5. BULL POPE GRIGORY XIII FROM FEBRUARY, 24TH 1582 CE.

Bull Pope Grigory XIII from February, 24th, 1582 CE is among those few basic documents which have generated a modern civilization of Mankind. Bull it is stated in Medieval Latin faultless, amazingly shining and refined on stylistics.

It contains 19 parts and never in a literal kind completely on Russian was translated [behind an exception only a preamble and the name «Inter gravissimas» (in transfer as- «Among the major»)] because of difficulties of transfer with medieval Latin in general and because of difficulties of a translation into Russian with absolutely exact medieval high English style of an authentic variant from the Latin made in the Vatican.

The basic semantic parts bull in an exact Russian translation is resulted for the first time in the given work. Those parts bull which have lost a historical urgency (an order of an actual excommunication from church and the legend to an anathema for disobedience, penalties collection in 100 Ducats gold, instructions of Vatican to courts under the account of 10 days which are withdrawn in a new calendar, references of the Pope to Catholic Christian sovereigns of Europe with the request for assistance in introduction of a new religious Catholic Christian calendar in "sponsored" it the countries, and other purely technical questions, to a calendar of the relation not having) are lowered only.

 

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Pope Grigory XIII (1572-1585CE).

 

For the first time the Gregorian calendar has been entered by Pope Grigory XIII in the Catholic countries on October, 4th, 1582 CE instead of old Julian: in the next afternoon after Thursday on October, 4th there was a Friday on October, 15th.

It is important to understand that Pope Grigory XIII absolutely meaningly aspired to co-ordinate the church "Gregorian" Catholic calendar with equinoctal year of a star the Sun, instead of with tropical year of a star the Sun. The Pope pursued the aim purely theological, instead of secular. The Pope needed to define precisely church Date of Easter, instead of exact secular astronomical Date of Seasons of year and historical events of Mankind.

All modern church religious tradition is adhered to a concrete Easter phase of the Moon, instead of to a secular history of Mankind. For this reason the Mankind is in great need in truly secular calendar first in the history. The Gregorian calendar basically cannot be a secular calendar on the church and religious Easter design.

Four seasons of year are direct function of tropical year, instead of Easter, not equinoctal year, not function of movement of the Moon. Interests of the Science and interests of Religion in this case do not coincide, but also do not disturb each other. The religious calendar without effort can be transformed to scientific absolutely exact tropical variant, but not on the contrary.

For this main reason Vatican never will give United Nations consent to translate a church "Gregorian" calendar in the secular. Vatican never will pass itself under the initiative with approximately equinoctal year for exact tropical secular year.

As proof of the given statement we more low literally quote the Vatican authentic English original bull «Inter gravissimas» (in transfer as - «Among the major») Pope Grigory XIII from February, 24th, 1582 CE. [It was resolved by Pope XIII. Necessary correction has been entered magnificent bull «Inter gravissimas» «270» under the account of the Pope Gregory XIII (1572-1585CE) from February, 24th, CE, 1582.]

[«Among our serious pastoral duties, not the last is that we care to complete those sacred rites reserved by the Council of Trent [i] [1545÷1563CE], with the guiding assistance of God. One notes in examining this that it is necessary to rule at the same time on three points to restore the celebration of Easter according to rules fixed by the previous Roman pontiffs, particularly Pius I [ii] [ca. 140 ÷ 154CE] and Victor I [iii] [ca. 189 ÷ 198CE], who established Easter's celebration on Sunday, rather than 14 Nisan favored by the «Quartodeciman» bishops of Asia], and by the fathers of the councils, in particular those of the [first] great ecumenical Council of Nicæa [May 20 - August 25, CE 325, deciding [iv] the following rules].

[Namely: First, the precise date of the vernal equinox, then the exact date of the fourteenth day of the moon which reaches this age the very same day as the equinox or immediately afterwards, finally the first Sunday which follows this same fourteenth day of the moon. Therefore we took care not only that the vernal equinox returns on its former date, of which it has already deviated approximately ten days since the Nicene Council, and so that the fourteenth day of the Paschal moon is given its rightful place, from which it is now distant four days and more, but also that there is founded a methodical and rational system which ensures, in the future, that the equinox and the fourteenth day of the moon do not move from their appropriate positions ».]

