Ecclesiastical Computation of Isaac Argyros the Monk

In the third year of the reign of Darius the Mede, by Jesus son of Josedech and Zorobabel son of Salathiel, the temple in Jerusalem was restored, in the year of Adam 5011. ^A

In the second year of the empire of Augustus Caesar the indictions began to be counted, and the Roman months were then first devised by them, in the year of Adam 5460.

In the forty-third year of Augustus is born, according to us who are above us, the only-begotten Son of the Father, Jesus Christ; and by nature for our sake he becomes a perfect man, who by nature was perfect God in himself, in the year of Adam 5501.

In the fifteenth year of Tiberius he is baptized in the Jordan, and bestows upon those who believe adoption by the Holy Spirit, in the year of Adam 5530. ^B

In the nineteenth year of the same Tiberius he comes to the saving passion, bestowing from it impassibility upon our nature, in the year of Adam 5534.

In the twentieth year of Constantine, the Council of Nicaea was held, in the year of Adam 5816, of Christ 316.

In the second year of Theodosius, the council was celebrated at Constantinople, in the year of Adam 5872, of Christ 372.

In the thirteenth year of Theodosius the Younger, the first Council of Ephesus was held, in the year of Adam 5913, of Christ 413.

In the first year of the empire of Marcian, the Council of Chalcedon was celebrated, in the year from Adam 5945, from Christ 443. ^C

In the twenty-sixth year of the empire of Justinian, the Fifth Council was held, in the year from Adam 6045, from Christ 545.

OF ISAAC THE MONK ARGYROS TO THE MOST WISE LORD ANDRONICUS OF OENOE, who requested the logical methods for solar and lunar cycles and those things that follow from them.

Proemium.

Since you have overlooked the others who live today, most excellent of men, and have judged us to be suitable, ^D

to hand over to you the methods for the matters to be discussed, and moreover the causes of the arguments concerning these—for, not unaware that we are not so well-furnished, you have boldly imposed such a task upon us; yet you rightly thought it should be attributed to friendship, which by nature usually exercises a kind of tyranny over those who yield themselves wholly to it, and corrupts their votes and judgments—therefore we, as you commanded, now set our hand to the work, having revered your judgment of us, and, as is consistent, taking no account of our own weakness for accomplishing that which is proposed. Whatever success may come to us, I shall acknowledge myself a debtor of gratitude to you, because, whether I execute the matter to my mind, you gave me the opportunity to perform it well; or whether I am unable to attain it, you judged one as unequal as me to be suitable for the task.

On the solar cycles.

^A I. First of all, we have before us for examination the subject of the so-called solar cycles: how it is that they are called cycles, and why, proceeding to the twenty-eighth, they take their beginning again from the first; and furthermore, by what method we might be able to accurately determine, for any given year, which cycle it belongs to.

Concerning this we say that they are named cycles because each one of them contains an annual interval; for since we measure the year by the motion of the sun, and the motion of the sun, performed in a circle and restored from the same point to the same, defines the year, the interval of time that the solar circuits circumscribe deserved to be called a circle; and with the appellation of "year" also joined to it, it is named an "annual circle."

Furthermore, these annual circles of the sun make a progression from one to the twenty-eighth, and thence begin again from one, for such a reason:

Why the solar cycles proceed up to twenty-eight.

^B II. Since the Creator of all things, God, brought this whole visible creation into existence in six days and rested on the seventh, as the book of Genesis records, all time following that week, being continuously propagated with the succession of things, is measured according to that week by the septenary number. And because the sun, completing its own orbit, is restored from one point to the same point in three hundred and sixty-five days and approximately a quarter, and makes the annual circle consist of the same number of days—which days, partitioned by the cycle of the week, produce fifty-two weeks and one day and a quarter—it is manifest that the beginnings of all years do not start from one and the same day of the week, but make a transition from the first to the second, from the second to the third, and so on in order.

^A This will become clearer by an example, bringing forward two or even more years that immediately followed the foundation of the world. For since the first year began from the first day and completed three hundred and sixty-five days with a quadrant, the following second year began necessarily from the second day, and the third likewise from the third, and so on in order. For the fourth part of the day, which is added to such 365 days to complete the circuit of the year, is by no means counted in every year, but one day is collected every four years. For four times a quarter makes one. And this completed day is added to the 365 days; and that year becomes 366 days, which we in the Roman tongue call bissextile. Therefore the fifth year from the creation of the world did not begin from the fifth day, following those before it, but from the sixth, because of the addition of one day, which, having been collected from the quadrants, was added to the fourth year that had passed. Then next, the sixth, ^B seventh, and eighth years duly, as the series required, followed the days of the week, that is, the seventh, and the first again, and the second; but the ninth had not the third, but the fourth: which happened in the following years by the same reasoning.

From these things it appears that if the year consisted only of 365 days, without any addition of a quadrant, the beginnings of each would return to the same day of the week after no more than seven years. But since a quadrant is added, for which reason every fourth year there becomes one remaining day, which is made from four quadrants, therefore after twenty-eight years the beginnings of the years are recalled to the same days of the week. For if you multiply 7 by 4, this is the changes of the beginnings ^C of the years which, as we have said, also occur every four years, there will be twenty-eight. Which is the reason why the annual cycles of the sun are explained in neither more nor fewer years than 28.

It remains that a certain method be handed down, through which we may be able to grasp the cycle of the sun for any given year. And it is of this kind.

Method of finding what the cycle of the sun is

[ξ] 3. We divide the years from the beginning of the world up to the current year by 28, and whatever is below 28, that is the cycle of the sun which is sought. For example, the years from the foundation of the world to this year are numbered 6881, from which 28 having been subtracted as far as is possible, 21 remain. ^D Whence the cycle for the current year 6881 is 21.

But since out of every method the more expeditious one is to be preferred, changing that same one into an easier form, we shall treat it thus:

After the completion of any century, taking the cycle of the last year as a root, we add it to the years of the following century: and dividing the sum by 28, in the same way as above, we obtain what is sought. For the sake of example in the proposed year, since in the past

^A century was completed, that is the 6800th year, the cycle of the sun was 24; we hold these 24 as a foundation for the entire incoming century. And adding to these the 81 years up to the present time, we divide the resulting sum by 28, and the remainder, which is 21, we say is the cycle of the sun.

