Ecclesiastical Computus of Saint Maximus

Pliny the Roman. Strabo. Æschines. Aristophanes. Plutarch. Thucydides. Pollux. Simplicius. Herodotus. Suidas. Synesius. Divus Cæsar.

SAINT MAXIMUS THE MONK AND MARTYR, BRIEF ENARRATION ON THE CHRISTIAN PASCHA, Explaining the method of the described register, Translated by Dionysius Petavius

^B170 To the illustrious Patrician, worthy to be celebrated by every kind of praise, Lord Peter, Maximus, a humble monk.

1. Since I know you, most heralded and God-guarded lord, to have made wisdom, the food and companion of your life, the very highest, and to be endowed with that heaven-sent power and faculty, fertile in divine prudence and virtue, which leads you to conceive and perform all that is honorable—the foundation of which is the pious and orthodox faith, which correctly perceives that it alone provides through hope, as if by a certain divine grace, a display of those things which are not visible as if they were already visible, even before they have been unfolded, to those who—as you do—continually and without interrupted progression await and sustain the advent of the Word to himself—I have decided, among your other deeds, and encouraged by this singular act of condescension toward me, to describe a certain register full of august religion, and to send it to you, who are superior to all in praise and commendation. In it is contained, first, the approach of the fasts, then the solemnity of the Lord’s Resurrection: likewise another feast, which was formerly celebrated but is now obscure and unknown among us, so that we take nothing from it for celebration, nor do we care for it at all: I mean the tenth of the seventh Jewish month, through which, once every year, according to the prescription of the Law, the high priest used to enter into the Holy of Holies; into which also happened the revelation to Zacharias concerning John, the great one of the truth—and—

-proclaimer and herald. Therefore, for the sake of those who love learning, I have not shrunk from marking this, but rather I have been eager, knowing that from this an easy comprehension ^A of the matters sought by you concerning each of the items in the intercalation, and the dates of the Resurrection, and in that very tenth day of the seventh month, will follow for you. I have made, along with the written diagram of the aforementioned table, a brief explanation of it, which is as follows. ^B

OF THIS TABLE, AND WHAT ITS POWER CONTAINS. II.

[TABLE OMITTED]

^C

^D

II. Propositi canonis expositio ad hunc modum se habet. ^A The explanation of this present table is as follows. It is divided into three parts, as it contains within itself the entrance of the holy fasts and the fourteenth day of the first month according to the Hebrews, I mean Nisan, upon which their Passover is celebrated, and the tenth day of the seventh month, that is to say, Thersi, upon which the high priest of old used to perform his entry into the Holy of Holies. The notation for the beginning of the holy fasts is written on the left side of this same table; the fourteenth day of the first month of the Jews is to the right, near the wheel inscribed in the middle. Immediately after this, to the right, comes that which we mentioned, the tenth day of the seventh month, so that nothing is written on the left side except the beginning of the fasts, while on the right side both are present, namely the fourteenth of the first month and the tenth of the seventh. The power of that entire description is of this kind.

What the table inscribed on the left signifies.

What the table situated to the right of the wheel signifies.

What the wheel described in the middle contains.

^A year follow in sequence up to the twenty-eighth, inscribed with their epacts, with the suitable ones recorded proportionately for each year. These things also regarding the middle wheel.

Why does the table on the left take its beginning from the nineteenth year of the moon, and not from the first.

VI. But the reason why the inscription of the lunar years on the left is not set from the first year, but from the nineteenth, and thus the first, second, and those following, is as follows: ^B The intercalation in the penultimate month—that is, in the eleventh—of the current year occurs perpetually, just as the Paschal feast occurs in the first month of the coming year. Since, therefore, the holy days of the fasts occupy two years, or rather are comprehended by two, so that their beginning depends upon the current lunar year, but their end upon the one which immediately follows, it is necessary that we measure the beginning of the fasts from the nineteenth year, when the saving Pascha is celebrated in the first lunar year. Thus, from the first year the beginning of the fasts is directed as often as the feast of the Resurrection falls in the second; and from the second, when it falls in the third; and from this, again, when it falls in the fourth, and so on. In this manner is the description of the lunar years on the left arranged.

What are the embolismic months, and what are the lunar epacts.

VII. The seven intercalary months are collected from those which we call the lunar epacts, after two or three years, when they arrive at the number of thirty days. Moreover, the epacts arise from the deficit of the lunar year compared to the solar year. For the solar year consists of 365 days; the lunar of 354. Hence it has its own proper month of 29 and a half days. But any lunar year whatsoever, being gifted with one of the seven intercalary months, possesses thirteen months in total; so that to the twelve, this last one is added for that purpose and is called intercalary; therefore it collects not only 354 days, as the other lunar years do, but 384, with the month of thirty days added besides. ^D

Regarding the Hebrew months which have been observed on the left part of the canon, and the count of their days, what sort they are, and why they are different.

VIII. Hence, therefore, the penultimate month of that lunar year is not the eleventh, but the twelfth, from which we rightly measure the introduction of the holy fasts, on account of the excess of one month, which forbids the observance to proceed from their eleventh, so that a greater number of days than those which actually exist may not be imputed to them, from which error might creep into those performing the calculation. However, in both months—I mean the eleventh and the twelfth—on the left side of the canon, the same number of days is not ascribed in the region of the Roman months, just as in the right part, for one month

^A but another and another. For instance, in the eleventh month, called Sabat by the Hebrews, we see its seventeenth day marked in those years that lack the embolismic month; but in the twelfth, called Adar, its eighteenth day is marked in those years which do, plainly, possess the embolismic month. And this is indeed consistent with reason. For since we reckon the twelve lunar months alternately as twenty-nine and thirty days, so as not to have to divide days by assigning twenty-nine and a half to each (for it so happens that in any lunar year we count three hundred and fifty-four days in full, without any fraction of a day; and it naturally follows from the established order and succession of the months that, after the first, the eleventh has twenty-nine days and the twelfth has thirty): since these things are so, we correctly noted the eleventh month and its seventeenth day in the simple years in our canon. For from this day until the fourteenth of the first Hebrew month of the coming year, the days attributed to the holy fasts are equal in number, namely fifty-seven. For precisely that many days do eight weeks contain: namely, the remaining thirteen of that eleventh month—if, as I said, it is reckoned at twenty-nine—and the thirty of the twelfth, and the fourteen of the first month. And it is clear that thirteen, and thirty, and fourteen make fifty-seven days. But in the years that have the embolismic month, we observed the twelfth month and its eighteenth day. For from this day until the fourteenth of the first month, the same fifty-seven days are gathered: namely, the thirteen remaining of the twelfth month, since it has thirty days, and the thirty of the embolismic (or thirteenth) month—for every embolismic month contains just that many—and finally the fourteen of the first month. Hence, in the last year, which is the nineteenth, we have also noted the sixteenth day of the eleventh month that corresponds to it; because we reckon its twelfth month at only twenty-nine days, just as we do the eleventh itself; but we will explain the reason for this later. ^B Therefore, for these reasons we also observe the same alternation on the left side of the canon for these aforementioned months—namely, the same number of days—for the sake of the intercalation of the fasts, just as we do on the right side of the canon for the fourteenth of the first month for the sake of the Resurrection; passing from Sunday and Monday and the subsequent days to the coming holy Sunday. Having said this much regarding these matters, we must now turn our discourse to the precise calculation of the lunar years.

On the distinction of the lunar years, and the sum of the days.

