Dissertation on the Chronology and Computations of the Greeks

Our learned Bonfrerius ^A observes in his commentaries that the same thought which had entered my mind has previously pleased him, a fact in which I rejoice greatly. Hence, all the points brought against us are completely resolved. If, they say, during the days of Unleavened Bread the ears of the barley harvest were still green, the wheat could not yet have been ripe. We answer that "green ears" are used here to mean fresh, not thoroughly unripe and immature. For in Hebrew, Leviticus 11:14 reads "fresh ears, roasted," and "bruised corn from full ears." So, in IV Kings 4:42, the interpreter translates it as "new grain." Nor is it absurd that at the time when barley is harvested, the wheat—if not yet ready for harvest—was at least ripe enough for eating. Josephus (Antiquities, Book III, ch. X) seems to be of the opinion that the sheaf offered as first-fruits was from the barley harvest. But this does not prevent it from being ^B from other crops offered when a late Passover occurred. He further adds that once the sheaf was offered, it was permitted to reap both publicly and privately: "and then it is allowed for all to harvest, both publicly and privately," as if this had not been permitted before. But there exists no prohibition of this matter in the sacred books. We, however, are speaking of the ancient times of the Jews, not the age of Josephus. Yet even from this very passage of the same historian, it is evident that the harvest was completed in Judea by the time of the Passover: which is sufficient for us. For if it was lawful to harvest the grain after the offering of the sheaf had been sampled, then the crops, even the wheat, were already ripe for harvest, and those at least ready for food. Whence it follows that the Sabbath *deutero-proton* (second-first) occurred on the seventh day after the offering of the sheaf; it is not at all absurd that at that time the apostles were rubbing green ears with their hands, which was what was proposed to be shown against a recent criticism.

A DISSERTATION ON THE ERAS AND CALCULATIONS OF THE GREEKS

CHAPTER I

On the triple era of the Greeks, which is reduced to two. The Chronology of Panodorus is illustrated. Maximus, George Syncellus, Theophanes, and others are noted.

It is agreed among scholars that there are three primary calculations and eras of the Greeks. For they say that one counts 5493 years from the beginning of the world to the arrival of Christ in the Virgin's womb or His birth; another counts 5501 years, and a third constitutes 5509 years. Some call the first the Antiochian, the second the Aethiopic, and the third the Alexandrian. Scaliger, in his *Excerpta Maximi*, names the first the lunar, the middle the oriental, and the last the paschal, the origins of which names he disputes at the beginning of Book V of *De emendatione temporum*. But you have them confuted by us and thoroughly refuted in the ninth chapter of *De doctrina temporum*, Book V, where we demonstrated most fully that the man had seen nothing in these calculations, nor understood what he was writing; and we gathered many things on that subject, unknown to the masses and most worthy of being known, and highly necessary for the matter now at hand; points which by no means need to be repeated in this place. Nevertheless, we omitted not a few things there that belong more to this present project and the paschal method of the Greeks than to the illustration of the calculations of Maximus, Isaac, and others, which have been related in the *Auctarium*. We have undertaken to explain this, the beginning of which dissertation will be drawn from that triple era and the common opinion of scholars regarding them, and from the error of being led to teach that there are three distinct eras thus far, when in reality there are only two.

^C For the diversity of calculations, or the annals of the Greeks, consists in this: that distinct totals converge upon one and the same year as a common terminus. But if the same titles are inscribed for different years, the calculation should not for that reason be called distinct; because a higher number extends to more distant years, while a lower number, drawn from the same beginning, extends to closer ones. For, to make what I say clearer by an example, let some calculation be taken from the creation of the world to the first year of the Christian era, such as our own of 3983 years, and in it let the same epoch—namely, the founding of the City—be bound to different years: one placing it in the year that is 754 before the Christian era, another thirteen years later; following Pictor, or some other ancient author I know not, say in the one that is 743 before Christ. Therefore, those two, while agreeing in the same sum at the rise of Christ, but differing in the years of the same epoch, such that the former counts 3230 years from the same beginning of the world to the founding of the City, and the other 3243, are certainly not said to establish distinct calculations from Adam: which they would only do if to that same year, which is 754 before the Christian era (the 3960th year of the Julian period), one counted 3230 years from Adam and the other 3243; and consequently, one reached 3983 years to Christ, and the other 5996.

Thus, therefore, for those three Greek calculations to be distinct and separate from each other, it would be necessary for them to converge upon one and the same terminus, and not to descend thereto from the same beginning and starting point. Such

however, is not the case. For of the three, two are those which assign the Incarnation—which, in Greek, is

τὴν ἐνανθρώπησιν

That those who count 5493 years to it and those who count 5501 do not assign the Incarnation of the Lord to the same year is evident from the authors who were either the inventors or the proponents of each. Georgius Syncellus attributes the former reckoning to a certain Panodorus, a learned and ancient monk. For he lived during the reign of Arcadius and, in addition to this, brought the use of astronomical matters to the science of exact chronology; of which, in Georgius’s account of him—which we shall set forth in the following chapter—there remain traces that are certainly not obscure. And Panodorus set the year of the

ἐνανθρώπησις

γέννησιν

Εὐαγγελισμός

χρηματισμός

χρηματισμός

ἀποδεικτικῶς

years less. We, in the ninth chapter, book IV, of *De doctrina temporum*, have attributed those two years of the *chrematismos* of Zachary and the Incarnation, as conceived by Maximus, to the years 4710 and 4711 of the Julian period, which are assigned the same Dominical letters. But the 12th indiction, which Maximus assigns to the birthday of Christ, suggests the year 4722, in which year the Roman indiction was current from January, or from March or April, as Maximus thinks. Thus, a similar flaw in the computation is seen in the baptism and passion of Christ, since chapter XXXII records that Christ was baptized by John in the year 5530, on the fourth day of the week. Therefore, with the Dominical letter D, adding thirty to that year in which He was incarnate—whence this year is 4752 of the Julian period—the year of Maximus, 5530 and 5531, resulted ^A from that. He also casts the passion of Christ into the year 5534, and on the day of Preparation, which indeed fell on the 23rd of March, as will soon be proven from George, with Dominical letter G. Therefore, it was the year 4755 of the Julian period. Such an opinion of the Greeks differs no less from history than it does from the evangelical narrative. For Christ was baptized when Tiberius was in approximately his 15th year; and He suffered in the third or fourth year thereafter, while the same emperor was reigning: but from the computation of Maximus it follows that the Lord approached the baptism of John when Gaius was reigning, in his second year, and suffered in the second year of Claudius: which things are most absurd.

George Syncellus is in the same case as Maximus. For he also wants Christ to have suffered on the 13th of March, and to have risen on the 25th, at the completion of his 33rd year, in the year 5533 from Adam ending, and 5534 beginning, from the 8th day before the Kalends of April. Furthermore, he counts the year of the death of Alexander the Great as 5170 of the world, and says this is ^C confessed among all. But he says the first year of the monarchy of Augustus, which began in the second year after the death of Julian, is 5458; and the death of the same occurred between 5514 and 5515, when Christ was nearing the fifteenth year of his age. The interval from the year 5170, into which the Thoth of Philip falls, to 5458 of the monarchy of Augustus, is 288 years; if you add these to the year of the Julian period 4590, in which Alexander the Great died, the year 4678 will be obtained, which is the eighth after the death of Julius Caesar, which occurred in the year 4670. Again, the interval from the year 5458 to 5515, the death of Augustus, is 57 years; if you add these to the year of the Julian period 4678, there will result the year of the Julian period 4735, in which the passing of Augustus occurred. But it is agreed that he died in the year of the Julian period 4727. Therefore, George delays the true epoch by eight years.

Hence it happens that in the annals of George Syncellus, and of Theophanes who continued his work—and also of the interpreter of this *Miscellanea*—they collect eight years of Christ less than the common era: nor did Maximus establish them otherwise. For he composes the 21st year of Heraclius with the 635th year of Christ to the 641st year of the common era, as will be told a little later. George and Theophanes indicate as the first year of Diocletian of the world ^B those who want 5777, and 277 of the Incarnation; but from their mind it ought to be 285 of our era. Thus, Theophanes refers the death of Constantine to the year of the world 5829, of the Incarnation 329: which it is certain fell in the year 337 of the common era. Again, he assigns the beginning of Valentinian to the year of the world 5857 ending; with 5858 immediately following, since he was created emperor in February; ^C he counts 357 of Christ likewise ending. A difference of 7 years, just as in the year of the Council of Chalcedon, which he teaches was celebrated in the year of the world 5944, at the 5th indiction, in the year of Christ 444. But it was held in the year 451 of our era, so Theophanes sometimes strays by one year from the established chronology: and he makes a difference of only seven years; whereas, from his own calculations, it should be eight years: the reason for which will be sufficiently clear from those things which will be said concerning the beginnings of the years in the Julian year.

From all these things, what we posited at the beginning is brought about: that of the three eras, two attach the Incarnation of Christ to different years; and of the former, which thinks it is 5501, the last corresponds to the ninth year of Christ; but of the other, which numbers 5493, the first year: and so there are not two annual computations, but they are one and the same. For each computation numbers 5493 as the first year of the Christian or Dionysian era: and each likewise reckons 5501 as the ninth year of the same era. Indeed, to arrive from the current year of the world to the year of the Incarnation, the former subtracts 5500 years; the latter 5492, from which a greater calculation results for this one than for that one. For example, the year of the world 6333 according to Maximus, subtracting 5500, will be 633 from the Incarnation. But subtracting 5492, it will be considered 641, to which in truth Maximus always paid regard. Thus, both computations agree in the years of the world, but disagree in the years of the Incarnation. Therefore, the eras or computations of the world ought not to be deemed two, but merely one, whose ^D *enanthropesis* is attached to different years of Christ. Wherefore there are only two, as we were saying, of the principal eras of the Greeks, namely one, to whose year 5493 the first year of the Dionysian or common era corresponds: and the other, which numbers the same year from the foundation of the world as 5509. The monk Isaac uses this latter; who in chapter III counts the year of the world 6881, which began from September of the year of Christ 1372, in which year occurred the 26th of October, the 3rd day of the week; accordingly, the letter was C, and it was certainly the year of Christ 1372. Subtracting 5508 from 6881, 1573 remain, since indeed the Greek year 6881 for the greater part of itself concurs with 1373 of Christ; or it begins from the spring month of the same year according to their common method, as I shall show soon. Furthermore, Gaza in the book *De mensibus*, chapter XXI, says that the year of Christ 1470 according to Roman calculation is considered by the Greeks to be 6978 of the world. Subtract 5508, 1470 remain. Wherefore, since, as we showed at the beginning of this chapter, those different computations are considered which collect different sums of years for the same year of our era; they are two, not three.

of the Greeks; of which the former and most ancient counts the year which is the first of our era from Adam as 5493; the posterior as 5509, nor do the ancient Greek computers record more: they call the former *kata Alexandreas*, which let us call the Alexandrian; the latter *kata Romaious*, which—from the new Rome—we call the Constantinopolitan. Thus Theophanes in more than one place. But Scaliger has devised certain new methods of using these eras, as well as new names, both of which are foreign and absurd, which we have refuted in book IX of *De doctrina temporum*, chap. V, and we shall speak of them again shortly. Some might think a third should be added to those two, which calculates the year ^A of the Incarnation and birth from Adam as 5507, which some authors use, such as the Alexandrian Chronicle; Cedrenus (p. 144) uses one less, ^B claiming that this occurred in the year 5506, and he contends that those who assign the birth of Christ to the year 5500 and the passion to 5533 are in error. But I do not believe this era differs from that which is Isaac’s, but rather that in the same computation they have assigned a different year to the Incarnation, that is, they place the common era two years earlier.

Although we can conclude nothing certain regarding the opinion of both of them, because they ignorantly mix everything together and employ characteristics that cannot be reconciled with one another. Concerning the error of the Alexandrian Chronicle, it has been said in book XII of *De doct. temp.*, chap. IV: Cedrenus is somewhat more consistent with himself. He asserts that Christ was born in the year of the world 5506, on December 25, the fourth day of the week. Therefore, the Sunday letter was F, and it may be the year of the Julian period 4711, two years before the common era. Later, he relates that He was baptized in the first indiction, in the year of the world 5536, January 6, the fifth day of the week; therefore the letter was B, and by this reckoning the year of the Julian period 4742; but the indiction then was from September and January 11. However, so that it might begin to be reckoned from the spring month, the first in use was January 6; in the following year 5037, he says the fourteenth moon of March was the 27th, the seventh day of the week. Therefore the letter was C, the cycle of the moon 10, and consequently the year of the Julian period 4741, *ano potamon*, as they say. Finally, he says the year in which Christ suffered was 5539, which was March 21, the sixth day of the week; and the fourteenth of March was the 24th, the seventh day of the week. ^C Therefore, the letter was G, but the cycle of the moon was 13: these correspond to the year of the Julian period 4744, in which Christ truly suffered according to the opinion of Epiphanius and other ancients. Wherefore Cedrenus placed the Incarnation of the Lord three years before the common era; but the passion in the thirty-first year of that same era. Whence it is not surprising that he considers that year of the Incarnation to be 5506 in the Constantinopolitan era. For if you add three years, it becomes the year 5509 Constantinopolitan; the Julian period 4714, which is the very same as the common era. There is therefore no other computation; but he has set different years of the same computation for the Incarnation. From these it is easy to perceive what the origin of these computations was: which will be declared in the following chapter.

CAPUT II. Concerning the annalistic computation of the ancients, especially the one freed from the bonds of Africanus’s method. The chronicle work of the same Africanus is illustrated. A dark passage of Photius is explained. The *Cesti* of Africanus.

The ancient Christians in Greece, in counting the years from Adam, looked only to the intervals which exist in the sacred books; nor did they have their annalistic computations entangled in any paschal method *schesei*, or by artifice: which the industry of those who followed them formed later. Therefore, just as each one terminated those spaces in more or fewer years, so he applied greater or lesser sums, regarding which variety of opinions mention has been made in the ninth volume of *De doctrina temporum*, chap. II.

Among these was that celebrated one of the ancient and learned chronologer Africanus, ^D who asserted that Christ was incarnate in the year of the world 5501, and suffered in 5531, at the age of about thirty-one: and gave origin to that era which we have said is called Alexandrian. Those who came after, having obtained his computation, interpolated the chronology of their author, and disturbed it most foully. For they distorted the Incarnation of Christ the Lord by about ten years from his epoch, and by eight from the Dionysian. It will not be irrelevant, I think, to investigate how this was done and what his sincere opinion was; nay, his inquiry will bring much utility for understanding the Greek eras.

Jerome writes in chapter IX of Daniel that Africanus opined that the passion of Christ the Lord occurred in the fifteenth year of Tiberius Caesar: which Georgius Syncellus affirms that he numbered as 5531 from the foundation of the world, at the age of Christ of about thirty-one. But Photius reports in number XXXIV that Africanus published a historical work divided into five books; in which he embraced all times, from the first origin of things to the *parousia* of Christ, that is, the Incarnation, so that he omitted nothing necessary, describing everything in summary. From there, however, he touched upon the remainder down to the reign of Macrinus, and in this work he testified that he had comprised 5723 years from Adam. The same is testified by Georgius Syncellus in the *Excerptis* of Scaliger (p. 39), where he reports from Africanus that 903 archons were numbered by the Athenians down to the olympiad, starting from Creon, who presided over the 19th olympiad; but the last was Philinus in the olympiad I have mentioned, in the consulships of Gratus Sabinianus and Seleucus; who were consuls 725 years after Brutus. Finally, it is gathered (p. 37) that 5723 years reach to the third year of Antoninus and Avitus. He understands Marcus Antoninus Alagabalus. In the same Scaligerian excerpts a fragment of Africanus is read concerning the olympiads (p. 317): in which he says that 248 *Stadionicae* are counted among the Athenians in the list of olympiads, and Scaliger lists the year of the olympiad of the *Stadionica*. If you take 5500 from the 5725 years from Adam, there will remain 223 from the Incarnation of the Lord according to Africanus.

^A Moreover, the year 5530 of the world ending, which 5531 followed around April; and finally the Passion occurring in the year of the world 5531, or the following 5532 from the Passover, or April of the common year of Christ 30, the 16th of Tiberius, and the second year of the 202nd Olympiad not yet begun from the summer months; but *κατὰ πρόληψιν* (by way of anticipation) from January, or some other beginning of the civil year, which Africanus used. For we know that the Olympiads—and as a consequence, the years of the Olympiad tetraeteris—are not always counted by many from the actual time of the games, but from the civil beginning of their own states; so that the games themselves coincide with the ends of the years. Thus the year of the world 5723, or 223 from the Incarnation, will be the common year 221, and the third of Alagabalus. But the third year of Alagabalus is the common year 221, in which the 250th Olympiad also falls. Africanus therefore preempted the Dionysian epoch by two years, and assigned the Incarnation of the Lord to the year 4712 of the Julian period.

This is furthermore deduced from the year of the Passion established by the same Africanus. For if He suffered in the fifteenth year of Tiberius, namely when the two Geminii were consuls (which was the opinion of most ancients), and that year is numbered 5531 from the world by Africanus, since he had attributed the Incarnation to the year 5501, as George Syncellus is our authority: since the consulship of the Geminii corresponds to the year of the Christian era 29, or the Julian period 4742; with 30 years subtracted from 4742, the remaining year of the Incarnation will be 4712 in the Julian period. Truly, in Eusebius *De demonstratione* (p. 243) another fragment of Africanus occurs concerning the Passion of the Lord and the weeks of Daniel; in which the Passion is conjectured to be in the second year of the 202nd Olympiad, the 16th of Tiberius. ^B

This very passage of Africanus was transcribed almost word-for-word by Jerome in his *Commentaries on Chapter IX of Daniel*, and yet he posited the fifteenth year of Tiberius; not the sixteenth, as it is in the Greek. Although, were the year of the Olympiad not appended, we could easily reconcile those things, for the 15th year of Tiberius, if it is counted from the month of August, in which he began to rule, began in the year of Christ 28, in the fourth year running of the 201st Olympiad, and lasted until the month of August of the year of Christ 29, in which the two Geminii were consuls. Thus Christ will have suffered in the 15th year of Tiberius, if it happened during those consuls. If, however, as is the custom of the ancient Christians and others, Africanus began the years of Tiberius from the preceding Paschal month, the same could be considered the 16th year of Tiberius. ^C But since the second year of the 202nd Olympiad began in the year of Christ 30, and was spread out until the summer of the following year, it is necessary that Christ be said by Africanus to have suffered in the common era year 30, or 31; depending on whether he used the 16th year of Tiberius as current or anticipated from the Paschal month. If he intended Christ to have suffered in the Dionysian year 30, it is necessary that the Incarnation be placed by him in the year 4713 of the Julian period, one year before the common era, as is placed even today. But if he assigned the Passion to the year of Christ 31, in the consulship of Tiberius V and Sejanus, clearly as Epiphanius and other ancients have thought, the Incarnation will correspond to the first year of the Christian era; just as we know it pleased Panodorus and the Greek and Latin computerists. But by this reasoning, the year of the world 5723, which ought to be 223 from the Incarnation according to Africanus' calculation, will be the common era year 223; in which the son of Alexander Mammaea began the second year of his reign: although Africanus expressly wrote that the third year of Antoninus Alagabalus fell in that year of the world.