It would also be reasonable to resolve here, as far as it is possible, the doubt that arises regarding what has been said. It is this: we said that the first year of the solar cycles begins from the first day, that is, Sunday, as it also first began from the creation of the world. But now, when the first solar cycle arrives, it begins from the second day, not the first. And what could be the cause of this difference regarding this first day? This is the point of doubt; it would be resolved as follows, according to reason.

Since the true beginning of the year, which is the day on which God, the creator of all things, began their creation, is unknown as to which of the days of the year it is—with some assuming one beginning for the year and others another—because when our September arrives, which we take as the beginning of the year, it is the 7th day; but when October arrives, it is the 2nd day; we consider the beginning of October a more suitable beginning for the solar cycles than that of September. We shall make this clearer in the method of finding the day of the week, which is as follows.

Method of finding the day of the week

IV. When we wish to find on which day of the week a certain day of the current month falls, we first take the number of past solar cycles, excluding the current one; and to this we add the days from the beginning of that month up to the very one sought, and also ^B the epact of that same month, which for October is 1. For from the true beginning of the year, which is the 1st day, the shift to the 1st of October, which falls on the 2nd day, causes a change of one day.

^C

And this is what we said above, resolving the problem that arises. And next, the 4th day corresponds to November. Because, since October has 31 days, 3 days remain after the 4 weeks; these, added to the 1 ^D that precedes October, make 4. And December gives 6; because, after the 4 weeks, November leaves 2 days, and these, with the previous 4, make 6. And likewise for January 2. For the

6 previous days and the 3 remainder of December that exceed the weeks make 9; from which, if one week is removed, 2 days remain. And so, in succession, February 5, March 5 again (because February, consisting of only 28 days, leaves no epacts for the following month). April 1, May 3, June 6, July 1, August 4, and September 6.

By collecting these together—that is, the days of the current month up to the day in question, the epacts of the same month, and in addition to these the past solar cycles, as has been said, and the quadrants of these, omitting anything less than 4—and finally dividing the sum by 7, the remainder, which is less than 7, shows on what day of the week the day ^B of the month in question falls.

For example: let it be our task to find on what day of the week the 26th of the current month of October falls. Take first the 26 days that have elapsed from its Kalends; add to these the epact of that same month, as well as 20 for the as many past cycles; and also their quadrants, which are 5. For we omit up to 4, as I said above. The sum collected is 52; dividing this by 7, we have 3 as the remainder: so that the proposed day, the 26th of October, will be the third day of the week (Tuesday).

In the same way, when the beginning of any month is sought, we take its Kalends, with the epacts of the same, and the cycles of the sun, and the quadrants of these: adding all of these together, we divide by 7, so that we may have what is sought. And since, so that the method may be more expeditious, we are about to present a table, we shall explain ^C what is relevant regarding it.

But this must necessarily be held in mind: as often as a cycle occurs that has a bissextile (leap) year, such as 4, or 8, and the rest, we do not take the quadrants of the cycles from the beginning of that cycle's year, but from the 29th of February, which then consists of 29 days: since for the sake of this very thing we take the quadrants of the solar cycles for the method of computing the day of the week.

TABLE FOR FINDING THE DAY OF THE WEEK

For, taking the solar cycle that is current from the first part of the table—which contains all of its cycles—and then in the uppermost and first row which displays the months, searching for the month whose Kalends are being sought, whatever day of the week is placed in the middle of the table at the current cycle, and lies directly beneath the month already found, that we shall say are the Kalends of that month about which we are inquiring. Having found this, we shall obtain any of the following days of that same month. For, counting all the days from the 1st up to the number of the day corresponding to the Kalends, excluding that day itself, and adding to these the days from the Kalends up to the day we are seeking, and dividing the total sum of all of these by 7, we shall have the result sought. ^B

Example

Let it be that by way of an example we seek the Kalends...

These things, then, concerning the solar cycles and what necessarily follows from them to be considered, this would be as far as it was possible to give a complete account of them. We must now move on to that regarding the moon, for the inquiry here will be conducted in a way and method similar to those things regarding the sun.

On investigating the lunar cycles, and what is consistent with them.

V. Since the moon was brought into existence by God, the creator of all things, on the fourth day of the creation of the world in a full and perfect form (for the Book of Genesis hints at this, saying: “And God made the two great luminaries, the greater luminary for the beginning of the day, and the lesser luminary for the beginnings of the night”); for since the sun rises, the moon could never on the same day rise at the beginning of the night, or rise at nightfall, unless it were positioned diametrically opposite to it, and thus be full; if we imagine it to have arrived at the full moon from a previous conjunction, then since on the fourth day of the world’s creation it was the fifteenth day of the moon, it is manifest that at the very beginning of the world it was [the twelfth day]. ^C Therefore, its epacts were eleven. These eleven days, indeed, through the admirable and altogether immense ingenuity of divine providence, are what is lacking each year after the twelve lunar months are completed, which collect 354 days and a little more than a third part of a day, toward making equal the solar year, which consists of 365 days and almost a quarter. Hence, in two years, these eleven days doubled make twenty-two epacts; in three years, thirty-three. From this sum, 30 having been subtracted, which constitutes approximately one lunar month, three remain. Whence the third year has three epacts; and so on in succession up to nineteen: ^D for in the nineteenth year the revolution of lunar months is completed, so that in the twentieth year the moon again has eleven epacts.

^A This revolution occurs when nineteen years and eleven are multiplied together. For these multiplied together produce 209 days, which are approximately seven lunar months. This is the reason why we say the cycles of the moon are neither more nor fewer than nineteen. However, it must be known that this is stated for the sake of ease and brevity, as are the other things which we shall explain regarding the moon in due course. For if we wish to speak accurately, the matter stands otherwise, as will be demonstrated later when we speak of Easter.