«Among our important vicarial duties, not the last is that circumstance that we care to execute sacred requirements imposed on us the Universal Christian Cathedral in Trent [1545-1563CE], with the Divine help supervising us.

Therefore, considering that for appropriate celebrating of a holiday of Easter, under instructions of sacred fathers and the Roman high priests of an antiquity we should restore Easter celebrating according to the rules based by earlier Roman high priests, especially Pius I [ca. 140 ÷ 154CE] and Victor I [ca. 189 ÷ 198CE] which have established Easter celebratings on Sunday, instead of together with Jews of 14 Nisan that Christian bishops of Asia] do till now "Quartodeciman" and also to be guided by decisions of fathers of church on Cathedrals, in particular, on [the First] Universal Cathedral in Nikkei [on May, 20th ÷ on August, 25th, AD {CE} 325], accepted below-mentioned rules.

Namely: first, it is necessary to return calendar Date of a spring equinox on its exact astronomical place, secondly, calendar Date of the fourteenth Moon (a full moon, 14 Nisan), should coincide precisely with an astronomical full moon and reaches this age (phase), in (same) day put to it as it should reach to it put a spring equinox, or at once after it next day, and at last, thirdly, Easter Resurrection should follow the fourteenth Moon at once.

Therefore we have taken care not only of, that day of a spring equinox has come back to the former Date in a calendar, which it already has deviated approximately for ten days after the Nikkei Cathedral (325 year CE), but also about, that the fourteenth day of the Easter Moon too has taken the lawful calendar place, from which it has kept away now for four days and more, and also, how to create such methodical and rational system which would provide effective correction of a calendar which would not give to day of a spring equinox and the fourteenth day of the Moon to move with places corresponding to them a calendar» in the future.

As methodical rational system «which would provide in the future effective correction of a calendar which would not give to day of a spring equinox and the fourteenth day of a full moon of the Easter Moon to move with places corresponding to them a calendar» in bull it has been made two proposals in point 9 and point 10.

[9). «Then, lest the equinox recede from XII calends April [March, 21st] in the future, we establish every fourth year to be bissextile (as the custom is), except in centennial years; which always were bissextile until now; we wish that year 1600 is still bissextile; after that, however, those centennial years that follow are not all bissextile, but in each four hundred years, the first three centennial years are not bissextile, and the fourth centennial year, however, is bissextile, so the years 1700, 1800 and 1900 will not be bissextile. Assuredly, the year 2000, as with our custom, will have a bissextile intercalation, February will contain 29 days, and the same rule of intermittent bissextile intercalations in each four hundred year period will be preserved in perpetuity».]

9). «In order that the equinox never receded from XII calend April [on March, 21st] in the future, we have established that each four years which were leap-years as usually, remain those, except some centuries which before reform always were till now leap; we wish that 1600 still was leap-year; after that, however, those century years which all follow it not will be leap, but in each period for four hundred years, first three centuries will not be leap-years, and only the fourth year of century will be leap, that is years, 1700, 1800 and 1900 [CE] will not be leap-years. Undoubtedly, 2000 will be leap-year and its February will contain 29 days, and the same rule of faltering leap-years will be kept in eternity in each period for four hundred years».

(Reformers change a rule of leap years of the Julian Egyptian calendar, correcting a solar cycle. The number of leap years decreases with 100 to 97 for the period in each 400 years. Therefore the Gregorian calendar began to overtake Julian a calendar for 3 days for each 400 years. By a new rule those years which share on number "400" without the rest instead of an old rule on which all those years which shared on number "4" without the rest were leap will be leap only.)

[10). «Our dear son Antonio Lilio, professor of science and medicine, brought to us a book, written at one time by his brother Aloysius [Luigi], in which this one showed that, by means of a new cycle of epacts which he had devised, and who directed his own particular Golden Number pattern and accommodated the entirety of any solar year, every [defect of] the calendar collapsed, and the constant calculations would endure for every generation. He was, thus, able to restore and explain how the calendar itself will never need published any further change.

