Abbo of Fleury
Abbo or Abbon of Fleury known as Saint Abbo or Abbon, was a monk and abbot of Fleury Abbey in present-day Saint-Benoît-sur-Loire near Orléans, France. Abbo was brought up in the Benedictine abbey of Fleury, he was educated at Paris and Reims, devoting himself to philosophy and astronomy. He spent two years in England in the newly founded monastery of Ramsey, assisting Archbishop Oswald of York in restoring the monastic system, he was abbot and director of the school of this newly founded monastery from 986 to 987. Abbo returned to Fleury in 988, but another monk, who had secured the support of the King and the Bishop of Orléans, contested the choice, the matter assumed national importance. It was settled in favour of Abbo by Gerbert of Aurillac; the new abbot was active in contemporary politics: He was present at the Synod of St. Basolus, near Reims, at which Arnulf, Archbishop of Reims was tried for treason and deposed, to make way for Gerbert. In 996 King Robert II sent him to Rome to ward off a threatened papal interdict over Robert's marriage to Bertha.
On the way to Rome he met Pope Gregory V, a fugitive from the city from which the Antipope John XVI had expelled him. Between the Pontiff and the Abbot the greatest esteem and affection existed; the royal petition for a dispensation was rejected. Abbo succeeded in bringing about the restoration of Arnulf to the see of Reims, he was influential in calming the excitement and fear about the end of the world, widespread in Europe in 1000. In 1004 he attempted to restore discipline in the monastery of La Reole, in Gascony, by transferring some of the monks of Fleury into that community, but the trouble increased. He concealed the wound and reached his cell, where he died in the arms of his faithful disciple Aimoin, who has left an account of his labours and virtues; the miracles wrought at his tomb soon caused him to be regarded in the Church of Gaul as a saint and martyr, although he does not seem to have been canonized by Rome. His feast is kept on 13 November; when in England Abbo learned of the martyrdom of Saint Edmund, he wrote a passion in Latin on it.
He wrote a Latin grammar for his English students, three poems to St Dunstan. Among his other works are a simplification of the computus, the computation of the date of Easter. Around 980 to 985, he wrote a commentary on the "Calculus" of Victorius of Aquitaine, before the introduction of Arabic numerals, when calculations were quite complex; the wide range of Abbo's thought is reflected in the commentary, covering the nature of wisdom, the philosophy of number, the relationship of unity and plurality, the arithmetic of the Calculus. Abbo drew on his knowledge of grammar and cosmology to illustrate his arguments, set it all in the broader context of his theology of Creation. Most of Abbo's works can be found in the Patrologia Latina. There is one contemporary biography, written by his disciple Aimoin of Fleury, in which much of Abbo's correspondence was reproduced, it is of great importance, among other things as a historical source of information for the reign of Robert II of France with reference to the Papacy.
Richard W. Pfaff sums up his achievement as follows: "One of the most versatile thinkers and writers of his time, Abbo put his mark on several areas of medieval life and thought, but none more so than in transmitting much, valuable from the tradition of reformed French monasticism to the nascent monastic culture of late tenth-century England." Cora E. Lutz, Schoolmasters of the Tenth Century, Archon Books 1977. Abbo 1 at Prosopography of Anglo-Saxon England Chisholm, Hugh, ed.. "Abbon of Fleury". Encyclopædia Britannica. Cambridge University Press; this article incorporates text from a publication now in the public domain: Herbermann, Charles, ed.. "St. Abbon". Catholic Encyclopedia. New York: Robert Appleton. Abbo in the Christian Cyclopedia
Alexander of Villedieu
Alexander of Villedieu was a French author and poet, who wrote text books on Latin grammar and arithmetic, everything in verse. He was born around 1175 in Villedieu-les-Poêles in Normandy, studied in Paris, taught at Dol in Brittany, his greatest fame stems from the Doctrinale Puerorum. He died in 1240, or in 1250, he was a Master of the University of Paris. His Doctrinale puerorum, a versified grammar, soon became a classic, it was composed around 1200, was all written in leonine hexameters. After several centuries, with the advent of printing, it appeared in countless editions in Italy and France, it was based on the older works of Priscian. Alexander wrote a short tract on arithmetic called Carmen de Algorismo—the Poem about Arithmetic, which achieved a wide distribution. A typical line from his Carmen de Algorismo, runs like this: Extrahe radicem semper sub parte sinistraWherein he instructs his students: "always extract the square root by starting from the left"; the poem is not long, only a few hundred lines, summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, as he calls the new Hindu-Arabic numerals.
