | | 400 " "

19 | Vernal | | | 500 " "

| equinox | | 46 B. C., |

| at the | | | 600 " "

| Council | | to the |

| of | | | 800 " "

| Nice, | | place it |

| 325 | | | 900 " "

| A. D. | | occupied |

| | | | | 1000 " "

| | | | | 1200 " "

| | | | | 1300 " "

| | | | | 1400 " "

| | | | |

ays, Coincidence 0 Restored in | | 1600 " "

| | |

rs behind time, 18 | 6 in advance. | | 1700 " "

| | |

" " 12 | 12 " " | | 1800 " "

| | |

" " 6 | 18 " " | | 1900 " "

| | |

Coincidence 0 Restored. | | 2000 " "

| | |

rs behind time, 18 | 6 in advance. | | 2100 " "

| | |

" " 12 | 12 " " | | 2200 " "

| | |

" " 6 | 18 " " | | 2300 " "

| | |

Coincidence 0 Restored. | | 2400 " "

the solar and civil year is restored very nearly every 400

Appendix I.

CHAPTER VI.

A SIMPLE METHOD FOR FINDING THE DAY OF THE WEEK OF EVENTS WHICH OCCUR

QUADRENNIALLY.

The inaugural of the Presidents. The day of the week on which they have

occurred, and on which they will occur for the next one hundred years:

April 30th, 1789, Thursday, George Washington.

March 4th, 1793, Monday, " "

" " 1797, Saturday, John Adams

" " 1801, Wednesday, Thomas Jefferson.

" " 1805, Monday, " "

" " 1809, Saturday, James Madison.

" " 1813, Thursday, " "

" " 1817, Tuesday, James Monroe.

" " 1821, Sunday, " "

" " 1825, Friday, John Q. Adams.

" " 1829, Wednesday, Andrew Jackson.

" " 1833, Monday, " "

" " 1837, Saturday, Martin Van Buren.

" " 1841, Thursday, Wm. H. Harrison.

" " 1845, Tuesday, James K. Polk.

" " 1849, Sunday, Zachary Taylor.

" " 1853, Friday, Franklin Pierce.

" " 1857, Wednesday, James Buchanan.

" " 1861, Monday, Abraham Lincoln.

" " 1865, Saturday, " "

" " 1869, Thursday, Ulysses S. Grant.

" " 1873, Tuesday, " "

" " 1877, Sunday, Rutherford B. Hays.

" " 1881, Friday, James A. Garfield.

" " 1885, Wednesday, Grover Cleveland.

" " 1889, Monday, Benjamin Harrison.

" " 1893, Saturday, Grover Cleveland.

" " 1897, Thursday,

" " 1901, Monday,

" " 1905, Saturday,

" " 1909, Thursday,

" " 1913, Tuesday,

" " 1917, Sunday,

" " 1921, Friday,

" " 1925, Wednesday,

" " 1929, Monday,

" " 1933, Saturday,

" " 1937, Thursday,

" " 1941, Tuesday,

" " 1945, Sunday,

" " 1949, Friday,

" " 1953, Wednesday,

" " 1957, Monday,

" " 1961, Saturday,

" " 1965, Thursday,

" " 1969, Tuesday,

" " 1973, Sunday,

" " 1977, Friday,

" " 1981, Wednesday,

" " 1985, Monday,

" " 1989, Saturday,

" " 1993, Thursday.

Any one understanding what has been said in a preceding chapter concerning

the dominical letter, can very easily make out such a table without going

through the process of making calculations for every year. As every

succeeding year, or any day of the year, commences one day later in the

week than the year preceding, and two days later in leap-year, which makes

five days every four years, and as the Presidential term is four years, so

every inaugural occurs five days later in the week than it did in the

preceding term.

Now, as counting forward five days is equivalent to counting back two, it

will be much more convenient to count back two days every term. There is

one exception, however, to this rule; the year which completes the century

is reckoned as a common year (that is, three centuries out of four),

consequently we count forward only four days or back three.

Commencing, then, with the second inaugural of Washington, which occurred

on Monday, March 4, 1793, and counting back two days to Saturday in 1797,

three days to Wednesday in 1801 and two days to 1805, and so on two days

every term till 1901, when, for reasons already given, we count back three

days again for one term only, after which it will be two days for the next

two hundred years; hence anyone can make his calculations as he writes,

and as fast as he can write. See table on 61st page.

SOME PECULIARITIES CONCERNING EVENTS WHICH FALL ON THE TWENTY-NINTH DAY OF

FEBRUARY.