IX. The total number of lunar years is, as has been said, nineteen. Seven of these have embolismic months—that is, they consist of three hundred and eighty-four days—while twelve are called common, that is, they consist of three hundred and fifty-four days. Moreover, all the days of the same nineteen lunar years, totaled together...

by the addition of the intercalary months, are equal in number to the days which result from nineteen solar years, each of which consists of 365 days. For this reason, the seven intercalary months are added to the lunar years, so that they may cause these, in the number of their days, to be without deficiency in relation to the years of the sun; and that there be not only [365] days, but also 354. For such is the number of days of the nineteen solar years, without there being even one extra day in the cycle of the entire nineteen-year period.

How the one day that is superfluous in the intercalary months is cut off, and in what year. [X]

X. For although we add seven months, that is, 210 days, to the lunar count on account of the falling short of each of those years from the solar ones, since those eleven days only make 209, the nineteenth year is mutilated by one day and consists of 354 days, so that it consumes the day that is superfluous from the intercalary months. For this year begins on the 4th of April and ends on the 22nd of March. Whence it comes about that the intercalary months retain their full number of days, and the cycle of the entire nineteen-year period does not exceed the nineteen solar years by even a single day.

That even those who multiply the lunar years five and six times remove the superflous day, although they do so in a different year. [XI]

XI. Whence those also who multiply the years of the moon five and six times, since they themselves calculate the same day, yet in the year which is the eleventh in their calculation, which we consider the fourteenth, they have a day taken from the intercalary months. For that eleventh year is set at only 352 days, beginning on the thirty-first day of December and ending on the eighteenth of the same month. And this is evident from the fact that their twelfth year has the fourteenth [day] on the Kalends of January, according to their calculation; whence it follows that the new moon falls on the 19th of December.

For what reason they multiply the lunar years five times; and how they err in the same calculation; and why their annual computation differs from ours. [XII]

XII. Furthermore, they attempt to inculcate this "exemptile" day in this way. For since they establish a lunar month of thirty days, and for each day take and aggregate one sixtieth part, it seemed good to them, for that day also which is diminished in every eleventh year, to gather five sixtieths each year, so that, by aggregating these through the twelfth year, they might transfer to the first day of this year the last day of that one; which we have already said is the 19th of December; and the same is the end of one year and the beginning of another according to that false calculation and method, but not according to the truth of the matter.

For how shall one day make two, or do the work of two? This is indeed not only impossible, but even foolish. Hence, therefore, their lunar calculation errs, which they cannot accurately attain because of the imperfect addition of the five sixtieths. ^A For the fact that they institute thirty-day lunar months, insofar as they gather one sixtieth for each day, as has been said, brings no redundancy, nor does it represent a month of as many days; since in the final analysis of the sum, not into 30, but into 29 and a half, the half-day added through the sixtieths is exhausted, and it does not allow for more than 29 and a half days. But the gathering of five sixtieths every year, for the sake of making up for the one day that is omitted in the eleventh year, entirely causes those who calculate to err. And this is evident from the fact that even after this one day is gathered from the remaining seven years of the moon, at the end of the nineteen-year cycle, they are cast away as superfluous. ^B And everything cast aside foolishly by those who calculate at any point in this manner indicates the disruption of the entire cycle and convicts those who use it of error. For their goal, as I said, is only to gather that one day; they have no concern at all for accuracy. Moreover, the year calculated by them differs from the one which we reckon according to ecclesiastical tradition; for they add sixteen years to the years from Adam. But having touched upon these matters by the way, let us return to the subject at hand, explaining the reason for the addition of the intercalary months in the lunar years.

What it is, and for what reason the addition of the intercalary months in the lunar years is made. Also from where the first year always begins, and where the last ends.

XIII. We say, therefore, that they were intercalated for this cause, so that the calculation might be regulated to proceed from one and the same month, which it cannot do otherwise in these years, since each year falls short by eleven days, ^C for which deficiency the solar year is devoid; it always proceeds from the same beginning, namely the vernal and equinoctial month; which indeed the Romans call April, the Macedonians Xanthicus, the Egyptians Pharmuthi, the Hebrews Nisan—to which the sign of Aries in the Zodiac corresponds. For since the moon each year falls short of the solar year by the eleven days mentioned, and by the same number of days travels ahead of the point from which it began, it therefore does not have a certain and fixed beginning in any one month among the others, but begins equally from all of them, because it traverses them all one by one by moving backward, so that it is obscure, and its beginning is defined by no certain mark from which its new moons arise. ^D Hence this artificial calculation and method is employed, in which seven intercalary months are added: so that it may...

be regulated and calculated ^A just as from one and the same month, just as the sun is. And not only this, but also that the entire nineteen-year cycle may be shown to be equal and similar to itself in every respect, like a single solar year, proceeding from the same point and ending at the same point; for example, in its first year always taking its beginning on the twenty-third of the month of March; but in the last, that is to say the nineteenth, receiving its end on the twenty-second of the same month. Thus far concerning the lunar epacts and the embolisms.

Of the Paschal Month

XIV. Next, that month must be noted which contains within itself the new moons of all the lunar years: which for this reason some call the Paschal [month], [and we must note] whence it takes its beginning and where it ends. We say, therefore, that its beginning is on the eighth of March, and its end on the fifth of April. For from that day to this, a solid lunar month of twenty-nine days intervenes. ^B By just as many days do the new moons of those years travel to and fro, on account of the eleven days by which these same years fall short, and the thirty days which are added through intercalation. Thus also the fourteenths, which proceed from the new moons, and on which the Pascha of the Jews falls, being moved up and down in the same way, taking their beginning from the twenty-first day of March, terminate on the eighteenth of April. For from the latter to the former twenty-nine days are counted. The lunar year which has the most remote new moon and fourteenth day of all is the sixteenth: that which has the nearest is the eighth.

Of the transgression of the legal Pascha, and of the Christian solemnity, from which day to which it is permitted to defer it according to the canon

XV. Wherefore we, who by grace ^C are permitted in the « unleavened bread of sincerity » (1 Cor. v, 8) to celebrate the Pascha of Christ our God, defer it by only one day, whenever we see the twenty-first day of March falling on a Sabbath, on which the fourteenth of the moon corresponds; but we pass over seven days, whenever we find the eighteenth of the month of April falling on a Sunday, into which likewise the fourteenth [day] of the first month of the Jews falls. By thirty-five full days (for just as many are those lying between the twenty-second of March and the twenty-fifth of April) we determine the time for celebrating the saving Pascha: so that we do not observe it before that [limit], nor beyond this one, because of the ecclesiastical canon ^D and tradition which retains the Pascha within these days. Thus, therefore, we calculate both the new moons and the Paschal limits in the lunar years.

^A From which Roman month do they derive the method for the Paschal order, and the calculation of the lunar years, by multiplying these by five and by six? And whence, and how, and in what years do they calculate the fifteenth and sixteenth day for that which is in reality the fourteenth day of the first month according to the Hebrews?

