OUR •CALENDAR -o-^3*@^ BY REV. GEORGE N. PACKER, Wallsbarp, Fa, OUR CALENDAR The Juliaii Calendar* and its errors. How corrected by the Grregorian which is now in use almost throughout the civ- ilized World. Also Rules for finding the Dominical Letter and the day of the week of any event, in any Year, from the commencement of the Chris- tian Era to the Year of our Lord 4000. ILLUSTRATED BY VJ\LUJ{BLE TABLES Afi/D Cf/A/fTS BY REV. GEORGE NICHOLS PACKER. WELLSBORO, PA. REPUBLICAN ADVOCATE PRINT. Zl "Z.Q f ?') 1890. X^HingTOM. **3 THE LIBRAEY OF CONGRESS WASHINGTON Entered according to Act of Congress, in the year 1890, By Rev. George Nichols Packer, In the Office of the Librarian of Congress, at Washington, D. C. 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. PffKFJlGE MANY years ago while engaged in teaching, the writer of this little volume was in the habit of bringing to the at- tention of his pupils a few simple rules for finding the dominical letter and day of the week of any given event within the past and the present centuries ; further than this he gave the subject no spe- cial 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, ts history, and the method of disposing of the fraction of a day found in the time required for the revolutio n 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 co- incidence of the solar and the civil year is preserved, it was suggest- ed to him that he should put the story of the calendar, its correc- tion by Gregory, and the theory and results of intercalation, in 6 writing. It was urged that this would give increased interest to the study, help the writer to forget his pains, and probably en- able 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 rinding 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 ap- pears 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 arrang- ed in accordance with a plan of his own devising, so that the out- line and the execution of the work may be truly said to be origi- nal. Of its value the world must judge. It has been prepared in weakness of body and in suffering, which have been to some ex- tent 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 it perusal. The Author. CONTENTS. PART FIRST. DEFINITIONS— HISTORY. Pages. Chapter I. — Definitions 9-10 Chapter II. — History of the divisions of time, and the old Roman calendar 10-15 Chapter III —History of the reformation of the calendar by Julius Csesar 16-17 Chapter IV —History of the reformation of the Julian cal- endar by Pope Gregory XIII 18-25 PART SECOND. MATHEMATICAL. Chapter I.— Errors of the Julian calendar . . . 26 28 Chapter II. — Errors of the Gregorian calendar 28-29 Chapter III— Dominical letter 30-35 Chapter IV. —Rule for finding the dominical letter 36-41 Chapter V -Rule for finding the day of the week of any given date, for both Old and New Styles 42-51 Chapter VI.— A simple method for finding the day of the week of events, which occur qudrennially ; 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 52-54 Some peculiarities concerning events which fall on the 29th of February 55-59 Appendix 60-68 OUR CALENDAR PART FIRST. DE FIN [TION 3. 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, eic. b — An hour is the subdivision of the day into twenty-four equal parts. c — 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. d — The week is a period of seven days having no reference whatever to the celestial motions, a circum- stance to which it owes its unalterable uniformity. 9 10 e — The month is usually employed to denote an arbitrary number of days approaching a twelfth part of a year, and has retained its place in the calendar of all nations. f — 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 eclip- tic to the same point again, and consists of 365 days, 5 hours, 48 minutes and 49.62 seconds of mean solar time. Appendix A. 6 —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, AISTD THE OLD ROMAN CALENDAR. Day— The subdivision of the dayinto 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 man- ner of distributing the hours. Europeans in general, 11 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. Astrono- mers after the example of Ptolemy, regard the day as commencing with the Sun's culmination, or noon, and find it most convenient for the purpose of com- putation to reckon through the whole twenty-four hours. Hipparchus reckoned the twenty-four hours from midnight to midnight. 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. 392, it has been employed from time immemorial in almost all Eastern countries ; and as it forms neither an ali- quot 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 aparticnlar 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 wa? consecrated to Saturn, that planet would also have the 8th, the 15th, and the 22d hours ; the 23d 12 would fall to Jupiter, the 24th to Mais, and the 25th or the first hour of the second day would belong to the Sun. In like manner the first hour of the 3d day would fall to the Moon, the first hour of the 4th to Mars, of the 5th to Mercury, of the 6th to Jupiter, and the 7th to Venus. * The cycle being completed, the first hour of the 8th day would again return to Saturn and all the others succeed in the same order. It will be seen by the table at the close of this chapter, and it is also re- corded by Dio Cassius, of the 2d Century, that the Egyptian week commenced with Saturday. On their flight from Egypt the Jews, from hatred to their an- cient oppressors, made Saturday the last day of the week. It is stated that the ancient Saxons borrow- ed the week from some Eastern nation, and substi- tuted 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. Dies Solis. Dies Lance- Dies Martis. Dies Mercurii. Dies Jovis. Dies Veneris. Dies Saturni, SAXON. Sun's Day. Moon's Day. Tiw's Day. Woden's Day. Thor'sDay. Friga's Day. Seterne's Day. ENGLISH. Sunday. Monday. Tuesday .