The civil year and the day must be regarded as commencing at the same

instant. We cannot well reckon a fraction of a day, giving to February 28

days and 6 hours, making the following month to commence six hours later

every year; if so, then March, for example, in

1888 would commence at 12 m. night.

1889 " " " 6 a. m.

1890 " " " 12 m.

1891 " " " 6 p. m.

1892 " " " 12 m. night,

again, and so on.

Instead of doing so, we wait until the fraction accumulates to a whole

day; then give to February 29 days, and the year 366. Therefore, events

which fall on the 29th of February cannot be celebrated annually, but only

quadrennially; and at the close of those centuries in which the

intercalations are suppressed only octennially. For example: From the year

1696 to 1704, 1796 to 1804, and 1896 to 1904, there is no 29th day of

February; consequently no day of the month in the civil year on which an

event falling on the 29th of February could be celebrated. Therefore, a

person born on the 29th of February, 1896, could celebrate no birthday

till 1904, a period of eight years.

In every common year February has 28 days, each day of the week being

contained in the number of days in the month four times; but in leap-year,

when February has 29 days, the day which begins and ends the month is

contained five times. Let us suppose that in a certain year, when February

has 29 days, the month comes in on Friday; it also must necessarily end on

Friday.

After four years it will commence on Wednesday, and end on Wednesday, and

so on, going back two days in the week every four years, until after 28

years we come back to Friday again. This, as has already been explained,

is the dominical or solar cycle. For example: February in

The year 4 has five Fridays.

" " 8 " " Wednesdays.

" " 12 " " Mondays.

" " 16 " " Saturdays.

" " 20 " " Thursdays.

" " 24 " " Tuesdays.

" " 28 " " Sundays.

" " 32 " " Fridays.

So that after 28 years we come back to Friday again; and so on every 28

years, until change of style in 1582, when the Gregorian rule of

intercalation being adopted by suppressing the intercalations in three

centurial years out of four interrupts this order at the close of these

three centuries. For example--1700, 1800 and 1900, after which the cycle

of 28 years will be continued until 2100, and so on. The cycle being

interrupted by the Gregorian rule of intercalation, causes all events

which occur between 28 and 12 years of the close of the centuries to fall

on the same day of the week again in 40 years; and those events that fall

within 12 years of the close of these centuries, to fall on the same day

of the week again in 12 years; after which the cycle of 28 years will be

continued during the century. See following table:

1804 February has five Wednesdays.

1808 " " " Mondays.

1812 " " " Saturdays.

1816 " " " Thursdays.

1820 " " " Tuesdays.

1824 " " " Sundays.

1828 " " " Fridays.

1832 " " " Wednesdays.

1836 " " " Mondays.

1840 " " " Saturdays.

1844 " " " Thursdays.

1848 " " " Tuesdays.

1852 " " " Sundays.

1856 " " " Fridays.

1860 " " " Wednesdays.

1864 " " " Mondays.

1868 " " " Saturdays.

1872 " " " Thursdays.

1876 " " " Tuesdays.

1880 " " " Sundays.

1884 " " " Fridays.

1888 " " " Wednesdays.

1892 " " " Mondays.

1896 " " " Saturdays.

1900

1904 " " " Mondays.

1908 " " " Saturdays.

1912 " " " Thursdays.

1916 " " " Tuesdays.

1920 " " " Sundays.

1924 " " " Fridays.

1928 " " " Wednesdays.

It will be seen from this table that in 1804 February had five Wednesdays;

and then again in 1832, 1860 and 1888; then suppressing the intercalation

in the year 1900 suppresses the 29th of February; so opposite 1900 in the

table is blank, and the 29th of February does not occur again till 1904,

and the five Wednesdays do not occur again till 1928--that is, 40 years

from 1888, when it last occurred.

Again taking the five Mondays which occurred first in this century, in

1808, and then again in 1836, 1864 and in 1892, you will see, for reasons

already given, that it will occur again in 12 years, that is, in 1904; and

so on with all the days of the week, when it will be seen what is peculiar

concerning the 29th of February.

But attention is particularly called to the five Thursdays, which occur

first in this century 1816, and then again in 1844 and 1872, the last date

being within 28 years of the close of the century. Suppressing the

intercalation suppresses the 29th of February; consequently the five

Thursdays do not occur again till 1912, that is 40 years from the

preceding date, after which the cycle will be continued for two hundred

years.

