Paschalistic treatise

“Easter Treatise” was written by St. Maximus the Confessor in 640-641. and represents the defense of the Alexandrian Paschal with the beginning of chronology in 5492 BC. The essay describes in detail the methods of Easter calculations. The treatise consists of a preface and 3 parts. Part 1 contains tables for determining the date of Easter and the beginning of Lent, as well as the day on which the revelation to Zechariah about the conception of John the Baptist was given. Part 2 contains a critical analysis of the alternative Easter method of “multiplying by five and six.” Part 3 includes descriptions of various practical methods of calendar calculations. This double issue of “Theological Works” is a jubilee one and is dedicated to the 50th anniversary of the collection, which turned 50 in 2010. The first issue of BT was published in June 1960. By 2012, 46 issues had been published (42 numbered, one double and two special ones, dedicated to the anniversaries of the Moscow and Leningrad Theological Academies). The appearance of such a publication was a great achievement of the Russian Orthodox Church in Soviet times. Permission to publish the collection was obtained from the Soviet authorities under the pretext of developing international contacts of the Church, which was supposed to have its own periodical. The first attempts to revive the journals of the Moscow and Leningrad theological academies dated back to 1945 and 1953. However, due to a lack of strength at the academies, in 1953 it was decided to organize a joint, more or less regular publication 1 . The BT collection became such a publication, the preparation of which began in 1957 2 . For many years, the collection (besides the ZhMP, where small articles, usually of a popular nature, were published) was the only publication that published serious works on the main scientific and theological disciplines. Hence the thematic diversity of materials presented on the pages of BT: works on dogmatic theology and patristics, church history, liturgics, canon law, biblical studies, hagiography and other church historical disciplines. The publication of archival materials occupied a significant place. Immediately after its release, BT became a bibliographic rarity. In Soviet times, the collection had not only scientific, but also enormous educational significance. In the 1990s. In connection with the reorganization of the Publishing Department of the Moscow Patriarchate, as well as after church academies and institutes began to have their own publications, the magazine began to experience certain difficulties, which affected the decline in the quality and variety of materials published in BT: research on the history of the Russian Church mainly predominated. At the meeting of the Holy Synod on December 26, 2001, Metropolitan Philaret of Minsk and Slutsk, chairman of the Synodal Theological Commission, was appointed chairman of the editorial board of the collection, which marked a new stage in the development of BT. Since the 38th issue, published in 2003, a constant increase in the quality of publications began, which was repeatedly noted in reviews 3 . At present, in the changed conditions of church-scientific life, the main task of the journal continues to be to consolidate the best forces of both church and secular scientists, to provide them with the opportunity, in particular, to publish works of a fundamental nature, without being constrained by the framework of ordinary journal articles, and, most importantly, to raise the level of scientific research. Thus, BT are intended not only to reflect the scientific and theological potential available in the Russian Church and to stimulate the growth of church science, but also to provide an opportunity for secular scientists to publish the results of their research. Indeed, to this day, due to the ideological prohibitions of the past, many secular academic publications remain closed to a number of areas of humanitarian scientific knowledge related to church history (and, more broadly, Christian) topics. At the same time, the collection does not abandon its traditional educational role in Russia and continues to publish research exclusively in Russian (or in Russian translation, if we are talking about the works of Western scientists), as well as translations of sources, as can be seen in the example of this issue. A symbol of the revival of BT was the restoration (starting with issue 41) of the hardcover, lost in 1992 (issue 31). It is also worth noting the creation in 2010 of the BT website (www.btrudy.ru), on which the archive of the magazine is gradually being posted in electronic format. Studying the history of the development of Paschalistic calculations in Byzantium and in the Christian East in general requires the development and scrupulous analysis of the entire complex of sources available to modern researchers. The number of theoretical works devoted to the technique of calculating the date of Easter and other calendar issues is very small. A special place among them is occupied by the Paschalistic