My conjecture, in this doubtful and perplexed question, is that Christ, in Africanus’ opinion, was incarnate in the year of the Julian period 4712, two years before the common era; in the year of the world 5501, and baptized ^D shortly before the year 5530 of the world ended, which 5531 followed around April; finally, having suffered in the year of the world 5531, or the 5532 following from the Passover, or April of the common year of Christ 30, the 16th of Tiberius, and the second year of the 202nd Olympiad not yet begun from the summer months; but *κατὰ πρόληψιν* from January, or some other beginning of the popular year, which Africanus used. We know indeed that the Olympiads sometimes, and, what follows as a consequence, the years of the Olympiad tetraeteris are not counted by many from the actual time of the games, but from the popular beginning of their own states: so that the games themselves coincide with the ends of the years. Thus the year of the world 5723, or 223 from the Incarnation, will be the common year 221, and the third of Alagabalus. And so those who report that Africanus attributed the Passion of the Lord to the year of Adam 5331, and the 31st of the Incarnation, ought to have understood it concerning the years that were ending. For it is probable that the baptism was conferred by him in the consulship of the Geminii; since it is established from Luke that this happened in the 15th year of Tiberius; or if, as Epiphanius and certain others thought, he came to the baptism in the month of November, not the sixth of January, in the year of the common era 28; certainly from here a solid year was interposed to the Passion, since Africanus imputed at least one year to *tῇ κηρύξει* (the preaching). Yet, if He had suffered in the month of March under the very Geminii consuls, the preaching of Christ would have lasted only three months from January, or five from November. Which I do not think Africanus felt.

But there are two things in that calculation of Africanus, which we have relayed from Photius and George, worthy of observation: one, what Photius writes, that Africanus extended his work to the reign of Macrinus, encompassing a history of 5723 years; this appears faulty, and for *Μακρίνου* one should read *Μ. Ἀντωνίνου*. Although another thing occurred, that Africanus found the *Stadionica* in the archives of the Athenians no further than the 248th Olympiad, which Olympiad extends to the year of Christ 217, and the reign of Macrinus; yet he extended the memory of historical events for five years beyond, that is to the year of Christ 221; Photius, however, thought the same was the end of the history, which was that of the Olympiad victors. See what we have said in *De doctrina temporum* XII, ch. XL. The other is that the argument for the new chronology is not firm enough; which from that passage of Africanus concludes that a slip of five years had crept into the Roman fasti and annals, and that there are as many superfluous ones in them. The force of the argument is of this kind. The year of Africanus 5723 is of the year of Christ 221, of the Julian period 4934, in which Gratus and Seleucus were consuls. But this pair of consuls, he says, is 725 years from the expulsion of the kings. Therefore, with the 243 years of the kings added to 725, there are 968 years, from which 221 being subtracted, there remains the year of the City 747: in which Christ was born according to the calculations of the common era; five years earlier than is commonly held, or rather six. For the birth is placed in the year of the City 752.

But Africanus, in numbering the years of consuls or annual magistrates, excluded those years in which there were no consuls: these are notably recorded in the Fasti as the five years from the year of the City 378 to the year 382, during which a continuous interregnum and anarchy existed. However, it is the custom for most writers, when calculating the times of annual magistrates or kings, to omit the years in which there were none: a practice which Onuphrius argues was observed by certain ancients in arranging the Roman annals, when he discusses the variety in the computation of the years of the City.

Finally, I should not omit the fact that it is evident from Photius and Eusebius in the *Chronicle* that a double work was produced by Africanus; of which one was historical, divided into five books, and carried up to the third year of Alagabalus, the year of the world 5723, which we have spoken of so far; the other, which he entitled *Cestoi*, was digested into fourteen books. Eusebius, however, in his *Chronicle*, numbers them as nine books. The subject matter of this work was manifold. For he dealt in those books with medical, physical, agricultural, and even chemical matters: both of which are the work of the same Africanus, as Photius attests, and not as Scaliger supposed, who undeservedly distinguished them as separate authors. Therefore, the *Cestoi* had nothing to do with chronology, contrary to what had escaped us in book XII of *The Doctrine of Times*, chapter XLII, where we confused the *Cestoi* with the five-book history.

CHAPTER III. Concerning the eras of the Greeks, insofar as they are involved with paschal methods and cycles. The computation of Panodorus is illustrated. From it the other, which Maximus uses, has been propagated.

Hitherto they calculated the mere sums of years from the creation of the world without any method of cycles, as has been said. Later, when the use of this artifice had already begun in the Church, it was decided to adjust the years of the world to it in such a way that, were the sum divided by the period of the cycles, the proper cycle of each year would exist. To this end, they altered the earlier computations, partly by the subtraction or addition of a few years; partly, while retaining them, they relegated the Incarnation and Passion of Christ to years other than those to which they were initially ascribed. I shall proceed to explain how this happened little by little, as my conjecture suggests.

Panodorus, the leader, insofar as one may suspect, in Egypt, when he perceived that the years collected from Adam according to the opinion of Africanus did not, if divided by 19, produce the current cycle of the moon, did not hesitate, for the sake of seizing that opportunity, to shave off a few years from the sum. For the lunar cycle of the Alexandrians had already become prevalent, which, taking its beginning from almost the paschal month, was the same as ours, that is to say, the first of the one was likewise the first of the other: inasmuch as the entire Roman paschal method is deduced from the source of the Alexandrians. Theophilus, Bishop of Alexandria, instituted a cycle of 437 years, of which he described the first hundred years, starting from the consulate of the Emperors Gratian V and Theodosius, which is the year 380 of the Christian era, with lunar cycle I. This year, according to the calculation of Africanus, was reckoned as 382 from the Incarnation, and 5882 from Adam (book II of *The Doctrine of Times*, chapter LXIX). If 5882 is divided by 19, 2 remains, yet it was required for the paschal method that the remainder be 1. Panodorus, therefore, by subtracting ten, numbered that same year as 5872 from the creation of the world, wherein, if divided by 19, that first cycle of the moon which was sought comes into being. Furthermore, he shifted the Incarnation and birth of Christ by a period of two years from the epoch of Africanus; and reduced it to the year which was reckoned as the three hundred and eightieth before the beginning of the cycle of Theophilus. If the year 5872 of the world were considered the three hundred and eighty-second from the incarnate Lord, the Incarnation would fall in the year 5491 from Adam. Panodorus, however, considered that to be the year 5493. Therefore, the first year of the cycle of Theophilus, counted by him from the era of the Incarnation, was the three hundred and eightieth: which is a significant observation of the antiquity of the common era, which they call Dionysian, although we anticipate the Incarnation by one year, for the reason that the era of Christ does not include the Incarnation itself, but begins from the Kalends of January nearest to the Nativity. Nevertheless, this notation of the common era is clear; which certain more recent chronologers insult and hold up to ridicule as being not at all ancient, and first discovered by Dionysius Exiguus. But we prove that its author was at least Panodorus from the fact that Georgius Syncellus asserts that the Incarnation of Christ was referred by him to the year of the world 5493; which we shall demonstrate to be the very same as the first year of the common era in the following chapter, where we shall transcribe that fragment of Georgius. If you divide 5493 by 19, the result is lunar cycle 2, which accordingly corresponds to the first year of our Christ. From this conformation by Panodorus, it happened that this recent computation from the creation of the world was eight years less than the previous one, which was that of Africanus. For although Panodorus took ten years away from it, he nonetheless moved the common term forward by two years. If he had retained the original year of the Incarnation published by Africanus, which is 4712 of the Julian period, two years before our epoch; the first year of the cycle of Theophilus, 380 of Christ, would have been year 382 of the Panodorus era of the Incarnation; 5872 of the world. With 382 years removed from 5872, the remainder is 5490. Therefore, the year 5491 would be the year of the Incarnation. The same first year of Theophilus according to the computation of Africanus was 5882 from Adam, 382 from the Incarnation of Christ. With 382 subtracted from 5882, the remainder is 5500; consequently, in the year 5501 Christ was incarnate, according to the reasonings of Africanus. The difference between this computation and the other is 10 years. But Panodorus, as has been said, commanded the Incarnation to be lowered by two years to the year 5493: whence there is a difference of only eight years born between the two. Access to that opportunity of the lunar cycle, and of the indiction

^A which neither of them established. Hence it appears that I correctly taught in Chapter I that this annual and Alexandrian computation does not differ from the computation of Panodorus, if you look to the years of Adam or of the world. Then it is established that this differs much from the era of Africanus. Wherefore, what I said in *De doctrina temporum*, Book IX, Chapter IV, that of the three common computations of the Greeks, the palm of antiquity is owed to the one which counts 5500 complete years to the Incarnation of Christ, is to be understood broadly; and [hi]*kath' holou*, if the sum of the years alone is considered, of the kind that Africanus established. But if it is considered in the way in which it was later adapted by the masters of computation, when the use of the paschal method was interwoven with it, it is the same as that of Panodorus; but its treatment and ^B conformation are more recent.

After this, another computation was devised, falsely and contrary to the faith of the Gospels, and corrupted by the Greeks in the manner which I have exposed; which, having excluded this former one, has been received in most of all Greece and the East in this age. This adds sixteen years to the epoch of the Incarnation of Panodorus; it adds eight years to the Alexandrian computation of Maximus and Georgius, if you look at the simple number of years. For, if the year itself is looked at, it is by those eight years, by which it seems to exceed the Alexandrian computation, less: that is, it anticipates the Incarnation of Christ by that many years; and it agrees with the epoch of Panodorus. Thus to the year of the world 5509 it attributes the rise of the Incarnation of Christ. ^C And I am persuaded that three causes for its establishment have existed entirely. The first was so that when years were divided by 15, the indiction might be had without the addition of a unit. With 5509 divided by 15, the indiction IV remains, such as it was in the first year of the Christian era: which they dedicated to the Incarnation. The second was the convenience of the Jewish cycle, which Victorius and the Latins and most Westerners have followed. Although I believe that the Greeks expressed nothing beyond the order of the cycles or golden numbers; nor did they adhere in the celebration of Easter to those errors which a perverse imitation of the Jews transferred into the sect of Victorius and the Latins. The Jewish and Latin cycle differs from the Alexandrian and Roman triad, in that what is the first of the Romans is the 18th of the Latins; what is the third of the former is the 19th of the latter, as I have explained in more detail in *De doctrina temporum*, Book VI; especially in Chapter V, where I have bordered the days of the Roman Calendar with the golden numbers of both on either side. There, to the Kalends of January, the Roman number 11 is ascribed, and 19 to the Latins. A ratio of cycles of this kind has a place in that last computation. For with 5509 divided by 19, 18 remain, which was the Jewish cycle, and the Latin, in the first year of the Incarnation, or of the common era: to which with three added, the golden number of the Alexandrians, 2, is produced, which occupies the same year. Thus in the year of Christ 1373, the Alexandrian golden number was 6. Isaacus calculates this year as 6881 of that era; which number, when divided by 19...

leaves 3. Therefore, this Jewish cycle was of 1373 years; the latter could furthermore have been added for this reason, so that they might wipe away that stain with which later craftsmen had infected the calculation of Panodorus; and might remove this immense anachronism, by which the baptism and passion of Christ is spread beyond the death of Tiberius into the times of Gaius and Claudius. ^A All these things they attained by adding sixteen to the sum of Panodorus. The year of the Incarnation was held by Panodorus to be 5493; when divided by 19, the remainder is the lunar cycle 11; when divided by 15, it leaves 4: with 16 added to 5493, the sum 5509, when distributed through both cycles, results in indiction 4, with lunar cycle 18, which was the proposition.

CAPUT IV

^B That Africanus, in calculating 5551 years (to the passion of Christ), and not 5553 as the sincere narrative of the Gospels contains, falls short by two years, is manifest. For it is plain that in the year Christ was entering upon his age—that is, a little more or less than thirty, according to those words of the evangelist Luke, chapter III: "Jesus was about thirty years of age"—he was baptized, and taught, and healed every disease and every infirmity for three years. So that the whole time, which extends from his divine conception, and from the year of the world 5501, and the first day of Nisan, the first of months, March 25, unto the life-giving resurrection—which falls upon the same March 25—is 33 years and one day, which began the year. Whence, forty days intervene until his divine assumption with ^C his body into heaven. If, however, anyone places less faith in those things which are mentioned in more than one place concerning that day—namely, that our Lord, having trampled upon death, poured forth life for us from the dead on that primary day—let him look to the eleventh period of the 532 cyclic years, in the 213th year of the same period, at the fourteenth day of the Passover among the Hebrews, and you will find it on the 23rd day of the same month of March, falling on a Friday, on which he voluntarily suffered for us the saving passion; and having been buried by Joseph, who was from Arimathea, and Nicodemus, he rose again on the third day after the same Friday, the first of the Sabbaths, and the first day of the first month, Nisan, among the Hebrews; which always falls on one and the same March 25.

Africanus, therefore, when he had assigned the divine Incarnation to the year of the world 5501, in accordance with apostolic tradition, erred by two years in recording the passion and the saving resurrection, having gathered both into the year of the world 5531. ^D But Eusebius Pamphili, gathering into one sum the years from Adam to the birth of Abraham, which is commonly agreed to fall in the times of Ninus and Semiramis, kings of Assyria, followed the Hebrew codices, and did not include second Cainan, who lived 130 years before he begat, in the series; whom the divine evangelist Luke mentions in his genealogy, as has been stated elsewhere. Furthermore, this our chronology from Adam to the birth of Abraham comprises 3132 years: which number agrees as much with the history of Moses as with the evangelical genealogy of Luke. And from the birth of Abraham unto...

^A Eusebius has collected 2048 years from Abraham to the life-giving cross and the vivifying resurrection, whereas they are reckoned by us, according to more accurate calculations, to be 2221. In total, the years from Adam amount to 5553, but according to Eusebius, 5232, which completely deviates from the apostolic tradition. This error crept into his work through such a hallucination, in that, besides the 130 years of Cainan, he omitted 111 years of the Allophyli [the Philistines], which are contained in the Book of Judges; as well as 40 years of anarchy and peace, and 11 years of Darius, who is also called Astyages.

Panodorus, one of the monks of Egypt, a historian not unskilled in accurate chronology, who flourished in the times of the Emperor Arcadius and Theophilus, Archbishop of Alexandria, although he held the truth in most matters, erred by seven years when ^B he arrived at the life-giving incarnation, attributing it to the year 5493. The cause of his error happened in this way. For the first year of Philip Arrhidaeus, who succeeded Alexander the Macedonian as king of the Macedonians—the same year in which Claudius Ptolemy fixed the calculation of his "Handy Tables" as the beginning of the Egyptian and Greek year, on the first day of the month called Thoth by the Egyptians, which corresponds to August 29—is universally acknowledged to be the year 5170 of the world. From the same first year of Philip until the defeat of Cleopatra, 294 years are calculated according to astronomical canons. Thus, from the foundation of the world and Adam to the end of Cleopatra, there are 5463 years. But he disagrees with the ecclesiastical tradition regarding the 43rd year of Augustus Caesar, in which our Lord assumed flesh. For it is said among mathematicians that Augustus reigned no more than 43 years after the defeat of Cleopatra and the subjection of Egypt. If this were granted by us, it would follow that ^C Augustus departed this life in the year of the world 5505. But that same year would be the fifth year of the Savior's age, which is openly false. However, that the Lord was about fifteen years of age at the time of the death of Caesar Augustus, and nearly thirty years of age in the 15th year of Tiberius Caesar, as holy Scripture teaches, is doubted by no one. Thus the death of Augustus Caesar would occur between the year of the world 5514 and 5515, while the beginning of his entire reign would be in the year 5458. Panodorus, however, following the mathematical canon, assigned the beginning of the reign of Augustus to the year of the world 5451, his death to 5506, and finally the birth of Christ to 5493, which is clearly shown to have been wrongly done by him. ^D

Appendix to Eusebius' Chronicle - Fragment of a Note by George Syncellus

Christ on March 23, the sixth day of the week; and that he rose on the 25th: which it is agreed was also the opinion of Epiphanius. See what we have explained concerning this partly in the *Animadversiones* on heresy LI, and partly in the XII books *De doctrina temporum*. Since, however, it was demonstrated in the previous chapter that Maximus, whom Georgius and others who held the Alexandrian era followed, placed the passion of Christ in the year which is the 42nd of our era, the 4755th of the Julian period, it happened admirably in that year that the fourteenth of the Paschal moon fell on the 23rd day of March. For the new moon, according to the Parisian tables, occurred on March 10, the sixth day of the week, at 10 hours, 21 minutes in the Jerusalem horizon. Wherefore the fourteenth corresponds to March 23, the sixth day of the week; the mean full moon, however, to the beginning of March 25, namely at 4 hours, 55 minutes after midnight. But in the Nicene time, the new moon fell on March 9, and in the fifth cycle the fourteenth fell on March 22: which agreed very well with the mind of the Greek computists, who asserted that Christ suffered both on March 23, and on the sixth day of the week, and at the full moon. But these things, however much they seem to agree with their reasons, are so discordant and utterly alien to evangelical truth. But neither did they attain the opinion of Epiphanius; which we have abundantly explained in those books. Finally, it is established from the decree of Maximus and Theophanes that Christ the Lord preached for three years and about three months from his baptism, as one who was baptized in the beginning of his 30th year, on January 6, at the end of the 5530th year of the world, so that the 5531st succeeded in the following spring: in the year 4752 of the Julian period, and that he suffered in the 30th year of his age completed, and three months besides; at the end of the 5533rd year of the world, the 4755th of the Julian period. In which they depart from the opinion of Epiphanius. See Maximus, ch. XXXII. It should be known, furthermore, that the Greeks count the age of Christ from the Incarnation itself, not, as reason demands and we commonly do, from the time he came out of the womb. The Roman computists also embraced their institution, Dionysius Exiguus, Bede, and almost all of the lower age. Hence Georgius begins to count the years of the life of the Lord from March 25. ^B

Consideration on the eleventh.

The first among the Hebrews.

Africanus, then.

Eusebius of Caesarea.

Panodorus, furthermore.

Regarding the first year.