Moreover, so that the fourteenth day of the Paschal moon is given with precision and that the age of the moon is presented with precision to the faithful in accordance with the antique use of the Church, to take note of it each day with the reading of martyrology[v], we order that once the Golden Number is withdrawn from the calendar, one substitutes the cycle of the epacts for it which, thanks to its very precise rules mentioned above for the Golden Number, makes so that the new moon and the fourteenth day of the Paschal moon always hold their place. And this is seen clearly in the explanation of our calendar, where are also presented Paschal tables in conformity with the ancient habits of the Church and which make it possible to find more surely and more easily the sacred date of the Easter ».]

10). «Our expensive son Antonio Lilio, the professor of medicine, has come to us with the book, written it together with the brother Aloizi [Luigi] in which they have shown that, by means of new cycles epacts the Moon (epacts) which they have conceived, instead of old Gold Number (Meton), epacts the Moon are placed now in full on any solar year beforehand. Therefore the old [insufficient] calendar has failed, and the new calendar (now it is easy) will sustain constant calculations for each new generation.

They, thus, managed to restore and explain, how such new (eternal) calendar never any more is not required to change and publish and any further changes in it never will be. Moreover, as for the fourteenth day (month Nisan) the Easter Moon is given with (high) accuracy, and that the age of the Moon is presented with such accuracy for believers that is in full conformity with ancient rules, which uses Church every day, since reading of Martyrology Sacred martyrs (the catalogue of martyrs and sacred located according to order of their holidays, i.e. According to a calendar).

The gold Number (Meton) is deduced from the use of a calendar and replaced with a cycle of Epacts instead of it which, thanks to the very accurate rules mentioned above for Gold Number, does so that in a new moon, and in the fourteenth day of a full moon the Easter Moon always holds precisely place in a new calendar. And it we saw clearly in that explanation for our new calendar in which Easter tables according to ancient church rules and on which it is easy to find more confidently sacred Date of Easter» are presented also.

(Reformers enter the new scheme of correction of a cycle of the Moon. In the scheme of Meton on forecasting of solar eclipses they have replaced a rule of Gold Numbers with new more exact tables of Epacts of the Moon (age or the Moon phase on the beginning of year) that allows to keep in a calendar a real astronomical Easter full moon at 14 bottom of month Nisan.)

 

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Pope Grigory XIII (1572-1585CE).


 

Difference of dates Julian and the Gregorian calendars

Difference, days

The period (on Julian to a calendar)

The period (on a Gregorian calendar)

10

On October, 5th 1582 –

on February, 29th 1700

On October, 15th 1582 –

on March, 11th 1700

11

On March, 1st 1700 –

on February, 29th 1800

On March, 12th 1700 –

on March, 12th 1800

12

On March, 1st 1800 –

on February, 29th 1900

On March, 13th 1800 –

on March, 13th 1900

13

On March, 1st 1900 –

on February, 29th 2100

On March, 14th 1900 –

on March, 14th 2100

14

On March, 1st 2100 –

on February, 29th 2200

On March, 15th 2100 –

on March, 15th 2200

15

On March, 1st 2200 –

on February, 29th 2300

On March, 16th 2200 –

on March, 16th 2300

 

Julian dates till October, 5th (15), 1582 too can be counted on a Gregorian calendar, but it is not accepted to do it. Usually dates before introduction of a new calendar are resulted on Julian to a calendar, and after - on Gregorian. In the countries which have accepted a Gregorian calendar not at once, for the period since October, 5th (15), 1582 and before its introduction two dates - on old (Julian) to style and, in brackets, on new (Gregorian), for example often name: «Pushkin Alexander Sergeevich [26.5 (6.6).1799, Moscow, - 29.1 (10.2).1837, Petersburg], the Russian writer, the founder of the new Russian literature».