• Wednesday. Thursday. Friday. Saturday. 13 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 Decern, ten; July and August were also denominated Quintilis, and Sextilis; from Quintus, live; 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 4o0 B. C, when the Decemvirs (ten magistrates) changed the order placing February after January, making March the 3d instead of the 1st 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 14 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 1461 years. Hence 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 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 inser- ted 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 conse- quently 366 1-4 days, so that the year was then too long by one day. The year was then reduced to 365 14 days by suppressing the intercalation in every period of eight years. 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 the autumn and the autumnal into summer. Appendix B. 15 A table of the order and the names of the planets in the Egyptian astromomy illustrating the origin of the names of the days of the week : =5 8 I Jupiter, *° j Thursday. 'Mars, 01 | Tuesday. .Si! if 4 ll 11 11 1 ^ 1 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 i 22 23 24 1 2 3 4 ! 5 6 7 8 9 10 il ; 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 16 CHAPTER III. HISTORY OF THE REFORMATION OF THE CALENDAR BY JULIUS CESAI*. Forty-six years before Christ, Julius Csesar called on the astronomers, especially on Sosigones of Alex- andria, to assist him in this very desirable under- taking. The mean length of the year was fixed at 365 1-4 days. It was decided that every year should consist of 365 days excepting the fourth, which should have 366. In order to restore the vernal equinox to the 24th of March the place it occupied in the time of Nurna, two months, together consisting of 67 days were in- serted 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 Febru- ary 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 commodius arrangement in the distribution of the days through the several months. It was decided to give to January, March, May, July, September and 17 November each thirty-one days ; and the other months thirty, excepting February which in com- mon years should have only twenty-nine days, but every fourth year thirty ; so that the average length of the Julian year was 365 1-4 days. A ugustus Csesar interupted 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 Oc- tober and December. In the Julian calandar a day was added to Febr- uary every fourth year, (that being the shortest) which is called the additional or intercalary day ail is inserted in the calendar between the 24th and 25th of that month. In the ancient Roman calen- dar the first day of every month was invariably call- ed the calends ; February then having twenty nine days, the 25th was the 6th of the calends of March, — sexto calendas ; the preceding which was the ad- ditional or intercalary day was called bis-sexto calen- das,— twice the sixth day. Hence the term bissexti- le as applied to every fourth year commonly called leap-year. 18 CHAPTER IV. HISTORY OF THE REFORMATION OF THE JULIAN CALENDAR BY POPE GREGORY THE XIII. True enough the year, in which Julius Caesar re- formed the ancient Roman calendar, was the last year of confusion, and the method adopted by him, a commodious one, and answered a ver y good purpose for a short time but as the years rolled on and century after century had passed away, astronomers be- gan 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 retro- grading 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 this time amounted to seven or eight days. Thus the errors of the calendar continued to in- crease until 1582, when the vernal equinox fell on the 11th instead of the 21st of March. Gregory 19 perceiving that the measure (of reforming the cal- endar) was likely to confer great eclat on his ponti- ficate undertook the ]ong 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 sub- stituted that which has since been received in al- most 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 of that month, thus suppressing ten days, and mak- ing the year 1582 to consist of only 355 days. So we see that the ten days that had been gained by mcorrect 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, name- ly the 21st of March. The Pope was promptly obeyed in Spain, Portu- gal, and Italy. The change took place the same year in France, by calling the 10th the 20th of De- cember. Many other Catholic countries made the change the same year, and the Catholic states of Ger- many 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 20 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 to be 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 in- tercalation. Russia is the only Christian country that still ad- heres 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 b : ween solai and civil time does not effect the day, for, as has already been shown, the mean length of the day is twentv-i\ ur hours, and is marked by one revolution of the Earth upon its axis. Nor does it effect the week, for the week is uni- formly 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 reckon- ed as many days, and as many w r eeks, 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 adopt- ed the New Style. Columbus sailed from Palos, in Spain, on Friday, August 3d, 1492, Old Style, which 21 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 cen tury is nine days ; during the 16th and 17th centur- ies, 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 Feb- ruary the 22d. As regards Russia, she could have made the change last year by calling the 20th of De- cember, 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 Decem- ber twenty days. The methods of computation, both old and new styles, will be explained in another chapter. To persons unacquainted with astronomy the dif- ference between Old and New Styles would probably be better understood by the diagram on the 23d page. The figure represents 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 22 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 inter- calation, the other by suppression. i\ppendix 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 trifl- ing error in his calendar. The error which was 11 min:ites and 10.38 seconds every year was hardly perceptible for a short period, but still amounted to three days every 400 yea v s. 