425. This is the epoch of the *Procheiron Kanon* of Ptolemy. From this the years are 294 Egyptian years complete to the Actian Thoth, or the new moon of Thoth, which in that year when Cleopatra died fell on the 31st of August; and it brought its beginning in the year of Nabonassar 719, in the year of the Julian period 4684, and according to Varro of the City 724. The interval from the death of Alexander to the Actian Thoth is 294 solid years. Again, the death of Augustus happened in the year of the City 767, in the 19th of the eponymous month, the 5th of the epagomenae, in the year of Nabonassar 762; the Thoth of which began on the 20th of August, in the year of the Julian period 4727, the fourteenth of the common era. But at that time the fixed year, and the Actian year, were in progress; the Thoth of which was perpetually begun from the 29th day of August. Therefore, from the Actian Thoth to the Thoth of the death of Augustus, there are 43 complete years. These are indubitable, and are demonstrated by us in the later volume of the work *De doctrina temporum*. From these another interval can be calculated, which is numbered between the first beginning of Augustus and the death of Cleopatra, of 13 years. For the empire of Augustus began in the year of the City 711: the Thoth of which coincides with the 3rd of September, in the year of Nabonassar 706, and the 4671st of the Julian period. From this to the Actian Thoth of the year of the City 724, 43 years are completed. Wherefore from the Thoth of the empire of Augustus to the Thoth of his death, there are 56 current Egyptian years. Panodorus assigned the incarnation of Christ to that year which had the lunar cycle 2 from the Alexandrian Thoth, if you consider the civil beginning, as we taught in Book IX of *De doctrina temporum*, Chapter IV. It was therefore the 4713th year of the Julian period, into which the vulgar birthday also fits, or 4714, if he followed the popular epoch; in the year of Nabonassar 748. Hence from the Thoth of the first Actian year, or of Nabonassar 719, to the Thoth of the year 748, 29 solid years are reckoned, so that in the beginning of the thirtieth Actian year the Lord was incarnate. Therefore, from the year of Nabonassar 425 beginning, or from the Thoth of Philip, to the year 748 of Nabonassar beginning, in which Christ was incarnate, there are 323 complete years intervening; so that in the 324th year from the beginning of Philip the *enanthropēsis* [Human-becoming] of Christ occurred. Having therefore subtracted 324 years from 5493, there remain 5169; and thus in the year 5170, from the autumn beginning, the Thoth of Philip occurs, as Georgius writes: who also in vain refutes Panodorus, because he tied the first beginning of Augustus Caesar ^D to the year of the world 5451. For he did that consistently with the chronology established by himself. For he deduced the first beginning of the emperor Augustus from the 4671st year of the Julian period, and the 711th from the founding of the City, from which most have begun his empire, as we said a little before. Now truly, from the Thoth of Philip, or from the year of Nabonassar 425 beginning, to the year starting 706 of Nabonassar, there are 281 current and complete years: which added to the year of the world 5170, to which the Thoth of Philip corresponds, make the year of the world 5451, the first of the monarchy of Augustus. Thence, with 56 years of that same empire added, there arises the year of the world 5507, in which Augustus passed away, although he died not yet having begun 5507, but still in the course of 5506. Moreover, that mathematical canon… ^B counting 337 current years, so that it began 338 from the death of Augustus. Now if you add those 338 years to the 424 of Nabonassar, the year 762 of Nabonassar will be formed, which is the Thoth of the death of Augustus. Add these same 338 to the year 5169 of Panodorus, and the year of the world 5507 will arise. Most rightly, therefore, Panodorus not only linked the Thoth of Philip with the year of his own calculation 5170, but also the first year of the Augustan empire with 5451; if indeed he attributed the incarnation of Christ to the year 5495, and the lunar cycle 2 proceeding from the popular epoch of the Alexandrians, that is, the 29th of August, from which they began their year; or from the vernal time, as computers are accustomed. Indeed, from these traces of the chronology of Panodorus, a conjecture can be made that he was an accurate and diligent man in these matters; and not a little more intelligent than his detractor Georgius, whose infinite errors in this genre exist; but none greater than that he undertook to criticize things which he himself neither grasped, and which had been consistently written by the very person whom he was accusing.

"As the year toward the month." These things are erroneous. For he seems to say that Christ came into the world in the 43rd year of Augustus; and yet he extends those same forty-three years from the death of Cleopatra to the death of Augustus. Both are true; and from the first beginning of Caesar Augustus to the year of the Incarnation, which Panodorus established, there are 43 years; and from the death of Cleopatra, the same number to the passing of Augustus. For in the third Julian year, the 4671st of the Julian period, Augustus began to rule, according to the opinion of Panodorus. Christ was incarnate in the Julian year 46. With two subtracted, there remains the 44th year from the beginning of Augustus: which begins from the Thoth of the 46th Julian year, the 4714th year of the Julian period. Thus, in the course of the 43rd year of Caesar Augustus, in the month of March, the *enanthropēsis* occurred. Again, Augustus died in the Julian year 59, the 19th of the month of Augustus, in the 4727th year of the Julian period. The interval from the 16th Julian year, in which the first Actian year began, to the 59th, is 43 years. Augustus therefore died at the end of the 44th year ^C from the overthrow of Cleopatra, that is, the Thoth of the death of Augustus was the forty-fourth from the Actian Thoth. These things are true. But Georgius speaks perplexedly and obscurely.

CHAPTER V.

The civil year of the Constantinopolitans began from the Kalends of September along with the indiction. Thence therefore it was fitting for the cycles of the sun and moon to begin, just as the Alexandrians, who three days before the beginning of the popular year had them, and from thence began their cycles. But since they used the description of the Roman years and Roman months, the Greeks sometimes used its beginning as well. Then, because for the sake of the Pasch, cycles were assumed by the Christians: since that feast is celebrated in the vernal time; hence most [did] not only [calculate] the beginning of the cycles, but also the beginning of the year itself...

^A in the vernal month; namely, in April or March.

This great variety of beginnings confused the writers themselves, no less than their readers, in an admirable perturbation. For they generally mix two different principles of years and cycles, and do not hold consistently to what they have once embraced. For example, some begin years from September or October; but the cycles of the sun and moon from March or April. Some begin the cycle of the sun from October; but those of the moon from the vernal month. Many command the civil year to begin from September, and the ecclesiastical from April; some even trace this from the month of March and a certain day thereof.

That the obscurity and confusion of this method may be clarified by the light of our own dissertation, it will be necessary to speak in order of each of the Greek computations and to search out the judgment of those who were the principals in handing them down, so that it may be declared by a certain standard how the cycles were constituted by them, in what manner they agree with the Romans, and finally what is the beginning of the year they adopted. We must therefore begin with Maximus, who is both more ancient ^B and an observer of a more ancient era, of whose year, and the epoch of whose year, we shall first decide.

Maximus shows that he holds the Alexandrian computation in his Paschal method in chapter XXXII, when he says the year of the Incarnation and the nativity of Christ was the year of the world 5501, and that he wrote in the year 6133, as he signifies in chapter XVII of Heraclius, whose year he uses as a common example. The first year of Heraclius began in the month of September of the year of the Christian era 610, but the thirty-first [began] in the year 641, with the Constantinopolitan indiction 14 after September, the cycle of the moon 14, of the sun 5, and the Dominical letter BA. He hands down this further method of his year in chapter XVII, so that when the sum is divided by 28, if there is any remainder, it is taken for the solar cycle, or if nothing remains, the cycle itself is complete. He prescribes the same for the lunar [cycle] and the indiction. In this way, when 6133 years are divided, the cycle of the sun is left as 1, of the moon 15, and of the indiction 10. It remains to be inquired whether that year 6133 begins from September of the year of Christ 640, or from the Paschal month of the year 641.

^C That he does not at all date his years from September can be concluded from chapter XXXII. For there he joins March and December to the same year of the world, namely 5501, in which he also says the feast of March 25 fell, and the twenty-fifth of December. Certainly Georgius Syncellus, who expresses the same era, begins his years from March 25; and so does the Alexandrian Chronicle, although it sometimes uses it otherwise, concerning which see chapter III, book IX, *On the Doctrine of Times*; but it declares clearly that the cycles of the sun and the moon begin from the same March, not from September, in chapter VI, where, detailing the table which was published on page 315, he says that January and February belong to the preceding year; but Easter, and its 14th day, to the following year. Furthermore, that is known from chapters X and XIII ^D: where he knows both the lunar year and the cycle, as well as the solar, starting from the vernal month; the one indeed from those days on which the golden numbers are affixed; but the solar from the equinoctial month. Likewise in chapter XXIII, he imputes the cycle of the sun 28, and of the moon 14, to April and the Pascha of that year which preceded the Pascha of the year 6133; the cycle of which was solar 1, lunar 15; in which chapter he also started the cycle of the sun 28 from the Kalends of April. Whence he defines the epacts of the sun as that day which is March 31, just as the lunar epacts [define] the age of the moon on the same day. From which it follows that the beginning of the year is constituted by him on the Kalends of April. Therefore, Maximus auspiciously begins his year in a double way, namely from the Kalends of April and from the Paschal new moon: a inconsistency of which we also observe in our own, that is, Latin, writers of computations in book VI *On the Doctrine of Times*, which, because it is very slight, is easily overlooked; while it is certain that Maximus fixed the epochs of his years in the vernal month, not the autumnal.

Although these things are well-explored, yet from the same computation of Maximus, a difficult knot occurs. Indeed, he joins the year 6133 and the indiction 14 with the 31st year of Heraclius, and thus with the cycle of the sun 1 and moon 15. Yet the 31st year of Heraclius began in October of the year of Christ 640. He died, however, on March 11 of the year 641. If the year of the world 6133 took its beginning from April, or the Paschal new moon, it would not have reached the 6133rd year [of Heraclius]: since the Paschal new moon in that year occurred on March 19; the fourteenth day of the Pascha on the Kalends of April, the first day of the week. Whence the Pascha is spread into April 8; with the cycle of the moon, both Roman and Constantinopolitan, 15; but the cycle of the sun 1 Constantinopolitan, 6 Roman, letter G, which Maximus expressly asserts in chapter XIX. This can be dissolved by no other reason than that which was brought forward in chapter III, book IX, *On the Doctrine of Times*, where we showed that Maximus confused the double epoch of the civil and ecclesiastical year: whereof the former proceeded from September; the latter from April, or the vernal month. Consequently, he compared the civil year 6133 sometimes with the 31st year of Heraclius and indiction 14; at other times he deferred the same year into the Paschal month, using an ecclesiastical beginning.

Before Maximus, Panodorus seems to have initiated his years from the vernal time; although Georgius, in that fragment which we cited in the previous chapter, where he refers the Incarnation and the birth [of Christ] bound to it in the year of the world 5493, gave no certain indication of that fact, yet thus the early Christians mostly, at least in ecclesiastical computations and histories, were accustomed to act, as we taught concerning Africanus in chapter I of this book.

These things having been noted, it will now be easy to investigate the cycles of the sun and moon of this era, and those proper to the computation, and to compare them with our own, in which Scaliger has blundered most foully, as will be shown a little later. For first, as far as concerns the solar cycle, with the years divided from the foundation of the world, the remaining number

^A is the solar cycle: or rather the whole number 28; moreover, from the fact that 6133 divided by 28 leaves 1. Hence, the solar cycle 1 coincides with the year of Christ 641, in which the Roman solar cycle was 6, with the Dominical letter G: from which the subsequent ones can be derived, as will be clear in what follows. The lunar cycle of Maximus is exactly the same as the Roman one. Since, by dividing 6133 by 19, the remainder is 15, and in truth in the year of Christ 641, the lunar cycle was 15. Which ought not to be surprising. For from the same source of the Alexandrians, Greeks and Latins alike drew their lunar cycle; and they began its use from the same epoch, namely from the spring month.

In calculating the indiction, Maximus is perplexed, and in that point the method, which divides by the entire cycle of the ^B *omados*, fails. For if you divide 6133 by 15, 10 will remain as the indiction. Yet 14 began in September of the year 640, and it continued still in the following year, 641; nor should any other have been composed with the year of the world 6133. But in the method, a unit is added to the world years, so that you may obtain the current indiction: just as chapter XXXIII of the *Computus* advises. For he established a twofold indiction: one, which exists from the world years divided by 15; and it is less by a unit than that which is actually used at that time; the other, which began from the second year of the reign of Augustus: which he indeed asserts made the first in that year out of the 15th indiction. These are fabrications. But the true origin was revealed above in chapter III. See also chapter I, where we dealt with the indictions of Maximus.

In the other computation, which Georgius calls "according to the Romans," and which exhibits the year of the Incarnation 5509, the epoch of the years is likewise ambiguous. Isaac Argyrus, in chapter III of the former *Computus*, says the civil year begins from September; the solar cycle, from the Kalends of October. But he begins the lunar cycle in chapter IV from January, and he asserts that by dividing the world years by the number congruent to each cycle, the current cycle is gathered in the last year. He writes in chapter III that the year in which this began was 6831 of the world, which when divided by 28, the solar cycle becomes 21, then he writes in the following chapter that on the 26th day of October the weekday was 3. This is the year of Christ 1372, with Roman cycle 9, Dominical letter DC. Yet the same man in chapter VI, ^C saying "up to the beginning of the beginning January," says the years are 6880: which when divided by 19, the cycle of the moon 2 is completed. These can be reconciled thus. Isaac was writing in the month of October of the year of Christ 1372; when the Constantinopolitan year 6881 had already begun from September with indiction 11, but in terms of the solar cycle from the Kalends of October: furthermore, in terms of the lunar cycle, it was to have its beginning on the following Kalends of January, so that the year 6880 would persist in the year 1372. Therefore, he says that at the Kalends of January of the year 1373, 6880 years had elapsed: the last of these occupied the year 1372, the cycle of which he is thus investigating. Divide 6880 years by 19, there remains 2, the Jewish and Latin lunar cycle; which corresponds to the Roman 5, by which he writes that the lunar cycle is begun, confirms our conjecture, which we explained above in chapter III; [confirming] that the Greeks in this type of computation received the golden numbers of the Latins and Jews. For it was proper ^D to the Latins to begin the lunar cycle from January; whereas with the Romans the same cycle would proceed from April, or the Paschal month. From these, it is concluded that the civil Constantinopolitan year corresponds to two lunar cycles, so that the year 6881 beginning from September of the year 1372, toward the end of December corresponds to the 2nd Constantinopolitan cycle, or the 5th Roman: from the Kalends of January, to the third cycle, and the 6th Roman, which must also be said concerning the Roman solar cycle. With the 6881 years distributed by 28, there remains the solar cycle 21 from October of the year 1372: during which the Roman 9 was still in use: but in the following year, 1373, the cycle was 10. Wherefore the first solar cycle of the Constantinopolitans was the Roman 17 from October; but the following January, 18; and the first Roman cycle corresponded in the month of January to the Constantinopolitan cycle 12. But to the 13th cycle in the following October. The lunar cycle was the same as that of the Latins; and it was less by three than the Roman and Alexandrian. But, as I have already warned, neither in the matter of the embolisms nor in the celebration of Easter did they differ from the institutions of the Alexandrians and Catholics. Finally, the indiction was the same as the Roman, or imperial, as they call it, from September; or ecclesiastical from January. Therefore, it was greater by a unit than that which emerges from the Alexandrian computation: which comes to be noted especially so that from it we may detect the errors of the Greek chronologers, and especially of Theophanes. We shall offer a specimen of this observation.

Theophanes says that Heraclius began in the year of the world 6102, of the Incarnation 602, October 4, weekday 3, indict. 14. This is the Dionysian year of Christ 610, endowed with solar cycle 3, letter D. Thus there is an error in Theophanes, and it should be read as weekday 1, not 3. The Constantinopolitan indiction from September was 14. Therefore, Theophanes thinks this year is 6102 from Adam; 602 of the Incarnation. But Cedrenus [says] of the world 6105, of the Incarnation 609. But as far as the years of the Incarnation are concerned, the *Miscella*, which is composed from the history of Theophanes, has the same year 602, so that it is certain that it was written by Theophanes thus: of the world 6102, of the Incarnation 602: with 6102 divided by 15, there remain 12; with a unit added, by the precept of Maximus, the indiction 13 is gathered, that is, the Roman from January to the end of December; but the Constantinopolitan from January to the end of August. Wherefore the indiction of the Alexandrian computation, which arises from the division of the *omados*, if we do not order the years from September, requires a unit to be added to itself from January to September; from September to the end of the year, it must be increased by two, so that it may attain the civil Constantinopolitan. Therefore, in the year of Christ 610, it was the 6102nd year of the Alexandrian era begun from the spring month. But from September

the civil year began to be reckoned [as 6103]: which Cedrenus followed; but before him, Theophanes, who, as we have seen, is accustomed to begin years from Easter in the ecclesiastical manner. ^C The same Theophanes narrates that Leo the Isaurian died in the twenty-fifth year of his reign, while it had already begun by two months and twenty-two days, on the 14th of the Kalends of July, in the 9th indiction, which the *Historia Miscella* also reports at the end of the twenty-first book. "The year was, from the creation of the world according to the Romans, 6248 from Adam; according to the Egyptians, that is the Alexandrians, 6232; and from Philip according to the Macedonians, 1063." Which Paul the Deacon translated thus: "When the year was from the creation of the world, according to the Romans, six thousand two hundred and forty-eighth from Adam; according to the Egyptians, however, or Alexandrians, six thousand two hundred and thirty-second; from Philip, according to the Macedonians, one thousand sixty-third." This was the 741st year of the common Christian era, in which was the 9th indiction, both Roman and Constantinopolitan, in the month of June. With 5508 ^D subtracted from 6248, there remain 740 years of the Dionysian era. Theophanes therefore calculates one year too few; which the indiction also demonstrates: for if you divide the years 6248 by 15, an 8th indiction results. Yet in the Constantinopolitan era, which Scaliger calls the Paschal and Theophanes calls the Roman, once the years from Adam are divided, [the correct] indiction remains. Thus, if the 6232nd year of the Alexandrian computation is divided by 15, a 6th indiction remains, two years less than the Roman, although it should exceed the Roman by only one. Likewise, if 5492 is subtracted from 6232, the result is 740, whereas 741 should remain. Finally, the year of Philip was passing in that same time as 1064, not 1063. Therefore Theophanes is incorrect, and it should have been written by him that after the death of Leo, Constantine succeeded in the year 6249 according to the Romans, 6233 according to the Alexandrians, and 1064 from Philip.

CHAPTER VI.

Now let us come to examine those things which Joseph Scaliger produced concerning these Greek computations in his work *De emendatione temporum* (p. 749 of the latest edition), which, to speak generally, are of such a kind that they demonstrate the man to have been completely unskilled and unlearned in the science of them. Nor shall we merely repeat those things which we have already refuted in the two chapters discussed by him in book V of *De emendatione temporum*, and in chapter V of the ninth book of *De doctrina temporum*. Here, only those things must be touched upon which he argued in the seventh book regarding the computation of Maximus; which had been reserved for this place and for this edition of the *Computus*.