For drawing, including, the maximum harm of Russian Orthodox Church (ROC) which uses the Julian calendar, atheists-communists led by V.I. Lenin have entered in the Soviet Russia the Decree of Council of National Commissioners from January, 26th, 1918 CE a Catholic Gregorian calendar «for the purpose of an establishment into Russia identical almost with all cultural [that is Catholic] the people of computation of time».

This Decree of Council of National Commissioners has been cancelled the Order of the commander of a shooting regiment the colonel Gravitsky on "white" garrison of a city of Kharkov from June, 25th, 1919 which said: «Chronology to conduct on Old style. Tomorrow, on June, 13th, at 12 o'clock, hours to put on Petrograd time» in connection with cancellation of a Catholic Bolshevist Gregorian calendar and return transition on orthodox Julian a calendar».

However the President of Russia D.A. Medvedev has confirmed Vladimir Lenin's decision, the atheist and enemy Russian Orthodox Church (ROC), from January, 26th, 1918, acceptance in modern Orthodox Russia a Catholic Gregorian calendar as the Federal law of the Russian Federation from June, 3rd, 2011 N 107-FZ "About computation of time" which «Is published: on June, 6th, 2011 in" RM "- in Federal release №5496» and «Enters in силу:7 August, 2011» and «It is accepted by the State Duma on May, 20th, 2011» and «It is approved by the Federation Council on May, 25th, 2011 CE».


 

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The decree of Council of National Commissioners from January, 26th, 1918 CE of V. Lenin is about Gregorian calendar introduction in Russia from January, 26th, 1918.

 


 

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The theme of introduction of a Gregorian calendar is displayed on the Bas-relief on a tomb of Pope Grigory XIII in St. Peter's Cathedral in Rome.

 

6. UNIFORM UNIVERSAL MATHEMATICAL MODEL OF CALCULATION OF DURATION OF YEAR FOR ALL EXISTING TYPES OF CALENDARS

Now there are three separate independent mathematical models from each other for lunar, solar and luni-solar calendars. The uniform universal mathematical model for all three types of calendars does not exist. The author offers in the given work the uniform universal mathematical model for all existing types of calendars.

[Keywords: mathematical model of a solar calendar, mathematical model of a lunar calendar, mathematical model of a luni-solar calendar, the uniform universal mathematical model of calculation of duration of year for all existing types of calendars, Calendar constant.]

 

In existing the following mathematical model is offered «the theory of a solar calendar»:

«Duration of tropical year [in round figures] makes 365.24220 days. Thus, calendar year in a solar calendar should make or 365 days - usual year, or 366 days - leap-year. That average duration of calendar year was close to duration of [astronomical] tropical year, the insert system leap years is necessary. For its definition it is possible to spread out a fractional part of duration of tropical year in chain fraction:


 

0.2422 = \frac{2422}{10000} = \frac{1}{4 + \frac{1}{7 + \frac{1}{1 + \frac{1}{3+\frac{1}{4 + \frac{5}{8}}}}}}

 

Breaking this fraction at different stages of division it is possible to receive following rules for introduction of leap-years of different accuracy:

 

\frac{1}{4}; \frac{7}{29}; \frac{8}{33}; \frac{31}{128}; ...

 

Where the denominator is specifies number of years in a calendar cycle, and numerator - number of leap years in this cycle. Thus, average duration of calendar year in days in these calendar systems turns out equal:

 

365,25000;\,365,24138;\,365,24242;\,365,24219;...

 

Arguing on accuracy of a calendar usually speak about a point of a spring equinox. So, one of requirements to a solar calendar is that fact that the moment of passage by the Sun of a point of a spring equinox after a calendar cycle should have for the same date.

The first system leap years with one insert for four years existed in Julian a calendar. As average duration of year Julian a calendar for 0.00780 days more tropical for 128 years the error in 1 day and [calendar] days of a spring equinox collects are displaced by the winter. The 29-year-old cycle was never used. The 33-year-old cycle has been offered by the Omar Khayyam and has laid down in a basis of the Persian calendar.

The 128-year-old cycle has been offered by Johann Medler in 1864, but has not been accepted in one calendar.