*Table of the propagation of lunar epacts.*

| Epacts of years | Years of cycle | Collected epacts | | :--- | :--- | :--- | | XI | I | 11 | | XXII | II | 22 | | III | III | 33 | | XIV | IV | 44 | | XXV | V | 55 | | VI | VI | 66 | | XVII | VII | 77 | | XXVIII | VIII | 88 | | IX | IX | 99 | | XX | X | 110 | | I | XI | 121 | | XII | XII | 132 | | XXIII | XIII | 143 | | IV | XIV | 154 | | XV | XV | 165 | | XXVI | XVI | 176 | | VII | XVII | 187 | | XVIII | XVIII | 198 | | XXIX | XIX | 209 |

Method of finding the lunar cycle

^B VI. The method of finding the current cycle of the moon, with its epacts, is of this kind: Taking the years from the creation of the world until the current year itself, and dividing them by nineteen, we say that the remainder which is less than nineteen is the cycle of the moon. Its commencement is taken not from September, nor from October, as in the solar [cycle], but from the beginning of January, for a reason which we shall state.

Let the proposed year be used as an example. From the creation of the world to the beginning of the January which next follows, there are 6880 years. These, divided by nineteen, leave two; therefore, we pronounce the cycle of the moon to be two. ^C

Another, more expeditious method

Furthermore, we shall present this method, which is slightly easier: since in the coming year 6800 the cycle of the moon would be seventeen, by adding this same number to the years of the following century—as now [we add] 80—and dividing the sum total by 19, we shall declare the remainder to be the cycle of the moon. Doing the same at the completion of any century, we shall find its cycle with greater speed.

How lunar epacts are to be investigated

VII. The epacts of the current lunar cycle [are found] in this way.

We shall take the cycle of the moon, multiply it by eleven, and add to it the three days of which we shall speak shortly; then, from the sum total, we shall subtract all the thirties that may occur, as each of these completes a lunar month. The remainder, which is less than thirty, we shall declare to be the epacts of that lunar cycle. Example: As in the demonstration, since it is shown that the current cycle of the moon is two, multiplying this by eleven and adding the three days to the twenty-two that arise, we have twenty-five as the epacts of the current lunar cycle.

The addition of the three days occurs for this reason: Such an addition seems to cause difficulty. The cause here, too, is ignorance of the true beginning from which the year starts. For not knowing this precisely, since those who first devised such methods found the moon in its first cycle to be full—that is, fifteen days old—on the first of January; and since it was also well-established that it was fifteen days old when it was ^B created, and for this reason there were three days before its creation, while eleven had elapsed since the previously assumed conjunction, they assumed the beginning of January and the beginning of the lunar cycles, but added the three days, so that in all things there might be conformity with that day on which it was created.

By what method might one discern by simple observation whether the moon is approaching conjunction or is full?

VIII. Since the moon receives its light from the sun, its illuminated part always faces the sun. Therefore, as it grows by receding from the sun—that is, after its conjunction with the sun—and moves toward the full moon, the part that receives light verges toward the west, the sun being then more to the west, while its horned and unilluminated part faces the east. But when it becomes positioned directly opposite the sun, and stands apart from it by the interval of a diameter, then it shines entirely, the whole moon looking toward the sun. When I say "the whole," I understand the part of it that faces us. But when after the full moon it begins to wane ^C and to approach conjunction with the sun, then again its illuminated part faces the east, the sun being then more to the east, while the unilluminated and horned part faces the west. Therefore, when the illuminated part of the moon faces the west, the moon is from conjunction and is moving toward the full moon; but when its illuminated part again faces the east, the moon is from the full moon and is moving toward conjunction.

How we might find the age of the moon, that is, its interval from conjunction, on any day of the month we might seek this.

IX. Since we have also received the argument regarding this, it would follow that we should ^D add how it is necessary to take the age of the moon on any given day of the proposed month.

^A For if we have the epacts of the current lunar cycle and add them to the day of the month on which we seek to find the age of the moon; and counting from March and the following months—excepting the month itself—if the month has 31 days, we take one and a half days, but if it has 30, half a day: and then, if the sum exceeds thirty, we subtract thirty, the remainder will show the age of the moon on the day we are inquiring about.

For example: let it be the 26th of the current month of October. To this we add the 25 epacts of the current solar cycle that we have already discovered, and we add the 7 days that have accumulated from March up to the end of September. From the sum, which is 58, subtracting thirty, we have the moon on the ^B proposed day, that is, the 26th of the current October, being 28 days old.

Nor should we easily pass over the reason why we begin to take those one-and-a-half or half days from the months starting from March, and not from January, since from the latter the beginnings of the lunar cycles and their corresponding epacts are derived. (For this seems to cause doubt concerning what has been said.) Then, one may also ask why we add those days. It is manifest, therefore, that since a lunar month consists of approximately 29 1/2 days, it is less by a day and a half than those months to which 31 days belong, and by half a day less than those which are thirty days long; ^C and for this reason we add this remaining one and a half, or half day, to the days of the following month, up to the day proposed. And we do the same thing in February, adding the remainder from January to the days of February. But because February, having 28 days, is even more deficient of the lunar month by a day and a half, for this reason the remainder from January—the one and one-half—compensates for the deficiency in February; and thus we add nothing to the days of March, being content with these alongside the addition of the epacts to find what is sought. After March, and thereafter, we do as we stated above. It must therefore be considered that these surplus days gathered from each month from March to the end of December, when added to the 11, make ^D the same epacts as those derived from January. Which, when added to the epacts of the preceding lunar cycle, constitute the epacts of the coming year.

On Easter

X. Having noted these things, which pertain to the calculations of the sun and moon, and having explained them as best we could, we shall next begin a discussion on Easter. For these are its elements, since the proposed table, by whose aid Easter is readily found, is constructed from the cycles of the sun and moon. Both methods—both that which existed before us and that which has been devised by us—rely upon these same cycles.

Table for the discovery of the venerable and great Pascha, composed by our holy and God-bearing Father John of Damascus.