XVI. Those who multiply these by five and six start their method from another Roman month, namely from December. For beginning from the fourth of this month, which is the month that contains all the new moons within itself, they reach the end at the Calends of January. For there are twenty-nine days from the former to the latter. Therefore, the first day of January is the rule for all lunar years. And from this same day, however many days are numbered until the end of March—which are ninety—these same days beget sixty-part fractions for the Paschal calculation; then, after the resolution of these three months into thirty-day units, ^B half a day remains, which is thirty sixtieth-parts. To these thirty sixtieth-parts, therefore, they add those parts gathered from each of the five sixtieth-parts mentioned above, and thus they make one whole day; and sometimes even more, if, as is likely, they extend the calculation even to some days of April, due to the delay of Pascha. For it is necessary to take in the sixtieth-parts of these as well. It happens in any case that by the addition of that one day or two, they set the fourteenth day of the first month according to the Hebrews at the fifteenth or sixteenth. And indeed, the fifteenth [comes] in these years, for example the fifth, and sixth, and seventh, and eighth, and ninth, and tenth, and eleventh, and twelfth, and thirteenth, and fourteenth, and fifteenth, and ^C sixteenth, and nineteenth; but the sixteenth [they set] in the year that corresponds to the same, for example in the sixteenth [year]. Yet they agree with us only on the fourteenth, and they themselves calculate the fourteenth day in only these five years: the first among them—which is clearly the fourth for us—and the second, and the third, and the fourth, and the eighteenth. These things being so, whenever the fourteenth day of the first month according to the Hebrews falls on a Sunday, it is necessary consequently for the Christian Pascha to be extended to the twenty-second or twenty-third day of the lunar month, the cause of which we have given before. So much for now regarding the Paschal month and the error regarding the fourteenth day committed by those who calculate in this way.

Concerning the years from Adam, how many they happen to be, and how the current year of the sun, moon, and leap year is found through them.

XVII. Furthermore, it is necessary in addition to these to know the number of the years from Adam. There are, therefore, according to the ecclesiastical calculation and tradition, up to the present fourteenth indiction, the thirty-first year of the reign of Heraclius, our most pious emperor, six thousand three hundred and thirty-three years. ^D For by dividing and resolving these, the current year of the sun, and of the moon, and of the leap year is known; for example, the year of the sun is obtained if you divide the total by twenty-eight; for there are just as many, namely twenty-eight, in the cycle of solar years: the year of the moon will be obtained if you divide by nineteen. For the same number, as has been said, belongs to that year. Thus the year of the leap year

^A is found by division by four. For through the analysis of the years from Adam, each of these is clearly known, corresponding to the number of the remaining years; for instance, if one remains through the division by twenty-eight, or nineteen, or four, the current year of the sun, or of the moon, or of the leap year according to this enumeration is said to be the first; if two, the second; and in this way also the third, if three; and so on up to twenty-eight, or nineteen, or four years. And thus the appropriate and current year of each of these is discerned.

How otherwise the year of the sun, and of the moon, and of the leap year is calculated.

XVIII. It is possible, however, for these to be calculated in another way. For if one divides those same 6,133 years by 552 years, he will find eleven periods of 552 years completed, and in the twelfth period 281 years. If one distributes these 281 years in a similar way to obtain the years of the sun—or by nineteen for the lunar year, or by four for the leap year—the ^B current year for each of these will definitely be known. But these 552 years come about through the interlacing of the years of the moon with those of the sun, and of those of the latter with the former. For whether the nineteen are added to the twenty-eight, or the twenty-eight multiplied by the nineteen, they complete the number of 552 years. Since, therefore, it has been shown through the years from Adam—that is, the 6,133—and through the current twelfth period, that is, the 281 years, the manner by which the current year of the sun and of the moon and of the leap year is known, it remains to point out the method by which, which is the main point, we may find the day of the beginning of the fasts and of the ^C resurrection in the same table.

It is demonstrated by what principle the Pascha can be computed annually from that table.

XIX. When, by that method of calculation which we have clearly proposed above, you have found both the current year of the moon (as, for example, the fifteenth next to come: this is indeed the one in which the saving Pascha will be celebrated by those who observe the feast with a spiritual rite) and in like manner the solar year following it, such as the first, which we are about to enter: search in the right part of the table ^D for the line in which that same fifteenth year is inscribed. And if it is desired to know the day of the resurrection itself, the Roman month should be sought which is situated opposite the fifteenth lunar year in the next wheel table: which you will find to be April. Then to the day attached to it, namely the Kalends, you will gather [the result], since indeed on the Kalends of April the fourteenth moon of the Jewish Nisan will fall in the fifteenth year of the moon. To this first day of April must be added the epacts of the next following solar year, which we said is the next: these are eight, noted for the first year of the sun. Thus, having divided eight by seven, remaining

one day, we know the day of the week, as the first of the month of April is established, that is, holy Sunday; and from this, we defer the feast of the Resurrection ^A to the following holy Sunday, which is clearly the eighth of April. Since it is necessary that we entirely pass over the Hebrew Passover—I mean the fourteenth of the moon, or rather the first month according to the Hebrews—whether it reaches a Sunday, moving to the other holy Sunday, or a Monday, or a Tuesday, to the same Sunday in the same way. But if it is on a Saturday, on the following day, which is then the fifteenth of the moon, indeed. Just as, when we again make the postponement from a Sunday to the following holy Sunday, having the twenty-first. For as it has already been said, we make the transition from the Hebrew Passover—that is, the fourteenth of the moon—only up to seven days; but no further. And these things are concerning the Passover.

Concerning the tenth of the seventh month, and how this also is found.

XX. If anyone should wish to know the tenth day of the seventh month according to the Hebrews, in the right-hand part—that is, in the canon inscribed after the Passover—let him look for the Roman month corresponding to the fifteenth year of the moon (I speak of September); and holding the day-number adjacent to it and existing therein, he will know that the tenth day of the same month according to the Hebrews always falls on that day in the fifteenth year of the moon. But if he should also wish to learn the day of the week, to the same day-number of the month of September—I mean the twenty-first—let him add the days added from the silk [column], along ^B with the epacts of the current solar year; for example, as in the first year, the seven days; and thus, having divided the total—that is, the thirty-four—by seven, [the remainder], six, will be taken, and from the remainder of six, he will know that the week-day is also the same, that is, Friday. According to these things, one shall also use each year of those set down on the right of the canon; wishing to learn either the fourteenth of the first month according to the Hebrews, or the tenth of the seventh, which tenth of the seventh month we have inscribed, not so much because we perform any solemnity on it (for it is not celebrated by us in any way), but for the sake of the curiosity of those who wish to be ignorant of nothing. Thus far concerning the right side of the canon ^C as well as those things set down to the right of the canon.

Concerning the entry of the holy fasts, how this also is known every year.

XXI. If anyone should also wish to know the entry of the holy fasts, from the table on the left, he shall look with accuracy at the year of the moon fourteen, which stands opposite the fifteenth year of the moon which is found on the right. We said, indeed, that the entry of the holy fasts is celebrated at the end of the current year. Therefore, finding the same fourteenth year of the moon, he shall look at the adjacent Roman month, that is to say, February, and taking the day-number set near it ^D, he shall note it fourth. To this, let him add the regulars fourth, written in silk, along with the first of the following solar year

^A — I mean the first of the 15 days, that is to say the epacts. He shall divide the sum, being 5 and 10 in number, by 7. And from the one remaining day, he shall know the first day of the week, that is to say, the holy Sunday, which is the fourth of the month of February, as mentioned. From this, moving forward to the other holy Sunday, which falls on the eleventh of the same February, he will find the obligatory feast being celebrated in every case according to the 14th of the lunar year. For it is proper to maintain the same displacement also in the case of the interpolation in the left-hand canon, that which occurs for us in the right-hand one, according to the 14th of the first month of the Hebrews, from Sunday, and Monday, and Tuesday, and simply from ^B Saturday, moving over to the next holy Sunday. Thus, therefore, it is necessary to calculate the interpolation for every other lunar year; for example, so that it may be briefly and again explained, one must keep the year, then the Roman month, and its day-number. And one must add to it the supplementary days placed near it in silk, and indeed also the epacts of the present solar year. For it is proper to use the same epacts of one and the same solar year, namely those of the current year, for all the months set out in this canon, adding these to their day-numbers along with the supplementary days in silk; and thus to divide by 7, and from what remains to know the day of the week; and from this to proceed thereafter to the next holy Sunday.