Hence it may be seen that the dominical or solar cycle of 28 years is so

interrupted at the close of these centuries by the suppression of the

leap-year, that certain events do not occur again on the same day of the

week under 40 years, while others are repeated again on the same day of

the week in 12 years; also the number of years in the cycle, that is 28 +

12 = 40.

And again the change of style in 1582, causes all events which occur

between 28 and 8 years of that change, to fall again on the same day of

the week in 36 years, and all that occur within 8 years of that change to

be repeated again on the same day of the week in 8 years, after which the

cycle of 28 years is continued for 100 years; also that the number of

years in the cycle, that is, 28 + 8 = 36.

CHAPTER VII.

RULES FOR FINDING THE DAY OF THE WEEK OF EVENTS THAT TRANSPIRED PRIOR TO

THE CHRISTIAN ERA.

First, it should be understood that the year 4 is the first leap year in

our era, reckoning from the year 1 B. C., which must necessarily be leap

year; so that the odd numbers 1, 5, 9, 13, etc., are leap years. Hence

every year that is divisible by four and one remainder, is leap year; if

no remainder, it is the first year after leap-year; if 3, the second; if

2, the third, thus:

45 / 4 = 11, remainder, 1,

44 / 4 = 11, no remainder,

43 / 4 = 10, remainder, 3,

42 / 4 = 10, remainder, 2,

41 / 4 = 10, remainder, 1,

and so on, every year being divided by four and 1 remainder is leap-year

of 366 days. It should be borne in mind that the same calendar was in use

without any correction from the days of Julius Caesar 46 B. C. to Pope

Gregory XIII in 1582; consequently the method of finding the dominical

letter is, in some respects, similar to the one already given on the 44th

page. But in some respects the one is the reverse of the other, for we

reckon backward and forward from a fixed point--the era; that is the

numbers increase each way from the era. Also the dominical letters occur

in the natural order of the letters in reckoning backward, but exactly the

reverse in reckoning forward. See table on the 73d page, where the

dominical letter is placed opposite each year from 45 B. C. to 45 A. D.

Now we use the same number 3, because C, the third letter is dominical

letter for the year 1 B. C., the point from which we reckon. But instead

of taking the remainder, after dividing by 7, _from_ 3 or 10, to find the

number of the letter, as in Part Second, Chapter IV, (q. v.) we add the

remainder _to_ 3; hence we have the following:

RULE.

Divide the number of the given year by 4, neglecting the remainders, and

add the quotient to the given number, divide this amount by 7, and add the

remainder to 3, and that amount will give the number of the letter,

calling A, 1; B, 2; C, 3, etc.; except the first year after leap-year,

(which is the year exactly divisible by 4), the number of the letter is

one less than is indicated by the rule.

This rule gives the dominical letter for January and February only, in

leap-year, while the letter that precedes it, is the letter for the rest

of the year. If the amount be greater than seven, we should reckon from A

to A or B again.

It has already been stated in Part First, Chapter III, (q. v.), that a

change was made by Augustus Caesar about 8 B. C., in the number of days in

the month; and, as this change effects the day of the week on which

certain events fall, it becomes necessary that they should be presented as

they were arranged by Julius Caesar, and as corrected by Augustus. Julius

Caesar gave to February 29 days in common years, and in leap-year 30. This

arrangement was the very best that could possibly be made, but, as has

already been shown, it was interrupted to gratify the vanity of Augustus.

The left hand column in the table on the 72d page represents the number of

days in each month from the days of Julius Caesar to Augustus, a period of

37 years. The right hand column represents the number of days in the

months as they now are, and have been since the change was made by

Augustus, 8 B. C. Consequently the rule for finding the day of the week on

which events have fallen for the 37 years prior to the last mentioned

date, is not perfectly exact, and needs a little explanation here.

The rule itself is given, and fully explained in Part Second, Chapter V,

(q. v.) but cannot be applied to the 37 years without some correction. In

all the months marked with a star, events fall one day later in the week

than that which is indicated by the rule. This should be borne in mind,

and make the event one day later in the week than that which is found by

the rule. For example, Julius Caesar was assassinated on the 15th of March,

44 B. C. By giving to February 28 days the first day of March would fall

on Wednesday, and, of course, the 15th would be Wednesday. But Caesar gave

to February 29 days, so that the first day of March fell on Thursday, and

the 15th was Thursday.