The specified way of the organization of a calendar is not unique. Attempts to improve Julian a calendar were repeatedly undertaken. The new calendar should be, on the one hand, more exact, and on the other hand - it is not enough to differ from Julian.

In a Gregorian calendar the sequence leap years is left practically without changes: it is added corrected that those century years which number of centuries shares on 4 without the rest [1/4 = 100/400] are leap only. Thus, from [a cycle in] 400 years are thrown out 3 superfluous days, that is the full calendar cycle of a Gregorian calendar makes 400 years, and its average duration [in days] is equal:


 

\frac{97 \cdot 366 + 303 \cdot 365}{400} = 365,24250

 

[Where is in numerator there the sum of all leap days equal to product: (97*366 days), - and all not leap days in a cycle from 400 days, equal: ({400-97=303} *365 days)].

Other variant of improvement Julian a calendar became New Julian a calendar offered by Milutin Milankovich. In this calendar sequence leap years same, as well as in Julian a calendar, but the additional rule according to which century year is considered leap if at its division on 900 in the rest remains 2 or 6 is entered.

The full cycle of such calendar makes 900 years on which extent 7 superfluous days are thrown out. Average duration of year of this calendar makes 365.24222 days. This calendar is used by a number of orthodox churches.

And Gregorian and New Julian calendars possess one essential lack. The insert of leap-year in them is made rather non-uniformly. Because of it, despite exact enough average duration of year, day of a spring equinoxin different years in a calendar cycle can get for different days [and not to correspond to a real astronomical situation].

In existing the following mathematical model is offered «the theory of a lunar calendar»:

«Duration synodic year on the average [with a rounding off] makes month 29.53059 days. Thus, calendar month can consist either from 29, or from 30 full days, and months in a year alternate that days of month as it is possible got on the same phases of the Moon is better.

In lunar calendars duration of year is accepted to equal 12 months. According to it, duration of lunar year turns out equal 12 months × 29.53059 days = 354.36708 days.

Means, in calendar year can be or 354 days - simple year, or 355 days - the continued (leap) year; and that average duration of calendar year was close to duration of [astronomical] lunar year the insert system leap years is necessary. For its definition it is possible to spread out a fractional part of duration of lunar year in chain fraction :}

 

0.36708 = \frac{36708}{100000} = \frac{1}{2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1+\frac{1}{1 + \frac{1}{1+\frac{500}{1027}}}}}}}

 

Breaking this fraction at different stages of division, it is possible to receive following rules for introduction of the continued years of different accuracy:

 

\frac{1}{2}; \frac{1}{3}; \frac{3}{8}; \frac{4}{11}; \frac{7}{19}; \frac{11}{30};...


 

Where the denominator is specifies number of years in a calendar cycle, and numerator - number of the continued years in this cycle. Thus, average duration of calendar year in days in these calendar systems turns out equal:

 

354,50000;\, 354,33333;\, 354,37500;\, 354,36364;\, 354,36842;\, 354,36667...

 

The cycle 3/8 has received the name "Turkish" and is used in a Turkish lunar calendar; other cycle 11/30 used in a Muslim calendar and often is called as "Arabian".

The lunar calendar is not adhered at all to year movement of the Sun, therefore annually lunar calendar is displaced concerning solar for 365.24220 days - 354.36708 days = 10.87512 days. Approximately for 33÷34 solar year runs one superfluous lunar year.

Change of phases of the Moon is one of most easily observable the heavenly phenomena. It is not surprising that set of the people at an early stage of the development used a lunar calendar. However, in formation of a settled way of life the lunar calendar ceased to satisfy requirements of the population as agricultural works are adhered to change of seasons, that is Sun movement. Therefore lunar calendars, with rare exception (for example, an Islamic calendar), were inevitably replaced with luni-solar or solar calendars.

It is necessary to carry that fact to inconvenience of a lunar calendar that duration synodic year continuously varies month in limits from 29d6h15m to 29d19h12m. The reason for it is difficult enough movement of the Moon on an orbit.