treatise, written by one of the largest Byzantine Church Fathers, the famous fighter for Orthodoxy against the heresy of monothelitism, the ascetic and confessor of St. Maxima (580–662). This work is the oldest treatise that has come down to us, describing in detail the methods of Paschal calculations. The treatise consists of a short preface and three parts. From the preface we learn that the first part was written in March 641, apparently in North Africa, and was addressed to illustration 4 and to the patrician Peter, who held the high post of exarch in Byzantine Africa 5 . Tables are given here to determine the date of Easter and the beginning of Lent, as well as the day corresponding to the 10th of Tishri - the date when, as the author explains, the revelation to Zechariah about the conception of John the Baptist was given. All calculations are based on the classical Alexandrian Easter. The second part of the work, probably written shortly after the first, is devoted to a critical analysis of the alternative Paschalistic technique of “multiplying by five and six,” briefly mentioned in the 1st part. Part 3 includes descriptions of various practical methods of calendar calculations. Finally, the treatise of Maximus the Confessor is accompanied by a chronological table of emperors, brought up to the middle of the 10th century. Since only the 1st part has the author's introduction and conclusion, the question arises about the authenticity of the remaining parts and the degree of intervention in their text by later editors. As the analysis shows, the text of the entire treatise and tables gives no reason to doubt the authorship indicated in the title. This also applies to the chronological list of emperors, which, although brought up to the beginning of the reign of Constantine VII Porphyrogenitus and Roman I (920), may well be based on a similar list compiled by St. Maximus the Confessor. This treatise, accompanied by a Latin translation, was first published by the famous French Jesuit scholar Dionysius Petavius ​​(Denis Peto, 1583–1652), who used it, among other materials, in his fundamental work on chronology 6 . In 1857, the text and translation from Petavia’s edition was reproduced in the Greek series of his Patrology by Jacques-Paul Min 7 . The Latin translation is far from scrupulous, and the circular table given in the publication, describing the system of “multiplying by five and six” (“wheel” No. 2), contains a lot of errors that need correction. In 1905, the German philologist and Church historian Eduard Schwartz 8 turned to the text of the treatise. His work contains an updated table describing the method of “multiplying by five and six,” as well as a facsimile of “wheel” No. 2. The text published by Petavius ​​is derived from a manuscript in the Vatican Library 9 . A new edition of the treatise, taking into account earlier manuscripts (for example, the 10th century codex from the Leiden collection of Scaliger Lugdunensis Scaligeranus gr. 33), is currently being prepared by J. Lampire and B. Markesinis 10 . The first of them recently devoted an article to a comparison of the Paschalistic methods of St. Maximus and another contemporary Byzantine Paschalist, monk and presbyter George 11. Of particular interest is the study of two Paschal systems, which are reflected in the treatise of St. Maximus the Confessor: Alexandrian and alternative method of “quintuples” (πενταπλοῦντες). Following Scaliger and Petavius, they were studied in the 18th century. Dutch scientist John van der Gagen 12, and at the turn of the 19th and 20th centuries. - the mentioned E. Schwartz, who saw in Maxim’s treatise features characteristic of Western Easterists. In particular, Schwartz correlates the system of “multiplying by five and six” with the Carthaginian Easter eggs of the 5th–6th centuries. His conclusions were accepted by Venance Grumel, who proposed his hypothesis about the reason for the emergence of the “quintupling” technique 13 . Our reconstruction of this methodology, carried out during the work on the Russian translation, made it possible not only to develop the observations of these scientists, but also to critically reconsider a number of provisions put forward by them. As the main Paschalistic, calendar and chronological scheme in the treatise of St. Maximus the Confessor stands for the classical Alexandrian system, developed in its main features by the monk Annianus at the beginning of the 5th century. According to R.H. This system is characterized by the following features: 1) era of the era - the beginning of the creation of the world, Sunday March 25, 5492 BC. e. 14, exactly 5500 years before the Incarnation (Annunciation); 2) Nativity of Jesus Christ - December 25, 5501 from the creation of the world (March style), 9 AD. e.; 3) Resurrection of Christ - March 25, 5534, 42 AD. e. 4) the “natural” (natural) beginning of the year is March 25, the date to which the three main events of Sacred history relate: the beginning of creation, the Incarnation and the Resurrection of Christ. Along with this system, which St. Maxim directly