A.—PAGE 12.

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 all calculations have been made accordingly.

B.—PAGE 19.

There is an apparent discrepancy among authors in regard to the intercalary day. While one asserts that it was between the 24th and 25th of February, another equally reliable, says that the 25th was the sexto calendas and the 24th was the bis-sexto calendas of the Julian calendar. Now it should be borne in mind that the Julian calendar is the basis of our own, and is identical with it in the number of months in the year, and in the number of days in the month. Also when the method of numbering the days from the beginning of the month was adopted, the intercalation was made to correspond with the intercalary day in the Julian calendar.

As in the Julian calendar there were twice the sixth day, so in the reformed calendar there were twice the 24th day, which was equivalent to 29 days in February. When the calendar was again corrected, making the 29th the intercalary day, then the 24th corresponded with the bis-sexto calendas of the Julian calendar. This reconciles the apparent discrepancy. While one author refers to the calendar in which the Julian rule of intercalation is adopted, another refers to the calendar when so corrected as to make the 29th of February the intercalary day. See following table:

C.—PAGE 20.

The city where the great council was convened in 325 is not in France, as some have supposed, that being a more modern city of the same orthography, but pronounced Nees. The city which is so frequently referred to in this work is in Bythinia, one of the provinces of Asia Minor, situated about 54 miles southeast of Constantinople, of the same orthography as the former, but pronounced Ni´ce, and was so named by Lysimachus, a Greek general, about 300 years before Christ, in honor of his wife Nicea.

D.—PAGE 23.

Between the 23d and 24th of February, 46 years before Christ, there was intercalated a month of 23 days according to an established 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.

CÆsar 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 revolutions, 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 regulated. Both are set with the regulator at 8 a. m. 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 10 minutes by stopping the clock until it is 6 by the regulator. By this means the coincidence is restored, 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 CÆsar reformed 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 February 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 year, 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 10 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 reckoned in the preceding hours. It may be seen by this illustration, how Gregory corrected the Julian calendar, the Julian year was too long, consequently behind true or solar time, so that when the correction 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 fraction, CÆsar 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 Gregorian 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 23.

As the day and the civil year always commence at the same instance, so they must end at the same instance; and as the solar year always ends with a fraction, not only of a day, but of an hour, a minute and even a second; so there is no rule of intercalation 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.

F.—PAGE 26.

It has been stated that by adopting the Julian rule of intercalation, time was gained; it has also 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 year.

As an illustration let us take the case of the supposed 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 suppressing 92 days, calling the 1st day of October, 92, the 1st day of January, 93, then October, November and 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; gained to the 364 years to the total loss of the year 365, passing from 364 to 366; 365 disappearing from the calendar.

G.—PAGE 50.

An era is a fixed point of time from which a series 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 the era of the Hegira, employed in Turkey, Persia and Arabia, which is dated from the flight of Mahomet from Mecca to Medina, which was Thursday night, the 15th of July, 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. Now there are different Mundane eras—the common Mundane 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 appeared among us, as one of us, in the person of our Lord Jesus Christ, to accomplish the great work of redemption; then His name was introduced as the turning point of the ages, the starting point of computation.

This was done by Dionysius Exiguus 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 eighth century. “It was a great thought of the little monk (whether so called from his humility or littleness of stature is unknown), to view Christ as the turning point of the ages, and to introduce this view into chronology.”

All honor to him who introduced it, and to the nations which have approved, for thus honoring the Great Redeemer. Dionysius probably 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 57.

It is recorded that, in the time of Numa, the vernal equinox fell on the 25th of March, and that Julius CÆsar 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 equinox fell on the 21st in 325, and was restored to that place by Gregory in 1582; since which time it has been made to fall on the 21st by the Gregorian rule of intercalation. Again it is stated by the same author that the discrepancies in time from CÆsar to Gregory is thirteen days, from the Council of Nice to Gregory ten days; now 10 + 4 = 14. While our author states it is thirteen days, he also states it is fourteen days; a discrepancy of one day. The mistake evidently is in making the 25th instead of the 24th, the date of the vernal equinox in the time of CÆsar, consequently a difference of four days instead of three from CÆsar to the Council of Nice.