Dietrich Reichling editor, Das Doctrinale des Alexander de Villa-Dei. Carmen de Algorismo Franaut entry Alexander de Villa Dei - Mathematics and the Liberal Arts Image of 1463 printed edition of the Doctrinale Bibliographical page Musicological page
In astronomy, the new moon is the first lunar phase, when the Moon and Sun have the same ecliptic longitude. At this phase, the lunar disk is not visible to the unaided eye, except when silhouetted during a solar eclipse. Daylight outshines the earthlight; the actual phase is a thin crescent. The original meaning of the term new moon, still sometimes used in non-astronomical contexts, was the first visible crescent of the Moon, after conjunction with the Sun; this crescent moon is visible when low above the western horizon shortly after sunset and before moonset. A lunation or synodic month is the average time from one new moon to the next. In the J2000.0 epoch, the average length of a lunation is 29.530588 days. However, the length of any one synodic month can vary from 29.26 to 29.80 days due to the perturbing effects of the Sun's gravity on the Moon's eccentric orbit. In a lunar calendar, each month corresponds to a lunation; each lunar cycle can be assigned a unique lunation number to identify it.
The length of a lunation is about 29.53 days. Its precise duration is linked to many phenomena in nature, such as the variation between spring and neap tides. An approximate formula to compute the mean moments of new moon for successive months is: d = 5.597661 + 29.5305888610 × N + × N 2 where N is an integer, starting with 0 for the first new moon in the year 2000, and, incremented by 1 for each successive synodic month. To obtain this moment expressed in Universal Time, add the result of following approximate correction to the result d obtained above: − 0.000739 − × N 2 daysPeriodic perturbations change the time of true conjunction from these mean values. For all new moons between 1601 and 2401, the maximum difference is 0.592 days = 14h13m in either direction. The duration of a lunation varies in this period between 29.272 and 29.833 days, i.e. −0.259d = 6h12m shorter, or +0.302d = 7h15m longer than average. This range is smaller than the difference between mean and true conjunction, because during one lunation the periodic terms cannot all change to their maximum opposite value.
See the article on the full moon cycle for a simple method to compute the moment of new moon more accurately. The long-term error of the formula is approximately: 1 cy2 seconds in TT, 11 cy2 seconds in UT The moment of mean conjunction can be computed from an expression for the mean ecliptical longitude of the Moon minus the mean ecliptical longitude of the Sun. Jean Meeus gave formulae to compute this in his Astronomical Formulae for Calculators based on the ephemerides of Brown and Newcomb; these are now outdated: Chapront et al. published improved parameters. Meeus's formula uses a fractional variable to allow computation of the four main phases, uses a second variable for the secular terms. For the convenience of the reader, the formula given above is based on Chapront's latest parameters and expressed with a single integer variable, the following additional terms have been added: constant term: Like Meeus, apply the constant terms of the aberration of light for the Sun's motion and light-time correction for the Moon to obtain the apparent difference in ecliptical longitudes:Sun: +20.496" Moon: −0.704" Correction in conjunction: −0.000451 daysFor UT: at 1 January 2000, ΔT was +63.83 s.