The beginning of month in lunar calendars is necessary on Neamenia, that is on the first occurrence of the young Moon in beams of the coming Sun. This event is easily observed, unlike a new moon. Neamenia lags behind a new moon on time for 2-3 days. And this time varies seasonally, widths of the observer and current duration synodic month.

Because of it is impossible how to conduct the same calendar based on supervision of the Moon, in the different countries, and to use a simple calendar from 29 and 30-daily allowances of months. A calendar entered on any system, it will be inevitable to disperse from real movement of the Moon though, with this or that accuracy, on the average to this movement will correspond ».

 

In existing the following mathematical model is offered «the theory of a luni-solar calendar»:

«Duration synodic year on the average [in round figures] makes month 29.53059 days, and tropical year - 365.24220 days. Thus, one tropical year comprises 12.36827 synodic months [365.24220 days/29.53059 of days = 12.368266 (26220472) months]. But fractional months do not happen.

Therefore, calendar year can consist or from 12 full (usual year), or from 13 full (embolismic year - from other-grech. ἐμβολή - intrusion) calendar months, and the number of days in months in a year alternates so that the same days of month as it is possible got on the same phases of the Moon is better.

That average duration of calendar year was close to duration of [astronomical] tropical year, the system of an insert of additional months is necessary. For its definition it is possible to spread out a fractional part of duration of tropical year in synodic months in chain fraction:

 

0{,}36827 = \frac{36827}{100000} = \frac{1}{2 + \frac{1}{1 + \frac{1}{2 + \frac{1}{1+\frac{1}{1 + \frac{153}{2543}}}}}}

 

Breaking this fraction at different stages of division, it is possible to receive following rules for introduction of the continued years of different accuracy:

 

\frac{1}{2}; \frac{1}{3}; \frac{3}{8}; \frac{4}{11}; \frac{7}{19}; \frac{123}{334};...

 

Where is the denominator specifies number of years in a calendar cycle, and numerator - number embolismic years [for 13 full months] in this cycle. Since ancient times cycles of 3/8 and 7/19 were used.

The eight-year cycle (3/8), or «octaeterid», was used in ancient Babylon, and Greece where has been, apparently, independently offered by Ancient Greek astronomer Kleostrat, and also in other countries.

In octaeterid the cycle of 8 tropical years that makes approximately 2922 days [365.2422 days * 8 years = 2921.9376 days] which in turn are approximately equal 99 synodic to months [2921.9376 days/29.53059 of days = 98.94613 (009763774) months] is accepted.

Actually, exact duration full 99 synodic is equal months to 2923.53 days [99 months * 29.53059 days = 2923.52841 days] that gives a calendar error in 1.59 days for 8 years [2923.52841 days - 2921.9376 days = 1.59081 days = 1 days of 14 hours of 10 minutes of 45.984 seconds]. The error in 1 full days therefore to approximate calculation runs for 5 years (8 years/1.59 of days = 5.031446540880503 years).

Nineteen-year cycle (7/19) often name «Meton», by name Ancient Greek astronomer Meton who has offered him though the cycle was known long before Meton in Babylonia and China.

In a cycle from 19 tropical years there are approximately [365.2422 days * 19 = 6939.6018 days] which in turn are approximately equal 235 full months [234.997 (0589818896)] synodic to months [6939.6018 days/29.53059 of days = 234.997 (0589818896) months].

Actually, exact duration full 235 synodic is equal months to 6939.68865 days [235 months * 29.53059 days = 6939.68865 days] that gives a calendar error in 0.08685 days for 19 years [6939.68865 days - 6939.6018 days = 0.08685 days].

The second variant of calculation:

In 7 embolismic year 2687.28369 days [7 years * 13 lunar months of *29.53059 lunar days = 2687.28369 days] contain.

In 12 simple years (19-7 = 12) 4252.40496 days [12 years * 12 lunar months of *29.53059 lunar days = 4252.40496 days] contain.

Total 6939.68865 days for a cycle in 19 years [4252.40496 days + 2687.28369 = 6939.68865 days] that completely coincide with the first variant of calculations.