I.—PAGE 59.

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 100 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 represented on the chart 6 hours in advance.

Now this discrepancy of 18 hours for the 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 100 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 years, though no account is made of it in computation. 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 hours behind the solar in 1900, consequently they must coincide again in 1867.

Discrepancy between Julian and solar time in—1 year is (365d. 6h.) - (365d. 5h. 48m. 49.62s.) = (11m. 10.38s.)

Discrepancy between Gregorian and solar time in—

Discrepancy between corrected Gregorian and solar time in—

J.—PAGE 89.

Lilius, author of the “Extended Table of Epacts,” says, when the full moon falls on the 10th of March, the following moon, which happens 29 days later, is the paschal moon, making the 18th of April its latest possible date. For, says he, because of the double epact that occurs on the 4th and 5th of April that lunation has only 29 days. It may have been very convenient for Lilius, in his peculiar method of determining the date of the paschal moon, to give to that lunation only 29 days; but nevertheless, when he did so, it was at the expense of accuracy, for he makes a difference of 12 days in the date of the paschal moon of that year, and the year preceding, and only 10 days difference between that year and the succeeding year; whereas the difference is uniformly 11 days from year to year through the whole cycle of 19 years.

By referring to the table on the 93d page, it will be seen that, in fixing the date of the paschal moon, six times in a cycle of 19 years the full moon falls before the 21st of March, and in every instance except this one the following moon is reckoned by Lilius 30 days later. By this uniform method of determining the date of the paschal moon, we make the 19th of April instead of the 18th, its latest possible date; so it should be borne in mind that whenever the 19th of April is the date of the paschal moon, as indicated in the tables commencing with the 93d page, that Lilius, and probably most, if not all other authors, have the 18th.

Now it is admitted that notwithstanding the cumbersome apparatus employed by Lilius in his calculations, the conditions of the problem are not always satisfied, nor is it possible that they can be always satisfied by any similar method of proceeding. We admit that none of these calculations are perfectly exact, but the sum of the solar and lunar inequalities is compensated in the whole period, or corrections made at the end of certain periods, not by interrupting the order of a uniform method during the cycle of 19 years.

Now the table of epacts was introduced by Lilius himself, making the excess of the solar year beyond the lunar, in round numbers 11 days. Then why interrupt this order every 19 years, for a period of 114 years; that is from 1596 to 1710, by making the epact 12 days for one year, and the following year only 10? After which, from 1710 to 1900, a period of 190 years, according to Lilius’ own calculations, the epact is uniformly 11 days, coinciding exactly with the calculations made in this work.

Then again after the year 1900, he gives to that particular lunation, in every lunar cycle for a period of 304 years, only 29 days; and having done so, he is under the necessity of giving only 29 days to another lunation in the same cycle, and also to all the cycles in the period to avoid the absurdity of making the paschal moon fall twice on the same day in the course of a lunar cycle.

By reference to the 101st page, opposite the year 1905, it will be seen that the date of the paschal moon is the 19th of April. Lilius, by giving to that lunation only 29 days, makes its date the 18th; and then again in the year 1916, lest he should make the paschal moon fall twice on the 18th of April in the course of a lunar cycle, (a thing which cannot really occur) he for the first time in more than 400 years, gives only 29 days to a second lunation in the same cycle and of course to all the cycles in the period of 304 years. Now the epacts for a lunar cycle of 19 years are represented thus:

The number 26 placed over the 25 shows Lilius’ first error in giving to that lunation only 29 days. He thereby makes a difference of 12 days between the epact 14 and 26, and only 10 between 26 and 6. He now has two epacts of the same number 26. In order to get out of the dilemma he makes that 27, by giving to another lunation only 29 days.