^A The aforementioned table is constructed in the following manner. In the first column are placed the nineteen cycles of the moon, and then, in another column, the Jewish Pascha, which indicates the day of the full moon for each lunar cycle; for it is on this day that the Jews celebrate their own Pascha, as the Mosaic law among them commands it to be celebrated. The Jewish Pascha is placed before the Christian Pascha because it was so decreed by the holy Fathers at the first holy and ecumenical council: that one should observe on which of the days of the week the Pascha of the Jews happens to fall, and that Christians should celebrate their own holy Pascha on the Sunday following, following the timing of the saving Passion and the life-giving Resurrection. For our Lord Jesus Christ was led to the Passion on the very day of the Jewish Pascha; after which He immediately rose from the dead; and we Christians, celebrating that Resurrection, call it the new Pascha. ^B After the second column, which contains the Jewish Pascha, as has been said, the holy Pascha of the Christians is set out in seven columns, occurring on different days of the week for the Jewish Pascha, yet on the same day of the month which is set alongside the lunar cycle; conversely, the Christian Pascha is celebrated on the same day of the week, that is, Sunday, but on different days of the month. Above these seven columns, in four rows and six spaces for each, the twenty-eight cycles of the sun are set out.

^A But if any person should perhaps inquire why we have not described the numbers of the cycles in order, we shall answer this question by showing that the proposed table of the correct order of numbers differs in no way from the other, which contains those same numbers in a perturbed state. For let them be described in this place in a double table, first in a confused order, as in the upper chart; then in the correct order: and let them be of this kind.

^B It is necessary, therefore, to observe how, in the second table, which has the numbers of the cycles arranged [in order], every fourth little space is left void of numbers: which fact indicates that, after four years, the year exceeds the previous order by one day; moreover, the position of the numbers in any row is the same in both, and one can see in the first rows the same numbers of the cycles arranged. For even if in the former table XVIII is placed above XII; yet since both are contained in the same row, no difference arises from that perturbed arrangement. The same appears in the remaining rows, so that, although it seems to be tumultuous, it does not, however, exceed the order. But such an arrangement of the cycles was devised by him who first constructed the chart for the purpose of filling up those intervals which, in the second description, are left empty for the reason we have stated.

Wherefore, that a more orderly description may be made by us in the four rows, let these be placed in the first, I, VII, XII, XVIII; in the second, let the same remain; let the third have these, III, VIII, XIV, XXV; the fourth, IX, XV, XX, XXVI; the fifth, IV, X, XXI, XXVII; the sixth, V, XI, XVI, XXII; in the seventh, let nothing be changed.

^C At the top of the chart, the cycles of the sun, as has been said, are placed for the purpose of showing upon what day of the week the Jewish Pascha falls in its own respective cycle of the moon. For not only is the Pascha of the Christians described in seven rows through every cycle of the moon; but along with the number of that month, say March or April, which is the Paschal index, the day of the week is also inscribed, on which day the Jewish Pascha is celebrated.

^D

Method of finding the Pascha.

XI. Therefore, when we wish to know into which of the two months, March or April, the holy Pascha of the Christians falls, we take the present cycle at hand

^A of the sun, lying at the top of the table, and we proceed down the column beneath it until we reach the row where the cycle of the moon is written at the beginning. The entry lying directly opposite—in depth to the current solar cycle, and in breadth to the lunar—we shall declare to be the holy Pascha of the Christians; and the day written beneath it, the day of the week on which the Jewish Pascha occurred at that time.

For example, let it be that we seek in the present year, which is the 6881st year from the creation of the world, on what day of March or April the Pascha of the Christians will be. And since it has been shown that in this year the solar cycle is the 21st, and the lunar the 3rd, we take them in the table, the solar from the top and the lunar from the side; and because, at the intersection in the middle of the table, the 17th of April is inscribed, we shall say that our Pascha will take place on the 17th of April. And the day ^B under this number, which is the 1st—that is to say, the first—we shall say is the day of the week, the first, before this Christian Pascha, on which the Jewish Pascha will be. For it is clear that since the 10th of April, which is placed beside the third lunar cycle, indicates the Jewish Pascha, and since this will occur on the first day of the week, if we add seven days to the 10th of April to reach the coming Sunday, we will arrive at the 17th of April, on which day the Pascha will be Christian. And thus, the thing sought is easily obtained from the table; and it is also obtained by the following method.

Another method of finding the Pascha

^C Since we have for each of the lunar cycles all the full moons that occur in March and up to the 17th of April, which indicate the Jewish Pascha, by means of the day-finder, as we have mentioned previously, we find on which day of the week the legal Pascha, which is placed by the current lunar cycle, will fall. By adding to this the remaining days of the week until the Sunday itself, we obtain the holy Pascha held by us, and on which of the days of March or April it occurs. Let the aforementioned example serve as our illustration. Since the 10th of April is placed ^D beside the current third cycle of the moon, indicating that the Jewish Pascha falls on that day, if we add to this the epact of that same month, which is 1, and also the 20 past solar cycles, and their 5 days from the quarter-days, the sum of 36 days, if divided by 7, leaves 1, which we shall say is the day of the Lord, upon which the 10th of April falls. And by adding to this the 7 days of the following week until the next Sunday, we will have, in this way as well, our holy Pascha occurring on the 17th of April.

On finding the beginning of the fast

XII. After obtaining the day of the month on which the holy Pascha falls, we add to it three days; or, if it is a bissextile year, four. In this way, we find the day of January or February

^A in which the Apokreo [Carnival] will occur. For if Pascha is in April, the Apokreo will be in February; but if it is in March, up to the 28th in a bissextile year, or the 29th in a non-bissextile year, the Apokreo will be in January. For if Pascha falls on the 28th of March, or the 29th, or the 30th, or the 31st, the Apokreo will be in February; as will be evident from the example of the coming Pascha, God willing. For since, as was said above, it will be on the 17th of April, by adding to it three (for the current year is not a bissextile one), we shall have as the result the 20th of February for the Apokreo.

Another method concerning the Apokreo and Pascha.