Among these, the first thing is most openly false: that these three computations have the nativity of Christ Dionysian as Christ's birth, [specifically] the Oriental [computation] in the year 5501, the lunar in the year 5495, and the Paschal in the year 5309. (For we warned at the beginning of this book that those three eras are so called by Scaliger.) Whence he advises that to the Dionysian years of Christ one should add the years 5494, 5508, and 5500, and recommence the beginning of the year from September: a thing which he makes clear by this example. In the year of Christ 1596, the year 1597 began from September in the Greek manner; add the years 5500, and you will have the year 7097 of the Oriental era; add 5508, and it will become the year 7105 of the Paschal era; add 5494, and it will be the year 7091 of the lunar era. But in this last lunar computation he orders two to be subtracted by a perpetual method, so that consequently that year is reckoned as no more than 7089. This, I say, is a most false precaution, as those things which I explained in the previous chapters about these three computations demonstrate. For two of them, the lunar and the oriental, provide the same year for the common nativity and the Dionysian, which is 5493 beginning from September. Maximus, George, and Theophanes used the era which Scaliger calls oriental. This is beyond controversy. For they assert that Christ was incarnate and brought into the light in the 5501st year of their own computation. Therefore the year of the Dionysian era 641, in which the 31st year of Heraclius was running in the Paschal month, ought to have been computed by them as 6141, with 5500 added to 641. Yet they establish that same year as the 6133rd year of the world, which is the number in the lunar computation; if to the 641 years you add 5492, as the method demands... Thus the first year of Heraclius, which began in October of the year of Christ 610, ought to have been reckoned as 6110 of the Oriental era, but it is reckoned by Theophanes as only 6102; which is the number of the era which Scaliger calls lunar.

Furthermore, George and Theophanes, as we noted above, call the Oriental era of Scaliger *according to the Egyptians* or *Alexandrians*, but the Paschal, *according to the Romans*. That passage of Theophanes is notable, which we cited in the previous chapter, where he says that the year of Christ which corresponded to the 24th year of Leo the Isaurian, and was 740, is 6248 according to the Romans (that is, in the Paschal era, as Scaliger calls it); *according to the Alexandrians* 6232. With 5508 subtracted from 6248, there remains the year 740; but with 5500 subtracted from 6232, as Theophanes is accustomed to do, the remainder is the year 732 from the Incarnation, eight years less than the Dionysian era, as is consistent: but with 5492 years removed, the residue is the 710th year of the Incarnation according to Panodorus. It appears, therefore, that each computation—one of which Scaliger named lunar, and the other oriental—collects an equal number of years from Adam. And if the Scaligerian method were true, and 5500 were to be added to the year of the Dionysian era for us to arrive at the year of the oriental computation (or *according to the Alexandrians*, not indeed the world [year] *according to the Alexandrians*, 6232), the year 740 ought indeed to have been reckoned as 6240.

Furthermore, it was also written falsely by Scaliger that the era which he calls lunar shows the nativity of Christ in the year 5495: but that according to the method it is truncated by 2 years: which is a fiction that is irrational and ridiculous. For it relies neither on reason nor on authority: rather, it is refuted by both. Since indeed the author of that computation was, as it appears, Panodorus. But he

attributes 5493 ^A to the Incarnation, as George attests. Truly, the calculation of cycles and indictions explained by us so far condemns this same error.

One person cited by him as a proponent of this opinion is Cedrenus, whom he claims follows that era which aligns the birth of Christ with the year of the world 5495; therefore, by the perpetual method, 5494 must be deducted from his era. Although that author is trivial and does not adhere to a consistent calculation, yet Scaliger (p. 365) did not even truly state regarding him that the common birth year aligns with his own era of 5495.

He explicitly affirms here first (p. 144) that he considers the years of the Incarnation to be 5506, and a little earlier he writes that the 15th year of Tiberius is the year of the world 5536. The 15th year of Tiberius is the 29th of the common era. Deduct 5494 years from 5536, as Scaliger commands, and the 42nd year of the Christian era will arise; but it will be the 30th if you remove 5506.

Besides these, he reports that the city of Constantinople was rebuilt in the 11th indiction in the year of the world 5858 (p. 233), on May 11th, the 2nd feria. Thus, the letter was D, and the year of Christ 330. If you take 5494 from 5858, the year of Christ will become 344. You see the *alogistia* (irrationality).

Again, the same Cedrenus says (p. 269) that Nectarius held the see of Constantinople in the year of the world 5888, the 594th of the Incarnation. The year in which Nectarius began to sit is 381 in the Dionysian era; he died, however, in 398. When 5494 years are deducted from 5888, 394 remain. But that is not the year of the Dionysian era, for which he knew 5494 years had to be removed to reach his method.

Fourth, he places the first Synod of Ephesus in the year 5915 (p. 276). Deduct 5494 years, and it becomes the year of Christ 421. Yet it was the year 431.

Fifth, he narrates that the Roman emperor, who held the empire with Constantine Porphyrogenitus, was reduced to order by his son Stephen in the year 6453, the 3rd indiction (p. 522). The year was 944 of Christ, in which the 3rd indiction was already in use on December 16th, when that happened. Now, if you take 5494 from the years 6453, the year of Christ 959 will remain. With 5508 years subtracted, however, the year 944 emerges. Wherefore Cedrenus held not the lunar, but the Paschal calculation in this place.

Sixth, he writes that Patriarch Theophylact died (p. 526) in the year 6464, the 14th indiction, on February 27th. ^D This year is the 956th of the common era, from which, when 5494 years are taken from 6464, 5508 remain. Consequently, Cedrenus uses the Paschal era here as well.

Finally, the same writer thereafter uses that calculation toward the end of the work which adds 5508 to the Christian era, as is evident from the indiction: such as in the year of the world 6468, 6471, 6497, 6508, 6518, 6524, 6526, 6527, and so on in the others. What rashness can therefore be equal to this, what greater negligence, than to assert that Cedrenus continuously calculated the years of the world in such a way as to add 5494 to the Christian era, when he adds 5508 almost everywhere; or, if he establishes it otherwise, that the numbers ^B are clearly faulty? There is, I suppose, one place in which he treats matters equally with the common era, which is thought to be 6021, in which he reports Justinian began; and which was the Dionysian year 527. If other examples were to abound besides this (for there is no leisure to search through everything), they are by far very few, and cannot be compared with the number of those we have produced. Wherefore none of the ancient Greeks ever thought of this era; no one curtailed it by two years: which is inept and frivolous.

But in the cycles of the sun and moon, which he adapted to Greek calculations in his *Notes on the Eclogues of Maximus*, what blindness! He asserts that the 15th cycle of the sun of the Paschal era, and the 9th of the Eastern, correspond to the first solar cycle of the lunar era; and to these three the letters B A, so that it might be the 5th Roman cycle: and that to the lunar cycle of the era, which he calls lunar, there correspond similarly in the Paschal calculation the 17th cycle, in the Eastern the 9th, when the 14th of the Paschal moon was April 5th. It is a wonder that when he wrote these things, he did not consult his Maximus, and notice the example of the Pasch described by him in the year of the world 6133, where Maximus imputes the first solar cycle and the 15th lunar cycle; and the Paschal limit, or the 14th day, he says fell on the Kalends of April, the first feria. If anyone else used the Eastern calculation, as Scaliger calls it, it is certainly Maximus. For he teaches that Christ was incarnate in the year 5501. Therefore, in the cycle of that era, there corresponds, from Scaliger’s table, the 21st solar cycle in the lunar era, and the 9th in the Paschal: but the 25th Roman cycle, that is the letters ED. Yet it is established that the letter G held with the 6th Roman solar cycle in that year: namely, in the year of Christ 642, in which the first feria fell on the Kalends of April. Also, from Scaliger’s table, the 14th of the Paschal [moon] in the 15th lunar cycle ought to have fallen on March 30th, so that the golden number would be 7, when nevertheless it is certain that in the same year the golden number was 15, and the Paschal limit occupied the Kalends of April. I could pursue these things more broadly, and expose with more examples the ignorance than which none greater can happen in this subject. But from the whole preceding discussion which we instituted concerning these calculations and their ratio among themselves, it is possible for anyone, even one instructed with a mediocre use of such things, to convict this infinitely patent hallucination.

CHAPTER VII

Regarding the Paschal method of the Greek calculations, and first regarding the *hemerophesion*, or the investigation of the feria. All its elements are set forth; what the regulars, the concurrents, and the *prosthetai* are. Scaliger's error in these. Maximus and Isaac are examined.

The Paschal method is divided into two parts; for the cycles are as many, of the sun and of the moon. Therefore, the prior part is the character of the day, or feria, which Maximus calls *hemerophesion*, the posterior points to the fourteenth Paschal day; and to the Christian Pasch itself; and it pertains to investigating the age, as they call it, of the moon. The reason for both will be explained by us, and we shall take our beginning from the former, which is proper to the sun. There are three things necessary for investigating the feria in the Greek calculation; *prosthetai*, or the epacts of the months...

annual epacts; and the *postata*, or the quota day of the month. Scaliger thinks (p. 749) that the *prosthetai* are what our calculators call "regulars"—in which he is greatly mistaken. Indeed, the very name of "regulars" can be accommodated to the *prosthetai*, just as we have done in the Latin version; but you should understand that the regulars of the Greeks are not the same as those of the Latins. We have treated these regulars and concurrens in Book VI of *De doctrina temporum*, Chapter XXVIII; the teaching of that chapter must be assumed for a knowledge of Greek calculation, for it is necessary. But it should by no means be repeated here, since it is readily available there, with the exception of a very few things, and those more critical, without which what I have proposed cannot be explained. ^A The Latin regulars are the feriae of the days in the year of the solar cycle which has the letter G, calculated from April, or even from January. If one calculates from January, the regulars of that month and the following are less by one than if we had started from April. When the Sunday letter is G, then the Kalends of April have feria 1, the Kalends of May, 3; and so on up to December, the feria is the initial letter of each month from A, plus one. But in the preceding January it was feria 2, and on the Kalends of February feria 5, where, with the addition of one, the letters display the feria of the Kalends. However, the calculation after the bissextile year begins more properly in March or April. Wherefore the regulars of January and February should be taken from the feria which their Kalends have in the following year, when the letter is F, then January begins on feria 3, February on 6. These are the regulars of the Roman months, which are perpetual and immutable; and they should not properly be called epacts of the months. For epacts, properly speaking, are the days preceding the beginning of a year or month, whose epacts they are called: just as lunar epacts are the days remaining from the completed last syzygy to the end of the civil year, for instance from December or March, if the lunar year begins in January or April; the annual solar epact is the feria of the last day of the whole year, as Maximus says. Therefore, the Kalends themselves of the month whose epacts are named do not come into the number of the epacts. ^D Therefore the *prosthetai*, that is, the epacts of the months, are not the same as the Roman regulars. Hence the annual solar epacts, or concurrents, provide the feria of the Kalends with the solar regulars. But the *prosthetai* with the solar epacts do not yield a feria unless the *postata* is added. Example: let the year of Christ be 1641, and let the feria of the Kalends of September be sought. The concurrents in that year were 7, and according to the method of Chapter XXVIII of Book VI of *De doctrina temporum*, when added to the regulars of September 6, they give feria 7, which in truth it was in that year, when the Sunday letter was G and the Roman solar cycle was 6. On the Kalends of April, the same concurrents 7 with the regulars give feria 1. But, according to the method of Maximus, the solar epacts 7 in the first year of the Constantinopolitan solar cycle, with the *prosthetai* of September 6, with the addition of the *postata*—that is, one day of September—produce feria 7; but the Kalends of April in the same year produce feria 1. The *prosthetai* of Maximus are the feriae of the last day immediately preceding the month whose *prosthetai* are spoken of in the first year of the Constantinopolitan cycle, which corresponds to the Roman cycle 6, when the letter is G; for then the last day of the preceding December falls on a feria. Therefore the regular of January of that year, from which the first year of the solar cycle immediately followed, at the beginning of April, is 1, and the feria of the last day of January is 4. This is the regular of February, and so on, as is evident from the table: which consists of four lines. ^B It includes: I, the Roman months; II, their epacts or *prosthetai*; III, the days of the individual months; IV, those same days added up, in order that the reason for propagating the regulars may be declared, which Maximus sets out in Chapter XXVI of the first part of his *Computus*. For he commands that all the days from April to the end of the month immediately preceding that whose *prosthetai* we are seeking should be divided by 7; and he says that the remainder, or if nothing remains, the number seven itself, is the epact of the following month. For example: it is desired to know what the epacts of September are. From January to the end of August there are 243 days; from which 90 being deducted, 153 remain from the Kalends of April, which if you divide by 7, the remainder is 6, the regulars of September. That the calculation begins from the Kalends of April does not follow from this, that the regulars of April are 1, but rather 7; a fact which escaped Scaliger, who incorrectly increases the *prosthetai* of Maximus by one. Thus he says that the epacts of March are 5, the epacts of April 1, the epacts of May 1; in a word, he asserts that the *prosthetai* are as many as the initial letter of any month plus one. In this way the *prosthetai* of September will be 7; of October, 2; against the protest of Maximus in the double table of Chapter I, whose left column in the first verse contains the *prosthetai* of March 4: which Scaliger falsely thought were set by Maximus as 5. ^C In the fourth verse the *prosthetai* are of September and October, of the latter 1, of the former 6: which Scaliger produces increased by one.

Table of *prosthetai* in the Julian and Constantinopolitan calculation.

| Months | *prosthetai* | Days of months | Collected days | | :--- | :--- | :--- | :--- | | January | 1 | 31 | 31 | | February | 4 | 28 | 59 | | March | 4 | 31 | 90 | | April | 7 | 30 | 120 | | May | 2 | 31 | 151 | | June | 5 | 30 | 181 | | July | 7 | 31 | 212 | | August | 5 | 31 | 243 | | September | 6 | 30 | 273 | | October | 1 | 31 | 304 | | November | 4 | 30 | 334 | | December | 6 | 31 | 365 |

It must be known, however, that although Maximus derives the reasoning of the *prosthetai* from April, in reality it starts from January. For from the Kalends of April to the last of December there are 275 days, which when divided by 7, 2 remain, which ought to be the *postata*...

tributes remain 2, which ought to be the ^A *prosthetai* [additions] of January, if they are calculated from April; but January only has 1. Hence, the calculation is propagated from January in this way. Since the letter G is in use, the Kalends of January are a Monday; the epacts of January are 1, and the last day of January is a Thursday. For, having divided 32 days, that is, 31 of January and its 1 epact, by 7, 4 days remain; thus, having divided 91, which is the number of days of three months, having added the epact of January, by 7, the remainder is 7, the epact of the month of April; thereafter, however, nothing is added from April, since the epact of April is 7, or 6. The observation of this matter, however slight, brings no small light to the calculation of the Greeks; by our ignorance of which, Scaliger is involved in a manifest error.

Isaac the monk, in chapter IV of his former computation, calls the *prosthetai* of the months *epaktai* [added days], the recapitulation of which begins from April; and he concurs with that former reasoning. For he establishes the epacts of any given month as the day of the week of the last day of the preceding cycle in the first cycle, when the letter is G. We showed above in chapter V that the solar cycle of Isaac corresponds to the XVII Roman cycle, whose letters are AG. Therefore, the Kalends of October fall on a Tuesday; and because the last day of September was a Monday, hence the epacts of October are 1, the epacts of November are 4, the epacts of January are 2, and the progress of the calculation is not interrupted in that month, as in Maximus. See chapter IV of Isaac the monk.

Next follow the epacts of the years, which the Roman computers call concurrents. The number of the remaining days from the preceding year is, however, the residue. As if the first year begins on a Sunday, the last day will be a Sunday; therefore the epacts of the second year will be 1, and so on. Maximus begins the solar cycle from the Roman VI, for which the letter is G. The first month is April. Therefore, in the first cycle, on the April Kalends, it is a Sunday. The epacts are 7; for the epacts are the day of the week of the last day of the preceding year. The epacts of the second year are 1, of the third are 2, of the fourth are 4, on account of the leap day. For when the epoch of the concurrents is taken from March or April, the epact is simple, but for years beginning in January, it is assumed as double, just as the letter is wont to be double in the Roman computation: ^B of which the former is used from the Kalends of January to the leap day; the latter from then until the coming year. Maximus describes the epacts corresponding to individual years throughout the whole cycle in a wheel, which we placed in the first chapter: and the method is not difficult.

Although Maximus begins the year and the solar cycle from April, and January is the tenth month, yet in the reckoning of the *hēmeroeuresia* [finding of the days] in this month, and in the two following, he assumes the epacts of that year which follows as new from April: as he indicates in chapter XIX. Example: the year of Adam 6133 began from the Kalends of April of the year of Christ 141, in the Roman cycle VI, letter G. Therefore, the Constantinopolitan cycle corresponding to it began on the same Kalends. Wherefore, for January, February, and March there still had the cycle of 28, whose epacts are 6. For the regulars of February, therefore, are 4; if the epacts 6 be added, and the *prosthetai* 1, it will become Thursday on the Kalends of February. But it was 5. Therefore, not the epacts 6 of the current year, but the 7 of the following year must be applied, so that Friday may result. The reason for this follows from the arrangement of the regulars. For if the progression begun from April were not interrupted in January, and the regular of this were, as was proper, 11, and of February 5, from the epacts of the old year, and the regulars of February with the *prosthetai*, it would be represented as a Thursday. Now, in January, a new beginning has surreptitiously occurred for the imprudent, so that the competent regular is made one less, namely 1. Because of this, the epact is one unit greater, that is, of the following year, and it anticipates.

Investigation of the annual epacts, or concurrents, is proposed differently by Maximus in chapter XXIX. He commands that as many days be taken as the cycle of the year is, which is sought, with one being subtracted. From thence, for all cycles, one should divide the number by 4, without any example: that which proceeds from the division is added to the previous sum, and the whole having been divided by 7, the remainder will be the epact, or if nothing remains, the number seven itself. For example, let the epacts of the 26th year be sought. Because of the 25 past cycles, I take 25 days, then I divide 26 by 4, and 6 result. They are totaled as 31. Likewise, you will finish the epacts of the 28 solar cycles. Take 27 days because of the same number of cycles already passed, then divide the 28 cycles themselves, with nothing subtracted, by 4, and add the seven which are born from there to 27, and divide the sum 34 by 7. The epacts of the last year will be 6. The reason for this method is clear. For days increase for the epacts of the last years. Therefore, since the 28th year is not yet completed, it contributes no common symbol. But because it begins after the leap day, therefore the leap day itself enters into the sum. For this reason, the division of all the cycles is performed without anything being subtracted. Isaac Argyros uses this method in chapter III. Lastly, I will not omit this; even if, according to the Roman custom, the leap day is fixed between the 24th and 25th of February; and from there, according to the Roman rite, a new letter succeeds; nevertheless, Isaac considers the leap day to be the one which is the 29th, and the last; just as he declares in the fourth chapter, which we remember others to have observed, ^D indeed, that it was transferred to March by some. Concerning which, see what we have noted in book IV, chapter XII, regarding the place in Galen described there.