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 w r hich 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 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 diagram that 23 21st. u *noistijnoo jo md£ ^SBjaq^,, pn-e 's£ep Qff=LQ-\-$z-\-9Q2 S I WW t v 's^p c^ jo isisaoo o; xea^ ^q^ Sappsui 's£ep vg\ 06 SuT^xBo.ia^ui iCq ;, q ft 9^ xeseeo snq / ^ \ -rif .£q pamioja^ 'jBpnaj'eo tremor aqx /^ * Y> eoi^sjog -cctng **si3 24 ten days were deducted from October in 1582 mak- ing it a short month consisting of only twenty-one days. The discrepancy between the Julian and Gregor- ian calander 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, and the winter solstice the place of the summer solstice. And in 43,700 years according to the Julian rule of intercalation there would be gained n arly 30"' 1-4 days, or one entire revolution of the Earth, ^o, 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 di- vided among the thousands of the preceding years. 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 25 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 sup- posed solar year of 365 days, there would be an ad- vance or gain of one day in that year and in 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. PART SECOND. MATHEMATICAL. CHAPTER I. ERRORS OF THE JULIAN CALENDAR. It will be necessary in the first place to under- stand the difference between the Julian and Grego- rian rule of intercalation. It' the number of any year be exactly divisible by 4 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 rale of intercalation, which is corrected by the Gregorian, by making every centu- 26 27 rial 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. EULE. Multiply the diflerenc 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 discrep- ancy 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.) = ( 1:L 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 (3-4x4)=(lx3)=3 ; the Julian rule gaining three days, the Gregorian suppressing three days in 400 28 years. (3xl0)=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 min- utes 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 deci- mal . 373, that being the fraction of a minute gained every year. To find how much time would be gain- ed 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. 29 SOLUTION. (24 X 60) -=-.373 =3, 861 nearly; hence the error would amount to only one day in 3,861 years. (.373X4, 000)-60=(24 h, 52 m,)=(l d, 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 and 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 errors in 4,000 years,) by this quotient, and you will have the dis- crepancy between the Gregorian and solar year for 100,000 years. By this improved method we sup- press 25 days, so that the error will only amount to 25 times 52 minutes. SOLUTION. 100,000-5-4,000 X(l A, 52 m,)=(25 d, 21 h, 40 m.) Now (25 d, 21 h, 40 m,)— 25 d,=(21 h, 40 m,) the error in 100,000 years. 30 CHAPTER III. DG VIJSTICAL LETTER. Dominical (from the Latin Dominus, lord,) indi- cating 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 al- phabet, 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 eight, and so on through the 860 days of the year. Now, if one of the days of the week, Sunday for example, is represented by F, Monday will be rep resented 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 Tues- day 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 dav of 31 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 one 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 fol- lowing year 1838 on Monday, 1839 on Tuesday, and so on. As the year consists of 52 weeks and one day, it is evident that the day, which begins and ends the year must occur 53 times; thus the year 1837 be- gins on Sunday and ends on Sanday, so the follow- ing year, 1838 must begin on Monday. As A repre- sented all the Sunda}^ 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 \ to G; so that G represents all the Sundays in 1838 5 A all the Mondays, B all the Tues- days, 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 P respectively. Therefore if every year consisted of 365 days, the dominical cyc- le would be completed in seven years; so that after 32 seven years the first day of the year would again oc- cur 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 b^ing leap-year, had two domini- cal letters, A andG; A for January and February, and G for the rest of the year. The year commenced on Sunday and ended on Monday, making 53 Sun days 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; i: they 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 Jan- uary 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 +* 33 week, their product would be 2x7=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 3x7=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 4x7=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 do- mincal letters return again in the same