What is the observance that ought to take place in the left-hand canon during every leap year.

XXII. ^C This one thing alone must be observed in the left-hand canon: that when a leap year occurs, we must make the days added in silk to the day-number of the months one fewer, when calculating for that year. For if the added day is one, we must not add it at all, but rather be content only with the day-number of the Roman month and the solar epacts, and thus divide the sum by 7. But if the supplementary days in silk are four, we must put three instead of four, but by no means four. This is done for the removal of the one extra day resulting from the leap year itself, so that we may not be led into error by it in ^D our search for the day of the week in the two Roman months, January and February. For with this finished, the leap day is counted, but by no means before it. And for those who are investigating the day of the week accurately from this canon, when it is not a leap year, there will be a grasp of one and the same day in both the canon of interpolation—I mean that which is placed on the left—and that of the Pascha, that is to say, Sunday and Sunday, or Monday and Monday, and likewise for the other days. But when it is a leap year, it is not the same and one, but the one after the one calculated in the left-hand canon. For example, Sunday in the left-hand one, say: in the right-hand one, Monday is certainly found. And Monday again there; Tuesday is clearly here. Thus also Sabbath.

^A in the right-hand canon; here it is necessary to advance seven days, from Monday, for instance, to Sunday; there, eight, from Sunday to the next Sunday. So that after eight days the beginning of the festival may reach Sunday in exact accordance with reason. For in this way the displacement of the left-hand canon differs from the transition occurring in the right-hand one: because the former extends as far as the eighth day on account of the bissextile day, while the right-hand canon does not exceed seven, but is limited to that number. So much for the caution regarding the bissextile year.

Concerning the epacts of the sun; what they are, and why we use them.

XXIII. And so that the solar epacts may not be left unexamined by us, we define the epacts of the solar years as the remaining days of the week after that which commences the solar year; which are reckoned backwards in the same week as far as the first, that is, Sunday. For example, let us take the present 28th ^B year. It began on the first of the month of April, from the seventh day, that is, the Sabbath. Therefore, it possesses as epacts the days elapsed of the same week up to Sunday: that is, Friday, the fifth, the fourth, the third, the second, and the holy Sunday itself, whence the epacts of it, the 28th year, exist. Wherefore, since the coming year, I mean the first, begins from the first, that is, the holy Sunday, it possesses seven days, that is, the epacts numbered from the Sabbath until the past holy Sunday. Similarly also the second year, because ^C it begins from the second day of the week, possesses one epact—that is, the holy Sunday—since it has no other day preceding it in the same week besides Sunday. We preserve these days, so that we may regulate every solar year hebdomadally, that is, from the beginning of the week, for the purpose of finding the day of the week, whenever we wish to learn the day of the month for each. For unless this is done in this way, and every solar year is regulated from the first day of the week, that is, the holy Sunday, it is impossible for those who wish to calculate ^D to determine the day of the week in any given month. For what do I mean? The present, as I said, 28th year, having taken its beginning on a Sabbath, if we had not added the six days before the Sabbath to the first of the month of April, we would not have known that it was a Sabbath. For since the past 27th year had its conclusion on a Friday, the days equal to Friday are technical days for that one, having finished on it; but of this one they are the epacts, because they are added to the first of April and by this indicate the day of the week from which the same 28th year began. For in this way the harmony according to the week is preserved.

^A How the day of the week for each day of the month may be found.

XXIV. The same method is to be used in seeking ^B the day of the week for any day of the month as we have used for the first of April. That is, one should add the epacts of the current solar year to the days from the first of April up to the day in any month that we are seeking; then divide the whole sum by seven, and from the remainder we know the day of the week. Or again, in another way, by taking the epacts themselves alone and the odd days remaining after the 28th of the months (such as the two of April, the three of May, and so on for the following months) up until the month for which we wish to calculate, we then add the day we seek in that month to these, and divide by seven again; from the remainder we surely find the day of the week. For this reason, then, solar years are said to have epacts, and for this reason we fittingly make use of them.

Which solar years are said to be without epacts.

XXV. Whence the years that begin on a Sunday, such as the first, the seventh, the twelfth, the eighteenth, are said to be without epacts, because they have no preceding day. For there can be no day prior to the first. For the solar year is not said to possess epacts in the same way as does the lunar year. ^C Indeed, it lacks nothing to complete 365 days, whereas the moon is deficient by eleven; and for this reason the nineteen-year cycle is increased by seven intercalary months; but it is solely for this one purpose: that we may arrive at the day of the week within the twelve months, just as the series of the hebdomadal cycle, accepted from the ancients, requires from the beginning of the solar years. So much for the solar epacts.

Concerning the additive or regular days: whence they have their origin.

XXVI. The regular days, marked in red color, which we have decided must be added to the solar epacts for the given days of the Roman months so that the division by seven might be made for the sums, have this rationale. Since we calculate the day of the week in two ways, as has been said: first, by collecting all the days from the Kalends of April up to the proposed day; second, by retaining those exceeding 28 from the sun: ^D from these very surpluses, which amount to 12 at the end of September, and to 15 at October, we subtract seven as far as we are able. From the 12, there remain six, which we have marked in red for September; and from the 15, after subtracting two weeks, that is to say 14 days, we have applied the remaining one in red color to October. For whether we retain 12 days or 15, or only six or one, to these same two months we apply

^A them in order to determine the day of the week; this is done clearly by adding together the epacts of the current solar year, whatever the year being calculated may be, whether the first or second, and so on simply up to the 28th year. The same reasoning applies also to the 4 days and the one day added from the red color, distributed in March, February, and January; even if, by the addition of days, their numerical result and rule has become contrary to the months regulated by subtraction, such as September and October, since those are calculated according to the past solar year and belong to it. For neither otherwise nor in any other way could the calculation be initiated from the current year, or rather from its epacts, so as to reveal the day of the week. For it is entirely advantageous, in order that the translation from Sunday, and the second, and the third, and the subsequent days to the other holy Sunday may be made, ^B in the right and left canon; the one, as has been said, for the entry of the holy fasts, that is to say the festival of Shrovetide, and the other for the saving Pasch. These things are said also regarding the days added from the red color, so that through a clarification of the explanation of everything contained in this canon, the comprehension of those things lying within it may become easy for the studious.

Concerning the epacts of the moon, how they are computed.

XXVII. ^C These things having been said, it is necessary to speak again regarding the epacts of the moon set down in the left canon, and to show how they are computed. One must therefore multiply the current year of the moon by eleven; for instance, as in the case of the present 14th year, and make the days collected from the multiplication 154, decreasing them by two days, because the first year of the moon has only 9 epacts; and thus distribute these into 30, on account of the intercalary months. I say that the 154, and the remaining days—that is to say, the two days—are known to contain the epacts of the year of the moon; which the same 14th year also has marked in this canon. Thus, therefore, it is proper to do for every other year of the moon. These epactal days happen to be what the year of the moon is found to have, in every case, on the thirty-first of the month of March. Since it is always necessary to know how many these are, for they are useful for calculating and finding the day of the moon in each of the twelve months.