CHAPTER VIII

For the diverse eras and computations of the Greeks from the creation of the world, there are likewise various cycles of the sun and the moon: and various methods of collecting epacts: which they could have reduced to a single, simple one; if they had added some years to the current year of their computation; and did not prefer to pursue that convenience, so that with the sum of their era distributed by the number of cycles, the cycle of each year proper

to everyone might arise. For example, we reckon 3984 years from the foundation of the world to the first year of the common era, such that it is the last year, which we make the first year of Christ. Scaliger and others count 3950 years. If that convenience pleased us, so that the sum which we define from Adam was distributed precisely, the current cycles would be produced: by dividing 3984 by 28, the remainder would be solar cycle 6; by dividing by 19, the cycle of the moon would be 13, just as from the division of 3950 years, the solar cycle would be 2, and the lunar 17. But there was the Roman solar cycle 10, and lunar 2, in the first year of the Christian era, in which we both agree. Therefore, to find the cycles, Scaliger adds 8 to the sum; so that by dividing 3958 by 28, the solar cycle 10 is found; and to the lunar 4 is added; so that by distributing 3954 by 19, the lunar cycle 2 emerges. We add 2 to the method of the solar cycle, and 8 to the lunar cycle. Thus, by dividing 3986 by 28, the remainder will be 10; and by dividing 3992 by 19, 2 remain. ^B Thus they avoid the cumbersome variety of cycles and chronology in the Latin computation from the foundation of the world, which the Greeks followed too ambitiously.

Since there are two divergences in Greek computations, as it is often necessary for us to repeat, and since the leaders of both—Maximus and Isaacus—are published by us in this work, we will explain the method of each as derived from both, in a few words; and first that of Maximus. He follows the Alexandrian era, in which the first year of the Christian era is counted as 5493: by which sum, divided by 28, the solar cycle is made 5; and divided by 19, the lunar cycle is 2. But the Roman solar cycle was 10, and the lunar 2. Therefore the solar cycle in this era is effectively the Roman cycle of 6, while the lunar is one and the same in both. The epoch of the year and the lunar cycle (for I have already spoken of the solar one) is set by Maximus at the Kalends of April. The head of the enneadecaeteris is the golden number 1. The first new moon is March 23. Thus, in the first year of the lunar cycle, on the Kalends of April, the moon was the tenth day. ^D Hence it is that Maximus writes in chapter XXVII that the epacts of the first lunar year were 9, since the age of the moon was that much on March 30, which day is the canon for the epacts of the sun and moon, as he also says in chapter XXXI. Not noticing this, Scaliger, in his *Notes on the Eclogae of Maximus* (p. 753, last edition), thought the sum was reduced by two units: [falsely assuming] that from the time of Christ the moon had anticipated by two days, than which nothing more ridiculous could be devised. But more on this error another time.

^A Regarding our belief in chapter V of the ninth book, *On the Doctrine of Times*, that Maximus had obscured the lunar *proēgēsis* by one day from the time of the Council of Nicaea—because he had established the canon of the age of the moon and the epacts on March 30, not March 31, as was just, if indeed the epoch of the year is fixed by him at the Kalends of April—know that this is an error which the corrupted reading of Scaliger produced for us. Scaliger, who interpolated the *Eclogae* of Maximus (p. 739) to his own liking and did not care to express them even in his own words, had written: *“H postaia tēs selēnēs en tē triakadi tou Martiou mēnos,”* etc. *“H de postaia tēs hebdomados kata tēn triakada prōtēn tou autou mēnos, tas tou hēliou.”* But [this is] wrong. For it should be written *triakadi prōtē*. This *ecloga* is taken from chapter XXXI of the first part of the *Computus*: in which it is read as I have said.

This two-day defect creeps through the whole cycle of the moon; and the epacts are perpetually two less than the Roman ones. For example, in cycle 14, the common epacts are 4, but by Maximus 2, as he says in chapter XXVII. And this is the natural progression of epacts, and it retains the genuine condition of epacts: which is such that every year has as many epacts as there are days remaining from the new moon immediately preceding the head of the year. In lunar cycle 14, on the Kalends of April, the Nicene moon was the third day. Therefore the epacts are deservedly 2. In the Roman computation, the vulgar method counts 4. Regarding which, see book VI, *On the Doctrine of Times*, chapter XXIII.

^C This method of finding the epacts is handed down by Maximus, chapter XXVII: multiply the lunar cycle by 11; divide the sum, minus 2, by 30; you have the epacts of the year whose cycle was multiplied. For lunar cycle 14, if you multiply 14 by 11, you get 154. Subtracting 2, and dividing the remainder 152 by 30, there remain 2, the epacts of cycle 14. So that the whole matter may be subjected to the eyes, we shall transcribe a table in which the lunar cycle has attached to it both the Roman and common epacts, as well as the truncated ones, and those proper to Maximus.

Table of the Lunar Cycle.

Years | Roman Epacts | Maximus' Epacts I | 11 | 9 II | 22 | 20 III | 3 | 1 IV | 14 | 12 V | 25 | 23 VI | 6 | 4 VII | 17 | 15 VIII | 28 | 26 IX | 9 | 7 X | 20 | 18 XI | 1 | 29 XII | 12 | 10 XIII | 23 | 21 XIV | 4 | 2 XV | 15 | 13 XVI | 26 | 24 XVII | 7 | 5 XVIII | 18 | 16 XIX | 0 | 28

From these, the age of the moon in any month, and on the day of [the month]...

is apprehended. Maximus in chapter XVIII, thus regarding this matter, is precipitous. Having found the epacts, add all the days from the Kalends of April to the one whose age is being sought, divide the sum by 29 1/2, and the remainder will be the age of the moon. This method is troublesome, and not useful for the general public, on account of the accompanying hourly epilogisms. It will be more expeditious if you use regulars: the propagation of which Isaac explains in chapter IX. Moreover, they can be propagated in a dual manner: the first is that the number of each month is the regular, which, when added to the annual epact in the first cycle, completes the age of the moon on its Kalends. In the first year of the cycle, the lunar epacts are IX; there is one unit lacking to complete the age of the moon on the Kalends of April, for it is the tenth; and thus it is for the remaining months. The second mode subtracts one from the prior kind of regulars. Or, it is the number remaining after the golden number has been set on the Kalends of April in the preceding month, starting from April. In the third cycle of the moon, the new moon falls on the last day of March. Therefore, there is no day remaining in that month after the new moon, and so the regular for April is 0. But for April, after the new moon, which happens on the 29th day, one day remains. Therefore, the regular for the following May is 1. Between both kinds, this is the difference: that the earlier regulars exhibit the age of the moon with the current epact, without the addition of any day, whereas the latter ones add the Kalends themselves. The following table will comprehend both kinds.

Table of the double lunar regulars in the cycle of Maximus.

Example: In the lunar cycle XIV, it is desired to know what moon it is on the Kalends of September. Epacts 2 with the prior regulars 7, or with the posterior 6—but with a unit added—give the ninth moon, as is indeed the case in the Nicene cycle XIV. Again, let the age of the moon be sought for lunar cycle V on the Kalends of October. Epacts 23, with regulars 7, or 6, provide the thirtieth moon from the Kalends themselves.

CAPUT IX. Concerning the method of the moon in the Constantinopolitan computation.

Now the method of another era and computation, which they call Constantinopolitan, must be opened, from the book of Isaac, who diligently explains it. In it, the procedure for investigating the cycles is the same as in the superior one; so that when the sum is divided by any cycle, the competent cycle of the year which is sought will be revealed. But since the number of years of this era exceeds the prior one by sixteen, hence, by adding 16 to the cycles of the other, the cycles of this one will arise. The first year of the Christian era is 5493, which divided by 19 gives 2. Adding 16, there will be a lunar cycle of 18, which the year 5509 of the posterior computation gives, if it be divided by 19. Thus, adding 16 to the solar cycle 5 of the prior era, there will result the solar cycle 21 of the year 5509. Likewise, to the indiction, which is left over from 5493 divided by 15, if 16 are added, the remainder will be the indiction 17 of the year 5509. Isaac repeats the head of the solar cycle from October, and the lunar from the following January, in chapters I and VI. Therefore, the lunar cycle 1 Constantinopolitan is the fourth of the Roman; and the first of the Roman is the XVII of the Constantinopolitan. For since the lunar cycle of the Alexandrine computation is the same as the Roman; if you add 16 to this cycle, suppose to 1, you will make 17 of the other. The progression of the epacts effectively, as in the Roman, is accomplished by the multiplication of the golden number by eleven: and it is not mutilated by two days, as Maximus institutes, but those epacts are what we called Roman in the schedule of the lunar cycle in the previous chapter. But the first year of the Roman cycle, or that which Maximus uses, and which has epacts XI, is the fourth of the Constantinopolitan. Epacts are the remaining days which are applied to the days of any month to complete the lunation. In the first year of the cycle, the epacts are eleven. This is the fourth of the Roman year: to which the third preceding left the remaining eleven days from the last syzygy, which began on December XXI. Therefore, the natural and proper perception and appellation of the [ἐπακτῶν] in this cycle is this.

The method of the regulars is equal to that in the previous one, and, as there, they are of two kinds. Some accomplish the age of the moon with the epacts, others are smaller by a unit. Then, because the epoch is led from January, not from April, no small variety occurs. We will describe both kinds in the following schedule.

Schedule of the double lunar regulars in the cycle of Isaac.

Example: For lunar cycle XIV, let the age of it be sought on the Kalends of September. The epacts are IV, which with the regulars of September VIII, or with VII, with a unit added, give the twelfth moon: which is indeed the case in the Roman cycle XVII, Constantinopolitan XIV. Isaac in chapter IX for the regulars, the remainder

the Julian months, with the whole syzygies subtracted, receives. The lunar syzygy is 29 1/2 days, from which, if 31 are deducted, 1 1/2 remain in February to be added to the annual epacts. But we have pointed out above that this method is by no means suitable for the common use. Therefore the Romans have by no means used it. Wherefore, ^A in order to adapt the epilogisms of the Greeks to the use and arrangement of our own Calendar, we combine the regulars with the epacts. One thing, however, must be observed: that with the cycle of Isaac, epacts VIII, not VII, are to be compared; and with the cycle XVIII, epacts XIX, not XVIII. The reason for which is to be sought from the *τομή* (section) of the moon; from which arises what they call the leap of the moon. For this indeed happens in the Roman cycle XIX, and the Latin cycle XVI, as we have said in Book VI *On the Doctrine of Time*, chapter XVII. Because of this *τομή*, the golden number XIX anticipates its position by one day in the description of the Roman Calendar. For this reason, not XI but XII must be added to the epact of the preceding year XXVI; so that the seven epacts of year XVII are formed, to which if XI are added, the nineteen epacts of year XVIII arise, which is the second of the Roman cycle; and finally, when XI are added, the epacts of year XIX become 0, which is the third of the Roman cycle. This, however, has its place in the arrangement of the golden numbers; of which kind is our own Calendar. For if it were established otherwise, another method of calculation would have to be entered into.

CAPUT X. On the lunar *προήγησις* (anticipation), and the shift of the terms. The ridiculous method of Isaac Argyrus. Why the Pascal fourteenth day is called the full moon. Various errors of Isaac; the equinox falsely observed by him.

Up to this point we have expressed the lunar method of both computations; which, in the Nicene century and thereafter, could be applied to the anticipation of the new moons: a method which, since older masters of calculation, such as Maximus, Dionysius, and Bede, did not know, they believed that that was the perpetual art of inquiring into the lunar age. Isaac, however, as he was skilled in astronomy, by the aid of celestial observations and astronomical calculation, noticed the anticipations of the sun and moon in the Julian year. Therefore, he discovered that both the Nicene—or rather, the Alexandrian—Paschal new moons, as well as the terms, were useless in his own age; as they had gradually made a transition from the first month to the second. Whether Isaac was prudent or imprudent in establishing this *προήγησις* of three days in collecting the age of the moon, or whether he acted either ^D ridiculously or maliciously in transmitting the method for it, is in question. For in chapter VII, teaching how the lunar epacts are to be investigated, he orders three to be added to the epacts. For example, in the year of Adam 6880, the cycle of the moon was 2, the epact 22, which with the regulars of January 1, return the twenty-third moon on the Kalends of January: as it was in the Nicene century. But in Isaac’s time it was the twenty-fifth. For the new moon in the year of Christ 1372 occurred on December 8, whose character—feria 2, 3, 28', 27"—at Constantinople, with the Roman cycle of the sun VIII, letter E. Hence the 25th moon fell on the Kalends of January of the following year 1373. Thus, on October 25th of the same year, the 25th moon occurred; which, by epact 12, the subsequent regulars 7, and the *προστασία*, that is, 26 days, is completed. The new moon fell on September 28, feria III. Whence the twenty-ninth moon corresponds to October 26, but Isaac thought the neomenia was from the 29th day of September. Therefore, he adds three to the epact 22, so that it may be 25: which, with the epacts of the months 7, and 26 days of October, produce the twenty-eighth moon, with 50 subtracted. He offers a laughable reason why these three are added; because, since the full moon was created by God on the fourth day, it is fitting that three days should have preceded it. Furthermore, in the first cycle, the 15th moon was found on the Kalends of January; how much it was at that moment when it was created. Since, therefore, the first cycle has eleven epacts by the method; by this reasoning it is feigned that the new moon occurred eleven days before the Kalends: to which those three are added,*πρὸ τῶν αἰώνων* (before the ages), as the Greek computers name them; they assign the fifteenth moon to the Kalends of January. For this reason, an addition of three is always made to the epacts.

This method is not even worthy of a beginner in astronomy, and is most unskilled. It proceeds first as if the world were created in the month of December, which no one hitherto has supposed. Then, although this may be granted voluntarily, it can be resolved by nothing more. For if he says the truth, it was necessary for all past centuries, and even for the Nicene, that those three days be added: which nevertheless was not necessary, since the simple epacts represented the age of the moon. Furthermore, Isaac himself, in chapter XVI, where he discusses the anticipation of the lunar and solar year, believes that in 304 years, the syzygies depart from the precise, civil, and by no means precise ones by one day. Whence it is a consequence that, 1,047 years before the Nicene synod preceded the age of Isaac and the year 1372, it had been more than three days less than those days on which they were incurred in his age. But then, if you were to add three days to the epacts, you would by no means catch up to the moon. Isaac, therefore, is most absurd; for when he perceives that the new moons by no means adhere to certain days, he reasons in the meantime as if they always consist in the same day: which was done by him either unskillfully, if he prescribed it in good faith, or cunningly, if for the sake of avoiding envy, lest he seem to know more than his ancestors, he did not wish to openly profess what the truth of the matter was.

For the rest, Argyrus was not the first to detect that defect of the Paschal computation: but Nicephorus Gregoras, who flourished at the same time, who, as he testifies in chapter XVI, offered a Paschal table revised by himself to the emperor and the senate; in which the terms were placed two days before the Nicene ones. You have both canons—the old, namely the Nicene, and the more recent—in the computation of Isaac; the former indeed under the name of Damascene, who perhaps arranged it thus, in chapter X; the other at the end of the former computation, concerning which revision of Nicephorus we shall speak passim, which things look toward illustrating the description of the Gregorian Calendar, and defending it against the attacks of Scaliger and Calvisius, this is, toward Book V of

^A *De doctrina temporum* adapted. He who previously wished to correct the institution, ought to have advanced the new moons and the limits by a full three days. But by that method, the limits would very often have preceded the full moon by two days, as we said happened in the Nicene age. Wherefore it was better to advance them by two days toward the earlier time. Which Nicephorus also performed. But not so that he might remedy that inconvenience which I mentioned, to which the founders of the Gregorian Calendar had regard, but because they were persuaded that the Nicene Paschal limits had coincided with the full moons. This was the error of Theophilus, Cyril, and the remaining computers thereafter, up to Bede and his age, which we refuted in Book V of *De doctrina temporum*, chap. XXIII. Hence it is that they often call the full moon a limit, or the fourteenth day, as our Isaac also does, though not for the reason which Scaliger and Calvisius defend (even though Scaliger, having been taught by our Clavius, departed from his former opinion, as we demonstrated in the same book), as if it were never permitted to celebrate Easter on the fifteenth day from the new moon, but because it was necessary that it be spread even to the twenty-second moon: but rather for this reason, that the new moon was placed one day after the *akribes* (exact) new moon. So that beginners may better grasp this, I shall submit two tables of a single lunar cycle for both times, that of the Nicene Council and that of Isaac Argyros, so that by a comparison of both, the anticipation of the moon and the correction by Nicephorus may be easier to understand. In each table there will be seven lines. The first will give the golden number; the second, the Roman cycle of the sun; the third, the dominical letter; the fourth, the new moon of the first spring month with the character, at the meridian of Constantinople; the fifth, the civil day of the Julian month into which the new moon should have fallen; the sixth, the full moon of the same month; the seventh, the civil day of the full moon. And as much in the new moon as in the full moon, we observe that as often as the sum of the hours exceeds twelve—which are computed from midnight—the following civil day is assigned to both, a rule which the Nicene table did not at all hold. As in the first year, when the new moon occurred roughly at the sixth hour after noon of the 20th of March, the new moon was nonetheless set on the 20th day itself, which by astronomical calculation is poured over into the following day. We have excerpted these conjunctions from the table of chap. X, book VII of *De doctrina temporum*, which are no less accurate than those that are composed from the astronomical tables of book VIII, and they do not differ much from them, and on account of the attached character of the day of the week, they appear more convenient for the business.

^B He who arranged the Paschal new moons and limits in that register by which all Catholics have governed ecclesiastical times from the Nicene age to our own, diligently obtained accurate and celestial lunar movements in setting the new moons; whence it happens that they agree exactly with those days on which the new moons were brought about according to the *homalen kinesin* (mean motion); but the Paschal limits, or fourteenth-day moons, very often precede the full moons by two days, for the reason that the full moon happens most frequently on the sixteenth day from that which had the new moon—on which subject [see] Book V of *De doctrina temporum*, chap. XIII. Hence it not rarely happened before the Gregorian edition that Easter fell one day before the full moon. After many years, when the moon had already made an anticipation (*prohegēsin*) of one day, the old limits fell almost on the day before that on which the mean opposition was occurring, as is evident from chap. XIII, lib. V of *De doctrina temporum*. Thence after some centuries, the anticipation of another day moved the old mean opposition into the limits, so that then the Paschal limits were in truth the full moons, and they happened on the sixteenth day from the new moon, which indeed happened under the time of Bede. Furthermore, in the age of Isaac, the *prohegēsis* was little more than three days; in the age of Gregory XIII, it is almost finished at four days, so that the Nicene limits gave the eighteenth moon. To this inconvenience of the Paschal new moons and limits did the translation bring a remedy; which, with the additional removal of ten days, which the solar year demanded, was done so opportunely that they descended from the ancient and Nicene limits by only one day. Anno Christi 1523, the Paschal new moon in cycle 3 falls on March 20, together with the astronomical syzygy; the limit, April 5. In the year 1577, in the same cycle, the new moon was brought about on March 19, the Paschal fourteenth day, on the Kalends of April. With ten days expunged, the Nicene new moon fell on the 29th day of March; the limit on the 11th of April. But in this way the full moons would have happened about two days early, and frequently. So that this might not happen, the Gregorian correction moved both the new moons and the Paschal limits by one day; as if the moon had gathered an anticipation of only three days. Thus the golden number 1, which is posited as having made a progression to March 20, descended to March 30; so that the limit might tally with the 10th day of April. Hence it happens that the limits more ^C often concur on that day which is the day before that on which the full moon happens. Of which matter it has been treated more fully in [Book] V of *De doctrina temporum*. To return the discourse to the age of Nicephorus and Isaac, when the moon then had an anticipation (*prohegēsis*) of three days; if...

one considers the first table of the Nicean enneadecaeteris and the second of the time of Isaac Argyrus, it appears how great the *prohegēsis* of the moon has become since the Nicean time down to Isaac, namely, more than three days and 8 hours. For let the year 324 be taken, in which the Roman cycle was 2, and the Constantinopolitan 18, with the new moon on March 12, at 2 hours, 38 minutes after midnight, at Constantinople; and let there be compared with it the year 1369, endowed with the same cycles; the new moon of which falls on March 9, at 0 hours, 23 minutes after midnight. And since the former year is leap year, and the latter is the one immediately following a leap year, therefore, with 6 hours subtracted from the latter, let 8 days, 18, 23' be subtracted from the former new moon; the difference will be 3 days, 8 hours, 16 minutes. Such is the anticipation of the moon from the Nicean time to the age of Argyrus.