order, on the same days of the month. Thus, for the year 1801, the domincal letter is B; 1802 C; 1803, B; 1804? A and G; and so on, going back five places every four years for 28 years; when the cycle being ended, D is again dominical letter for 1829, C, for 1830, and so on every £8 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 interrupt ed, however, at the end of every century, for the leap-year is not suppressed in every fourth Centur- ial year, consequently the cycle will then be con- tinued for two hundred years. It should be here stated that this order continued without interrup- 84 tion from the commencement of the era until there- formation 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 pro- duct 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 complet- ed, 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 Feb- ruary. It has already been shown that every addi- tional day causes the dominical letter to go back one place. As there are 366 days in leap-year, the let- ter must go back two places, one being used for Jan- 35 uary 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 confus- ion 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 un- til the intercalation is made in June or September, and the second to me 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 com- putation), 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 Gr, 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 num- ber 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. 36 CHAPTER IV. KULE FOR FINDING THE DOMINICAL LETTER. Divide the number of the given year by 4, neglect- ing the remainders, and add the quotient to the given number. Divide this amount by 7 and if the re- mainder be less than three, take it from 3; but if it be 3 or more than 3, take it from 10 and the remaind- er 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 0, 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 41st page. It should be understood here, that in leap-year the letter found by the pre- ceding rule will be 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 Jan- uary and February. EXAMPLES. To find the dominical letter for 1365, we have 1365-^4=341+ ; 1365+341 = 1706 ; 1706-^7=243, re- mainder 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-j-4=405; 1620+405=2025; -2025--7=289, re- 37 mainder 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 pro- cess of finding the dominical letter is very simple and easily understood, if we observe the following order: 1st. Divide by 4. 2nd. Add to the given number. 3rd. 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 41st 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 consists 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 38 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 1-1-7=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=E. P. 6-f-7=0 " 6 " 10— 6-=-4=D. 7^7=1 " u 3— 0=3=C. 8-f-7=l " 1 : < 3— 1=2=B. 10-=-7=l " 3 " 10— 3=7=G, A.. ll-f-7=l " 4 " 10— 4=6=F. 12-5-7=1 kt 5 " 10— 5=5=E. 13^7=1 " 6 " l()— 6=4=D. 15-^7=2 " 1 u 3— 1=2=B, C. From this table it may be seen that it is these re- mainders 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, in- variably, there would be no change in the dominical letter from year to year, but the letter that repre- sents Sunday in any given year would represent Sun- day in every year. Did the year consist of ( nly 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 add- ed to that number, thus removing the dominical let- 39 ter 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. As seven is the number of days in the week, and the object of these subtractions is to remove the do minical letter back one place every common year, and two in leap-year, why not take these remaind- ers from 7? We answer, all depends upon the day of the week on which the era commenced. Had Gr, 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 p]ace 40 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 com- mence 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 the 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 the year 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 41st page that the small er numbers run from 1 to 6 ; the larger ones from 7 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 i 41 1582 to 1700 from 6 or 13. 1700 to 1800 from 7. 1800 to 1900 from 1 or 8. 1900 to 2100 from 2 or 9. 2100 to 2200 from 3 or JO. 2200 to 2300 from 4 or 11. 2300 to 2500 from 5 or 12. 2500 to 2600 from 6 or 13. 2800 to 2700 from 7 2700 to 2900 from 1 or 8. 2900 to 3000 from 2 or 9. 3000 to 3100 rrom 3 or 10. 3100 to 3300 from 4 or 11. 3300 to 3400 from 5 or 12. 3400 to 3500 from 6 or 13. 3500 to 3700 from 7. 3700 to 3800 from 1 or 8. 3800 to 3900 from 2 or 9. 3900 to 4000 from 3 or 10. 4000 to 4100 from 4 or 11. 4100 to 4200 from 5 or 12. 4200 to 4300 from 6 or 13. 4300 to 4500 from 7. 4500 to 4600 from 1 or 8. 42 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 fall or will fall, from the commence- ment of the Christian era to the year of our Lord 4000 may be calculated. (Appendix (r.) 