identifies with church tradition, at the beginning of the 7th century. The so-called was rapidly gaining popularity. Byzantine (“Roman”) Easter-chronological system, first attested in the treatise of the monk and presbyter George (639) in the 15th and 10th–11th centuries. which became the only official era in Byzantium. As V. Grumel showed, the Byzantine era is in direct connection with the era of the so-called “Easter Chronicle” (630). Both theoretically rely on a chronology that dates creation to March 5509 BC. e., but at the same time the Byzantine era has a September era, and the counting of years in Easter begins in March 5508. Thus, the 1st year of the Alexandrian era (5492 BC) turns out to be the 17th year of the Byzantine era. Hence the discrepancy in the counting of years in the cycles of the Byzantine and Alexandrian Easter: – in a 19-year (“lunar”) cycle: 1 year of the Alexandrian era (hereinafter referred to as Al. E.) = 17 year of the Byzantine era (hereinafter referred to as Byzantine era); 1 year visa. e. = 4 years al. e.; – in a 28-year (“solar” or weekly) cycle: 1 year al. e. = 17 visas. e.; 1 year visa. e. = 13 al. e. It should be noted that in the most common type of Byzantine Easter, created under Emperor Heraclius, the dates of the Paschal full moons and, accordingly, the Sunday Easter 16 completely coincide with the Alexandrian dates 17. The latter are based on the 19-year lunisolar cycle, which appeared in the first years of the 4th century. However, there were other versions of Easter, which offered other ways of calculating the lunar phases. One of them critically examines the treatise of Maximus the Confessor entitled “multiplying by five and six”; its supporters are called literally “quinquple and sixfold” (πενταπλοῦντες καὶ ἑξαπλοῦντες) 18. The mathematical principle of this method is the introduction of conditional lunar “days” 19. The difference between solar and lunar days is 1 mite (1/60th of a day, 24 minutes according to modern standards). The specificity of the method is that the lunar day is not equal to 59/60 solar days (as one would expect based on the equality of the lunar month to 29.5 days), but 60/61, because a solar day is taken to be 61/60 lunar “days”. The phase can be determined to the nearest mite by adding these increments and then subtracting the full 30-day months. The beginning of the countdown is taken to be January 1 of the 1st year of the 19-year cycle 20, therefore the epacts of the moon 21 are taken on the eve of this date, and at the beginning their number is 0 (which is only true in the Constantinople cycle). Traditionally, in Easter studies, the annual epacts of the moon were calculated as follows: the serial number of the year in the lunar cycle was multiplied by 11, and then entire 30-day months were subtracted from the resulting number. The number 11 is explained by the difference between the solar 365-day year and the lunar 354-day year. Leap days are not taken into account in this case, since the 19-year Easter cycle itself is calculated in such a way that after its 4-fold repetition, leap days are completely compensated, ensuring high accuracy of phase calculations (the error is 1 day in approximately 312 years, which is better than the classical cyclameton and similar to the 76-year Calippus cycle). Proponents of the new technique carried out multiplication by 11 in two stages - first by 5 and then by 6, which is where their name comes from. The meaning of the procedure is clear from the cited prp. Maxim 22 examples for determining the phase for April 17 - the date considered the Easter full moon (14th day of the moon) in the traditional Easter (16 Byzantine = 19 Alexandrian lunar cycle): 1) the serial number of the required day is determined, starting from January 1, 23: n=107; 2) the number of the year in the lunar cycle (N) is multiplied by 5: 16Å 5=80; 3) the addition in mites for the year and day is determined: nÅ 1/60+NÅ 5/60 = (n+N)Å 1/60 = (107+80)Å 1/60 = 187/60 = 3 7 /60, in integers – 3 (at this stage the convenience of separate multiplication by 5 and 6 is manifested); 4) year number N is multiplied by 6: 16Å 6=96; 5) the terms are summed up: 107 + 80 + 3 + 96 = 286, which after subtracting whole 30-day months (9Å 30 = 270) gives 16 (accurate to mites - 16 7 /60). It turns out that the age of the moon on April 17, according to this calculation method, turns out to be not 14, but 16 days. As Maximus the Confessor points out, the “quintuplings” receive, instead of the 14th day of the moon indicated in the Alexandrian Paschal, the 16th day in the 16th year of their cycle (which is proven by example) and the 15th in the 5th–15th, 17th and 19th years of the 24 cycle. Having simulated the described method on a computer, it is easy to verify that the phases of the dates taken as the 14th days of the moon in the Alexandrian Easter turn out to be the 15th in most cases. St. Maxim sharply criticizes this method, considering it incorrect. Indeed, a smooth phase increment of 61 lepta per day leads to the fact that the traditional 19-year cycle of 6935 days (=19x365) does