The term includes a tidal contribution of 0.5×. The most current estimate from Lunar Laser Ranging for the acceleration is:"/cy2. Therefore, the new quadratic term of D is = -6.8498"T2. Indeed, the polynomial provided by Chapront et alii provides the same value; this translates to a correction of +14.622×10−12N2 days to the time of conjunction. For UT: analysis of historical observations shows that ΔT has a long-term increase of +31 s/cy2. Converted to days and lunations, the correction from ET to UT becomes:−235×10−12N2 days; the theoretical tidal contribution to ΔT is about +42 s/cy2 the smaller observed value is thought to be due to changes in the shape of the Earth. Because the discrepancy is not explained, uncertainty of our prediction of UT may be as large as the difference between these values: 11 s/cy2; the error in the position of the Moon itself is only maybe 0.5"/cy2, or (because the apparent mean angular velocit
Computus is a calculation that determines the calendar date of Easter. Because the date is based on a calendar-dependent equinox rather than the astronomical one, there are differences between calculations done according to the Julian calendar and the modern Gregorian calendar; the name has been used for this procedure since the early Middle Ages, as it was considered the most important computation of the age. For most of their history Christians have calculated Easter independently of the Jewish calendar. In principle, Easter falls on the Sunday following the full moon that follows the northern spring equinox. However, the vernal equinox and the full moon are not determined by astronomical observation; the vernal equinox is fixed to fall on 21 March. The full moon is an ecclesiastical full moon determined by reference to a lunar calendar, which again varied in different areas. While Easter now falls at the earliest on the 15th of the lunar month and at the latest on the 21st, in some areas it used to fall at the earliest on the 14th and at the latest on the 20th, or between the sixteenth and the 22nd.
The last limit arises from the fact that the crucifixion was considered to have happened on the 14th and the resurrection therefore on the sixteenth. The "computus" is the procedure of determining the first Sunday after the first ecclesiastical full moon falling on or after 21 March, the difficulty arose from doing this over the span of centuries without accurate means of measuring the precise tropical year; the synodic month had been measured to a high degree of accuracy. The schematic model, accepted is the Metonic cycle, which equates 19 tropical years to 235 synodic months. In 1583, the Catholic Church began using 21 March under the Gregorian calendar to calculate the date of Easter, while the Eastern Churches have continued to use 21 March under the Julian calendar; the Catholic and Protestant denominations thus use an ecclesiastical full moon that occurs four, five or thirty-four days earlier than the eastern one. The earliest and latest dates for Easter are 22 March and 25 April, in the Gregorian calendar as those dates are understood.
However, in the Orthodox Churches, while those dates are the same, they are reckoned using the Julian calendar. Easter is the most important Christian feast, the proper date of its celebration has been the subject of controversy as early as the meeting of Anicetus and Polycarp around 154. According to Eusebius's Church History, quoting Polycrates of Ephesus, churches in the Roman Province of Asia "always observed the day when the people put away the leaven", namely Passover, the 14th of the lunar month of Nisan; the rest of the Christian world at that time, according to Eusebius, held to "the view which still prevails," of fixing Easter on Sunday. Eusebius does not say. Other documents from the 3rd and 4th centuries reveal that the customary practice was for Christians to consult their Jewish neighbors to determine when the week of Passover would fall, to set Easter on the Sunday that fell within that week. By the end of the 3rd century some Christians had become dissatisfied with what they perceived as the disorderly state of the Jewish calendar.
The chief complaint was that the Jewish practice sometimes set the 14th of Nisan before the spring equinox. This is implied by Dionysius, Bishop of Alexandria, in the mid-3rd century, who stated that "at no time other than the spring equinox is it legitimate to celebrate Easter", and it was explicitly stated by Peter, bishop of Alexandria that "the men of the present day now celebrate before the equinox...through negligence and error." Another objection to using the Jewish computation may have been that the Jewish calendar was not unified. Jews in one city might have a method for reckoning the Week of Unleavened Bread different from that used by the Jews of another city; because of these perceived defects in the traditional practice, Christian computists began experimenting with systems for determining Easter that would be free of these defects. But these experiments themselves led to controversy, since some Christians held that the customary practice of holding Easter during the Jewish festival of Unleavened Bread should be continued if the Jewish computations were in error from the Christian point of view.