XIII. ^B And this is the method that has reached us, set forth by those before us; but the one that we have discovered, possessing logical proof, and which we promised to explain here, is as follows. Since from the lunar conjunction that precedes January to the Jewish Pascha of that same year, in which the full moon conjunction occurs, there are three and a half lunar months, which comprise about 103 days; and in the Paschal calendar these Jewish Paschas appear to be two days in excess of the exact full moon, which we will demonstrate later through proof; if we add these two days to the aforementioned 103, and from the sum of 105 days we subtract the 48 days of the Quadragesima, and also the 7 days of the Tyrophagus ^C (that is, the week in which the eating of cheese is permitted), if we set out the remaining 50 days from the conjunction before January, it is evident that we will arrive at the week of the Apokreo from the Sunday of the Prodigal Son up to the Saturday before the Apokreo. By adding to these the subsequent days up to the Sunday, we will have the Apokreo; and consequently the Pascha will be manifest.

Let us also take the example of the coming Apokreo: Since at the beginning of the coming January, the third year of the lunar cycle beginning, the epacts are six, it is evident that the conjunction before it will be six days ^D before the beginning of January. Therefore, by setting out from such a conjunction the aforementioned 50 days, which comprise the 6 days before January, and the 31 days of January itself, and the 13 days of February (for these, when added together, make 50), we will arrive at the 50th day ending on the 13th of February; and we shall find the 13th of February falling on a Sunday, namely that of the Prodigal Son. To the 13 days of February, which as stated fall on a Sunday, by adding the 7 days of the Tyrophagus week, we will get the 20th of February, which will be the Sunday of the Apokreo; and consequently the Pascha will be obtained.

Another method concerning Pascha.

XIV. Since from the lunar conjunction that precedes January to the Jewish Pascha, or full moon, of the same year, three and a half lunar months intervene, which are about 104 days; furthermore in

^A the Jewish Paschal celebrations in the Paschal canon exceed the accurate full moon by two days, as we shall demonstrate by showing the reason for this later on. If we add these calculated days—105, or 106 if a leap year occurs,—except for the 16th, 18th, and 19th cycles, where we add only 105 days—to the lunar conjunction that occurred before January, assigning to each month its proper days, we will have the final day ending at the full moon conjunction of the Jewish Pascha. Having found, by means of an almanac, what day of the week this is, and by adding the remainder until Sunday, we shall also have our own holy Pascha, and on which of the days of March or April it will fall. ^B Indeed, it is necessary to know this also: that in the 5th and 16th cycles of the moon, the aforementioned number of 106 or 107 days is not arrived at by adding the epacts together with the rest of the following months; but in the 5th cycle, 136 days are collected, and in the 16th cycle, because of the leap year, 137 days are collected. From these, by subtracting thirty, the remaining number is brought to the same amount as in the other lunar cycles.

16. That the things said might become clear by an example, since in the beginning of the current year January is at the start of the third cycle of the moon, and the epacts are 6: it is evident that the lunar conjunction preceded the Kalends of January by six days. From that conjunction, counting 105 days and unfolding them—which indeed consist of the 6 days before January, and the 31 of January, and the 28 of February, and the 31 of March, and 10 of April—we discover that the last of these days falls on the 10th of April, which is also the day of the full moon and the Jewish Pascha. Having found by an almanac that this day is a Friday, we will have our own holy Pascha on the following Sunday, namely, the 12th of April. ^C

On the fast of the holy Apostles. 15. In addition to these points, we shall explain the method for the fast which is observed before the commemoration of the holy Apostles, resting it more firmly upon reason and demonstration. For since from that same holy and glorious Sunday of Pascha until the very Sunday of the holy Apostles, on which we also abstain from meats because of the aforementioned fast of the holy Apostles, there are as many days (there are 57), so too are the days from the 3rd of May until the 28th of June, on which day such a fast is completed; if we remove the days that are common from the 3rd of May until the Sunday of All Saints, we shall have a remainder and an equal sum of the days, namely, those from that glorious Sunday until the 3rd of May, and those from the beginning of such a fast ^D until the 29th of June.

From glorious Sunday to the 3rd of May; From the 3rd of May to the Sunday of All Saints; From the Sunday of All Saints to the 28th of June.

^A Let, then, line AB represent those 57 days that lie between the Paschal Sunday and the Sunday of the holy Apostles and of all Saints, and in place of the other 57, from the 3rd of May to the 28th of June, let there be line CD. Since, therefore, the interval from the 3rd of May to the Sunday of All Saints, which is BC, is common: if it be taken away from the extremes, it follows that the line AB, that is, the interval from the Paschal Sunday to the 3rd of May, is equal to the line CD, that is, the interval from the Sunday of All Saints to the 28th of June, on which the fast terminates; so that, once the day of the month on which Easter falls has been found, if we count how many days there are from that day to the 3rd of May, we shall have the same number for the total of the days of the fast of the holy Apostles. ^B

On the Correction of Pascha, or, on the Defect of the Table. XVI.

full moons showed the time of opposition most exactly, we could more accurately calculate how many years ago such a canon was composed. ^A In the meantime, however, so that we may know after how many years from the present two whole days, which depart from the check, will be completed, we have entered into a calculation of the Paschal full moon of the Jews assigned to the present cycle, which has the tenth day of April inscribed. We find, however, because of the inaccurate ratio of the lunar conversion, as has been said, which the canon employs, that it departs to the eighth of April, three equinoctial hours after the rising of the sun, which occurs on the same day. Since, indeed, three equinoctial hours are the eighth part of the time of night and day, if we take the eighth part of the three hundred and four Roman years—in which the exact conversion, which occurs in nineteen years, anticipates the inexact one by a whole night and day—that is, thirty-eight Roman years: we shall have [the number of] years from the present after which there will be two whole days deficient from the examination. After that conversion, again by other scruples of the day, an exact restitution will begin to anticipate the inexact. After thirty-eight years from the present, it will be the year 6919. ^B [Ac si, ut vulgaris est opinio, septimo millesimo vertente totius mundi futura consummatio est; frustra in castiganda lunari methodo labor adhibebitur; cum anni tantummodo de summa reliqui sint LXXXI, per quos emendata illa methodus in usu erit: cum in istis LXXXI annis non multum intersit, si ii, qui tum superstites erunt, veterem nec emendatam assumpserint. Sin, ut plerisque visum est aliis, quibus et assentior, ultimi occasus mundi tempus incertum est: congruum fuerit lunæ castigare methodum, et ad epactas addere, quas vulgata ratio præscribit, et explorat, ac sic ab recto non deflectere. Nam si laterculum Paschale sine emendatione maneat, majus illius vitium fiet. Cum enim biduæ duntaxat a veritate abhorreat hoc tempore, expleto anno 6919, tridui error incidet.]