not include exactly 235 lunar months, but 35 lepta more: 6935x 61/60=7050 7/12 = (235x30)+ 35/60. The fact is that the methodology is based on the condition that a 365-day solar year exceeds the duration of 12 lunar months by 11 days and 5 lepta (365 x 61 /60 = 371 5 /60 = (12x30) + 11 5 /60). Because of this, an annual amendment of 5 mites arises, which is taken into account when multiplying the serial number of the year of the cycle by 5. Already by the 12th year, the amendment reaches 60 /60 = 1 25, being an analogue of the “moon jump” 26 at the end of the traditional 19-year cycle (enneacadecaetherids). However, the accumulation of mites continues after the 12th year, reaching 35/60 by the end of the cycle; In order for the cycle to close and start again from scratch, the increment accumulated from the 12th to the 19th year must be discarded. Such mathematical inconsistency, as rightly indicated in the treatise of St. Maxim 27 demonstrates the theoretical inconsistency of the method under consideration as a whole 28 . It should be emphasized that, as can be seen from the context, the supporters of the new technique did not at all claim to change the traditional dates of the Easter full moon: their theoretical calculations were aimed only at bringing the calculated lunar phases closer to the observed ones. However, by doing so, the “quintuple and sixfold” ones willy-nilly came into conflict with the generally accepted rules for determining the holiday of Christian Easter. An analysis of these contradictions, which manifested themselves in cases where the traditional 14th day of the moon was considered by them to be the 15th or 16th and fell on Sunday or Monday, St. Maxim devoted the 2nd part of his treatise. The meaning of his reproaches boils down to the following: 1) if the date of the “Jewish Passover” is actually the 15th day of the moon, Sunday, or the 16th day of the moon, Sunday or Monday, then on this very day, according to the rules, Passover should be celebrated; otherwise, Easter falls on the 22nd or 23rd day of the moon, beyond the traditionally established interval from the 15th to the 21st lunar day inclusive; 2) in the case when “Jewish Passover” 29 turns out to be the 16th day of the moon, on Monday, according to the rules of Paschal, Christian Easter should be celebrated the day before on Sunday, the 15th day of the moon - but in this case it will precede the “Jewish Passover” - which is absurd. The immediate reason that led to the “quintuple and sixfold” reform is not precisely known, but one can hardly agree that in this case we are talking about idle “mathematical games,” as some researchers are inclined to believe 30 . The question of Paschal, which forms the core of Christian worship, touched on the fundamental foundations of the tradition and daily practice of the Church. To clarify the goals of the new technique, scientists have put forward a number of hypotheses. Thus, according to E. Schwartz, supporters of this method tried, like Victoria of Aquitaine in the 5th century, to reconcile Eastern Paschal practice with the requirements of the ancient Western tradition 31 . Since the time of the Council of Nicea in 325, in the Christian East, Easter was celebrated on the first Sunday after the Easter full moon (“Jewish Passover” on the 14th of Nisan), that is, Easter could fall on the interval from the 15th (if the full moon fell on Saturday) to the 21st (if the full moon fell on Sunday and should have receded by a week) day of the moon. But in the West, since ancient times, the tradition has taken root not to celebrate Easter on the day after the Saturday full moon, that is, on the 15th day, and therefore the interval of Easter dates was limited to 16–22 days of the moon. In addition, in the ancient Roman tradition, it was customary to celebrate Easter no later than April 21 (so that Lent would not fall on the feast of the founding of Rome). But this requirement is incompatible with the 19-year cycle of the Alexandrian type, and Western Easterists (for example, Victoria of Aquitaine) tried to at least bring the latest Easter a little closer (according to the Alexandrian Easter - April 25). According to Schwartz, the “quintuple and sixfold” wanted to achieve something similar. However, this opinion cannot be accepted, since their methodology does not at all eliminate the celebration of Easter on the 15th day of the moon, which, even taking into account the “leap additions” (see below), takes place in the 421st and 441st years of the complete 532-year Easter cycle (calculated phases, respectively, 15 13 / 15 and 15 23 / 30). And late Easter on April 24 and 25 in the system of “multiplying by 5 and 6” is allowed no less often than in the Alexandrian Paschal. V. Grumel proposed another explanation, based on the tasks of church chronography 32. In the Byzantine tradition, perhaps dating back to Anatoly of Laodicea (III century), the historical date of the Passion of the Lord is considered to be Friday, March 23, 31 AD. e. 33 For this year, the 11th in the Constantinople and 13th in the Alexandrian 19-year