At the First Council of Nicaea in 325, it was agreed that the Christians should observe a common date, independent from the Jewish method. The council agreed to two rules without explicitly stating them, that 14 Nisan was to occur after the vernal equinox, that Easter was to occur on the Sunday after 14 Nisan; the first prevented two Easters in one solar year, while the second prevented Christians from celebrating Easter at the same time as the Jews celebrated Passover. The council ignored the fact that the Christian vernal equinox was a day rather than an astronomical instant, that the Christian 14 Nisan was a different day than the Jewish 14 Nisan, that Alexandria and Rome used different Easter tables; the Patriarchy of Alexandria celebrated Easter on the Sunday after the 14th day of the moon that falls on or after the vernal equinox, which they placed on 21 March. However, the Patriarchy of Rome stil
Easter called Pascha or Resurrection Sunday, is a festival and holiday commemorating the resurrection of Jesus from the dead, described in the New Testament as having occurred on the third day after his burial following his crucifixion by the Romans at Calvary c. 30 AD. It is the culmination of the Passion of Jesus, preceded by Lent, a 40-day period of fasting and penance. Most Christians refer to the week before Easter as "Holy Week", which contains the days of the Easter Triduum, including Maundy Thursday, commemorating the Maundy and Last Supper, as well as Good Friday, commemorating the crucifixion and death of Jesus. In Western Christianity, Eastertide, or the Easter Season, begins on Easter Sunday and lasts seven weeks, ending with the coming of the 50th day, Pentecost Sunday. In Eastern Christianity, the season of Pascha begins on Pascha and ends with the coming of the 40th day, the Feast of the Ascension. Easter and the holidays that are related to it are moveable feasts which do not fall on a fixed date in the Gregorian or Julian calendars which follow only the cycle of the sun.
The First Council of Nicaea established two rules, independence of the Jewish calendar and worldwide uniformity, which were the only rules for Easter explicitly laid down by the council. No details for the computation were specified, it has come to be the first Sunday after the ecclesiastical full moon that occurs on or soonest after 21 March, but calculations vary. Easter is linked to the Jewish Passover by much of its symbolism, as well as by its position in the calendar. In most European languages the feast is called by the words for passover in those languages. Easter customs vary across the Christian world, include sunrise services, exclaiming the Paschal greeting, clipping the church, decorating Easter eggs; the Easter lily, a symbol of the resurrection, traditionally decorates the chancel area of churches on this day and for the rest of Eastertide. Additional customs that have become associated with Easter and are observed by both Christians and some non-Christians include egg hunting, the Easter Bunny, Easter parades.
There are various traditional Easter foods that vary regionally. The modern English term Easter, cognate with modern Dutch ooster and German Ostern, developed from an Old English word that appears in the form Ēastrun, -on, or -an; the most accepted theory of the origin of the term is that it is derived from the name of an Old English goddess mentioned by the 7th to 8th-century English monk Bede, who wrote that Ēosturmōnaþ was an English month, corresponding to April, which he says "was once called after a goddess of theirs named Ēostre, in whose honour feasts were celebrated in that month". In Latin and Greek, the Christian celebration was, still is, called Pascha, a word derived from Aramaic פסחא, cognate to Hebrew פֶּסַח; the word denoted the Jewish festival known in English as Passover, commemorating the Jewish Exodus from slavery in Egypt. As early as the 50s of the 1st century, writing from Ephesus to the Christians in Corinth, applied the term to Christ, it is unlikely that the Ephesian and Corinthian Christians were the first to hear Exodus 12 interpreted as speaking about the death of Jesus, not just about the Jewish Passover ritual.