With these things thus observed, let us consider whether the day on which we celebrate the holy Pascha also errs in individual years, or whether it errs at one time and not at another. It will be manifest, indeed, that if our Pascha were determined only to one day of the month, and there were no necessity to celebrate it on Sunday, it would err every year, just as the Jewish Pascha does, and [would err] by those same two days, just as that one does. But because it is celebrated on Sunday, it happens that it would not err. Therefore, we must attend to the exactness. ^C Since the Jewish Pascha is subject only to the day of the month, when it occurs on a Sunday, or the second, or the third, or the fourth, or the fifth [day of the week], our Pascha does not err, being celebrated on the following Sunday; but when it occurs on Friday or Saturday, then it departs greatly from correctness. For when it falls on Friday, although we are obligated to celebrate Pascha on the immediately following Sunday, [and] we move to the Sunday after that; since the Jewish Pascha, which is understood in the canon, is shown to be more... ^D

^A exactly by two days, as we have demonstrated, than the Pascha customarily celebrated by the Jews. The same thing also occurs on the Sabbath for the same reason. For when we ought to celebrate Pascha on the Sunday which follows immediately after, we transpose it to the Sunday that comes after that. This error arises from the cause we have mentioned, namely, the lack of exact lunar restoration. There is also another case (although this happens rarely) due to the shifting of the vernal equinox. It, too, moves backward over the course of years for a reason we shall explain, which is as follows: it is commonly understood that the sun returns from the same point to the same point in an interval of three hundred and sixty-five days and a quarter. According to a more subtle calculation, however, it is less by a certain portion of a day—as Ptolemy states in his *Composition*, by one three-hundredth; but as the Persian mathematicians, ^B and we ourselves, observing the solstice over many days, have estimated, that portion is larger than one two-hundredth; or rather, we find this portion to be slightly less than one-hundredth, which must be subtracted from the three hundred and sixty-five days and a quarter. For Ptolemy says that in his own age the vernal equinox occurred on the twenty-first of March. We, however, find it to occur before the fifteenth of the same month in our own times, calculating it in this way from observations of the summer solstice. And since the Mosaic law instructs the Jews to celebrate Pascha immediately at the first full moon after the vernal equinox, during the time when the Paschal canon was constructed—when, specifically, the vernal equinox was completed on the twenty-first or twentieth of March—the days of the ^C full moons noted in the canon, which indicate the Jewish Pascha, are certainly the first full moons after the equinox. For you would not find the canon passing over the full moon of the twenty-first of March (which was the first after the equinox at that time) and choosing another one after it in its place. But now, with the vernal equinox having regressed by six or more days, other *first* full moons could occur before the twenty-first of March; and while the Jews celebrate Pascha according to these, we, following the full moons of the canon, fall short by an entire month from the proper time. For fifty years ago, when I was still young in age and living in a certain city of Thrace called Enos, I saw the Jews who had their residence there celebrate their Pascha on the twentieth of March, ^D while we celebrated our holy Pascha on the twenty-third of April, following the day of the paschal full moon from the canon, which is marked as the eighteenth of April. At that time, I was perplexed by the matter, not yet having grasped mathematical reasoning; but later, having learned the causes of such things through the science of astronomy, I perceived that this had happened for a logical reason. From these things, therefore, it is clear that as often as a full moon happens near the accurately determined vernal equinox—that is, from the fifteenth of March to the twentieth of the same—we deviate from the rule, because [the canon] transitions to the full moons of April.

- gressum facinus. ^A And indeed, before us, the most learned Gregoras wrote on this, having edited a book in which you could desire no degree of erudition, and having transformed the canon and restored it to the state that befits this age. He, having been commended by everyone—in the presence of the people, the emperor, and his own senate, as well as the most select men of the Church—was such that they, confuted by the very truth of his words, decreed that universal Pascha, if it were possible, ^B should be celebrated according to the emendation of the canon.

But we, for our part, have not undertaken this discourse with the thought that the solemn feast of Pascha ought to be changed; nor have we poured forth these words in order to incriminate those who originally established this canon, as if they had published its principle in an ignorant and faulty manner. Far be such words from us! And let not captious men brandish their slanderous tongues against us. For this alone have we striven to demonstrate: that in whatever way a canon is published, it cannot be avoided that it becomes faulty with time, causing the distinction of the nineteen-year cycle—which in a short time seemed small and imperceptible—no longer to be neglected in the process of many years, as it creates an error of significant account over many years.

The Paschal Table corrected by the philosopher Nicephorus Gregoras, concerning which Argyrus discoursed in the aforementioned method.

[index: entries preserved]

ANOTHER METHOD

^A Since you, most excellent of monks and most beloved brother to me in Christ, have wished to learn from me a certain method by which the cycle of the Indiction, as well as those of the sun and moon, may be easily investigated; and furthermore the foundation of this latter; and how many days it happens to be increasing and decreasing, and how many hours it shines on any given evening; and in addition to these, the Apocreos, the legal Pascha, and the holy Pascha of the Christians, together with the days of the fast of the holy Apostles: behold, fulfilling your request, I will explain to you point by point how each of these may be found by you without error and with ease. And since the beginning of the cycle of the Indiction precedes the cycles of the sun and moon, from it I shall also take my beginning.

On finding the Indiction.

^B I. The Indiction, which is also called the Fusion and Epinemesis, has its own cycle beginning from the Kalends of September, which is the beginning of the year. Furthermore, the same cycle ascends to fifteen years, and thenceforth takes a new beginning. If, therefore, you wish to find this, take the years from the creation of the world, which at this time are numbered as 6885, from which subtract, as much as you are able, the

^A number of the Indiction cycle, that is to say, 15; and whatever remains below, or even if it is 15 itself, is the current cycle of it.

