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how the date of Easter is calculated

As a moveable feast, the date of Easter is determined in each year through a calculation known as Easter is celebrated on the first Sunday after the Paschal full moon (a mathematical approximation of the first astronomical full moon, on or after 21 March itself a fixed approximation of the March equinox). Determining this date in advance requires a correlation between the lunar months and the solar year, while also accounting for the month, date, and weekday of the Julian or Gregorian calendar. The complexity of the algorithm arises because of the desire to associate the date of Easter with the date of the Jewish feast of Passover which, Christians believe, is when Jesus was crucified. It was originally feasible for the entire Christian Church to receive the date of Easter each year through an annual announcement by the pope. By the early third century, however, communications in the Roman Empire had deteriorated to the point that the church put great value in a system that would allow the clergy to determine the date for themselves, independently yet consistently. Additionally, the church wished to eliminate dependencies on the Hebrew calendar, by deriving the date for Easter directly from the March equinox. In The Reckoning of Time (725), Bede uses computus as a general term for any sort of calculation, although he refers to the Easter cycles of Theophilus as a "Paschal computus." By the end of the 8th century, computus came to refer specifically to the calculation of time. The calculations produce different results depending on whether the Julian calendar or the Gregorian calendar is used. For this reason, the Catholic Church and Protestant churches (which follow the Gregorian calendar) celebrate Easter on a different date from that of the Eastern and Oriental Orthodoxy (which follow the Julian calendar). It was the drift of 21 March from the observed equinox that led to the Gregorian reform of the calendar, to bring them back into line.

Easter commemorates the resurrection of Jesus, which Christians believe to have occurred on the third day (inclusive) after the beginning of Passover. In the Hebrew lunisolar calendar, Passover begins at twilight on the 14th day of Nisan. Nisan is the first month of spring in the northern hemisphere, with the 14th corresponding to a full moon. By the 2nd century, many Christians had chosen to observe Easter only on a Sunday. The Hebrew calendar does not have a simple relationship with the Christian calendars: it resynchronizes with the solar year by intercalating a leap month every two or three years, before the lunar new year on 1 Nisan. Later Jews adopted the Metonic cycle to predict future intercalations. A possible consequence of this intercalation is that 14 Nisan could occur before the equinox, which some third century Christians considered unacceptable (this cannot happen in the fixed calendar in use today). Consequently, it was decided to separate the dating of Easter from the Hebrew calendar, by identifying the first full moon following the March equinox. By the time of the First Council of Nicaea (AD 325), the Church of Alexandria had designated 21 March as an ecclesiastical date for the equinox, irrespective of actual astronomical observation. In 395, Theophilus published a table of future dates for Easter, validating the Alexandrian criteria. Thereafter, the computus would be the procedure of determining the first Sunday after the first ecclesiastical full moon falling on or after 21 March.

The portion of the tabular methods section above describes the historical arguments and methods by which the present dates of Easter Sunday were decided in the late 16th century by the Catholic Church. In Britain, where the Julian calendar was then still in use, Easter Sunday was defined, from 1662 to 1752 (in accordance with previous practice), by a simple table of dates in the Anglican Book of Common Prayer (decreed by the Act of Uniformity 1662). The table was indexed directly by the golden number and the Sunday letter, which (in the Easter section of the book) were presumed to be already known. For the British Empire and colonies, the new determination of the date of Easter Sunday was defined by what is now called the Calendar (New Style) Act 1750 with its annexe. The method was chosen to give dates agreeing with the Gregorian rule already in use elsewhere. The act required that it be put in the Book of Common Prayer, and therefore it is the general Anglican rule. The original act can be seen in the British Statutes at Large 1765. The annexe to the act includes the definition: "Easter day (on which the rest depend) is always the first Sunday after the Full Moon, which happens upon, or next after the Twenty first Day of March. And if the Full Moon happens upon a Sunday, Easter day is the Sunday after." The annexe subsequently uses the terms "Paschal Full Moon" and "Ecclesiastical Full Moon", making it clear that they approximate to the real full moon. The method is quite distinct from that described above in For a general year, one first determines the golden number, then one uses three tables to determine the Sunday letter, a "cypher", and the date of the paschal full moon, from which the date of Easter Sunday follows. The epact does not explicitly appear. Simpler tables can be used for limited periods (such as 1900–2199) during which the cypher (which represents the effect of the solar and lunar corrections) does not change. Clavius's details were employed in the construction of the method, but they play no subsequent part in its use. J. R. Stockton shows his derivation of an efficient computer algorithm traceable to the tables in the prayer book and the Calendar Act (assuming that a description of how to use the Tables is at hand), and verifies its processes by computing matching tables.