The error in 1 full days therefore to approximate calculation runs for 218.767990 years (19 years/0.08685 of days = 218.767990 [7887162] years) that disperses, however, from the astronomical data which mark an error in 1 full days for the period in 304 years. It means that the real error for a 19-year-old cycle of a solar-lunar calendar makes not 0.08685 days, but only 0.0625 days [19/304 = 0.0625 days].

Such error is connected by that the luni-solar calendar has inherited from a lunar calendar both its advantage, and its lacks. In spite of the fact that on Moon phases simply enough to keep count of time, but duration synodic continuously varies month in limits approximately from 29d6h15m to 29d19h12m. The reason for this approximate huge size to a divergence at 12 o'clock 57 minutes are difficult enough movement of the Moon on an orbit.

The beginning of month in luni-solar, as well as in lunar calendars, is necessary on Neamenia, that is on the first occurrence of the young Moon in beams of the coming Sun. This event is easily observed, unlike a new moon. Neamenia lags behind a new moon on time for 2-3 days. And this time varies seasonally, widths of the observer and current duration synodic month.

Because of it is impossible how to conduct the same calendar based on supervision of the Moon, in the different countries, and to use a simple calendar from 29 and 30-daily allowances of months. A calendar entered on any system it will be inevitable to disperse from real movement of the Moon though, with this or that accuracy, on the average to this movement will correspond».

 

In calendars today are used 11 kinds of years different in their duration (an eclipse year and anomalistic year for creation of mass calendars are not used and consequently here are not considered).

1). The longest is Galactic year(365.2563630 days = 365 days 6 hours 9 minutes 9.76 seconds). It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 20 minutes and «+» [(24,76 seconds/60 seconds = 0.41267 minutes), or on (20,41267 minutes)], or on [20,41267 minutes/60 minutes = 0.340211167 hours for a year].

[1 day of backlog run for 70.5444 years = 24 hours/0.340211167 hour for a year]

2). Year of a star contains Sirius in a variant of Pharaoh Ptolemy III Everget (Julian or Egyptian year) (365.2500000 days of =365 days 6 hours).

It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 11 minutes 15 seconds/60 seconds = 0.25 minutes (or 11,25 minutes) or [11,25 minutes/60 minutes = 0.1875 hours for a year/24 hour =0.0078125 days for a year = 365.2500 days - 365.2421875 days]

[1 day of backlog run for 128 years = 24 hours/0.1875 hour for a year].

3). Average year is equal in the Judaic [luni-solar] calendar:

A] 365.2468205 days = [365.2421875 days + 0.00463302631 days] = 365.2468205 (2631) days = 365 days 5 hours 55 minutes 25.2912 seconds)],

Where is 0.00463302631 days for a year = (0.0880275/19) days for a year = [0.00463302631 days for a year * 24 hours = 0.11119263144 hours for a year] (excess of duration of Judaic year over tropical by the first variant of calculations).

[1 day of backlog run for 215.84 (16406661848) year = 24 hours/0.11119263144 hour for a year]. The result does not correspond 215.84 years to the experimental astronomical data. Therefore lower under the specified data it is received:

B] 365.2454769 days = [365.2421875 days + 0.0032894 (736842105) days = 365.2454769 days = 365 days 5 hours 53 minutes 29.20416 seconds)],

Where is 0.0032894736842105 days for a year = (0.0625/19) days for a year = [0.0032894736842105 days for a year * 24 hours = 0.0789476 (38421052) hour for a year] (excess of duration of Judaic year over tropical by the second variant of calculations).

Last [B] is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 4 minutes and «+» (44.20416 seconds/60 seconds = 0.736736 minutes). Or on (4.736736 minutes), or on [4.736736 minutes/60 minutes = 0.0789456 hours for a year]

[1 day of backlog run for 304. (0068097525385) year = 24 hours/0.0789456 hour for a year]. This result is [304.0 years] in the full consent with the experimental astronomical data.