^A But in order that what we previously showed regarding the accuracy (*akribeia*) of Isaac may appear more certainly, certain things written by him in Book XVI of his *Computus* regarding the lunar *prohegēsis* must be called to examination. He suspects that the Paschal Canon, which he proposed in Book X, and whose Paschal limits Nicephorus anticipated by two days, was composed 608 years before he published these things, because it delays the Jewish Passover by two days, even though he asserts that those two days are not yet completed; but that three hours still remain of them; to exhaust which, eight and thirty years are required. Indeed, at the beginning of cycle 3 (this is the Dionysian year of Christ 1373), the Jewish full moon falls three equinoctial hours after the sunrise of the 5th day of April. It was the Constantinopolitan year 6881. And so, after three years

sixty-eight, that is, in the year 6919, which is the Dionysian year 1411, it will be complete, he says, a delay ^A of two days. This is what Argyrus says. And indeed, in that very year which he names, A.D. 1373, the Jewish full moon occurred about three hours after sunrise. It was the Jewish year 5135, whose Tishri began on August 30, feria 2, in A.D. 1372, with the Roman solar cycle 9, and the letter DC. For its character was 2, 3, 606. But the Nisan character was 5, 20, 757, feria 6, on account of the shift; March 25, Roman solar cycle 10, letter B, A.D. 1373. The full moon, therefore, occurred on April 8, feria 6, at 15, 73 1/2, that is, at 15 hours and 73 minutes after sunrise. For the Jewish epilogism begins at sunset. But what a miraculous stupefaction of the man, who did not perceive that the Paschal limits, which are described in the table inscribed with the name of Damascenus as its author, are none other than the Nicene ones, which Theophilus, Cyril, and the Catholics thereafter used. But Isaac believes that about the year of Christ 765 that canon was compiled and the limits set. For having subtracted 608 from 1573, there remains the year ^B that could have been the creator of that artificial table; yet he did not establish the limits. But Argyrus was deceived by that which I just noted. He thought that at the time when the Paschal limits were fixed, the full moon arrived upon them. This, in truth, happened in the age of saint Damascenus, since in the earlier Nicene age the full moons anticipated them by nearly two days. This is the source of the error: to which another is to be immediately added, that he wishes the Jewish Pasch to be perpetually the fourteenth Paschal day of the Christians; nor does he understand that the Jews generally celebrated the Pasch before the vernal equinox: then indeed, on account of various shifts, they spread it to the fifteenth or sixteenth day.

Moreover, the same Argyrus writes ridiculously about the vernal equinox, having himself detected it before the fifteenth day of March from an observation of the solstices: as if it were not much safer and easier to observe the equinoxes themselves by themselves, ^C than to deduce them from the ratio of the solstices. For the observation of these is difficult and slippery on account of the very slow and insensible progression of the sun from the tropical points, so that it is more to be wondered at that any astronomer at all should have attempted a method of knowing the vernal equinox, rather than that, having used this method, he should have been off by about three days. For the true vernal equinox in the year of Christ 1372 occurred at dawn on March 10, feria 6, at 1 hour 11 minutes after midnight, in Constantinople. But how can it be possible that in the time of Isaac the equinox should have fallen on the fifteenth or fourteenth day of March, when nearly a hundred years later, that is, in the year 1488, Joannes Regiomontanus discovered the same on March 11, at 3 hours, 40 minutes after midnight? Which, having been extracted for him from our tables, was accommodated ^D to the time of Isaac. Then Copernicus, in the year 1516, observed it on March 11, at 4 hours, 20 minutes. Therefore, that observation of Isaac Argyrus is false; just as when he adds that Ptolemy observed the equinox on March 21; since it is most certainly established that it was noted by him as having been after noon on March 22. Finally, as to what he writes—that fifty years ago in the town of Aenus in Thrace the Jews celebrated the Pasch on March 20, when the Christian one was celebrated on April 23; the letter was A, and it was the year of Christ 1318, solar cycle 11, moon 8, Tishri new moon in the year 1317, solar cycle 10, letter B, September 8, feria 4, 8, 286, Nisan March, feria 6, 12, 724, in the year 1318, letter A. But because of the shift, the new moon was March 4, feria 7, 5, 40 1/2. If the new moon happened on March 4, the Pasch was celebrated on the evening of the 17th day, and the day following was the first day of unleavened bread. Wherefore it does not correspond with Isaac's description. Furthermore, the Christian and Nicene Pasch was celebrated on April 21, when the limit fell on the 18th, a Tuesday.

CHAPTER XI

There is also another method for Greek computers for knowing the Paschal limits, or Jewish Paschs; which the Alexandrian Chronicle follows: not much different from it is that which is read in the computation of Andrew of Jerusalem; whose explanation will contribute not a little to those things which have been explained in the preceding chapters.

Through 19 years distributed from Adam, it commands one to find the cycle of the moon and the epacts congruent to it. To these, then, it commands one to add the twelve days *tas pro ton phosteron*, which precede the luminaries; likewise, *proselinous*, the seven before the moon, then to add as many days as are necessary to complete the fourteenth. The method is easy. From the Kalends of March to the eighth day, which is the first Paschal new moon, there are seven days. For this reason they are called *proselinous*; because they precede the first new moon. Then to March 20, there are 13 days, which are called *pro ton phosteron*, in that they immediately precede the first and last of all the fourteenths, which is March 21. Moreover, the opposition of the luminaries was thought by them to occur at the Paschal limit. And so, without complications, a rule could be given for adding 20 days to the epacts. Now, if the sum of the days accrued is less than 44, add the remainder until you complete 44: whatever has been added will be reckoned after March 20, and will show the Paschal limit. But if 44 days or even more are counted, add only as many days as is necessary so that, with the days removed, fourteen remain, and begin to reckon that very thing which you added after March 21. For any fourteenth falling before March 21 is not yet Paschal: for that reason the calculation must be carried further. Let the Constantinopolitan cycle be 14, which is the Roman 17; the epacts are 4, which, with March 20, make 24. But in order that, with the complete syzygy removed, there may remain 14 days, add 20. There are 44 days. Furthermore, those added 20 arrive from March 21 to April 9; on which day the 14th Paschal occurs, with the Nicene moon cycle 17. Again, with the Constantinopolitan cycle 5, Roman 8,

are 25, which with the 20 days of March give 45; and so that 74 days thus exist, one must add 29. These will amount to 49; from which, subtracting the 31 days of March, 18 days of April remain, on which day the Nicene Paschal limit falls in cycle VIII.

But Saint Andrew of Crete, so that ^A the same limit may be obtained, multiplies the current lunar cycle by 11. He adds six days to the total sum, which he calls *ap’ aiōnos* [from of old], up to cycle XVI; but for cycles XVII, XVIII, and XIX, he adds seven. After this, he commands that the entire sum be divided by 30, and that the remainder be counted from March, and that as many days be added to it as are necessary so that the number may reach fifty, to such an extent that even something from April is added. The last of these days will be the Jewish Passover. We have found the reason for this decree, lest anything elude our diligence.

Lunar epacts, which precede March, are the days remaining after the last syzygy: their completion to 30 completes that lunation which cannot yet be the Paschal one. Therefore, the Paschal one will be that which, after it, has the fourteenth day on March 21, or next after the 21st. Therefore, one ought to add to the epact, beyond its completion that perfects the entire syzygy, 45 days, to obtain the fourteenth day of the following syzygy. If indeed this arrives at March 21, it will certainly be the Paschal one. If not, 30 days must be added to the whole, so that the Paschal limit may be had.

Wherefore that rule ^C could be shaped in this way: To the epact, add as many days as are needed to complete 44, from the first cycle to the 16th, or to complete 43 in the three remaining cycles, with the beginning made on the Kalends of March, if what is lacking is greater than 20. But if it is less, accommodate the entire remaining syzygy of 30 days. Whatever has been added, if it is counted from the Kalends of March, will end on the Paschal limit. This rule ends up the same as that of Saint Andrew. For he who adds six days to the epact, and then counts the remainder from the Kalends of March until he completes fifty, does the same as if, without adding those six days, he were to complete only 44. But when the lunar leap, and the *hypotomē* of one day occurs in cycle XVI, which is the Alexandrian 19th, as we have demonstrated above; for this reason the golden numbers of the following cycles offer the new moon one day earlier. For that reason, ^D Andrew adds seven days to the epacts in cycles XV, XVIII, and XIX, and we command the sum to be completed only to 43; so that the new moon may be fixed one day earlier than in the others. Let the Paschal limit for cycle V be sought as an example. The epacts are 25; when 6 are added to them, they become 31; and by throwing away 25, 1 remains. Therefore, add 49 days, and from them take away the 31 days of March, and 18 days of April will remain. Therefore, the Paschal limit falls on the 18th day of April in the Constantinopolitan cycle V, which is the Alexandrian VIII. But according to our rule, since 19 days are lacking to reach 44, which number is less than [the 20 we set], therefore they become 49 days; from which, 31 days of March having been subtracted, 18 remain. Again, let the limit be sought for the lunar cycle XVII. The epacts are 18, to which 7 are added, so that they become 25, to which 25 are lacking for fifty. Therefore, the fourteenth day of the Paschal cycle falls on the 25th day of March in the Roman cycle XV, and the Constantinopolitan XVI. The same thing will happen if you count the complement of 17 epacts to 43, that is, 25, from the Kalends of March. This method has a place in the Nicene syzygies. But if anyone wishes to conduct the lunar calculation by *proēgēseōs* [advancement], he can add to the epact as many days as the moon anticipates the former epoch, as Isaac Argyrus instituted.

Chapter XII.

It remains for us to experience the use and fruit of the Greek computation, which we have accurately set forth up to this point, in finding their principal feasts, which we commonly call movable. These are described furthermore in their Horologion, which was published at Venice in the year of Christ 1555. At the end of which a series of 40 years is described, from the year of the world 7044 to 7083: in which the times of each of the feasts and religious days are contained.

In the first place is placed *hē paramonē tōn Christou gennōn*, that is, the Vigil of the Lord's Nativity, whose feast day is noted.

Secondly, *Kreōphagia*, that is, all the days from the Nativity to the Sunday of Sexagesima, which is called *Apokreōs*, during which the eating of meat is allowed.

Thirdly, *to Triōdion*, which is the Sunday immediately preceding the Sunday of Septuagesima.

Fourthly, *Apokreōs*, the Sunday of Carnisprivium [the beginning of the meat-fast]; which we call Sexagesima, from which the Greeks begin to abstain from meats, 56 days before Pascha; which they do for this reason, that the fast of 40 days may be established in solidum. For since they consider it wrong to fast on a Saturday, with the one exception of the day before Pascha, if you strike out the 8 Sundays and 7 Saturdays from the 56 days of the 8 weeks, the remaining days will be 41. For the last Saturday is set apart for the fast outside the order.

Fifthly, the legal Pascha.

Sixthly, the Christian Pascha.

Seventhly, *hē Kyriakē tōn Hagiōn pantōn*, the Sunday of All Saints; which corresponds to Trinity Sunday.

Eighthly, the fast of the holy Apostles, which the Greeks celebrate from the Sunday of All Saints, up to June 28. For on June 29 they celebrate the feast of the holy Apostles Peter and Paul. The day after is *tēn synaxin tōn dōdeka apostolōn*. Moreover, this fast is sometimes longer, sometimes shorter. For it is of as many days as are counted from Pascha to the third day of June, as Isaac teaches in chapter XV. For from Pascha to the Sunday of the Apostles, there are 55 days. From the third of May to the 28th of June, there are just as many days. Wherefore, having taken away the interval from the third of May

to the Lord’s Day of the Apostles, there will be one and the same space from Easter to the third of June, which is from the Lord’s Day of the Apostles ^A to the twenty-seventh of June.

That all these things may be settled by the superior method, we propose the *diagraphē* of one year from the Horologion. Year 7044, indiction 9. Solar cycle 16. Lunar cycle 14. Forefeast of Christ’s Nativity, day 6. Meat-eating, 56 days. The Triodion begins February 19. Legal Easter, April 9. Christian Easter, April 16. Lord’s Day of All Saints, June 11. Fast of the holy Apostles, 17 days.

<Anno 7044, indictione IX, solis cycli XVI, lunae XIV, exspectatio natalis Christi, feria VI, carnium esus diebus LVI. Triodium incipit Februarii XIX. Legale Pascha Aprilis IX. Christianorum Pascha Aprilis XVI. Dominica Sanctorum omnium Junii XI. Jejunium sanctorum Apostolorum, dies XVII.>

This year is the year of Christ 1536, as is apparent if 5508 be subtracted from 7044. The Roman solar cycle was 5, letters BA, moon 14. In the preceding year, 1535, the Roman cycle was 4, the letter was C: hence the twenty-fourth of December was a *feria* VI. Dividing 7044 by the competent cycles, there will result the solar 16, lunar 14, epact of the moon 4. Add 40 days, and, with the days of March subtracted, the ninth of April remains, which was the Paschal limit. The *concurrentes* for the sixteenth solar cycle were 5, which, with the *regulares* of April 1, and 9 days, make 15: subtracting 7, the remainder is *feria* I, on which the limit falls. Therefore, Easter was prorogued to the sixteenth of April. Now, if from the sixteenth day of April, that is, from the 107th day from the Kalends of January (for it was a leap year), you subtract 57, 50 will remain, or the nineteenth day of February, which happened to be the Lord’s Day of the *Apokreōs*. From the twenty-fifth of December to the nineteenth of February, there are 57 days: during which it is permitted to eat meat, before the Lenten fast. Add 56 days to the Paschal Lord’s Day, which is the sixteenth of April; the calculation will end on the eleventh of June, which will be the Lord’s Day of All Saints. The interval from Easter to the third of May is 17 days, as is also the interval from the eleventh of June to the twenty-eighth. Hence the fast of the Apostles had 17 days.

Besides these feasts, which are called movable, the *mesopentēkostē* is commemorated among the Greeks, ^{B D} which is the twenty-fifth day from Easter, and it always falls on a *feria* IV. Therefore, by adding 24 to the day of Easter, the *mesopentēkostē* will occur. Thus, in that year of Christ 1536, if 24 days are added to the sixteenth of April, we fall on the tenth of May, which was the *mesopentēkostē*. The Chronicon Alexandrinum makes mention of this feast, as we have observed concerning the Breviarium of Nicephorus; and likewise the Horologion of the Greeks.

Finally, we must not pass over the error of Maximus, which is in chapter VIII of the first part of the *Computus*. For there, from the Lord’s Day of *Apokreō*, which is also called the *Pareisasis* of the holy fasts, up to the *Pascha nomikon*, or the fourteenth Moon, he calculates 57 days. In this way, the legal Easter will be identical with the Christian.

CAPUT XIII. The method of the Greek *computus*, which has been transmitted by Maximus, Isaac, and others, is transmitted most expeditiously through tables and canons.

I believe it will not be unpleasant to those devoted to Greek history and to readers of annals; indeed, I am confident it will be a great aid to them if, out of all those things which we have hitherto explained copiously regarding Greek eras and *computi*, some method is fashioned, redacted into fixed canons; by whose aid what is sought may be most quickly investigated. For the sake of this matter, we have constructed six tables, whose rationale and use I shall briefly discuss. The first offers the years of the Greek *computus* with three cycles, which is exactly the same as the table of the Julian period described in book VII of *De doctrina temporum*, chapter IX. For although the Greek *computus* is not an artificial period, and is instituted solely for the method of cycles, as is the Julian: yet it is no less divided by whole cycles, and offers the proper [data] of any given year. Therefore, it differs from the Julian period only in that the latter provides the cycles of the sun, moon, and indiction of the Romans, and the particulars of the Julian year. The Greek *computus*, however, exhibits at least a different cycle of the sun. For the Alexandrian concurs with the lunar cycle, but has the indiction lower by one. The Constantinopolitan produces only the indiction as common with the Julian period; the lunar cycle is Jewish. Wherefore, for the sake of distinction, we call that the Julian period which serves the Roman cycles: but we call the Greek *computus*, or era, not a period. The second table contains the cycle of the sun with the *epactae* of the sun, or *concurrentes*; which have a place in both *computi*. We have also added the Dominical letters, not those which correspond to the Greek solar cycle, but those which are attributed to the Roman: which we have done to the end that the usage of the following method may be regulated by it as a rule. For just as the *feria* of any day is discovered through *concurrentes* and *regulares*, it must be verified by the Dominical letter, lest any error should creep in. The third table will give the monthly *epactae* of the sun, or *regulares*, in the Julian months: which *regulares* indeed we have so disposed, that with the same starting point proposed for both *computi*—namely, from January—that variety is avoided which is observed in the method of Maximus and Isaac. In the fourth table, we have joined the cycle of the moon with the *epactae* of Maximus and Isaac, and also the Paschal limits and the *regulares* of the days on which the limits fall. The first lunar cycle in the *computus* of Isaac, or the Constantinopolitan, is the fourth of the Roman cycle, which we have taught to be the same as the cycle of the Maximus and Alexandrian *computi*. Therefore, a different series of limits has been instituted. The fifth table contains *regulares* accommodated to both *computi*, so that they may be referred to the same starting point of January for the sake of ease. Finally, the sixth [table contains] the periods

^A of 532 years, which Maximus and other Greeks use in their *computi*.