*Example.* For example: Since there are, as has been shown, 6885 years from the creation of the world, we say that 15 times 450 make 6750, and 15 times 8 make 120; together they make 6870. There remain until the 6885 years 15 years, which is the current cycle of it. But since this method seems difficult for those who do not know how to calculate skillfully, another one has been devised, more clear and easier for those using it, which is as follows:

Take the small years; and by small we mean those below the thousands and hundreds, that is, those consisting of units and tens, or both, which today are 85. To these ^B add 5, and they become 90. Subtract from these 15, as often as you can, saying: 15 times 5 make 75; there remain until 90, 15; which is, as has been shown above, the current cycle of the Indiction; which you will also find reliably by using the methods mentioned, whichever you prefer.

*Concerning the finding of the solar cycle.*

II. The solar cycle begins on the 1st of the month of October, and advances up to 28 ^C years, whereupon it again takes its beginning. If, then, you wish to find this as well, take the years from the creation of the world, and divide them by the number of its cycles, that is, 28; the remainder below 28, or up to it, is the solar cycle.

*Example.* But that what I say may be clearer here, I take the mentioned years, 6885, and I divide them by 28 in this way: 28 times 200 make 5600; 28 times 40 make 1120; and 28 times 5 make 140. The sum is 6860. There remain until the completion 25 years, which is the current year's solar cycle, being the twenty-fifth.

Since, however, not only to those just touching upon the arts, ^D but also to many who are more perfect in them, more clear things are preferable because they are more easily grasped, we shall demonstrate here the clearer path, which is as follows: To the small years, which are now 85, we add 24, and they become 109; these we divide by 28, saying: 28 times 3 make 84; there remain until 109, 25; so that the current cycle is 25, as has been shown before.

*How one must find the day of each month, in which of the 7 days of the week it falls.*

III. Let this also be known to you, that whatever day of the twelve months of the whole year someone may wish

to know exactly in which of the 7 days of the week it falls, he shall use the cycle of the sun and the leap years (bisextiles) that follow it, and furthermore the epacts of the months to find this. Therefore, you must know that the leap years ^A in the cycle of the sun are 7; for four times 7 make 28. For in the fourth year one day is gathered; but the epacts of the months, if they consist of 31 days, are 3; but if they have only 30, they are 2. They begin, at any rate, from the month of October, from which the cycle of the sun also begins. And they are: for October 3, for November 2, for December 3, for January 3, for March 3, for April 2, for May 3, for June 2, for July 3, and for August 3, and for September 2. If, therefore, you wish to find the day of any month (let us set for the sake of example the 29th of the month of March in the present year, on which also our holy Easter of the Christians will fall), it will be found in this way:

Take for me the current cycle of the sun, which is 25, and its leap years, which are 6; for four times 6 make 24. To these, which have reached 31, add the epacts of the months, namely the days of October, November, December, and January, which are 11; lo, there are 42. Add to these the 29 days of March, since you are seeking to learn on which day of the week the 29th of that month falls; and lo, they have reached 71. Divide these by the 7 days of the week, saying: 10 times 7 is 70; there remains 1, which is the first day of the week, or the Lord's Day [Sunday], on which the 29th of March, which we are seeking, falls. But if it should happen in another such inquiry and summation that, after the division by 7, two, or three, or four days, and so on up to 7, remain, then clearly the other days ^C of the week in order will be the result. By following the same method for the months before March, and for all those after it, you will certainly and without error find the day of any month of the whole year that you wish. However, February does not contribute anything at all to the said epacts of the months. For it is diminished, and consists of only twenty-eight days. And even if in a leap year it has twenty-nine, with one day added—this is indeed the day that after four years is added to the cycle of the sun in the leap year. Therefore, it should not be counted twice. Furthermore, in calculating the day of the week, it does not proceed into the sum before February has ended—that is, from the beginning of March and thereafter.

On the method of investigating the lunar cycle.

IV. The cycle of the moon begins from the Calends of January, and after being carried through 19 years, it takes a new beginning. If, therefore, you wish to find this also, divide the sum of the years from the creation of the world by the number of the lunar cycle, that is, 19; the remainder, which is below 19, or 19 itself, is the current cycle of the moon. Let this cycle of the moon be sought for the present year; we shall reason in this way: 19 multiplied by 300 makes 5700.

1140; 19 multiplied by 2 makes 38. The years total 6878. There are wanting to complete 85, 7 years, which is the lunar cycle of the present year. It is permitted to test this by another method with greater brevity: to the smaller number of years, that is 85, add 17, and they become 102. If you distribute these by 19, there will remain 7, just as in the prior method.

^B Someone surveying these methods with a bit more care might doubt why, in this latter method, when we are searching for the cycle of the indiction, we add 5; to the cycle of the sun, 24; and to that of the moon, 27; and whether these additions must always be made. We reply that they are not always reliable, but only for the year just completed; after that, a different number will have to be added—as, for example, 15 or 0 to the cycle of the indiction, 12 to the solar, and 3 to the lunar—by the aid of which all these cycles will emerge without error. But for the sake of a more abundant learning, I shall explain to you the reason for this matter, which perhaps many do not understand. At the turn of any centenary, whatever remains in the hundredth year from the calculation of the sun, moon, or indiction, always add this number to the smaller sum exceeding the incoming centenary, so that by an appropriate addition of each kind you may have the cycle, as we showed above: that is, the solar by adding the solar proper, the lunar by adding the lunar, and the indictional from that which is proper to the indiction.