cycle, the Easter full moon falls on Saturday, March 24. However, this, according to Grumel, could not suit experts in gospel history: after all, the crucifixion took place on Friday, on the very day of the 14th of Nisan (according to weather forecasters) or even the next day (according to John). Trying to solve this problem, modern St. Maxim, the author of the "Easter Chronicle", obviously familiar with the system of "incrementing mites", resorted to a not entirely correct technique: when calculating the lunar phase on March 23, he not only took into account the daily 60th shares (mepts) accumulated to this day from January 1 (having received 13 22 /60), but also added an amendment designed to take into account the leap cycle (45 /60 for a given year), which gave opportunity to get phase 14 7 /60. 34 Meanwhile, the 19-year cycle of the Christian Easter does not need to take into account the leap cycle 35. We can conclude that in this case we are dealing with the ad hoc application of a certain procedure reminiscent of the technique described by St. Maximus the Confessor. Grumel suggested that the “quintuplers” were puzzled by the same problem, but approached the matter systematically. Indeed, in their system the actual lunar phase for March 23 of the 11th year of the cycle is 14 12 /60, which provides the required result: Good Friday March 23, 31 AD. e. coincides with 14 Nisan 36. Thus, according to Grumel, the goal of the authors of the new system was to obtain compensation for the “moon jump” precisely at the end of the 11th year of the 19-year cycle. However, it is not difficult to verify that this phenomenon is just a spontaneous consequence of the elementary principle according to which solar days are taken to be equal to 1 1/60 lunar days. In addition, the advance by the day of the usual Easter required to move the full moon from Saturday to Good Friday is achieved after the 5th year of the cycle, and the “moon jump” is compensated for later than the supposedly sought date of Nisan 14. Moreover, the calculations in the “Easter Chronicle” are fundamentally different from the system of “quintupling and quintupling” in that its author calculates the addition of 1 mite per day from January 1 of each year without taking into account their accumulation in previous years of the lunar cycle. This gives a result that is virtually identical to the usual Alexandrian and Byzantine Paschals. A similar technique for smoothly incrementing lunar phases has been widely known since at least the 3rd century. With different amounts of daily or monthly fractional additions, it is used in the Western 84-year cycle, in the Jewish calendar, in the Passovers of Iron and the anonymous Karnthaler “B VIII” 37. The question naturally arises: why did the system of multiplication “by five and six” appear precisely at the beginning of the 7th century? This question can be answered by taking into account one circumstance, which, in our opinion, played a decisive role in this case. We are talking about the error of the Christian Easter in relation to the lunar phases. The average lunar month adopted in it is 0.00026 days longer than the synodic one, which causes a delay in relation to the observed lunar phases by about a day every 312 years. Approximately this period passed from the time of registration of the Alexandrian Paschal (about 300) to the era when the described reforms began (about 630). Thus, there is reason to assert that the “quintuplings and quintuplings” relied in their works not so much on some speculative constructions, but on the astronomical realities of the 2nd quarter of the 7th century. n. e. 38 In this regard, worthy of attention is the example of the combination of the “quintuple” system with the “leap of the moon” in the 17th year of the cycle, found in an anonymous treatise of 1079. 39 Such duplication of amendments to the cycle (“leap” plus annual addition) can easily be explained by the aggravation by the 11th century. astronomical error of the Calippan cycle: 625 years from the beginning of the 4th century. the error regarding the lunar phases exceeded 2 days 40 . Let's take as an example the year the treatise was compiled - 640 AD. e. According to the Alexandrian Easter, this is the 14th year of the cycle with the Paschal full moon on April 12. According to the “quintuplers,” this is the 11th year of the cycle, when on April 12 the moon phase is 15 31/60; therefore, the new moon refers to March 28, 8 hours 48 minutes. at midnight (phase 11/30 by the end of the day). Modern astronomical tables place the true new moon on March 28, 640, at 6:40 GMT 41, 8:40 longitude of Constantinople. Thus, the method, completely untenable from a theoretical point of view, perfectly corresponded to the observed celestial phenomena - which could not be said about the much more elegant Alexandrian Easter from a mathematical point of view. The mathematical model of the cycle of “quintuples and sixfolders” 42 that we have compiled allows us to critically reconsider the reconstruction of the principles of the system for calculating lunar phases proposed by E. Schwartz 43 and put forward the following theses. 