Table I of the cycles. In hundreds and thousands.

| Centenaries totaled | Cycle of the Sun | Cycle of the Moon | Indiction | | :--- | :--- | :--- | :--- | | 100 | XVI | V | X | | 200 | IV | X | V | | 300 | XX | XV | XV | | 400 | VIII | I | X | | 500 | XXIV | VI | V | | 600 | XII | XI | XV | | 700 | XXVIII | XVI | X | | 800 | XVI | II | V | | 900 | IV | VII | XV | | 1000 | XX | XII | X | | 2000 | XII | V | V | | 3000 | IV | XVII | XV | | 4000 | XXIV | X | X | | 5000 | XVI | III | V | | 6000 | VIII | XV | XV | | 7000 | XXVII | VIII | X | | 8000 | XX | I | V | | 9000 | XII | XIII | XV | | 10000 | IV | VI | X |

Table II of the cycle of the sun.

| Years | Concurrents | Roman letters | | :--- | :--- | :--- | | I | 7 | GF | | II | 1 | E | | III | 2 | D | | IV | 4 | C | | V | 5 | BA | | VI | 6 | G | | VII | 7 | F | | VIII | 2 | E | | IX | 3 | DC | | X | 4 | B | | XI | 5 | A | | XII | 7 | G | | XIII | 1 | FE | | XIV | 2 | D | | XV | 3 | C | | XVI | 5 | B | | XVII | 6 | AG | | XVIII | 7 | F | | XIX | 4 | E | | XX | 5 | D | | XXI | 6 | CB | | XXII | 1 | A | | XXIII | 2 | G | | XXIV | 3 | F | | XXV | 5 | ED | | XXVI | 6 | C | | XXVII | 7 | B | | XXVIII | 6 | A |

In decades.

| Years | Cycle of the Sun | Cycle of the Moon | Indiction | | :--- | :--- | :--- | :--- | | 10 | X | X | X | | 20 | XX | X | V | | 30 | II | XI | XV | | 40 | XII | II | X | | 50 | XXII | XII | V | | 60 | IV | III | XV | | 70 | XIV | XIII | X | | 80 | XXIV | IV | V | | 90 | VI | XIV | XV |

^{A B C D}

Table III of the monthly epacts of the sun, or regular numbers in both computi.

| Roman Months | Epacts of the Alexandrian computus | Epacts of the Constantinopolitan computus | | :--- | :--- | :--- | | January | 1 | 2 | | February | 4 | 5 | | March | 4 | 5 | | April | 7 | 1 | | May | 2 | 5 | | June | 5 | 6 | | July | 7 | 1 | | August | 3 | 4 | | September | 6 | 7 | | October | 4 | 2 | | November | 5 | 5 | | December | 6 | 7 |

Table IV of the lunar cycle with epacts, and terms.

| | Epacts of Maximus | Epacts of Isaac | Paschal terms in the Alexandrian computus | Regulars | Paschal terms in the Constantinopolitan computus | Regulars | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | I | 9 | 11 | Aprilis V | 5 | Aprilis II | 3 | | II | 20 | 22 | Martii XXV | 1 | Martii XXII | 6 | | III | 1 | 3 | Aprilis XIII | 6 | Aprilis X | 4 | | IV | 12 | 14 | Aprilis II | 2 | Martii XXX | 7 | | V | 23 | 21 | Martii XXII | 5 | Aprilis XVIII | 5 | | VI | 4 | 6 | Aprilis X | 3 | Aprilis VII | 1 | | VII | 15 | 17 | Martii XXX | 6 | Martii XXVII | 4 | | VIII | 26 | 28 | Aprilis XVIII | 4 | Aprilis XV | 2 | | IX | 7 | 9 | Aprilis VII | 7 | Aprilis IV | 5 | | X | 18 | 20 | Martii XXVII | 3 | Aprilis XXIV | 1 | | XI | 29 | 1 | Aprilis XV | 1 | Aprilis XII | 6 | | XII | 10 | 12 | Aprilis IV | 4 | Aprilis I | 2 | | XIII | 21 | 23 | Martii XXIV | 7 | Martii XXI | 5 | | XIV | 2 | 4 | Aprilis XII | 5 | Aprilis IX | 3 | | XV | 13 | 15 | Aprilis I | 1 | Martii XXIX | 6 | | XVI | 24 | 26 | Martii XX | 3 | Aprilis XVII | 4 | | XVII | 5 | 8 | Aprilis XVII | 2 | Aprilis V | 7 | | XVIII | 16 | 19 | Martii XXIX | 5 | Aprilis XXIV | 2 | | XIX | 27 | 0 | Aprilis XVII | 3 | Aprilis XIII | 7 |

| ^A Table V of the regulars of the moon in either cycle.

| Maximi Regulars | Isaaci Regulars | | :--- | :--- | | January | 10 | 0 | | February | 11 | 1 | | March | 10 | 0 | | April | 0 | 1 | | May | 1 | 2 | | June | 2 | 3 | | July | 3 | 4 | | August | 4 | 5 | | September | 6 | 7 | | October | 6 | 7 | | November | 8 | 9 | | December | 8 | 10 |

Table VI of the Victorian periods.

| Number of periods | Collected years | | :--- | :--- | | 1 | 532 | | 2 | 1064 | | 3 | 1596 | | 4 | 2128 | | 5 | 2660 | | 6 | 3192 | | 7 | 3724 | | 8 | 4256 | | 9 | 4788 | | 10 | 5320 | | 11 | 5852 | | 12 | 6384 | | 13 | 6916 | | 14 | 7448 | | 15 | 7980 |

^B CANONS OF EACH COMPUTATION.

CANON I. Given any year of the Alexandrian or Constantinopolitan era, to know its proper cycles.

Seek the nearest smaller number from the first table in hundreds and thousands, and let the competent cycles of the sun, moon, and indiction be noted for them; then let the rest be supplied from the decades. Then let the individual cycles be collected in their own respective categories. To these, let the remaining years be added if any remain beyond the decades. In this way, the cycles proper to that year which is proposed will be made, by subtracting from the sum, as much as you can, the complete cycles.

Example: Let the year from Adam of the Alexandrian computation be 7034, and of the Constantinopolitan 7050. In this way you will collect the cycles of each.

In the Alexandrian: | | Cycles | Sun | Moon | Indict. | | :--- | :--- | :--- | :--- | :--- | | | 7000 | 24 | 8 | 10 | | | 30 | 2 | 11 | 15 | | | 4 | 4 | 4 | 4 | | | Sum | 6 | 4 | 14 |

In the Constantinopolitan: | | Cycles | Sun | Moon | Indict. | | :--- | :--- | :--- | :--- | :--- | | | 7000 | 28 | 8 | 10 | | | 50 | 22 | 12 | 5 | | | Sum | 22 | 1 | 15 |

Therefore, in the year of the Alexandrian era 7034, the solar cycle was 6, the lunar 4, the indiction 14; the Constantinopolitan 7050 had the solar cycle 22, the lunar 1, and the indiction 15.

CANON II. With any year of either computation given, to know what position it has in the Julian period, or the Roman era of Christ, and with what Roman cycles it is affected.

Three things are to be observed first: first, that both Greek eras from Adam to the first year of the Christian era exceed the sum of the years of the Julian period which are reckoned to the same era. The Alexandrian era exceeded the Julian period by 779 years, the Constantinopolitan by 795; second, that the first year of the Christian era is numbered as 5493 in the Alexandrian, and as 5509 in the Constantinopolitan; third, that the Constantinopolitan era exceeds the other by 16 years. Therefore, if you add 16 to any year of the Alexandrian era and its cycles collected by the canon, you will produce the years and cycles of the Constantinopolitan; with full cycles subtracted; or, if the calculation is precise, with the totals themselves assumed as proper to that year. From the first it follows that, in order to find the first years of both computations in the Julian period, one must add another complete period of 7980 years to the current period. The year 4714 of the Julian period is the first year of the Christian era; with the complete period joined, 12694 years are collected.

Having noted these things, if the proposed Greek year is greater than 779 in the Alexandrian era, with 779 subtracted from it, the remainder will give the year of the Julian period to which it corresponds. In the Constantinopolitan, however, it will emerge by subtracting 795; and so, by dividing the residue by 28, 19, and 15, you will obtain the Roman cycles. If the proposed year is less than 779 or 795, add 7201 to the Alexandrian year or 7185 to the Constantinopolitan; there will exist a year of the proleptic and antecedent Julian period; and by dividing this by the three cycles, you will attain what you wish.

Furthermore, you will produce the current year of Christ in either era if you subtract 5492 from the Alexandrian and 5508 from the Constantinopolitan.

Example: The year of the Alexandrian era 7034, with 779 ^D subtracted, is 6255; but with 5452 subtracted, it is the common year of Christ 1542. Thus the Constantinopolitan year 7050, with 795 removed, leaves 6255; with 5508 subtracted, it makes the remaining 1542 of Christ: from which you will extract the competent cycles if you compare the years of the Julian period in Table I, namely in this way.

| | Sun | Moon | Indict. | | :--- | :--- | :--- | :--- | | 6000 | 8 | 15 | 15 | | 200 | 4 | 10 | 5 | | 50 | 22 | 12 | 5 | | 5 | | 5 | 5 | | Sum | 11 | 4 | 15 |

Therefore, to each Greek year corresponds the Roman cycle 11, moon 4, indiction 15.

Again, I wish to know how many of the three Roman cycles correspond to the first years of either era. If you remove the years 5493 from 12694, the year 7201 will be left. Therefore, the first year of the Alexandrian era is 7202 of the Julian period, to which, according to the first canon, belongs the cycle of the sun 6, moon 1, indiction 11. Likewise, from 12694 subtracting 5509, 7185 remain. Therefore, year 7186 of the Julian period is the first of the Constantinopolitan era. Which consequently began with the Roman cycle of the sun 18, moon 4, indiction 1.

From these arises a summary of the canon of this kind: having found the cycle of the sun of the Alexandrian era, add 5, to the cycle of the moon 0, to the indiction 1: you will obtain the Roman cycles of that year; but for the Constantinopolitan, to the cycles add: to the solar 17, to the lunar 3, to the indiction 0, as this table teaches. For example, the Alexandrian year 7034 by the first canon ^A obtains a cycle of the sun of 6; add 5, you will have 11. The cycle of the moon is 4, adding 0 it is 4; the indiction is 14, adding 1 it is 15. Thus, for Constantinopolitan years, the cycle of the sun 22 increased by 17 becomes the Roman 11; the lunar, adding 3, from 1 becomes 4; the indiction is the same, 15.

However, we begin all those years of the Greek calculations from the Kalends of January; although the civil year and the indiction begin from the preceding September: but if any one should wish to imitate [this], he would reduce the Roman cycles by one at the beginning of each year. And he will do the same in the Greek ones themselves, so that the holidays, or the moon, follows the age in the first 4 months, according to the method of the following canons.

Canon III. Given the cycles of the sun and moon in Greek calculations, to find what year of the Victorian period it is; and also from Adam. Likewise, given an indiction in addition to those cycles, to elicit the year both of the Greek era and of the Julian period into which those three cycles fall.

Multiply the given cycle of the sun by 57, and reserve the sum. Then take the cycle of the moon from 19; or if it is 19, take 19. Multiply that number by 56. The sum that arises from this, combine it with the previous one, which arises from the cycle of the sun multiplied by 57. Whatever result is formed from both, split by 532, ^B the remainder will be the year of the Victorian period. If you add this to the year which you can suspect to be the next lowest to the proposed year of the Greek era, in table VI, you will have the Greek year, whose own cycles of the sun and moon are the ones placed in the question; which having been found, by the method of the preceding canon, you will seek in what year of the Julian period, or of Christ, it falls. But if an indiction is given in addition: what survived from the last division by 532, divide by 15, adjusting, if it is necessary in this way, by the complete period of 15 years: what remains from this, keep, and from it, subtract the proposed indiction, adjusting, if need be, by a whole number 15; multiply the remainder by 1064, and add to the sum what remained from the latest division by 532. But if the sum is greater than 7980, subtract this same number from it. Thus that year will be made, which is required, from which in the Julian period you will obtain its proper year.

Example: The Alexandrian Chronicle fixes the migration of the Israelites from Egypt to that year which had a cycle of the sun of 2, moon 19, with indiction 13. I ask how many that was of the Greek computation, and indeed the Constantinopolitan; for this author uses this one. First, the cycles of the sun and moon having been published, the cycle of the sun 2 must be multiplied by 57, making 114. Then, subtracting 19, the cycle of the moon, from the complete cycle of the same moon, I have a remainder of 19: which number I multiply by 56. There result 1064, which sum I add ^C to the previous 114, so that they become 1178: which being divided by 532, there are 114 remaining. Therefore the Alexandrian Chronicle ascribes the Exodus to year 114 of the Victorian period. But in order for you to know what it is of the Constantinopolitan computation, there is need of conjecture. Since, therefore, the Greeks, by the faith of the LXX Elders, commonly number the years from Adam to the flood 2242, and from the flood to Abraham 1172, the sum is gathered from Adam to the birth of Abraham of 3414 years. From here, however, until the migration from Egypt they think [it is] about 505 years. Wherefore there are gathered nearly 3919 years from Adam to the Exodus. So that you may have the accurate year of the Alexandrian Chronicle, look in table VI for the year next smaller than that which you suspect: which as it is greater than the year 3724, it is less than 4256. Wherefore, add the 114 years of the Victorian period to 3724, you will make the year 3838, which divided by the cycle of the sun and moon, will return for the sun indeed, cycle 2, for the moon however 19: which was the proposition.

Again, if in addition to those two cycles, the indiction 13 is given, I divide the number 114, which was at last left from the division by 532 instituted; by 15 I divide, 9 remain, from which, adding 15, I subtract 13, the remainder becomes 11; which if they are multiplied by 1064, the sum will be 11704, to which add 114, and out of the total 11818, since it exceeds 7980, these having been subtracted, there is left the year 3838, from which 795 having been taken away, there emerges the year in the Julian period 3043, which corresponds to the Constantinopolitan year 3838. In order for you to know that this is true, by the method of the preceding canon, to the Constantinopolitan cycle of the sun 2 add 17; to the lunar 19 add 3, and to the indiction 13 add 0, you will finish the Julian cycles, sun 19 with the letter E; moon 3, indiction 13: ^D all of which concur in the year of the Julian period 3043, which is the hundred and fortieth before the true epoch of the Exodus, which we have brought into the year of the Julian period 3183. And indeed, the Chronicle says that in that year, which it numbers 3838, the first holiday occurred on April 13, and on the same day the fourteenth of Nisan fell; so that there is the least doubt that the golden number was 3, and the cycle of the sun 19, and the Dominical letter E.

Another example: The same Chronicle numbers the conception of John the Forerunner from Adam 46

5505, if the year is begun from March, in the 14th indiction, on the 9th day of April, ^A just as it is read in the Greeks' Horologium: which we collect from our method to be most false. When 5505 years are divided by 28, the solar cycle of 17 remains; but when divided by 19, 14 will remain. Yet the indiction is not 14, but 15 arises from division by 15. Let that year be sought in which the 14th indiction concurs with the solar cycle of 17 and the lunar of 14. First, by multiplying the solar cycle of 17 by 57, 969 is produced; then, by taking 14 from 19, 5 remains, which multiplied by 56 yields 280. With 969 added, the sum 1249 is produced, which, distributed by 532, leaves 185. Therefore, that year 185 of the Victorine cycle is the one which the Chronicle signifies. But the indiction which it adds disturbs the calculations. For if you divide those remaining 185 by 15, there remain 5, from which, if the proposed indiction of 14 is taken away, there remain 6; which, multiplied by 1064, make 6384. Add 185, and the year 6569 will exist, which exhibits the cycles: solar 17, lunar 14, indiction 14. Moreover, this is the year of Christ 1856. You see the absurdity. Wherefore this method cannot be sufficiently recommended, which, for correcting the errors of the Greek annals, especially Theophanes, will bring greater fruit than one might think.

CANON IV. To proclaim the weekday of any proposed day in the Greek computation.

Having found the solar cycle in both computations by Canon I, apply to it the corresponding monthly epacts from Table II of the solar cycle; then seek the monthly epacts from Table III, and add those of their own kind, that is, to the concurrents of the Alexandrian era cycle add the regulars of the Alexandrian computation, but to the concurrents of the Constantinopolitan computation, the regulars of the same. Finally, to the sum obtained thence, add the [τὴν ποσταίαν], that is, the day of the month. What is produced from this, when divided by 7, will demonstrate the weekday.

Exemplum: Let it be proposed to investigate what is the moon's age on April 2 of the year 7034 of the Alexandrian computation and 7050 of the Constantinopolitan; whose lunar cycles we have produced by Canon I; of the former indeed 4, of the latter 1. Therefore with the cycle of the Alexandrian era 4, I find in Table IV the epact 12. In Table V, the regulars of April 0. Accordingly, the epact 12, and the day of April 2, offer the fourteenth moon on that day. But the year 7050 Constantinopolitan obtains the epacts in Table IV of 11; the regular in Table V, 1; they become 12, and with the two days of April, 14. Indeed, in the Greek Horologium, they compute the moon in that year as the fourteenth on the second of April.

CANON V. To discover the age of the moon in any given year and day of both computations.

After you have found the lunar cycle by the first canon, go with it to Table IV: seek in its first line your cycle; then look directly at the epact responding to it. If the cycle is found by you as proper to the Alexandrian computation, the epact responding to it is that which is called Maximus'. To the Constantinopolitan, indeed, is attributed the other, which is inscribed to Isaac. Betake yourself then to Table V of the moon's regulars, and extract the proper regulars of the computed month for whichever computation you propose. Increase the sum gathered from the epacts and regulars by the number of days, in the last of which the age of the moon is required. Those collected days, and divided by 30, will show what the moon is on that day.

CANON VI. To find the Paschal limit or Jewish Passover, and its weekday in both computations.

With the lunar cycle found by the first canon, go to Table IV. In the first line take your cycle, and directly therefrom the Paschal limit, with its regular as much in the Alexandrian era as in the Constantinopolitan. Then combine the concurrents of the solar cycle of the proposed year found by Canon IV, with the regular of the limit. The sum will give the weekday, with seven added. As in the same Alexandrian year 7034, the lunar cycle 4 has in the table the responding Paschal limit of April 2, whose regulars are 2, the solar cycle 6 gave the concurrents likewise as 6 by Canon IV; both collected exhibit the weekday 1. Thus in the Constantinopolitan year 7050, the lunar cycle 1 offers the limit in Table IV, April 11, with the regulars 3. The solar cycle from Canon I was 22, the concurrents 5, which with the regulars show the weekday 1, just as the Greek Horologium demonstrates.