How we find the day of the moon in each month.

XXVIII. ^D We collect all the days which have elapsed from the Kalends of April to the desired day in any Roman month. And to these we add the aforementioned epacts of the moon, which it is found to have, as I said, on the aforementioned thirty-first of the month of March. And we divide the whole total by 29 1/2; the remainder is the age of the moon. These things have been said concerning the method and discovery of the epacts of the moon.

On the epacts of the sun, and how they too are calculated.

XXIX. ^A We must next come to know about the solar epacts, and how they too are to be understood. One must therefore take the current solar year—let us say, for instance, the 28th year—and take a number of days equal to it, save for one single day only, that is to say, 27 days; and one must also take the equal number of quarters, omitting no quarter—that is, 28 quarters, which clearly make 7 days. And in this way, one must divide the whole sum, that is the 34 days, by 7 for the sake of the weeks. And the remainder, that is to say 6 days, is known to be the epact of this same 28th year; which we have also written down in this canon for it, just as we have done for the others, providing the appropriate values. Thus, therefore, it is possible for any other year to be calculated for the discovery of its epacts.

How solar years possess epacts that are equal to, or exceed, one another.

XXX. For every solar year leaves behind a day and a quarter as epacts. These are nothing else at all than that day and quarter which remain after the 52 complete weeks out of the 365 1/4 days. But it is said to have two, or three, or four, and up to seven, because they transfer that day and the quarter to one another and are increased. For the progression occurs in order up to a complete week, that is, seven days, and then begins again. But since it is a leap year, and one day is collected from the quarters of the quadrennium, ^B as this is added, the increase of the following year becomes greater by one day; for it receives two days instead of one, as can be seen in the 28th year itself. For whereas the year before it, that is, the 27th, has only four, this one claims for itself six because of the addition of the one day. Hence, that 27th year did not end on the same weekday on which it had its beginning. For if it had only 365 days, it would end on the same day as that from which it began, as from a Sunday to a Sunday, or from a second feria to a second. But since it consists of 366 days, it therefore ends on the day after that from which it began; that is, from a Sunday to a second feria, or from this to a third, and so on in turn. For this reason the epacts of the sun must be observed, and calculated in the manner that has been stated. Truly, to repeat the same thing again: as many days as the year of the sun is numbered, one less is to be taken; then the quarters corresponding to the whole number without deduction. The sum is to be divided by 7; what remains will give the epacts of the year we are seeking. Furthermore, we instructed that the number of days equal to the years should be reduced by one day, since the first year has no epacts ascribed to it, because it is defined from the first feria, that is, Sunday. For every year beginning from a Sunday, as has been said above, possesses absolutely no epact, except for such fragments as a quarter, or a half, or a half of a quarter, which ^D are [necessary] to investigate the hebdomadal feria.

...nothing whatever, until after four years they consume one day.

How it is possible to observe the epacts of the sun and the moon otherwise. ^A

XXXI. That I may speak briefly concerning the epacts of the sun and the moon, and their easy determination: as far as the moon is concerned, any given year of it always has as many epacts as the number of days of its age on the thirty-first day of the month of March. But the sun, or its current year, always possesses as its own epacts a number equal to the day of the week on which the preceding year ended on the day before the Kalends of April. These things, said concerning the epacts of the sun and the method of their determination, are sufficient. ^B

Concerning the birth of the Savior according to the flesh, and His baptism, and passion, in what year of the sun and moon they occurred, and on what day of the week.

XXXII. With these things thus established, it is firstly necessary to have noted the year of the creation of the world, in which our Lord and God Jesus Christ undertook that sublime incarnation in the world from the holy Theotokos and ever-virgin Mary. At that time, therefore, it was the year of the world 5501, as is evident from the holy scriptures and the computation of those years from Adam. For in this year the annunciation was made to the Virgin, and the incorrupt and most blessed birth from her took place. It was the 13th year of the sun, the 10th of the moon, and the day of the week at the time of the annunciation was the second, and of the birth the fourth. Thus also His most light-bearing and super-luminous baptism occurred in the year 5530. It was the 14th year of the sun, the 1st of the moon, ^C and it was the third day of the week. Again, His saving passion, that is, the crucifixion, was in the year 5534. It was the 18th year of the sun, the 5th of the moon, and it was clearly the day of Preparation. So that the years from His incarnation until the now present indiction—the 31st year of the reign of Heraclius, our most pious emperor—amount to 633 years. But how many years have passed from the saving baptism and passion it is manifest to all from these same [calculations], namely 604 years from the former, and 601 from the latter.

Why the year of the birth of the Lord is noted in a different indiction year, and how the indiction is calculated, and from where it is reckoned. ^D

XXXIII. In each of these we find a different indiction (for so it is called by the Romans), as if it were not calculated in one way from the years of Adam and from the time when it was first instituted. For it arose, as historians have recorded, in the first year of Augustus Caesar, in his second year of empire, which was the year of the world 5460. These divided by 15 exhibit the last year of the indiction, that is to say the fifteenth year, in which he, establishing its beginning, wished the last to be first. Thus it happens that the indiction, which is calculated from the second year of Augustus, advances by one year, [compared to] that which is led from the beginning of the world, when the years of Adam are divided by 15...

^A Hence, according to this calculation, it was the eleventh year of the indiction in which the birth of the Savior according to the flesh took place; but according to the other, it was the twelfth, because it precedes by one year. Thus also the most holy baptism has the tenth and eleventh [indictions], and the saving passion the fourteenth and fifteenth. For in the forty-third year of the reign of Augustus, He who was begotten of the Father as God before the ages was born among us as a man. Therefore, the years calculated as 6001, when divided by 15, show that the birth of the Lord occurred in the twelfth year of the indiction, as I have said; just as, conversely, 5991, when divided by the same number, again shows the eleventh year. It is possible, through both calculations, to find the present year of the indiction in the same way; for example, for those from Adam, because of the reason mentioned, by adding one year, it becomes 6154, and when divided by 15, we find it. For those who use only the years from the second year of Augustus, the number is 624, and when divided by the same number, 14 is obtained from both—this is the present year. It is appropriate for those who wish to examine the number of the year of the indiction for any year at all to do this. ^B

Regarding the announcement given to Zechariah, and the years of John and the Savior.

XXXIV. It is also appropriate, as the order now demands, to record with greater precision the announcement made to Zechariah regarding John the Baptist, which we mentioned earlier, in what year of the sun and moon it occurred. And for those who are attentive, it is clear that each [year] is the one immediately preceding the birth of the Savior; for instance, the sun was the twelfth [year], the moon the ninth, the month was September among the Romans, having the twenty-seventh day, on the fifth day of the week. For the birth of John himself was in the same year of the sun and moon as the birth of the Savior; but the month was clearly different, being June, with the twenty-fourth day, on the second day of the week. Again, the year of his completion—that is, his beheading—was the seventeenth year of the sun and the fourth of the moon; the month was August, having the twenty-ninth day, on the third day of the week. ^C Thus, the years of Christ in the flesh are thirty-three, plus eighty-nine days; the years of John the Baptist are thirty-three, plus sixty-seven days, being twenty-two days less than the days of the Savior.