How the root of the moon is to be investigated

How the age of the moon may be found

Take ^A the established foundation of this, and beginning from January, take from each month: from those that have thirty-one days, 1 1/2, and from those that have thirty, 1/2. As for the month in which you make the inquiry concerning the days of the moon, you shall introduce those specific days that it possesses, beginning until the sought-after day. Having added all these to the foundation, divide the resulting number by 29 1/2, and the remainder below this, or up to it, is the number of days from its birth. Therefore, beginning, as has been said, from January: if the year is bissextile, introduce the 1 1/2 days taken from it for the whole year, that is, until you reach the beginning of January in the following year; but if it is not bissextile, take this [amount] only until February is completed, and after it is completed, omit both. For the deficiency of one is compensated by the redundancy of the other. From then on, those two months remain free from that contribution for three full years, that is, until the next bissextile. Therefore, passing over these, as has been said, begin from March, and as we have prescribed, take those days up to the one that is proposed. ^C And so that the whole matter may become clearer by an example, let it be required for us to seek in the current year 6885 how many days old the moon is on the 20th of May. Holding, therefore, its foundation, which is 20, we pass over January and February, as they have nothing to contribute. Beginning from March, we take 1 1/2 from it, and 1/2 from April, which days, added to the 20 [of the] foundation, together with the 20 days of May, make 42. Having subtracted 29 1/2, 12 1/2 days are left. Therefore, we say the age of the moon on the said 20th of May is 12 1/2 days. How to find how many hours the moon shines on each evening. VII. Since you perhaps desire to know how many hours the moon shines each evening while increasing and decreasing, we shall demonstrate to you the account ^D concerning this, without sloth or delay. The moon, after it is born, increases until the 15th day; within which [time] both its increase and diminution occur. From this [day] it begins to decrease until the 30th, on which day, its light being completely gone, it begins to be born new again. Therefore, in the said 15 days of its increase, it shines for 4 scrupula on the first evening, for 8 on the second, and for 12 on the third; and successively in the same way until the 15th, such that each night adds 4 scrupula, which scrupula are reckoned as 5 to any given hour. Thus, when it reaches the 15th evening, which is also called the full moon because it is entirely illuminated by it, it shines for 60 scrupula on that night, which are 12 hours. For four times 15 is 60, and five times 12 is likewise 60. If, therefore, you wish to know how many hours the moon shines each evening while increasing,

find how many days old it is, and giving to these four ^A scrupula each, divide the sum obtained by 5, having converted it into hours; and you will have what you seek. Likewise, after the full moon, that is, after its 15th day, if you wish to find the hours of its shining, subtract from the 14 hours of the entire night each evening four scrupula for each elapsed day, and the remaining hours, or minutes with them, will be its shining. Or else, in another way: find how many days there are from its birth until the day after the full moon on which you are seeking the hours of its illumination. Subtract these days from 30, keep the remainder up to the full 30, and giving to these four scrupula each, divide all the minutes resulting from these by 5; and you will know precisely how many hours and minutes it is going to shine.

But since such a method uses in all the nights of the year the number of 15 hours—while for us the nights at the vernal and autumnal equinox, that is, in September and March, only have 12 hours each, at which time the aforementioned method is safe, as it equalizes the hours—after September, as winter arrives, the night exceeding 14 hours gradually advances up to 15. Likewise, also after March, progressing toward summer, it decreases to 9 hours. From this it appears that the hours obtained by the mentioned calculation, while they are similar and equal in both equinoxes, are in other parts of the year sometimes greater and sometimes smaller, which are also called "temporal" [καιρικαί]. Our goal, however, is, through the equal hours of each month as it occurs, with the night increasing and decreasing, to find the hours of its illumination each evening. A method ^B will be shown, by which this can also be safely accomplished, by converting the mentioned unequal temporal hours into those which are equal for us, called "equinoctial" hours. It is as follows.

Find first how many days old the moon is, and how many temporal hours it is going to shine, as you heard above. Multiply these, therefore, by the hours which the night of the current month has for us, the month, that is, in which the inquiry is made. And divide the number gathered from this always by 14, that is, the number of temporal hours; and thus you will have the conversion of such hours. For instance, for the sake of a test: let the moon be five days old in January, when the night has 14 hours for us. We wish to know for how many hours it is going to shine, and we find according to the recorded method 20 minutes, that is, 4 temporal hours. Wishing to convert these into equinoctial hours, we multiply these same 4 hours by the 14 hours which the night has, as has been said, in the month of January; we say: four times 14 is 56. These 56 we divide by 14, ^{C D}

and there belong to them 4 and 2/3, which are the converted hours, having become 4 and 2/3. And so much for these matters.

How the Legal Pascha is to be found.

VIII. You will find the Carnisprivium (forerunner to Lent), the legal Pascha, the holy Pascha of the Christians, and the days of the fast of the holy apostles in this manner: Multiply the current lunar epact by eleven; to the sum add 6 days—namely, the three that arise from the previously mentioned sesquiscruple and third part, which, as we have demonstrated, represent the lunar monthly cycle beyond the 29 days; and three other [days] to fulfill the number of fifty. From the 17th [year] and onwards, instead of three, add four, because a greater quantity of them occurs in these [years]. You will add these days from the first cycle up to the entire 16th. In the three remaining [years], instead of three, add seven. ^B From the sum subtract thirty, as often as you can. To the remainder, which is less than thirty, beginning with March, add as many days as shall be necessary until you complete fifty. If March is not sufficient, provide the remaining days from April until you make up the sum of fifty days, which I mentioned. Wherever that [sum] ends—suppose it be this or that day of March or April—know that the Jewish, that is, the legal Pascha, falls on that very day. And, having found on what day of the week it falls by the method of day-finding described above, know that the holy Pascha of the Christians is celebrated on the next Sunday of that same week. This having become known to you, if you add three each year to the date of the month in which you find the Pascha, you will have the Carnisprivium. If it is a bissextile year, instead of three, add four. Furthermore, if the holy Pascha of the Christians falls before the 28th day of March, January will hold the days of the Carnisprivium. ^C But if it should fall after the 28th day, it occurs in February. If the Pascha should fall on the 28th day itself: if the year is not bissextile, the Carnisprivium will be on the 31st day of January; for twenty-eight plus three make thirty-one. But if it is bissextile, it will fall on the Kalends of February; for twenty-eight plus four make thirty-two: of which, after leaving thirty-one to January, one remains, which is the first day of February. You will do the same for the 29th, 30th, or 31st of March, as you add three in the remaining years; but four only in the bissextile. Then, having cast out—as has been said—the 31st of January, assign that which remains to February; the last of these days is the Carnisprivium. ^D The number of days of the fast of the holy apostles will be made clear in this way.

How the days of the fast of the holy apostles are to be found.

IX. From that very holy day of Pascha, excluding that same day, compute how many days there are up to the 3rd of May; and having collected these, whether in March, or in April,