K.—PAGE 122-3.

It will probably be noticed that according to the showing in the tables the ecclesiastical year contains only 364 days. The reason for this is, that Advent Sunday, which is the first day of the year, happens one day earlier every year until it occurs on the 27th of November, its earliest possible date; then the first Sunday after the 26th of November, which is Advent Sunday, falls on the 3d of December, its latest possible date, so that the year begins six days later, making a year of 371 days. Then there is the loss of a day every year until Advent Sunday again falls on the 27th of November and so on. Hence, did the civil year always consist of 365 days, then the ecclesiastical year would always contain either 364 or 371 days. But as every fourth year contains 366 days, this order is so interrupted that sometimes the first Sunday falls on the 2d instead of the 3d of December; so that the year begins only five days later, making a year of only 370 days. Hence the ecclesiastical year may consist of either 364, 370 or 371 days. But five times out of six it will contain only 364 days.

L.—PAGE 83.

But why did the Pope, in correcting the Julian calendar in 1582, not correct the whole error of thirteen days? Why did he leave the three days uncorrected? This question has been asked an hundred times, but a correct answer has never yet been given. Some say that the Pope did according to his best ability, and would make us believe that neither he nor his astronomers knew what the error was. This is not true, for history records the fact of the error, and just what that error was. He simply did not want to correct the three days, and for good reasons, which we shall endeavor to show; reasons which every churchman ought to know.

When CÆsar formed his calendar, 46 B. C., the vernal equinox fell on the 24th of March. At the meeting of the Council of Nice, in 325, it had fallen back to the 21st, the error being three days in about 400 years. Now it should be borne in mind that the Julian calendar was the only one in use at that time, and for the next 1257 years, when in 1582, it was corrected by Pope Gregory XIII. Easter, and all the movable feasts, had been unsettled during the 1257 years intervening, from the Council of Nice to Gregory, on account of the errors of the Julian calendar. The Easter question had been the cause of a good deal of discussion between the Eastern and Western churches during the second and third centuries, as they could not agree on the day of the week on which that event should be celebrated.

The Western churches observed the nearest Sunday to the full moon of Nisan. The Asiatics, on the other hand, adopted the 14th of Nisan upon which to commemorate the crucifixion, and observed the festival of Easter on the third day following, upon whatever day of the week that might fall. Finally, the Council of Nice was convened, and the matter came before that council, and a reconciliation was accomplished. It was then and there agreed by the two parties that Easter should be celebrated on the first Sunday after the full moon that falls upon or next following the day of the vernal equinox, and that the 21st of March should be accounted the day of the vernal equinox.

It has already been shown that the error in the Julian calendar is three days in 400 years; so that in 400 years from the Council of Nice the vernal equinox had fallen back to the 18th of March; in 800 years it had fallen back to the 15th; in 1257 years, that is in 1582, it fell on the 11th. Still the 21st of March, by the only calendar in use at that time, was accounted the date of the vernal equinox, by which date Easter was determined, so that, in 1582, when it was the 21st by the calendar, the correct date was the 31st. Hence, the error had been increasing at the rate of three days every 400 years until in 1582 it amounted to ten days.

Again it should be borne in mind that the Pope was a churchman and wished to abide by the decision of that council in celebrating the festival of Easter, so he drops the ten days and restores the vernal equinox to the 21st of March, its date at the meeting of the Council of Nice in 325, the date by which Easter day was determined. He not only made the correction, but he so reformed the calendar that the solar and the civil year are now made to coincide very nearly. Had he dropped the thirteen days, the vernal equinox would have been restored to the 24th of March, its date in the time of CÆsar, and the 24th would still be its date. But the Council of Nice decided that the 21st should be the date by which Easter day should be determined. Hence the reason for dropping the ten days instead of the thirteen is evident; and it is also evident that the Pope acted understandingly when he made the correction in 1582.