1. The basis of this system is the 19-year lunar cycle of the Byzantine type, close to the cycle described in the “Easter Chronicle”, but with the count of years shifted by one (the so-called Syrian version of the 19-year cycle, preserved in the modern Jewish calendar and known to Dionysius the Less as “cyclus lunaris”) 44. 2. The starting point of the cycle is the new moon on the eve of January 1, 0 of the cycle, which also finds correspondence in the system of the “Easter Chronicle”, where exactly the zero (pre-cyclical) year is taken as the year of the creation of the world 45. 3. From this date, taken as the absolute zero of the age of the moon, the lunar phases are counted sequentially at 61 lepta per day. This rate of lunar growth corresponds to the average duration of the lunar period (month), taken to be exactly 29.5 days. 4. As a result, the following series of annual epacts spontaneously arise (the age of the moon on the eve of the beginning of the January year, on the night of December 31 to January 1): As you can see, epacts increase annually by 11 days and 5 mites (1/12 = 5/60); these 5 lepta by the 12th year form a whole day, which in whole numbers produces the effect of a “moon jump” (transition from epacta 1 to 13 instead of 12). In the 19th year, instead of 0, epact 7/12 appears, which must be eliminated for the period to acquire a cyclical character. Particular attention should be paid to how leap years were taken into account in this calculation system. As is known, the traditional Christian Paschal, first introduced by Anatoly of Laodicea in the 2nd half of the 3rd century, is a combination of the 19-year cyclameton with the 4-year Julian cycle, which gives it the character of the 76-year Calippan cycle 46 . At the same time, the calculation of the 19-year cycle is deliberately carried out without taking into account leap leaps, which, after 4 lunar cycles, completely “dissolve” in them. However, as the explanations to the table of Maximus the Confessor show, “those who multiply by five and six” when calculating the lunar phases resorted to taking into account leap years, adding ¼ of a day in the 1st year after a leap year, in the 2nd year - ½ of a day, in the 3rd - ¾ of a day, and in a leap year - a whole day. This technique made it possible to further increase the phases of the moon, bringing them closer to the observed ones. It is characteristic that in the treatise this incorrect procedure is not particularly condemned, however, the table distinguishes between leap additions and the accumulation of contributions using the method of “multiplying by 5 and 6”. As the analysis of the calendar-mathematical method, with which Maxim the Confessor argues, shows, there is no reason to talk about the use of the deliberate addition of 5 extra mites annually. Despite the fact that the method of “quintupling and increasing” has all the features of an artificial construction that violates the harmonious mathematical system of the classical Easter egg, one cannot see in this some kind of game and trickery, as Schwartz impartially assesses this system 47 . In our opinion, we are talking about an attempt at a very delicate solution to the fundamental Easter problem: the obvious discrepancy between the tabulated values ​​of the lunar phases and the observed ones. It is unnecessary to talk about the relevance of accurate calculations of lunar motion: both in antiquity and in the Middle Ages, lunar phases played a vital role in medicine, agriculture, navigation and other areas of practical activity, as evidenced, in particular, by calculations of the “lunar current”, well known in ancient Russian books. For the 7th century, when the discrepancy was measured in 1–2 days, this solution seemed the least painful. It is characteristic that St. Maxim, harshly criticizing the shortcomings of the method of “multiplying by 5 and 6,” realized the attractiveness of this system and considered it necessary to compile a table for the entire 532-year cycle (in a historical context - the 12th and 13th great indictions, from 345 to 1408 AD) in order to warn about possible errors in the calculation of Easter 48 . Subsequently, when by the XIII century. The discrepancy between the Alexandrian Paschal and the movement of the moon reached 4 days, it became clear that it was necessary either to undergo a radical reform of the Paschal, or to come to terms with its conditional nature. The Orthodox tradition, in contrast to the Catholic one, followed the second path 49 . Below is a commented translation of the 50th Paschal treatise of Maximus the Confessor, made according to the edition: PG. 19. Col. 1217–1280, taking into account corrections by E. Schwartz (Ostertafeln. S. 85–87). Like the saints of our father Maxim, monk and martyr, interpretation point by point for the saving Easter of Christ our God, explaining the written canons 51 52. For more details, see: Dionysius (Shlenov), abbot. “Theological Bulletin” of 1945: a new find on the way from pre-revolutionary to modern academic journal // BV. 2010. No. 11–12. Anniversary issue. pp. 921–934. See: Polishchuk E. S. Collection “Theological Works” (background, history and review of the contents of published issues) // BT. 2003. Vol. 38. P. 376–400, here: P. 382; It's him. Theological creativity in the Russian Church // ZhMP. 2011. No. 5 (May). pp. 88–95. See, for example: Zhurinskaya M. A. [Rec. on] Theological works. Collection 38 // Alpha and Omega. 