7. MATHEMATICAL MODEL

UNIFORM UNIVERSAL MATHEMATICAL MODEL [FORMULA] OF CALCULATION OF DURATION OF YEAR FOR ALL EXISTING TYPES OF CALENDARS

The uniform universal mathematical model [formula] of calculation of duration of year for all existing types of calendars looks, according to the author, as follows:

L = K + α ± β = K + µ,

µ = α ± β = const = 0.2421875 days = 5 hours 48 minutes 45 seconds [«the Calendar Constant» - «CC»]

[α ± β] - µ = 0

[α ± β] - 0.2421875 days = 0

|β| = const * {1 - α/const} = µ * {1 - α/µ},under boundary conditions: 0 ≤α ≤ µ

Max |β| = µ = const = 0.2421875 days, when is: α = 0.

L = K + 0 days ± |β| days = K ± |β| days [The sign «±» (plus-minus) shows direction the rotation of the Sothic Egyptian calendar in size the equal: |β| = µ = 0.2421875 days / each year.

The |β| is vector size.] {Sothic Egyptian cycle}

Min |β| = 0, when is: α = µ = const = 0.2421875 days

L = K + 0.2421875 days ± |0| days [Calendar rotation is absent.]

 

{This is the Kepler cycle - accepted by us in our Cycle of the Ideal Tropical active calendar}

 «The calendar constant» has for the calendar theory precisely same great value what Planck's constant for theoretical quantum Physics has a constancy of a velocity of light in the Relativity Theory.

 Where is «L» the general duration of year in integers and shares of days,

"K" base duration of year in integers of days [365 days in the Earth].

«α» [«accuracy of a calendar»] = [(number of leap years) / (the size of a cycle)] is a factor of correction of the first level, is calculated by a method of chain fractions.

«[1 / α] » - rate (frequency of leap corrections).

«β» (size of a systemic error) = [1 / (the quantity of years for which run 1 days of a system error with an advancing of tropical year or backlog from it)] is a factor of correction of the second level.

Changing parameters "K" and «µ», we easily receive a calendar for any planet of any Sun in any star system. Thus in a science is created the new direction which it is possible while conditionally to name as «CALENDAR ASTRONOMY» that is doubtless actual for the epoch of piloted astronautics.

As a detailed example of calculations we will consider the following:

1). The full formula of calculation of duration of year of the Galactic calendar:

L = 365 days + [102.5452/400] days - [1/70.5444] days = 365 days + 0.2563630 days - [1/70.5444] days = 365.2563630 days - [1/70.5444] days = 365.2563630 days - 0.014175469633309 days = 365.2421875 (30366691) days.

Accuracy: α = [102.5452/400=31.814464/128] days = 0.2563630 days led to a cycle of 400 years or to an equivalent cycle of 128 years.

[Return recalculation for reception of number of leap years on a cycle in 400 years: 0.2563630 * 400 = 102.5452.]

Rate [1 / α] = 400/102.5452 = 3.900 (71890249373)

Size of a systemic error: β = [1/70.5444] days. [1 (full) day of "a negative error» run for 70.5444 years].

2). The full formula of calculation of duration of year of the Julian (Egyptian)calendar:

L = 365 days + [100/400] days - [1/128] days of =365 days + [32/128] days - [1/128] days = 365 days + [(32 - 1)/128] days = 365 days + [31/128] days = 365 days + 0.2421875 days = 365.2421875 days.

Accuracy: α = [100/400=32/128] days = 0.25 days.

Rate: [1 / α] = 400/100 = 128/32 = 4.000 - (frequency of leap corrections for a cycle or "rate")

Size of a systemic error: β = 0.0078125 days = 1/128 days [1 (full) day of "a negative error» run for 128 years].

0.0078125 days * 24 hours = 0.1875 hours * 60 minutes = 11.25 minutes = 11 minutes 15 seconds.

3). The full formula of calculation [] duration of year of the Judaic [luni-solar] calendar will make 400 years led to a cycle:

L = 365 days + [98.190 (76)/400] days - [1/304] days = 365 days + 0.2454769 days - 0.0032894 days = 365. 2454769 days - [1/304] days = 365.2454769 days - 0.0032894 (736842105) days = 365.2421875 days.