Exemplum: Let there be the year of the Alexandrian computation 7034, but the Constantinopolitan 7030. The cycle of the former, produced by Canon I, is 6. The concurrents from Table II are 6. I wish to know the weekday of the 9th day of April. The Alexandrian regulars are gathered from the table as 7, to which, when added to 6 and the 9 days of April, the sum is 22 (less 7, twice), the weekday is 1. Again, the cycle of the year 7050 Constantinopolitan was detected by the same canon to be 22, the concurrents 5. The regulars of April in the second line of the regulars of Table III are 1, which, with the 5 concurrents and 9 days of April, compose 15; and therefore the weekday is 1. Let us test whether the calculation was correctly placed by the characterism of the years of Christ, or the Julian period. The Alexandrian year 7034, with 779 deducted, is found to be 6255 of the Julian period, which also proceeds from subtracting 795 from the 7050 Constantinopolitan year by the method of Canon II. But these same years also conspire to the year of Christ 1542, if from the Alexandrian I cast off 5492, and from the Constantinopolitan 5508. But both the year of the Julian period 6255 and of Christ 1542 obtain the Roman solar cycle 11, to which letter A belongs from Table II. Therefore the weekday was most correctly pronounced.

CANON VII. To define the Christian Passover, and the principal feasts of the Greeks in any given year.

The Paschal limit having been noted by the above canon, if its weekday is Sunday, it is deferred to the following Sunday. If the limit falls upon another weekday, the next following Sunday will obtain the Christian Passover. Having found the Passover, you will find the remaining feasts with no trouble. This method is both easy in itself, and is contained in the preceding chapter of this book.

DISSERTATION ON THE ERAS OF THE GREEKS. CHAPTER XIV.

Concerning the Paschal computation and method of certain men, whom S. Maximus calls *pentaplountas* and *exaplountas*; we call them "undecuplatores" [the elevenfold-reckoners], whose method, which seemed almost inextricable to Scaliger, is explained very easily. Regarding the error of Scaliger concerning the same.

Maximus the Confessor, in the first part of his computation, chapters XI and XII, and throughout the entire second part, argues against certain artificers of computation, whom he calls *pentaplountas* and *exaplountas*; that is, those who multiply the years of the lunar cycle by five and by six; or, to express it in one word, by eleven: whose method he proposes in the second part, but obscurely and perplexedly, so that it is difficult to grasp what he intends. This is something which Scaliger also confesses in Book II of *De emendatione temporum*, in the chapter on the false Paschal cycle. We, however, with God inspiring us, and aided by the prayers of that same most holy confessor, have overcome all these intricacies and have explored that entire method: the knowledge of which we shall here impart to studious young men. For this labor is required of us not only to explain the computation of Maximus, but it is also most apt for illustrating the origin of the Constantinopolitan computation, which we set forth above. First, therefore, we shall gather into one place what Maximus has handed down in different locations concerning them.

^A "Hypotome of the moon," which the common people of the computus call the "saltum lunae" [leap of the moon], he says in the eleventh chapter of the first part is brought in by them in year XI of the cycle, which is the fourteenth of the Greeks and Romans, whose new moon began on the last of December; the year itself, however, ends on December XVIII. Hence the following year, which was affected by their cycle XII, and the Catholics' XV, began from the XIX day of December; so that the fourteenth moon would fall on the Kalends of January. Wherefore, the eleventh year of the cycle among these "undecuplatores" (for so I may be permitted to call them), or the fourteenth of the orthodox, consisted of three hundred and fifty-three days, and the suppression of one day took place on December XIX, if indeed the new moon of their cycle XII, our XV, was transferred from December XX, to which it had been fixed according to the prescription of the Alexandrians, to the XIX; so that that one day might be dissimulated. And then the nineteenth of December was accustomed to be counted twice, so that it would be considered as much the last of the preceding year as the first of the following one. ^B Again, chapter XII relates that they, to compensate for the exemptible day, add five sixty-parts [sexagenaria scrupula] to each year; and thus, in twelve years, a full day is gathered, and finally they add those same years to the sixteen years of the computation of Adam.

In the second part of the computation, he explains their method with three figures, that is, one wheel, and a table or double canon: of which one was drawn to the left, the other to the right of the wheel. But because of the narrowness of the page, the wheel was placed above both. Furthermore, from this method it is clear that the lunar cycle among them was three years smaller than the Greek or Alexandrian, which was also the Roman: and the first of the "undecuplatores" is the seventeenth of these [Alexandrians]. ^C But the solar cycle is so ordered that the first of the "undecuplatores" is the thirteenth of the Greeks, who employ the Alexandrian computation, one of the two which I explained above. Therefore, since the Alexandrian cycle of the Greeks corresponds to the sixth Roman cycle, as we demonstrated in chapter V of this book, the first of the "undecuplatores" will be the eighth Roman one.

The caput of their method, and the primary matter for reproach in Maximus, consists in the Paschal terms, or fourteenths, because they frequently considered the same days, which were numbered as the fourteenths by the Alexandrians and Catholics, to be the fifteenths or sixteenths: which, that they might do by way and method, they proposed to multiply any year of their cycle first by five, then by six; that is, in total by eleven, then to add days to the sum gathered therefrom; which were counted from the Kalends of January to the Paschal term of that year. Again, as many as those days were from the Kalends of January to the term, they took sixty-parts of one day [sexagesimas unius diei]: to which they added five other daily sixty-parts as many times as the year of the cycle was deemed to be; they reduced the total of the sixty-parts, divided by 60, into days: which they added to the former ^D group of days. In this way, the same day, which was the Nicene fourteenth, was often counted as the fifteenth, that is, in the years XIII; it was counted as the sixteenth once, in the sixteenth year of the cycle; in the others, that is, in I, II, III, IV, and XVIII, it was considered the fourteenth. Hence it happened that Easter, according to their reckonings, was celebrated on the twenty-second or twenty-third moon; while at other times it agreed with the Catholic rule. Which, before it is demonstrated by usage itself and by examples, we shall render an account of the three schemata which Maximus described in Part II.

In the cycle of Victorinus, which is of 532 years (for this present one is assumed for the plan), there are twenty-eight lunar cycles, or enneadecaeterides; likewise there are nineteen solar cycles, or *eikosioctaeterides*. In the wheel, the zones or circles are nine. The first and outermost contains the solar cycle, such as is used by Maximus, which is proper to the Alexandrian computation of the Greeks; the second orb has the epacts of the sun, or concurrents; the third marks the bissextiles; the fourth, the years of the solar cycle of the "undecuplatores"; which correspond to the former in such a way that the first of the "undecuplatores" is the thirteenth, and so on in order. The remaining five orbs contain the years marked by solar and lunar cycles, in which the Paschal term, or the Nicene fourteenth, is deemed by the "undecuplatores" to be the fifteenth or sixteenth. Thus, therefore, are those interior five circles arranged, so that to every year of the solar cycle in the fourth circle, two cells are subject to each of the five circles. Of these cells, that which is to the left embraces the cycle of the sun; that which is to the right, the cycle of the moon. Indeed, the indices of the cycles of the sun in the fourth circle are one [set of] enneade-...

enneadecaeterides, to which correspond in the first circle the cycle of 17, that is, 17 of the Alexandrian computus. To this, in the fifth zone, two cells correspond, of which the one on the left holds the cycle of the sun as 9, that is 9; the one on the right, the lunar cycle 9, by which it is signified, in that year of the first enneadecaeteris, which is endowed with a cycle of the sun and moon of 9, the 14th Nicene moon is reckoned by the Undecuplantes as 15. Which, indeed, could happen for a double reason. For it happened either because of the intercalary bissextile day or because of the epilogisms of the daily sexagesimals: to those years in which a bissextile day did this, Maximus had ascribed a single point; but to those years which obtained the same for the reason of the fractions, he had noted two points; finally, three to those years in which the 14th was considered the 16th. But those points had been omitted by the copyists in the transcript; nor have we wished to bestow more anxious than useful labor in restoring them; especially since from our method it can be clear for any year whether its 14th is 15th or 16th. But Maximus warns that not all those years are noted in the inner circles, in which the 14th Nicene [moon] becomes, by the calculation of the Undecuplantes, 15 or 16. For this happens fourteen times in individual cycles; but only those [years] which bring the same term, supposed by one day or two days more, into the Sunday: which occurs at most in 5 years in any enneadecaeteris. From the two small tables underneath the wheel, the first one on the left has the embolismic years noted; the second, the years of the Alexandrian lunar cycle: whose fourth corresponds to the first year of the lunar cycle of the Undecuplantes; which is described in the third line. The fourth line contains the lunar epacts, such as Maximus employs. The fifth and last, the Paschal terms, or [the] 14th, which they often count as 15; but once as 16, five times 14. The right-hand small table serves entirely for the Catholic Pasch; for it offers the Paschal term, and its day, in the Julian months in the second and third lines; the epacts in the fourth; in the fifth, the lunar cycle. The regulars, or προσθέται, in the first are not described in their entirety by the copyist.

Now let us demonstrate the use of the method with one or two examples. Let this be the year which is subject to the first enneadecaeteris, or the first year of the solar cycle in the fourth border closely in the border V, whose cycle of the sun is 9, and the lunar likewise 9. First let us inquire from canon III of the preceding chapter, what year of the Greek computus this is. We will find it to be the year 5861 from Adam, which, divided by 28 and 19, leaves 9 in both cases. The lunar cycle 9 of the Undecuplantes is the Alexandrian 12, whose 14th [of the] Pasch falls on April 4, which is 94 days from the Kalends of January, as the right-hand small table shows. Furthermore, the Undecuplantes reckon the 14th of that year as the 15th: which can be seen in the left-hand small table, and 94 added to 99, which arise from the lunar cycle 9 of the Undecuplantes, will make 193 days, and moreover 94 sexagesimals (which are, clearly, the days that intervene from the Kalends of January), with 45 sexagesimals, which arise from the 9 years multiplied five times; for the reason that for each year five daily scruples are imputed; all, I say, those sexagesimals in the number 139, divided by 60, make two full days with 19 sexagesimals of the higher sum of days added, 195 days are formed, which, being distributed by 30, leave 15 days. Hence the fourth day of April, which is the 14th Nicene, becomes 15 by their subtraction. Let us see whether that term falls on a Sunday. By canon IV of the same chapter, the epacts of the sun in that year are 3, which added to the monthly epacts of April 1, and the 4th of April, show the first day of the week on that same day, April 5: which is also gathered from the doctrine of canon II. Indeed, the Roman cycle of the sun, with 17 added, becomes 26; but the lunar 12. The Dominical letter is C. Therefore, in that year, the 15th of the Pasch falls on the Sunday among the Undecuplantes; and the Pasch is celebrated [on the] 22nd moon. Therefore, this year had to be marked with two points, which, if you wish to find what year of the Julian period or of Christ it was, you will solve through the canon. For by subtracting 795 from 5861, there will remain the year of the Julian period 5063, which is 352 of Christ. But why the cycle of the Undecuplantes is referred by us to the Constantinopolitan computus, the following chapter will reveal.

Again, let that year be proposed which in the same first enneadecaeteris has the cycle of both sun and moon as 16. This will be the year 5868 of the Constantinopolitan computus by the canon of the preceding chapter. The Paschal term is April 17, which day the Undecuplantes reckon as 16 for the 14th, as it is in the left-hand small table. Let us test [it] by the scruple method. The cycle of the sun 16 in the Constantinopolitan computus is the Roman 5 by canon 11. Therefore the year is bissextile. Therefore from the Kalends of January, 108 days are counted, not 107. 16 multiplied by 11 makes 176 days, which with 108 gives 284; 108 scruples with 90, which are gathered by multiplying 5 by 16, make 3 days; and moreover 17 sexagesimals: these being neglected, and the three days added to the 284, and the sum 287 distributed by 30, 17 days remain. Hence the 17th of April was the 17th moon. Therefore, because of the bissextile day, the term was pulled back from the 17th of April to the 16th: which, since it was the first day of the week, the Pasch was spread to the 20th day, which was the 21st moon, about which matter Maximus taught us nothing: who wrote that they have regard for the bissextile day; and as often as because of it the term falls on a 15th moon on the first day of the week, this year is marked with one point. For, when the moon is 16, for whatever reason it happens, it is designated with a triple point. Wherefore, a triple point had to be attributed to that year. From these it is evident that the hallucination of Scaliger, who

explicating that cycle of those who add eleven (lib. *De emend.*), asserts that whenever the fourteenth of the Pasch was considered the sixteenth, if it fell on the second day of the week, they celebrated the Pasch on the preceding Sunday, their moon being the 15th, the Nicene the 13th, which is most false; and Maximus expressly affirms the contrary; namely, that their moon was the 22nd, and on the same day as the Catholics, they kept their Pasch; but that the Catholics reckoned the moon as the 20th, which those others thought to be two days more. But when the 16th moon fell on a Sunday, both groups postponed the Pasch to the following Sunday; which the "eleven-adders" wanted to be the 25th moon, [and] the Catholics the 21st. ^B

CHAPTER XV.

While we examine more attentively the things which Maximus reports concerning these "eleven-adders," we notice certain things not common; which neither, as it appears, were known to Maximus himself, nor to Scaliger, who attempted to explain Maximus, and which are primarily useful for illustrating the Greek computation. By careful observation, therefore, we discover: that those "eleven-adders" were the authors of one of the two computations, which we have named the Constantinopolitan; which offers as the first year of the Christian era that which is 5509 from Adam: which is gathered partly from the cycles of the sun and moon, [and] partly from the number of years certain. Maximus writes in chapter XI of part I, that the cycle of the moon, which they reckon as 11, is that which he himself computes as 14; the lunar cycle of Maximus is the same as the Roman, as we demonstrated above. Therefore, the first cycle of the "eleven-adders" is the Roman, or the fourth of Maximus; and the first of Maximus is their seventeenth. Therefore, the lunar cycle of the "eleven-adders" is the same as that of the Jews and of Victorinus, or the Latins' cycle: which we have taught is proper to the Constantinopolitan era. The first year of the Christian era had the Roman cycle: sun 10, moon 11. The year 5509 gives the cycle of the sun 21, moon 18. But the year 5493 [has] sun 5, moon 11. Thus, the lunar cycle of the Constantinopolitan computation differs from the Roman and the Alexandrian, which Maximus uses, by sixteen; so that the seventeenth Constantinopolitan coincides with the first Alexandrian. But the solar cycle of the Constantinopolitan is smaller than the Alexandrian by twelve, that is, the first of the former is the thirteenth of the latter. Wherefore, since the first of the Alexandrian solar cycle is the sixth of the Roman, the first of the Constantinopolitan cycle will be the eighteenth of the Roman; all of which agrees with the cycle of the "eleven-adders," so that there is no doubt they held to the Constantinopolitan computation.

Moreover, that which Maximus says, that they added sixteen to the years of Adam, confirms our conjecture uniquely. For in this Constantinopolitan computation, it is numbered in the Alexandrian as 5495; in the Constantinopolitan, 5509. The interval of years is 16. From this, it also appears that what we guessed before we thought about that cycle of the "eleven-adders" is true: that the Constantinopolitan computation had been accommodated by the Greeks to the method of Victorinus or the Latins, and to the Jewish lunar cycle. For not only does this lunar cycle itself persuade [us] of this, but also that they derived the lunations from the Kalends of January, as Maximus testifies: which Beda also reports the Latins instituted (lib. *De temporum ratione*, cap. LIV), and Victorinus himself in the Preface to Pope Hilary, which we have published in tome II of *De doctrina temporum*; so that the Kalends of January were the canon of the Paschal ^B syzygy. There is also added the cycle itself, which they used, of 532 years, of which Victorinus was the inventor. Whence it becomes likely that they borrowed all that Greek [computation] from the Latins under the age of Maximus. From which also this corollary follows, that we have rightly contended against Scaliger [that] the Constantinopolitan era is much more recent than either that of Panodorus of 5492 [years] up to Christ, or that of Africanus of 5500 years. For Maximus speaks thus of these computers, who added 16 to the era of Adam, as if they were as recent as his own time and almost his contemporaries.

These things being so, all that we wrote in the third chapter concerning the Constantinopolitan era must be understood about these "eleven-adders"; and especially that they borrowed the lunar cycle from the Latins or Jews in such a way that they by no means imitated their errors in celebrating the Pasch, but obtained the same day with the Catholics: differing in this one point, that they instituted the epilogisms of the neomeniae, or the limits, differently. And there is a strong suspicion that they added this meticulous calculation only for the sake of the method, and so that they might attain a truer calculation of the moon. For since they understood from the Jews, or from astronomical tables, that the new moons were anterior to the Paschal neomeniae, so that these fell into the lunar ^C *phasin*—that is, the second moon—and the limits (or the fourteenths) almost fell on the full moons: which was particularly noticed in the age of Maximus—they were eager to enter upon some method; which would render the age of the moon more accurately, and which would commute the limits (or the Nicene fourteenths) into fifteenths, of which sort they almost were at that time. To that end they devised the epilogisms of the sixtieths; which would increase the sum by one day, or by two days. We have discussed this *metanthesis* of the new moons and of the Paschal neomeniae at length in V *De doctrina temporum*, chapter XIII, where we compared the Nicene [calculations] with those which were in the time of Dionysius Exiguus. Hence, many things can be dissolved and explained which are argued against the "eleven-adders" by Maximus and Scaliger. For it is certain that they were free from fault and reprehension, if, as is our conjecture, they looked to this one thing in that method: that they might provide an exact calculation of the moon *technikos* and *methodikos*; not that in the Pasch ^D

they might innovate anything: a fact ^A to which those who cried them down were themselves witnesses. For what Maximus himself objects regarding the 22nd and 23rd moons, as if it were a sin to celebrate the Pasch on those days, has nothing to do with the matter. For it is one thing to disturb the terms themselves and dislodge them from their ancient station; it is quite another thing, while keeping to the same days, to count them one day or two days more: the former was not permitted without the public authority of the Church; the latter every man learned in astronomical matters, before the Gregorian reform, practiced privately without reproach. For this was nothing else than to derive the age of the moon from the celestial new moons, and not from the civil new moons, which had gradually withered away.

Furthermore, regarding the greater and Victorian cycles, which the *undecuplatores*, and indeed all the Greeks, used in their computations, it must be known that they begin from Adam: as is gathered from the end of Maximus's computation. For therein the eleven cycles are unfolded from the beginning of the world; and the years in which they each terminate are noted. The last of these is said to terminate in the 24th year of the reign of Constantine, which is reckoned as 5852. Subtracting 779 from the method of canon 2 of chapter 13, there remain 5073 years of the Julian period, which is the year of Christ 360. Wherefore for *Constantini* read *Constantii*; for it was the 24th of Constantius. Therefore in the Alexandrian era, the twelfth cycle of Victorinus began in the year of Christ 360. Now, indeed, having subtracted 795 from the 5852 years, there remains 5057 of the Julian period, which is the 344th year of Christ vulgaris, in which computation, according to the Constantinopolitan, the eleventh cycle of Victorinus was terminated; and the twelfth began in the following year, 345. Wherefore, if you apply only the years of the Greek eras, the Victorian cycles end and begin in the same years. But if you refer them to the year of the common era of Christ, they end sixteen years later in the Alexandrian than in the Constantinopolitan; since the sum from Adam to the beginning of the common era is greater by that many years in the latter computation than in the Alexandrian. ^{B C}