2004. No. 1 (39). pp. 363–367; Dunaev A.G. Russian theology between the past and the future // ZhMP. 2009. No. 12. pp. 76–83. Illustria (Latin illustris, Greek ἰλλούστριος) – the title of senator of the first rank. Magister militum per Numidiam. About him see: The Prosopography of the Later Roman Empire / Ed. by J.R. Martindale. Vol. III: 527–641. Camb., 1992. P. 1013 (Petrus, N 70); Duval Y. Le patrice Pierre, exarque d’Afrique? // Antiquites africaines. 1971, pp. 209–214. At the trial of 655, Maximus the Confessor was accused of the fact that 22 years earlier (i.e., in 633), on his advice, Peter, who at that time was the strategist of Numidia, did not, contrary to the order of the emperor, go on a campaign against Egypt, which led to the capture of this province by the Arabs. St. Maxim denied this accusation. Dionysii Petavii Opus de doctrina temporum. Parisiis, 1627. T. III: Uranologium. In his “Address to the Reader,” Petavius, polemicizing with the Protestant Joseph Justus Scaliger (1540–1609), points out: “Next [follows] the “Paschalia” of St. Martyr Maximus, which through my efforts was copied from the Vatican Library. Joseph Scaliger included some excerpts from it in his work “On the Correction of Chronology” and in extensive notes interpreted the doctrine of the Greek Paschal: which subject, like everything else in this work, he failed, as the enlightened reader will understand from our dissertation “On the eras and Paschals of the Greeks”” (PG. 19. Col. 745–746). Schwartz E. Christliche und jüdische Ostertafeln. B., 1905. S. 81–85. (Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Neue Folge, Bd. 8, N 6) (hereinafter: Schwartz. Ostertafeln...). It was apparently a parchment codex from the 12th century. Vatic. gr. 1502 or Vatic copied from it in 1520. gr. 505. See: Lempire J. Le calcul de la date de Pâques dans les traités de S. Maxime le Confesseur et de Georges, moine et prêtre // Byzantion. 2007. T. 77. P. 269–270 (hereinafter: Lempire. Le calcul...). Lempire. Le calcul... On the chronological controversy in Byzantium in the 7th century. see: Kuzenkov P.V. Disputes about the age of the world in Byzantium in the 7th–11th centuries. (About three world eras: Alexandrian, “proto-Byzantine” and Byzantine) // VV. 2007. T. 66. pp. 93–124. van der Hagen J. Observationes in Heraclii imperatoris Methodum Paschalem, ut et in Maximi monachi Computum Paschalem, necnon in Anonymi Chronicon Paschale ejusque chronotaxim et methodum paschalem. Amstelaedami, 1736. Grumel V. La Chronologie. P., 1958. P. 117–122 (hereinafter: Grumel. La Chronologie...). The traditional abbreviation "R. X." in this case, it can lead to misunderstanding, since it relies on a different dating of the Nativity, used in the era of Dionysius the Lesser. Diekamp F. Der Mönch und Presbyter Georgios, ein unbekannter Schriftsteller des 7. Jahrhunderts // BZ. 1900. Bd. 9. S. 14–51; Lempire. Le calcul... P. 271–272, 285–293. Here and further, in order to avoid confusion, Christian Easter (Resurrection of Christ) is written with a capital letter, the Old Testament (Passover from 14 to 15 Nisan) - with a lowercase letter. See: Grumel. La Chronologie... P. 54 (N VII), 101. Enarratio, I 11–12, 16; II (Col. 1227–1230, 1233, 1252–1264). Similar average lunar “days” (1/30 of the average synodic month) were already known in Mesopotamian astronomy, and were later used in medieval Indian astronomy. According to B. van der Weerden, they were also used by the astronomer Calippus. See: van der Waerden B. L. Greek Astronomical Calendar, 2: Callippos and His Calendar // Archive for the History of Exact Science. 1984. Vol. 29/2. P. 115–124; cf.: Neugebauer O. A History of Ancient Mathematical Astronomy. B.; Heidelberg; N.Y., 1975. P. 617. The January beginning of the “lunar circle” was traditional for Byzantine Easter. This is evidenced, in particular, by the treatises of Michael Psellos (XI century) (Redl G. La chronologie de Psellos // Byzantion. 