4). The Gregorian year is equal (365.2425000 days = 365 days of 5 hours of 49 minutes of 12 seconds). It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 27 seconds/60 seconds = 0.45 minutes (or 0.45 minutes/60 minutes = 0.0075 hours for a year).

[1 day of backlog run for 3200 years = 24 hours/0.0075 of hour for a year].

5). The Iranian year of the Omar Khayyam contains (365.2424242 days = 365 days of 5 hours of 49 minutes of 5.43 (36) seconds). It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 20.43 seconds/60 seconds = 0.3405 minutes (or 0.3405 minutes/60 minutes = 0.005675 hours for a year).

[1 day of backlog run for 4229 years = 24 hours/0.005675 hour for a year].

6). Equinoctal year consists from (365.2424000 days = 365 days 5 hours 49 minutes 3.36 seconds). It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 18.36 seconds/60 seconds = 0.306 minutes (or 0.306 minutes/60 minute = 0.0051000 hours for a year).

[1 day of backlog run for 4706 years = 24 hours/0.0051000 hour for a year].

7). Duration of New Julian year makes 365+218/900=365.2422222 = 365 days 5 hours 48 minutes 47.99 (808) seconds. It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 2.99808 seconds/60 seconds = 0.049968 minutes (or 0.049968 minutes/60 of minutes = 0.0008328 hours for a year).

[1 day of backlog run for 28818 = 24 hours/0.0008328 hour for a year]. 

8). Duration of year Maya has made 365.2422000 days = 365 days 5 hours 48 minutes of 46.08 seconds. It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) for 1.08 seconds/60 seconds = 0.018 minutes (or 0.018 minutes/60 minutes = 0.0003 hours per year).

[1 day of backlog run for 80000 years = 24 hours/0.0003 hour per year].

9). Year so-called «a mathematical calendar» Ahmad Birashk’s contains: 365.2421985 (815602837) days = 365 days + 0. 2421985 (815602837) days [24 hours*0.2421985 (815602837) = 5.812765957446809 hours (60 минут*0. 812765957446809 = 48.76595744680854 minutes) {60 second * 0.76595744680854 = 45.957 (4468085124) seconds}] = [365 days 5 hours 48 minutes 45.957 (4468085124) seconds].

It is longer than tropical year (365 days 5 hours 48 minutes 45 seconds) on [0.9574468 seconds/60 seconds = 0.0159574467] minute. Or on [0.0159574467 minutes/60 of minutes = {0.000265957} 445 hours per year]

[1 day of backlog run for 90240. (000613632) years = 24 hours/0.000265957 (445) hour per year].

10). Tropical astronomical [defined by experimental researches-supervision] year of I. Kepler(365.2421875 days = 365 days 5 hours 48 minutes 45 seconds). Precisely coincides with duration of offered new purely settlement uniform calendar digital mathematical quantum calendar [L = 365 days + (31/128) = 365 + 0.2421875 = 365.2421875 days].

11). The shortest is Lunar year. Duration of lunar year on the basis of calculation molad interval (round the Earth) for 12 months makes duration of 29.530594 days of one average turn of the Moon: L = 354.367128 days (12 х 29.530594 days = 354.367128 days=354 days 8 hours 48 minutes 39.8592 seconds).

Tropical year is longer than Lunar year for 10.8750595 days or for 1/0.0919535 (198864889) years.

[365.2421875 days - 354.3671280 days = 10.8750595 days = 1/0.0919535 (198864889) year].

Nine last years lagged behind tropical year, and Lunar year, on the countrary, advances Tropical year. One full (1) days of an advancing run for 33.58530475 days [365.2421875 days / 10.8750595 days = 33.58530475 (166596) days] or [33.58530475 (166596) days / 365.2421875 days = 0.0919535 (198864889) parts of year] Lunar year of Tropical year.

But these official values of duration listed above 11 basic calendar years are not exact in mathematical sense. They do not consider size of a systemic (regular) error. Low resulted is more the exact correct uniform universal mathematical model [formula] of calculation of duration of any calendar year, developed by the author, with the account of a system error (the analogue in the scientific literature resulted above, is absent).