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 repre- sented by the same letter, begin on Sunday ; Febru- ary, March, and November, for the same reason, begin on Wednesday ; April and July on Saturday ; May on Monday, June on Thursday, August on Tu~ esday, September and December on Friday. It is evident that every month in the year must commen- ce 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.lst4w0d " " " byD, Wed. Apr. 1st 4 w 3d " " " by G, Sat. 43 May 1st 4w2d " <• ^ by B % Mon. June 1st 4 w 3d " " " by E, Thur. July 1st 4w2d " " " by G, Sat. Aug. 1st 4w3d " <> « byC, Tues„ Sep. 1st 4w3d " " " by F, Fri. Oct. 1st 4w2d " " " byA 3 Sun. ISTov.lst 4w3d " u " by D, Wed. Dec, 1st 4 w 2d " " " 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 eule : Find the dominical letter for the year. Read from this to the letter which begins the given month, al- ways reading from A towards G, calling the domini- cal letter Sunday, the next Monday, etc., this will show on wJiat day of the week the month commenc- ed; 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 -f- 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 Gr to B the letter for May, we 44 have G Sunday, A Monday and B Tuesday; hence May commenced on Tuesday and the 29th was Tues- day. The change from Old to New Style was made by Pope Gregory XIII, October 5, 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 Mon- day; hence October commenced on Monday, and the 5th was on 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, J) Monday, E Tuesday, etc. Hence October corameiv ced on Friday, and the 15th was Friday. How is this says one? You have just shown by computation that October 1582 commenced on Mon day, you now say that it occured on Friday. You also stated, that the 5th was Friday; you now say that the 15th was Friday. This is absured, 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, 45 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 occured, there was no time for anything to occur; the edict of the Pope de- cided 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 w r hat 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 seven. 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 Janu- ary 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 w^e have for the year 3, 10 — 3 =7;therefore Gr being the seventh letter is dominical letter for the year 3. Now reading from G to A, the letter for January, we have Gr, Sunday, A Monday; hence January commenced on Monday. fh 46 On what day of the w^ek did the year 4 com- mence? Now we have a number that is divisible by four; so we have 4-^-4=1; 4+1=5; 5-^-7=0, remain- der 5. Then 10 — 5=5; therefore E being the fifth 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)-f-4=5, the dominical letter going back from Gr in the year 3, to F, for January and February, and from F to E for the rest of year, causing the follow- ing year to commence two days later in the week than the year preceding. The year 1 had 53 Saturdays the year 2, 53 Sun- days, 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 con- 47 cerning all the future years, and the reason for the divisions, additions and substractions 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 Feb- ruary 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 1885. Now reading from A to G, the let- ter for April, we have A, Sunday, B, Monday, C, Tuesday, etc. Hence April commenced on Satur- day, 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, S mday B, Monday C, Tuesday, and D, Wednesday: hence March will commence on Wed- 48 nesday, and the 4th will fall on Saturday. Colum- bus landed on the island of St. 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 ? It is evident that Columbus discovered America the 12th of October, Old Style, which may be seen by table on the 49th page, to be nine days behind the true time; consequently nine days must be added to the 12th of October to find the day of the month on which the anniversary will fall. We have then 1892^4=473; 1892+473=2365; 2365—7=337, re- mainder 6. Then 8 — 6=2; therefore B and C are the dominical letters for 1892, C for January and February and B for the rest of the year* Now read- ing from B to A the letter for October, we have B Sunday, C Monday, etc. Hence October will com- mence on Saturday and the 21st will be Friday. Although there w T as an error of thirteen days in the Julian calendar when it was reformed by Greg- ory in 1582, there was a correction made of only ten days. There was still an error of three days j from the time of Julius Csesar to the council of .Nice which remained uncorrected. Gregory restored the 1 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 Cseser, namely the 24th of March. Had he done so it would now fall on the 24l1i by adopting the Gregorian rule of intercalation. Ap- pendix H. 49 If desirable calculations may be made in both Oldnnd 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 num- bers one day. Now in order to make the calculation, proceed according to rule already given for find- ing the dominical letter; and for New Style take the remainders after dividing by 7 from the numbers in table on 49 page. From 400 to 500 from 4 " 500 " 600 " 5 600 " 700 '• 6 700 li 900 " 900 " 1000 " 1 1000 " 1100 " 2 1100 " 1300 " 3 1300 " 1400 " 4 1400 " 1500 " 5 1500 " 1700 " 6 by calculation that from the a or 11 " 12 " 13 7 " 8 " 9 10 11 12 13 a a a a It will be found year 400 to 500 the 500 600 700 900 1000 1100 1300 1400 1500 u u * . u u u u u 600 700 900 1000 1100 1300 1400 1500 1700 it a a (; << a a discrepancy is 1 day 2 3 4 5 6 7 8 9 10 U u u a u u u u u u u u t i u u u u u u it a u u u Hence the necessity, in reforming the calender in 50 1582, of suppressing ten days. (See table on 51st page.) On what day of the week did January com- mence 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 the year 450, Old Style, and January commenced on Sunday. For New Style we have 4 — 2=2 ; there- fore 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 commenc- ed 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 ^ame domin- ical 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 eight by the New. Vernal equinox in the time of Nnma, It is here seen that 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. 51 about 700 B. C. .March 24 46 B. C. 23. in.. 12.. 