But you, God-favored master, having this knowledge transmitted to you in summary form, observe the solemnities of the religious feasts of our Savior: so that a perpetual choir and celebration in heaven with the angels may receive you, through the prayers and intercessions of our all-glorious and immaculate Lady, the Theotokos and ever-virgin Mary, and all the saints. Amen. ^D

SAINT MAXIMUS' COMPUTUS, PART TWO. The second canon is explained.

1831. Since, as the order demanded, it has been declared in the previous exposition in which years and how many [they are] of...

the fifteenth or sixteenth moon of the first month according to the Hebrews—those who calculate these, I mean the years of the moon, by multiplying them by five and six—^A they, however, have not indicated which ones, according to them, and how many are of the Sunday. It happens, then, that they calculate such [years] to be the 15th and 16th (from which, furthermore, they fix the Christian Pascha at the 22nd and 23rd moon, which is in no way observed by us; for it is not lawful for us to exceed the 20th [day], just as it is not to fall below the 15th), I have decided to define this by a certain rule and show it as well, one that contains everything with precision, not only the years of one nineteen-year cycle, but of the 28 nineteen-year cycles, from which the entire period of 532 years is completed.

Of the Sun, Years, and Epacts; and Leap Years and Remainders

[index: entries preserved]

^A

The description of such a table is as follows. In the middle there is a huge wheel, flanked on either side by single columns. In the wheel there are zones, or nine circles, which are arranged in this way. The first circle, which is the outermost, displays the 28 years of the sun, beginning not from the first year, but from the 13th, 14th, and so on; the second [displays] the epacts of the years; the third, the bissextile years of the four-year Julian cycle; the fourth, the 28 years of the sun, once again starting from the first year. ^B For this reason, in the first circle the 28 years of the sun begin from 13, and in the fourth they begin from the first year, so that the studious reader might more easily understand that the year which they calculate as the first year of the sun is, for us, 13, and the second is 14, and the rest [follow] in this same manner of arrangement. In the upper part, however, we have placed the years as they are truly counted according to ecclesiastical tradition. ^C

These are [the contents] in the four circles of the wheel; but the five circles inscribed within them contain the years of the 28 nineteen-year cycles of the moon, in which the 14th of the first Jewish month falling on a Sunday is calculated by them as the 15th or 16th. Moreover, each nineteen-year cycle is indicated in order, the first by the first year of the sun, which is inscribed in the fourth circle; after which, as I have said, the lunar years are described; the second nineteen-year cycle by the following [year]. And thus each [is indicated] by the aforementioned years of the sun. For we have used these to signify the nineteen-year cycles of the same number, so that there would be no need for another circle for them; and we have attributed the years of each nineteen-year cycle to only five circles, since we observe that in no case at all does their 15th or 16th (which we calculate as the 14th) fall on a Sunday beyond the space of five years; this happens only in one, two, three, four, or five years, and it does not happen in more. ^D

^A For in every nineteen-year cycle, in both the thirteenth year [15th] and, in the one, the 16th [14th], as I have said, they calculate. But it does not fall on a Sunday. From which, even if they err concerning the day of the moon, yet at least they do not miss the Pascha according to Christ, ^B until the moon reaches the 21st, which they do not exceed, calculating the 15th from Monday, just as they do not the 16th from Tuesday. For if, I say, they were to calculate the 16th from Monday of the week, they would necessarily exceed it; for 16 and 6 make 22. For this reason, we have marked the years of the moon thus calculated by them with two points. Similarly, we have distinguished by three points those years in which they observe the 16th [15th] on the aforementioned Sunday. And this is the reasoning concerning the points which are in certain years of the moon, from which the distinction arises of those in which 22, and in which 23, and in which they erroneously regulate the Pascha according to Christ before the Jewish Pascha. For in those which have two points they do this, calculating for the Sunday which is for us the 13th, the 15th of the moon themselves. To which, as much as they ought to celebrate the Pascha according to the 15th itself—a thing the canon allows—they instead provide for it as a 14th on the following day, that is, on the second day of the week, before the Jewish Pascha, thus proving the Pascha according to Christ to be wrongly regulated and missing the true calculation. For if even the coincidence of the Pascha according to Christ with the Jewish [Pascha] is reprehensible, what would one say concerning that which precedes it? It happens again that they celebrate in the same way in these years. ^C For in these the fifteenth [. . .] is calculated on the Sunday [. . .] of the Pascha. That therefore [. . .] manifest these [. . .] to the readers [. . .] we have marked with a single point. There are therefore those which lack any point. [. . .] ^D and they output the Pascha according to their own calculation; the first and the 21st, existing according to the ecclesiastical method. Those, however, having one or two points, in which they regulate, not only for the 22nd, but also the legal Pascha according to Christ, even if to some, as if to those having one point, it is because of the leap year; and to those with two points, because of the days which are added to them from the sixtieths; and those marked with three points, in which they extend towards the 23rd, along with some other minute parts, which is absurd. In these mentioned five zones, or circles, we have arranged the solar years alongside the lunar years from the left, together with their 19-year cycles, namely the twenty-eight-year cycles, written from the silk [original].

^A Thus it may be known for every nineteen-year cycle which year is the 14th or the 15th, or the 16th and 14th; they calculate the 14th, as to how many years of the sun it is, and to which twenty-eight-year cycle it belongs, and from its own proper epacts, written according to the second circle of the same wheel, the day of the week may be determined, according both to the combination of the epacts of the solar year, and the postae of the Roman month, and the additions from the sericum, and the analysis of those summed up by seven. For this cause we have arranged the years of the sun along the left, next to the years of the moon. These are the details concerning the description in the middle wheel.

Concerning the table on the left

III. ^B Regarding the two tables by which the wheel is flanked on either side, the one that extends from the top downwards on the left contains the following: in the first row, the seven embolismic months inscribed in their proper years of the moon; in the second row, the nineteen years of the moon, which do not begin from the first year, but begin from the fourth; in the third row, the nineteen years of the moon, proceeding in order from the first year. This was done here too, so that the first year of the moon which they calculate might be shown as the fourth among us, and the second as the fifth, and so on, up to the nineteenth year. Again, those years we calculate according to the ecclesiastical count were placed before those calculated by them, because of the truth found in them. In the fourth row are the epacts of the years, that is to say, the days which they calculate that every lunar year possesses, according to the first of the month of January; in the fifth row, the excess days, or those equal to the fourteenth of the first month, which is to say, the 14ths themselves. These are the contents of the left table.

Concerning the table situated on the right

IV. The table which is on the right side displays in the first row the nineteen years of the moon, ^C starting from the fourth, in precisely the same manner as they are described in the second row of the table on the left, as they are usually calculated by us; in the second row are their epacts, that is, the days which are attributed to the age of the moon each year by ecclesiastical calculation on the day before the Kalends of April; in the third row are the two Roman months, March and April; in the fourth row are the days of those same months, on which the fourteenth of the first Jewish month falls, that is, the Passover which they celebrate every year; in the fifth row are the regulars painted in red color. These are found in the other table. Moreover, in this right-hand table is inscribed the Paschal canon, the rationale of which has recently been explained by us, so that by its aid any year of the moon, ^D as it is thought of by them, and as it is proposed by us in the second table, as well as in the intervening wheel and its five circles—so that, I say, one might find the year in which the Passover, according to their calculation, is the 15th or 16th, and, by accurately performing the calculations, prove whether the matter stands as we have taught in our discourse.

The use of the canon is demonstrated by an example

V. Come now, before everything else, let us do this very thing, and by calculating this with precision, determine if it is so as we have distinguished in the discourse.