In most of the non-English speaking world, the feast is known by names derived from Greek and Latin Pascha. Pascha is a name by which Jesus himself is remembered in the Orthodox Church in connection with his resurrection and with the season of its celebration; the New Testament states that the resurrection of Jesus, which Easter celebrates, is one of the chief tenets of the Christian faith. The resurrection established Jesus as the powerful Son of God and is cited as proof that God will righteously judge the world. For those who trust in Jesus' death and resurrection, "death is swallowed up in victory." Any person who chooses to follow Jesus receives "a new birth into a living hope through the resurrection of Jesus Christ from the dead". Through faith in the working of God those who follow Jesus are spiritually resurrected with him so that they may walk in a new way of life and receive eternal salvation. Easter is linked to Passover and the Exodus from Egypt recorded in the Old Testament through the Last Supper and crucifixion of Jesus that preceded the resurrection.
According to the New Testament, Jesus gave the Passover meal a new meaning, as in the upper room during the Last Supper he prepared himself and his disciples for his death. He identified the matzah and cup of wine as his body soon to be sacrificed and his blood soon to be shed. Paul states, "Get rid of the old yeast that you may be a new batch without yeast—as you are. For Christ, our Passover lamb, has been sacrificed"; the first Christians and Gentile, were aware of the Hebrew calendar. Jewish Christians, the first to celebrate the resurrection of Jesus, timed the observance in relation to Passover. Direct evidence for a more formed Christian festival of Pascha begins to appear in the mid-2nd century; the earliest extant primary source referring to East
A Runic calendar is a perpetual calendar, variants of which have been used in Northern Europe until the 19th century. The calendar is based on the 19-year-long Metonic cycle, correlating the Moon. Runic calendars were carved onto staves of wood, bone, or horn; the oldest one known, the only one from the Middle Ages, is the Nyköping staff from Sweden, believed to date from the 13th century. Most of the several thousand which survive are wooden calendars dating from the 16th and the 17th centuries. During the 18th century, the Runic calendars had a renaissance, around 1800, such calendars were made in the form of tobacco boxes in brass. A typical Runic calendar consisted of several horizontal lines of one above the other. Special days like solstices and celebrations were marked with additional lines of symbols; the calendar does not prove knowledge of the length of the tropical year or of the occurrence of leap years. It is set at the beginning of each year by observing the first full moon after the winter solstice.
The first full moon marked the date of Disting, a pagan feast and a fair day. On one line, 52 weeks of 7 days were laid out using 52 repetitions of the first seven runes of the Younger Futhark; the runes corresponding to each weekday varied from year to year. On another, many of the days were marked with one of 19 symbols representing the 19 Golden numbers, the years of the Metonic cycle. In early calendars, each of the 19 years in the cycle was represented by a rune; the new moon would fall on that day during that year of the cycle. For example, in the 18th year of the cycle, the new moons would fall on all the dates marked with Tvimadur, the symbol for year 18. Calendars used Pentadic numerals for the values 1–19. A version using Latin alphabet for weekdays and Arabic numerals for the golden numbers was printed in 1498 as part of the Breviarium Scarense; because this system needed 19 runes to represent the 19 golden numbers which stood for the 19 years of the perpetual calendar's cycle, the Younger Futhark, a Runic alphabet, was insufficient, having only 16 characters.
The solution devised was to add three special runes to represent the numbers above 16: Arlaug and Belgthor. A primstav is the ancient Norwegian calendar stick; these were engraved with images instead of runes. The images depicted the different nonmoving religious holidays; the oldest primstav still in existence is exhibited at Norsk Folkemuseum. Adherents of the Estonian ethnic religion have published Runic calendars every year since 1978. During the Soviet occupation, it was an illegal samizdat publication. Computus Runicus Germanic calendar Nationalencyklopedin Scythe sword Dominical letter Becker, Alfred: A Magic Spell "powered by" a Lunisolar Calendar, Asterisk, A Quarterly Journal of Historical English Studies, 15 Becker, Alfred: Franks Casket. Zu den Bildern und Inschriften des Runenkästchens von Auzon An article on rune calendars, with illustrations from gangleri.nl Alfred Becker – Franks Casket
For astronomy and calendar studies, the Metonic cycle or Enneadecaeteris is a period of close to 19 years, nearly a common multiple of the solar year and the synodic month. The Greek astronomer Meton of Athens observed that a period of 19 years is exactly equal to 235 synodic months and, rounded to full days, counts 6,940 days; the difference between the two periods is only a few hours, depending on the definition of the year. Considering a year to be 1⁄19 of this 6,940-day cycle gives a year length of 365 + 1⁄4 + 1⁄76 days, about 11 days more than 12 synodic months. To keep a 12-month lunar year in pace with the solar year, an intercalary 13th month would have to be added on seven occasions during the nineteen-year period; when Meton introduced the cycle around 432 BC, it was known by Babylonian astronomers. A mechanical computation of the cycle is built into the Antikythera mechanism; the cycle was used in the Babylonian calendar, ancient Chinese calendar systems and the medieval computus.