11 By suppressing 10 days, coincidence Restored in Hours behind time, 18 " 12 6 Coincidence, in advance 12 18 restored Hours behind time, 18 I 6 in advance " 12 12 " 6 1 18 " Coincidence, restored 100 A. 300 " 400 " 500 " 600 " 800 " 900 " 1000 " 1200 " 1300 " 1400 " 1600 " 1700 " 1800 " 1900 " 2000 " 2100 " 2200 " 2300 '« 2400 " By the Georgian rule of intercalation the coincidence of the solar and the civil year is restored very nearly every 400 years. Appendix I. 52 CHAPTER VI. A SIMPLE METHOD FOR FINDING THE DAY OF THE WEEK OF EVENTS, WHICH OCCUR QUADRENIALLY. 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, U U it u 1797, Saturday, John Adams u u 1801, Wednesday, Thomas Jefferson. u u 1805, Monday, u u 4 I u 1809, Saturday, James Madison. i £ u 1813, Thursday, u u U u 1817, Tuesday, James Monroe. U c. 1821, Sunday, »w U u u 1825, Friday, John Q. Adams. (. u 1828, Wednesday, Andrew Jackson. u u 1833, Monday, u u u u 1837, Saturday, Martin Van Buren, £ I " 1841, Thursday, Wm. H. Harrison. ( i >' 1845, Tuesday, James K. Polk. b( i. 1849, Sunday, Zachary Taylor. 1 4 1 4 1853, Friday, Frank Pierce. 1857, Wednesday, James Buchanan. J 861, Monday, Abraham Lincoln. 1865, Saturday, 1869, Thursday, Ulysses S. Grant, 1873, Tuesday, 53 1877, Sunday, Ruth'f d B. Hayes. 1881, Friday, James A. Garfield. 1885, Wednesday, Grove r Cleveland. 1889, Monday, Benjamin Harrison. 1893, Saturday, 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 , Any one understanding what has been said in a 54 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 live 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 conveni- ent to count back tw^o 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 Monday in 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, alter 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 o2d page. 55 SOME PECULIAKITIES CONCERNING EVENTS WHICH EALL ON THE 29th 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 in 1888 would commence at 6 a. m. 1889 " " " 12 m. 1890 " " " 6 p.m. 1891 ifc " " 12 m. again, and so on. Instead of doing so, we wait until the fraction ac- cumulates 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 annu- ally , but only quadrennially ; and at the close of those centuries in which the intercalations are suppressed only octenniaJly. 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 29th of February, 1896, could celebrate no birthday till 1904, a period of eight years. In every common year February has 29 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 56 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 : The year 4 has five Fridays. The year 8 u " Wednesdays. The year 12 " " Mondays. rp he year 16 " " Saturdays. The year 20 " " Thursdays. The year 24 wi " Tuesdays. Thevear28 " u Sundays. The year 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, Avhen the Georgian rule of intercala- tion being adopted by suppressing the intercalations in three centurial years out of four interrupts this order at the close of these three centuries. For exam- ple— 1700, 1800 and 1900. after which the cycle of 28 years will be continued till 2100, and so on. The cycle being interrupted by the Georgian rale of intercala- tion, causes all events which occur between 28 and 12 years of the close of these centuries to fall on the same day of the week again in 40 years ; and those 57 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 table on 57th page. 1804 February has five Wednesdays 1808 1812 1816 1820 1824 1828 1832 1836 1840 1844 1848 1852 1856 1860 1864 1868 1872 1876 1880 1884 1888 1892 1896 1900 Mondays. Saturdays. Thursdays. Tuesdays. Sundays. Fridays. Wednesdays. Mondays. Saturdays. Thursdays. Tuesdays. Sundays. Fridays. Wednesdays. Mondays. Saturdays. Thursdays. Tuesdays. Sundays. Fridays. Wednesdays Mondays. Saturdays. 58 1904 " " " Mondays. 1908 " . " " Saturdays. 1912 " " " Thursdays. 1916 " " " Tuesdays. 1920 •< " " Sundays. 1924 " u " Fridays. 1928 " " " Wednesdays. It will be seen from this table that in 1804 Febru- ary had five Wednesdays ; and then again in 1832, 1860 and in 1888 ; then suppressing the intercalation in the year 1900 would make it to occur again in 1900 or 12 years from the preceding date ; but sup- pressing the intercalation suppresses the 29th of February ; so opposite 1900 in the table is blank, and the 29th of Feburary does not occur till L904, and the five Wednesdays do not occur again till 1928 ; that is 40 years from 1888 when it last occur- red. 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 occured first in this century in 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 59 February ; consequently the five Thursdays do not occur again till 1912, that is 40 years from the pre- ceding 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 in 40 years ; while others are re- peated again on the sam^ 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 eight years of that change to be repeated again on the same day of the week in eight years, after which the cycle of 28 years is continued for one hundred years ; also, that the number of years in the cycle, that is, 28+8=36. APPENDIX. ^->^> A.— PAGE 10. Authors differ in regard to the length of the Solar year. One gives 365 days, 5 hours, 47 minutes and 51.5 seconds ; another, 365 days, 5 hours, 48 minutes and 46 seconds ; and still another, 365 days, 5 hours, 48 minutes and 49.62 seconds. In this work the last has been accepted as the true length of the solar year, and a,ll calculations have been made accord- ingly. B.— PAGE 14. Some authors say that the ancient Roman year of 355 days was increased to 365 by intercalating a month of thirty days every three years, so that the Romans would have lost nearly one day every four years. It is evident that by some means the year was too short, and consequently in advance of the true or solar time. C.