Hence, every event from March to September will fall one day later in the

week than the rule indicates. But the rule is applicable to September, for

it will make no difference whether there are 29 days in February or 31 in

August, there are the same number of days from February to September. But

the 31 days in September will cause all events to fall one day later in

the week during the month of October, but they coincide again during the

month of November. The order is interrupted again in December by giving 31

days to November. See following table:

_As Arranged by Julius_ | _As Corrected by Augustus,_

_Caesar._ | _8 B. C._

January, 31 | January, 31

February, 29 | February, 28

March, 31* | March, 31

April, 30* | April, 30

May, 31* | May, 31

June, 30* | June, 30

July, 31* | July, 31

August, 30* | August, 31

September, 31 | September, 30

October, 30* | October, 31

November, 31 | November, 30

December, 30* | December, 31

---------+-----+---------+-----+---------+-----+---------+-----

---------+-----+---------+-----+---------+-----+---------+-----

|B. C.| |B. C.| |A. D.| |A. D.

cb | 45 | b | 22 | b | 1 | c | 23

a | 44 | ag | 21 | a | 2 | ba | 24

g | 43 | f | 20 | g | 3 | g | 25

f | 42 | e | 19 | fe | 4 | f | 26

ed | 41 | d | 18 | d | 5 | e | 27

c | 40 | cb | 17 | c | 6 | dc | 28

b | 39 | a | 16 | b | 7 | b | 29

a | 38 | g | 15 | ag | 8 | a | 30

gf | 37 | f | 14 | f | 9 | g | 31

e | 36 | ed | 13 | e | 10 | fe | 32

d | 35 | c | 12 | d | 11 | d | 33

c | 34 | b | 11 | cb | 12 | c | 34

ba | 33 | a | 10 | a | 13 | b | 35

g | 32 | gf | 9 | g | 14 | ag | 36

f | 31 | e | 8 | f | 15 | f | 37

e | 30 | d | 7 | ed | 16 | e | 38

dc | 29 | c | 6 | c | 17 | d | 39

b | 28 | ba | 5 | b | 18 | cb | 40

a | 27 | g | 4 | a | 19 | a | 41

g | 26 | f | 3 | gf | 20 | g | 42

fe | 25 | e | 2 | e | 21 | f | 43

d | 24 | dc | 1 | d | 22 | ed | 44

c | 23 | | | | | c | 45

---------+-----+---------+-----+---------+-----+---------+-----

PART THIRD.

CYCLES--JULIAN PERIOD--EASTER.

HEBREW CALENDAR.

CHAPTER I.

THE SOLAR CYCLE.

Cycle, (Latin _Cyclus_, ring or circle). The revolution of a certain

period of time which finishes and re-commences perpetually. Cycles were

invented for the purpose of chronology, and for marking the intervals in

which two or more periods of unequal length are each completed a certain

number of times, so that both begin exactly in the same circumstance as at

first. Cycles used in chronology are three: The solar cycle, the lunar

cycle, and the cycle of indiction.

The solar cycle is a period of time after which the same days of the year

recur on the same days of the week. If every year contained 365 days, then

every year would commence one day later in the week than the year

preceding, and the cycle would be completed in seven years. For if the

first day of January, in any given year, fall on Sunday, then the

following year on Monday, the third on Tuesday, and so on to Sunday again

in seven years.

But this order is interrupted in the Julian calendar every four years by

giving to February 29 days, and consequently the year 366. Now the number

of years in the intercalary period being four and the days of the week

being seven, their product is 4 x 7 = 28; twenty-eight years then is a

period after which the first day of the year and the first day of every

month recur again in the same order on the same day of the week. This

period is called the solar cycle or the cycle of the sun, the origin of

which is unknown; but is supposed to have been invented about the time of

the Council of Nice, in the year of our Lord 325; but the first year of

the cycle is placed by chronologists nine years before the commencement of

the Christian era.

Hence the year of the cycle corresponding to any given year in the Julian

calendar is found by the following rule: Add nine to the date and divide

the sum by twenty-eight; the quotient is the number of cycles elapsed, and

the remainder is the year of the cycle. Should there be no remainder, the

proposed year is the twenty-eighth, or last of the cycle. Thus, for the

year 1892, we have (1892 + 9) / 28 = 67, remainder 25. Therefore, 67 is

the number of cycles, and 25 the number in the cycle.

Our Calendar The Julian calendar and its errors. How corrected by the Gregorian. Rules for finding the dominical letter, and the day of the week of any event from the days of Julius Caesar 46 B.C. to the year of our Lord four thousand; a new and easy method of fixing the date of Easter. Hebrew calendar; showing the correspondence in the date of events recorded in the Bible with our present Gregorian calendar. Illustrated by valuable tables and charts. · 第 1 / 8 章