It regulates the 19-year cycle of intercalary months of the modern Hebrew calendar. The start of the Metonic cycle depends on. At the time of Meton, axial precession had not yet been discovered, he could not distinguish between sidereal years and tropical years. Most calendars, like the used Gregorian calendar, are based on the tropical year and maintain the seasons at the same calendar times each year. Nineteen tropical years are about two hours shorter than 235 synodic months; the Metonic cycle's error is, one full day every 219 years, or 12.4 parts per million. 19 tropical years = 6,939.602 days. 235 synodic months = 6,939.688 days. 254 sidereal months = 6,939.702 days. 255 draconic months = 6,939.1161 days. Note that the 19-year cycle is close to 255 draconic months, so it is an eclipse cycle, which lasts only for about 4 or 5 recurrences of eclipses; the Octon is 1⁄5 of a Metonic cycle, it recurs about 20 to 25 cycles. This cycle seems to be a coincidence; the periods of the Moon's orbit around the Earth and the Earth's orbit around the Sun are believed to be independent, not to have any known physical resonance.
An example of a non-coincidental cycle is the orbit of Mercury, with its 3:2 spin-orbit resonance. A lunar year of 12 synodic months is about 354 days 11 days short of the "365-day" solar year. Therefore, for a lunisolar calendar, every 2 to 3 years there is a difference of more than a full lunar month between the lunar and solar years, an extra month needs to be inserted; the Athenians seem not to have had a regular means of intercalating a 13th month. Meton's discovery made it possible to propose a regular intercalation scheme; the Babylonians seem to have introduced this scheme around 500 BC, thus well before Meton. Traditionally, for the Babylonian and Hebrew lunisolar calendars, the years 3, 6, 8, 11, 14, 17, 19 are the long years of the Metonic cycle; this cycle, which can be used to predict eclipses, forms the basis of the Greek and Hebrew calendars, is used for the computation of the date of Easter each year. The Babylonians applied the 19-year cycle since the late sixth century BC; as they measured the moon's motion against the stars, the 235:19 relationship may have referred to sidereal years, instead of tropical years as it has been used for various calendars.
According to Livy, the king of Rome Numa Pompilius inserted intercalary months in such a way that in the twentieth year the days should fall in with the same position of the sun from which they had started. As the twentieth year takes place nineteen years after the first year, this seems to indicate that the Metonic cycle was applied to Numa's calendar. Apollo was said to have visited the Hyperboreans once every 19 years at the high point of the cycle; the Runic calendar is a perpetual calendar based on the 19-year-long Metonic cycle. Known as a Rune staff or Runic Almanac, it appears to have been a medieval Swedish invention; this calendar does not rely on knowledge of the duration of the tropical year or of the occurrence of leap years. It is set at the beginning of each year by observing the first full moon after the winter solstice; the oldest one known, the only one from the Middle Ages, is the Nyköping staff, believed to date from the 13th century. The Bahá'í calendar, established during the middle of the 19th century, is based on cycles of 19 years.
The Metonic cycle is related to two less accurate subcycles: 8 years = 99 lunations to within 1.5 days, i.e. an error of one day in 5 years. By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate more accurate cycles. For example, simple arithmetic shows that: 687 tropical years; this gives an error of only about half an hour in 687 years, although this is subject to secular variation in the length of the tropical year and the lunation. Meton of Athens