We present for consideration, for those who come upon this, the year xvi of the moon ^A according to them, and xix as calculated by us. For in this year, the fourteenth of the first Jewish month always and necessarily falls on the seventeenth of the Roman month of April. For this reason, we have set this seventeenth in our chart to the right in the month of April. Therefore, gathering the days from the first of the month of January up to the seventeenth of April, we make 107 days; then we add, by the multiplication of the 16 years by five, 80 days, and by the addition of the sixtieths of those same years, 3 days. Furthermore, the same 16 years multiplied by six accumulate 96 days. So that the total sum resulting is 286 days, which, when divided by 30, we find to be the sixteenth, which is in truth the fourteenth of the first month.

If one should wish to know on which day of the week this falls, it is fitting to observe the year of the sun in its first cycle, of the same number as the lunar, namely 16, which we have taught to be placed to the right of that lunar year, and near to it. We add its epacts, written in the second circle of the wheel, which are 6, to the 17 days of the month of April; and ^B we divide the total by 7. From the remaining 2, we know the day of the week on which the Jews observe the Passover. Wherefore, those counting 15 of the moon on the day preceding this—that is, on the holy Sunday—set the Christian [Passover] before the Jewish, which is unlawful. For this reason, then, as has already been said, we have marked the same year 16 with two dots.

And in this manner it is fitting to calculate in the first enneadecaeteris the same year 16 of the moon. But if, in the sixth enneadecaeteris, we should again wish to know on which day of the week they calculate the 16 in the same year—that is, in the 16, which carries with it the solar year from the left, namely the 27th of the fourth 28-year cycle—taking the epacts, which are 5, we add them to the same 17 ^C days of the month of April; and dividing the 22 days by 7, from the remainder we find Sunday to be the day of the week. From this year 16, calculated by them in the sixth enneadecaeteris, they set the Christian Passover on the 23rd of the moon, since it is transferred to the following holy Sunday according to ecclesiastical tradition; but from that calculated in the first enneadecaeteris, because it falls on the second day of the week, the Passover falls on the 22nd of the moon.

In this way, if anyone makes use of the lunar years that are inscribed in the five circles of the wheel, he will investigate in each the first day of the week, or the second; and as for those ^D written on the left, if he seeks the one that follows the first or the second, or the third day, and the rest in order, as the proposed enneadecaeteris year shall be, he will seek out the corresponding year of the 28-year cycle, and joining its epacts to the day of the Roman month on which the 14th of the first month falls in that year of the moon which is sought, and then making the division by 7, he will obtain from it the day of the week.

^A Having set these matters forth briefly by way of demonstration, we leave the rest to studious readers who are eager to investigate them in the same manner, and who seek to discover and obtain the exact truth in all things.

COMPUTE OF SAINT MAXIMUS: PART THREE

I. How the weekday of the week may be found and known by the studious in the preceding tables and methods—insofar as it was within our power—we have already explained: now it has seemed good to describe a proper and specific wheel for this same matter; by the benefit of which we may investigate any given day in any of the twelve Roman months, and in accordance with reason easily discern it in any year of the solar cycle whatever, and this not in one way only, but alternately; namely, whether we advance the calculation in the direct order from April to May, June, and the others, or if it pleases us to proceed backward from March to February, January, and the remaining months preceding them. ^{B C D}

On the description of the wheel.

On the description of the middle tablet.

By what method the calculation is to be instituted in direct order.

On the observation of the leap year.

December ^A — where the leap day was also noted. For this is the reason why we have attached March to these two months. Thus, therefore, from this wheel we discover the day of the week, when we continue the calculation in regular order, through April, as I have said, and May, and June, and those that follow.

On the retrograde calculation.

VI. But when we wish by a retrograde calculation, through March, February, and January, and the months preceding these, to explore the day of the week using the epacts of the present year, which is the 28th, not assuming the epacts of the previous 27th year: if indeed this is done in the two months of the first row, we add the 6 epacts we have already found to the given day of April or July, along with the epacts marked in red color for April, which are 6, in a common year, and we divide the sum by 7. In a leap year, however, we take those epacts which are rubricated for July, namely 5, together with the others, and divide the sum by the same number. For those rubricated days, which are placed on the left side, are used in a leap year when we compute in retrograde order. But in a leap year we use those on the right side simultaneously rubricated. These things we prescribe regarding the two months of the first row. Furthermore, in those which occupy the second row, May and January, in year 28, we add the actual epacts of these to the day we are seeking, together with the number painted in red for January, which is 7. We divide the sum by the same number. I mentioned ^B the number appended to January because the previous year, the 27th, had a leap year; but the rubricated numbers, as we have said, pertain to leap years.

On the observation of the leap year.

VII. In this method of retrograde counting, the rule regarding March is that it must always be calculated, both after a leap year and without a leap year, by the days of December; I mean the four days marked in red. And one should not, either by the days of September or by those of February, as we said, perform the calculation in the regular order and method, but only by those same four red days which are set against the month of December. And to speak simply, so that we may treat both calculations briefly: in the seven rows of the table for the twelve Roman months arranged on either side, we use the regular calculation for every year; but in the two rows correctly placed in red, we use the method of retrograde calculation, according to the combination, as I said, of the epacts of the current solar year, by which also the seven rows of the table mentioned are clarified. We have done these things ^C as much as possible to make the discovery of the day of the week easy for the readers, regarding both this wheel and the determination of the day itself.

How the age of the moon should be calculated.

VIII. 1. We take the epacts of the moon together with the

day of the current Egyptian month [proposed], and half of the count of the months that have elapsed from Thoth, along with one regular. If the sum amounts to thirty, or less than thirty, we determine the moon to have that many days. If it exceeds thirty, we subtract thirty from those, and we attribute the remainder to the age of the moon.

2. Otherwise. We take the years from Diocletian, minus one, and we divide them by 19; the remainder we keep, and to this we add—for one year, if this alone happens to be left over after the division, 10 days; for 2 years, 20; but for 3 years, not a single one, just as for 6 years not a single one, or for 9, or for 12, or simply for any third year. For 4 years, again, 10, and for 5 years, 20. And thus consecutively for 7, and for 10, and for 11; and for those which are calculated after the third-year increments. We keep, therefore, as I said, the remaining years and the days for these according to the revealed method; and the current day of the Egyptian month, and half of the months from Thoth, and one regular; and we divide the whole by 30.

3. Otherwise. We take the epacts of the moon, and for each Roman month, starting from September up to the one being calculated, one day; and the day of this current Roman month, and two regulars, and we divide the whole by 30.

4. Otherwise. We take the epacts of the moon, minus one, and the days from January up to the day we are seeking; and add the 60th part of these, and divide the whole by 30.

5. Otherwise. We take the epacts of the moon, and half of the months from April up to the one being calculated, and the initial thirtieths ^A, and the day of the current month; and we divide the whole by 30.

6. Otherwise. We multiply the lunar year by eleven, and we take all the days that have flowed from the first of January up to the day being sought, together with their 60th part; subtracting three as a regularity, we thus divide the total by 30.

7. Otherwise. We take the epacts of the moon; for January and February we use them in full; in March, only one less; but in April and those following, taking them in full, and one other [day] for each month up to the one being calculated; but only two for August, along with the day of the month; and we divide the whole by 30.

8. Otherwise. We take the epacts of the moon, and the days from the first of April up to the one being sought, and the day of the month; and we divide the whole by 29 1/2.

Regarding the years from Diocletian: how they are calculated, and by what method the lunar year is found through them.