1927–1928 [1929]. T. 4. P. 221–222; 1929 [1930]. T. 5. P. 229–286), Kirik Novgorod (XII century) (Simonov R. A. Mathematical and calendar-astronomical thought of Ancient Rus': According to medieval book culture. M., 2007. pp. 306–337) and the section on Easter in the “Alphabetic Syntagma” of Matthew Vlastar (XIV century) (Σύνταγμα τῶν θείων καὶ ἱερῶν κανόνων / Ἐκδοθὲν ὑπὸ Κ Ῥάλλη καὶ Μ. 1992 (repr.). T. 6. Σ. Lunar epacts are the annual increment in the age of the moon at the beginning of the year, which is approximately 11 days. Knowing the epacts of a given year, it is easy to calculate the lunar phase of a particular day. Enarratio, II 5 (Col. 1261). January 1 is the beginning of the Roman consular year. Enarratio, I 16 (Col. 1233); see also: Schwartz. Ostertafeln... S. 82. Apparently, it is precisely this type of 19-year cycle that we encounter in the Slavic Tolkova Paleia, a monument whose dating and origin are still the subject of scientific debate. In the calculation of lunar phases presented here, the beginning of the year is considered to be January 1, and the “moon jump” occurs precisely in the 12th year (Ivanova N.P., Tsyb S.V. Historical chronology. Barnaul, 2003. P. 131). This is especially pointed out by Maximus the Confessor (Enarratio, I 11; Col. 1228). Maxim. Conf. Enarratio, I 12. Col. 1229. The average lunar month for the “quintuplings” is 29.51 days, which is certainly worse than the parameters of the classical enneacadecaetherides (29.53085 days) in relation to the actual duration of the synodic month (29.5306 days). Hereinafter, “Jewish Passover” refers to the first spring full moon indicated in the Christian Paschal. It is not directly related to the dates of the celebration of the Jewish Passover, because they were calculated using a completely different system. Serruys D. De quelques ères utilées chez chroniqueurs byzantins // Revue de philologie, de littérature et d’histoire anciennes. 1907. T. 31. P. 182. Schwartz. Ostertafeln... S. 84, 88. Grumel. La Chronologie... P. 119. For the first time this date appears, apparently, in Eusebius of Caesarea in the unsurvived treatises “On the Paschal Celebration” and “On Unleavened Bread” (apud: Eliae metropolitae Nisibeni Opus chronologicum. Pars II / Ed. et tr. I.-B. Chabot. Textus. Parisiis; Lipsiis, 1909. (PO, III ser., 8 (t) = [62/2] = Syr. 22); Versio. Romae, 1910. (PO, III ser., 8 (v) = [63/2] = Syr. 24)). P. 51–52, 108 (textus); 73, 118 (versio). Chronicon Paschale/Ed. L. Dindorf. Bonn, 1832, pp. 414–415. The length of the classic Metonic cycle is 6940 days: for every 19 years of 365 days, 5 days are added to it (19Å 365+5=6940). In the Julian cycle, 4 days accumulate in 19 years due to leap years. Combining the Metonic cycle with the Julian year turns it into a much more accurate 76-year Calippan cycle: every 4 lunar cycles (76 years) completely “absorb” the increases of the 19 leap cycles. Grumel. La Chronologie... P. 120–122. Karnthaler F. P. Die Chronologischen Abhandlungen des Laur. Gr. Plut. 57, Cod. 42, 154–162 // Byzantinisch-Neugriechische Jahrbücher. 1933.Bd. 10. S. 1–64. As an example of a reform designed to improve the accuracy of calculations of lunar phases, one can point to the calendar system of George, a student of James of Edessa, where the dates of the new moons of the Easter month are better correlated with astronomical ones. See: Eliae metropolitae Nisibeni Opus chronologicum. Pars II/Ed. et tr. I.-B. Chabot. Textus. Parisiis; Lipsiis, 1909. (PO, III ser., 8 (t) = [62/2] = Syr. 22); Versio. Romae; Lipsiis, 1910. (PO, III ser., 8 (v) = [63/2] = Syr. 24). P. 132; 145–146 (versio). Mentz A. Beiträge zur Osterberechnung bei der Byzantinern. Königsberg, 1906. S. 86. On average, the discrepancy between the observed lunar phases and the Easter indications accumulated by a day over approximately 312 years; spasmodic shifts occur in the 4th–5th, 7th–8th, 10th–11th and 13th–14th centuries. For example, Michael Psellos (late 11th century) in his Easter calculations consistently calls the Easter full moon not the fourteenth lunar day, but the fifteenth (Redl G. La chronologie appliquée de Michel Psellos // Byzantion. T. 4. 1927/28 [1929]. P. 197–236; T. 5. 1929 [1930]. P. 229–286). Espenak F. Phases of the Moon. http://eclipse.gsfc.nasa.gov/phase/phases0601.html [electr. resource, access 03/26/2012]. Unfortunately, the resulting spreadsheet of the complete lunar cycle, which occupies 24,180 cells, cannot be published in printed form. Schwartz. Ostertafeln... S. 81–87. See: Bolotov V.V. Alexandrian Easter: logic and aesthetics // Calendar issue / Ed. A. Chkhartishvili. M., 2000. pp. 105–144. Therefore, the statement that the cycle begins with epact 11 (Schwartz. Ostertafeln... S. 83) is inaccurate. Schwartz. Ostertafeln... S. 10. Schwartz. Ostertafeln… S. 84 (“Nun kommt es ja vor dass Leute betrügen ohne rechten Zweck, nur weils ihnen Spass macht…”). In the enormity of the mathematical calculations carried out by St. Maxim, the author of these lines had the opportunity to verify for himself: even with the help of a computer, compiling a similar table took several days. Discrepancy between Easter (tabular) full moons and real ones in the 14th–18th centuries. did not bother the Orthodox scribes. For example, in Russian liturgical books, calculations of the “Lunar Current”, much closer to astronomical realities, coexist with Easter tables recorded nearby, indicating other dates of full moons (see: Pentkovsky A.M. Calendar tables in Russian manuscripts of the XIV-XVI centuries // Methodological recommendations for the description of Slavic-Russian manuscript books. Issue 3. Part. 1. M., 1990. pp. 167–171). The author expresses his heartfelt gratitude to St. Mikhail Asmus for valuable comments and corrections. The Greek term κανόνιον here and below means a short table.