-PAGE 18. The city w^here the great council was convened in 325 is not in France as some have supposed, that be- ing a more modern city of the same orthography, but pronounced Nees. The city which is so fre- quently referred to in this work is in Bythinia, one 60 61 of the provinces of Asia Minor, situated about 54 miles southeast of Constantinople, of the same or- thography as the former, but pronounced M-ce, and was so named by Lysimachus, a Greek general, about 300 years before Christ, in honor of his wife, Nicea. D.— PAGE 22. Sometime during the year 46 B. C, before Caesar reformed the old Roman calendar there was inter- calated a month of 23 days according to an establish- ed method, but still the civil year was in advance of the solar year by 67 days ; so that when the earth in her annual revolutions should arrive to that point of the ecliptic marked the 22d of October, it would be the 1st day of January in the Roman year. Caesar and his astronomers, knowing this fact, and fixing on the 1st day of January, 45 years before Christ and 709 from the foundation of Rome, for the reformed calendar to take effect, were under the necessity of intercalating two months, together, consisting of 67 days. Now, as the civil year would end on the 22d of October, true or solar time, it would be reckoned in the old calendar the 1st day of January ; so they let the old calendar come to a stand while the earth performs 67 diurnal revolu- tions, and thereby restored the concurrence of the solar and the civil year. As an illustration, let us suppose that in a certain shop where hangs a regulator are two clocks to be legulated. Bothare set with the regulator at 8 a. m. 62 to see how they will run for ten consecutive hours. It was found that when it was 6 p. m., by the first clock, it was 5:50 by the regulator, the clock having gained one minute every hour. To rectify this discrepancy we must intercalate ten minutes by stopping the clock until it is six by the regulator. By thes means the coincidence is re- stored, and the time lost in the preceding hours is now reckoned in this last hour making it to consist of 70 minutes. By this it may be seen how Csesar re- formed the Roman calendar. The Roman year was too short, by reason of which the calendar was thrown into confusion, being 90 days in advance of the true time, so that December, January, and Feb- ruary,- took the place in the seasons, of September, October, and November; and September, October and November, the place of June, July and August. To make the correction he must stop the old Roman clock (the calendar) while the Earth performs 90 diurnal revolutions to restore the concurrence of the solar and the civil years, making the year 46 B, C, to consist of 445 days. It was also found that when it was 6 p. m., by the regulator, it was only 5:50 by the second clock, it having lost one minute every hour. To rectify this discrepancy we must suppress ten minutes, calling it 6 p. m., turning the hands of the clock to coincide with the regulator, making the last hour to consist of only 50 minutes, too much time having been reck- oned in the preceding hours. It may be seen by 63 this illustration, how Gregory corrected the Julian Calendar, the Julian year was too long, consequent- ly behind true or solar time, so that when the cor- rection was made in 1582, the ten days gained had to be suppressed to restore the coincidence, making the year to consist of only 355 days. As the solar year consists of 365 days and a frac- tion, Caesar intended to retain the concurrence of the solar and the civil year by intercalating a day every four years ; but this made the year a little too long, by reason of which it became necessary, in 1582, to rectify the error, and by adopting the Gre- gorian rule, three intercalations are suppressed every 400 years ; so that by a series of intercalations and suppressions, our calendar may be preserved in its present state of perfection. E.— PAGE 22. As the day and the civil year always commence at the same instant, so they must end at the same in- stant; and as the solar year always ends with a fraction, not only of a day, but of an hour, a min- ute and even a second ; so there is no rule of inter- calation by which the solar and the civil year can be made to coincide exactly. But the discrepancy is only a few hours in a hundred years, and that is so corrected by the Gregorian rule of intercalation that it would amount to a little more than a day in 4,000 years ; and by the improved method less than a day in 100,000 years. * P.— PAGE 25. It has been stated that by adopting the Julian rule of intercalation, time was gained ; it has also 64 been stated that by the same rule time was lost. Now both are true. Time is gained in that there is too much time in a given year, in other words the year is too long ; but what is gained in a given year is lost to the following years. As an illustration let us take the case of the sup- posed solar year of 365 days, and the civil year of 366. The civil year would gain one day every year, or be too long by one day ; but the one day gained is lost to the following years, and if continued 31 years, when the Earth is in that part of its orbit marked the 1st day of January 32, the civil year would reckon the 1st day of December 31 ; so that in the thirty-one years would reckon thirty one days too much, and before the civil year is completed, the Earth will have passed on in its orbit to a point marked the 1st day of February. Now to reform such a calendar, we would have to suppress or drop the thirty-one days, by calling the 1st day of December the 1st day of January, and thus the month of December would disappear from the calendar in the year 31, making a year of only eleven months, consisting of 334 days. If this method be continued 92 years, there would be gained 92 days, to the loss of 92 days in the year 92. If the calendar be now reformed by suppress- ing 92 days, calling the first day of October 92, the first day of January 93, then October, November, aud December would disappear from the calendar in the year 92 ; and if continued 365 years there would be crowded into 364 years, 364 days too much ; gain- ed to the 364 years to the total loss of the year 365, 65 passing from 364 to 366 ; 365 disappearing from the calendar. G.