9. The years from Diocletian up to the present ^B indiction. ^{C D}

The fourteenth indiction, which falls in the thirty-first year of our most religious emperor Heraclius, is numbered 357. ^A And it is calculated in this way: we multiply the 15 years of the indiction by 22, and to these we add the 13 regulars together with the years of the indiction already begun. But if this fifteen-year period is itself completed, the multiplication must be done by 23, and again by 24, when that too has unfolded its fifteen years. And to speak generally, after the completion of any cycle of 15 years, we count one year for this, as if it were finished, and in this way we multiply, adding meanwhile at the end the 13 regulars. From which the number of years from Diocletian to this time will be easily understood by readers. We divide those 357 years by 19; and from the remainders we gather the current year of the moon.

Regarding the epacts of the moon, and by what method they should be used.

^B X. We measure the epacts of the moon in two ways: namely, up to March 31st, and up to August 28th. If we proceed by this latter method, we divide the years of Diocletian minus one by 19, and the remaining years, multiplied by 11, we distribute into 30: what is left will give the epacts. But if the lunar epacts are to be constituted from March 31st, we multiply the current lunar year by 11, and from the sum, having subtracted two, we divide the remainder into 30. Furthermore, since the method for calculating epacts is double in this way, both are to be applied to the methods of calculating the lunar days brought forward above: so that where March 31st is marked as the day, we use those epacts that pertain up to that date. But where the 28th is placed, those which are terminated by that same date are to be assumed. ^C Thus, the calculations will hold true without error.

Regarding the first month according to the Egyptians, which month it is among the Romans, and where it begins.

XI. It must be noted that the Egyptian Thoth is the Roman September, the beginning of which, or the new moon, always falls on August 29th.

Calculation and sum of times.

Adam was born . . . 230 Seth . . . 205 ^D 435 Enos . . . 190 625 Cainan . . . 170 795 Malaleel . . . 165 960 Jared . . . 162 1122 Enoch . . . 165 1287 Mathusala . . . 167 1454 Lamech . . . 188 1642 Noe . . . 500 2142 Sem . . . 100 Arphaxad . . . 135 Cainan . . . 130 Sala . . . 130 Heber . . . 132 Phaleg . . . 130

^A Ragau 132 Seruch 130 Nachor 270 Tharrha 70 Abraham 100 Isaac 60 Jacob 85 Levi 47 Caath 62 Aram 75 Moyses 80 In the desert 40 Jesus 32 Elders 50 Chusarathem years 50 3909 Othoniel 50 3959 Aeglon 18 3977 Aoth 50 4027 Semgar 20 4047 Jebusaei ^B 20 4067 Debbora 40 4107 Oriph and Ziph 7 4114 Gedeon 40 4154 Abimelech 3 4157 Thola 22 4179 Jaer 22 4201 Ammanitae 18 4219 Jephte 6 4225 Essebon 7 4232 Elon 40 4242 Abdon 8 4250 Philistaei 40 4290 Sampson 20 4310 Interregnum and peace 40 4350 Heli the priest 20 4370 Samuel the priest 20 4390 Saul 40 4430 David 40 4470 Salomon ^C 40 4510 Roboam 17 4527 Abia 3 4530 Asa 41 4571 Josaphat 39 4610 Joram 8 4618 Ochozias 1 4619 Gotholia 7 4626 Joas 40 4666 Amasias 29 4695 Ozias 52 4747 Joatham 16 4763 Achas 16 4779 Ezechias 29 4808 Manasses 55 4865 Amos 2 4867 Josias 31 4896 Joachaz 1 4897 Joacim 11 4908 Jechonias 1 4909 Sedecias ^D 11 4920 Nabuchodonosor 21 4944 Valamadarach 5 4949 Baltasar 3 4952 Darius and Astyages 17 4969 Cyrus the Persian 32 5001 Cambyses 8 5009 Darius the Mede 28 5057 Xerxes 21 5058 Artaxerxes 33 5091 Darius 19 5110 Artaxerxes 34 5144 Ochus 21 5165 Arses Ochus 2 5167 Darius 6 5173 Alexander the Macedonian 12 5185 Ptolemy of Egypt 24 5209

APPENDIX TO EUSEBIUS' CHRONICON

^A Ptolemy Philadelphus 30 5259 Ptolemy Euergetes 25 5264 Ptolemy Philopator 17 5281 Ptolemy Epiphanes 23 5304 Ptolemy Philometor 34 5339 Ptolemy Euergetes 29 5368 Ptolemy Physcon 16 5384 Ptolemy Siderites 9 5393 Ptolemy, who is also Alexander 3 5396 Ptolemy Soter 8 5404 Bacchus the Younger 28 5432 Cleopatra 22 5454 C. Julius of the Romans 4 5458 Caesar Augustus 57 5515 Tiberius 22 5557 Caius 4 5541 Claudius 14 5555 Nero 14 5569 Vespasian 10 5579 Titus 3 5582 ^B Domitian 15 5597 Nerva 1 5598 Trajan 18 5616 Hadrian 21 5637 Aelius Antoninus 20 5657 Marcus Antoninus 16 5673 Commodus 12 5685 Severus 18 5703 Antonius Caracalla 7 5710 Antonius the other 4 5714 Alexander Mammeas 14 5727 Maximinus 5 5730 Gordian 6 5736 Philippus 6 5742 Decius 1 5745 Gallus and Volusianus 2 5745 Valerian and Galienus 15 5760 Claudius 1 5761 Aurelian 6 5767 Probus 7 5774 ^C Carus and Carinus 2 5776 Diocletian 20 5796 Constantine 32 5828 Constantius 22 5852 Julian 3 5855 Jovian 1 5856 Valentinian 10 5866 Valens 4 5870 Theodosius 16 5886 Arcadius 14 5900 Theodosius the Younger 42 5942 Marcian 6 5948 Leo 18 5966 Zeno 17 5983 Anastasius 27 6010 Justinus 9 6019 Justinian 38 6057 Justinus the Younger 13 6070 ^D Tiberius 4 6074 Mauricius 20 6094 Phocas 8 6102 Heraclius 31 6133 Constantine 1 6134 Constantine 27 6161 Constantine 17 6178 Justinian 10 6188 Leo 3 6191 Tiberius, who is also Apsimarus 7 6198 Justinian again 6 6204 Philippus, who is also Bardanes 2 6206 Artemius, who is also Anastasius 5 6209 Theodosius Atramytenus 1 6210 Leo, who is also Conon 25 6235 Artavasdus 3 6238

Constantinus Leonis f. ^A 31 6269 Leo Constantini f. 5 6274 Constantinus Leonis f. cum matre Irene 10, m. 2 6284 Constantinus solus 6, m. 9 6290 Irene rursus 2, m. 6 6292 Nicephorus 8, m. 9 6301 Stauracius Nicephori f. 2 6305 Michael et Theophylactus 2 6305 Leo Armenius 7 6312 Michael 9 6321 Theophilus Michaelis f. 12 6333 Michael Theophili f. eum matre Theodora 14 Michael solus 11 Michael et Basilius 1 Basilius Thrax 19 Leo et Alexander Basilii filii 26 Alexander et Constantinus 1 Constantinus solus 7 ^B Constantinus et Romanus

Of the six millennia, when each of these concludes, and under whom it is completed.

Of the XI periods, consisting of 531 years each: when and under whom each of these is completed.

Explanation of the subject chapters, when and under whom each of them occurred.