Due to the discrepancies between the approximations of Computistical calculations of the time of the mean (northern hemisphere) vernal equinox and the lunar phases, and the true values computed according to astronomical principles, differences occasionally arise between the date of Easter according to computistical reckoning and the hypothetical date of Easter calculated by astronomical methods using the principles attributed to the Church fathers. These discrepancies are called "paradoxical" Easter dates. In his Kalendarium of 1474, Regiomontanus computed the exact time of all conjunctions of the Sun and Moon for the longitude of Nuremberg according to the Alfonsine Tables for the period from 1475 to 1531. In his work he tabulated 30 instances where the Easter of the Julian computus disagreed with Easter computed using astronomical New Moon. In eighteen cases the date differed by a week, in seven cases by 35 days, and in five cases by 28 days. Ludwig Lange investigated and classified different types of paradoxical Easter dates using the Gregorian computus. In cases where the first vernal full moon according to astronomical calculation occurs on a Sunday and the computus gives the same Sunday as Easter, the celebrated Easter occurs one week in advance compared to the hypothetical "astronomically" correct Easter. Lange called this case a negative weekly (hebdomadal) paradox (H− paradox). If the astronomical calculation gives a Saturday for the first vernal full moon and Easter is not celebrated on the directly following Sunday but one week later, Easter is celebrated according to the computus one week too late in comparison to the astronomical result. He classified such cases a positive weekly (hebdomadal) paradox (H+ paradox). The discrepancies are even larger if there is a difference according to the vernal equinox with respect to astronomical theory and the approximation of the computus. If the astronomical equinoctial full moon falls before the computistical equinoctial full moon, Easter will be celebrated four or even five weeks too late. Such cases are called a positive equinoctial paradox (A+ paradox) according to Lange. In the reverse case when the Computistical equinoctial full moon falls a month before the astronomical equinoctial full moon, Easter is celebrated four or five weeks too early. Such cases are called a negative equinoctial paradox (A− paradox). Equinoctial paradoxes are always valid globally for the whole Earth, because the sequence of equinox and full moon does not depend on the geographical longitude. In contrast, weekly paradoxes are local in most cases and are valid only for part of the Earth, because the change of day between Saturday and Sunday is dependent on the geographical longitude. The computistical calculations are based on astronomical tables valid for the longitude of Venice, which Lange called the Gregorian longitude. In the 21st and 22nd centuries negative weekly paradoxical Easter dates occur in 2049, 2076, 2106, 2119 (global), 2133, 2147, 2150, 2170, and 2174. Positive weekly paradoxical dates occur in 2045, 2069, 2089 (global), and 2096. Positive equinoctial paradoxical dates in 2019, 2038, 2057, 2076, 2095, 2114, 2133, 2152, 2171, and 2190. In 2076 and 2133, double paradoxes (positive equinoctial and negative weekly) occur. Negative equinoctial paradoxes are extremely rare. They occur only twice until the year 4000 in 2353, when Easter is five weeks too early and in 2372, when Easter is four weeks too early.

When expressing Easter algorithms without using tables, it has been customary to employ only the integer operations addition, subtraction, multiplication, division, modulo, and assignment as it is compatible with the use of simple mechanical or electronic calculators. That restriction is undesirable for computer programming, where conditional operators and statements, as well as look up tables, are available. One can easily see how conversion from day of March (22 to 56) to day and month (22 March to 25 April) can be done as 1=if (DoM > 31) {Day=DoM 31, Month=Apr} else {Day=DoM, Month=Mar}. More importantly, using such conditionals also simplifies the core of the Gregorian calculation.