— PAGE 43. An era is a fixed point of time from which a serie s of years is reckoned. Among the nations of the Earth there are no less than twenty-five different eras; but the most of them are not of enough importance to be mentioned here. Attention is particularly called to the Roman era which commenced with the building of the city of Rome 753 years before Christ. Also the Mahometan era, or era of the Hegira, em- ployed in Turkey, Persia, and Arabia, which is dated from the flight of Mahotnet from Mecca to Medina, which was Thursday night, the 15th of Ju?y, A. D., 622, and it commenced on Friday the day following. But there is a point from which all computation originally commenced, namely the creation of man. Such an era is called the Mundane era. How -there are different Mundane eras, — the common Muudane era 4,004 B. C, the Grecian Mundane era 5,598 B. C, and the Jewish Mundane era 3,761 B. C. All these commence computation from the same point;, but differ in regard to the time which has elapsed since their computation commenced. God's people used the Mundane era, until the great .. Creator ap- peared among us, as one of us, in the person of our Lord Jesus Christ accomplish the great w^ork of redemption; then his name was introduced as the turning point of the ages, the starting point of com- putation. 66 This was done by Dionysius Exiguns in the year of our Lord about 540, known at that time as the Dionysian, as well as the Christian era, and was first used in historical works by the venerable Bede early in the 8th century "It was a great thought of the little monk (whether so called from his hu- mility or littleness of stature is unknown), to view Christ as the turniug point of the ages, and to intro- duce this view into chronology.' 5 All honor to him who introduced it, and to the nations which have approved, for thus honoring the great Redeemer. Dionysius x>i obably did not know, neither is it now known for a certainty the year of Christ's birth, but it is evident, however, from the best authorities, that the era commenced at least five years too late, and probably more H. -PAGE 48. It is recorded that, in the time of Numa, the ver- nal equinox fell on the 25th of March, and that Jul- ius Caesar restored it to the 25th, when he reformed the ancient Roman calendar in the year 46 B. C. It is also recorded that in less than 400 years from that time, at the meeting of the Council of Nice in 325, it had fallen back to the 21st, — four days in less than 400 years. Now there is an error somewhere, for it is found by actual computation that the discrepancy between the solar and the Julian year is about three days in 400 years. It certainly is true that the vernal equi- nox fell on the 21st in 325, and was restored to that place by Gregory in 1582; since which time it has 67 been made to fall on the 21st by the Gregorian rule of intercalation. Now with these facts before us, we must come to the conclusion that the vernal equi- nox did not fall on the 25th of March in the time of Numa, nor of Julius Caesar, but the 24th. I.— PAGE 51. The concurrence of the solar and the civil year was restored by Gregory in 1582, or 1600 is the same in computation; but the discrepancy between civil and solar time is 11 minutes and 10.38 seconds every year, which in one hundred years will amount to 18 hours and 37.3 minutes; reckoned in round numbers 18 hours, and is represented on the Chart, Hours behind time 18. The intercalary day or 24 hours being suppressed in 1700, causes the civil year to be 6 hours in advance of the solar, and is re j resented on the chart 6 hours in advance. Now this discrepancy of 18 hours for tlie next 100 years, will cause the civil year in 1800 to be 12 hours behind; again suppressing the intercalation it will be 12 hours in advance. In 1900 it will be 6 hours behind, but the correction makes 18 hours in advance. The 18 hours gained the next one hundred years restores the coincidence in the year 2000, and so on, the solar and the civil year being- made to coincide very nearly every 400 years. From close examination it will become evident that the solar and the civil year coincide twice every 400 68 years, though no account is made of it in computa- tion. From 6 hours in advance in 1700, the civil year falls back to 12 hours behind the solar in 1800, consequently they must coincide in 1733. Again from 12 hours in advance in 1800, it falls back to 6 hour? behind the solar in 1900, conse- quently they must coincide again in 1867. Discrepancy between Julian and solar time in — 1 year is (365 d, 6 h.)— (365 d, 5 h, 48 m, 49.62 s)= (11 m, 10.38 s.) 100 years is (11 m, 10.38 s.)Xl00=(18 h, 37.3 id.) 400 " " (18 h, 37.3 m.)X4=(3 d, 2 h, 29.2m.) 4,000 *' (3d, 2 h, 29.2 m.)XlO=(31 d, h, 52m.) 100,000 u (31 d, Oh, 52 m.)X25 = (775 d, 21 h, 40 m.) Discrepancy between Gregorian and solar time in — 1 year is --------- - .373 m. 100 years is .373 m. xl00== . - - - - 37.3 m. 400 '' 37.3 m. X4= - - - 2 h, 29.2 m. 4,000 " (2 h, 29.2m.)xl0= 1 d, h' 52 m. 100,000 " (1 d, h, 52 m.)x25=25 d, 21 h, 40 m. Discrepancy between corrected Gregorian and so- lar time in — 4,000 years is (1 d, h, 52 m)— 1 dav = - 52 m- 100,000 - " (52 m. X 25 )= 21 h. 40 . > ' 6 i^S=7^ 022 008 929