Accuracy: α = [98.190 (76)/400=31.4210432/128] days = 0.2454769 days,

[Return recalculation for reception of number of leap years: 0.2454769 * 400 = 98.19076.]

The conditional "resulted" rate [1 / α] a Judaic calendar is calculated as: 400/98.19076 = 4.07370306533.

Size of a systemic error: β = [1/304] days = 0.0032894 (736842105) days. [1 (full) day of "a negative error» run for 304 years].

[Return recalculation for factor reception «β»: 1/304 = 0.0032894 (736842105), that is 1 (one) days run for 304 years, that is through each 16 cycles of Meton for 19 years (16*19=304).]

4). The full formula of calculation of duration of year of the Gregorian calendar:

L = 365 days + [97/400] days - [1/3200] days = 365 days + 0.2425 days - 0.0003125 days = 365.2425 days - 0.0003125 days = 365.2421875 days.

Accuracy: α = [97/400=31.04/128] days = 0.2425 days.

Rate: [1 / α] = 400/97 = 4.1237 (11340206186)

Size of a systemic (regular) error: β = 0.0003125 days = [1/3200] days, that is 1 full days of a "negative" systemic error run for 3200 years: (400/0.125=3200 that is through 8 cycles for 400 years [3200/400=8 cycles]).

Calculation is based on a parity 400/128 = 3.125 days = 3 full days + 0.125 fractional days. That is in a cycle from 400 years for each 128 years runs backlog three times at a rate of 1 full day and 0.125 more incomplete days remains in the rest in each cycle at a rate of 400 years.

These incomplete days are summarized and in 3200 years [1/3200=0.0003125 days] give exactly 1 more full day of a system error in addition. Therefore at the first level of correction (at level of 1 full day) 100 leap years but only 97 years (100 - 3 = 97) as indemnification of 3 days of backlog of calendar dates from real astronomical measurements turn out not.

And in 3200 years [8 cycles for 400 years, that is (0.125 day*8=1 days)] to them will be added 1 more full additional leap days, that is in one of leap-years ["super leap" year] will be 367 days (365+1+1=367 days), instead of usual «366» days (365 + 1 = 366 days).

5). The full formula of calculation of duration of year Iranian [Omar Khayyam's] acalendar will make:

L = 365 days + [8/33] days - [1/4229] days = 365 days + 0.2424242 days - [1/4229] days = 365. 2424242 - [1/4229] days = 365.2424242 days - 0.0002367 days = 365.2421875 days.

Accuracy: α = [8/33=96.9696969/400=31.03030303/128] days = 0.2424242 days,

Rate [1 / α] = 33/8 = 4.1250;

Size of a systemic error: β = [1/4229] days = 0.0002367 days. [1 (full) day of "a negative error» run for 4229 years].

The error in 20.43 seconds for 1 year (0.005675 hours for a year) between the Iranian year of the Omar Khayyam (365 days of 5 hours of 49 minutes of 5.43 seconds = 365.2424242) and Tropical year (365 days 5 hours 48 minutes 45 seconds = 365.2421875) gives an error in 1 whole days for 4229.07 (489) [24/0.005675=4229.07489], that is through 128 full cycles for 33 years everyone [4229/33=128.1515151 a cycle].

6). The full formula of calculation of duration of year Equinoctal the calendar will make:

L = 365 days + [96.96/400] days - [1/4706] days = 365 days + 0.2424000 days - [1/4706] days = 365. 2424000 days - [1/4706] days = 365.2424000 days - 0.0002125 days = 365.2421875 days.

Accuracy: α = [96.96/400=31.0272/128] days = 0.2424000 days,

[Return recalculation for factor reception «α»: 0.2424000 * 400 = 96.96]

Rate [1 / α] = 400/96.96 = 4.1254 (12541254125);

Size of a systemic error: β = [1/4706] days = 0.0002125 days. [1 (full) day of "a negative error» run for 4706 years].

The error in 18.36 seconds for 1 year (0.0051000 hours for a year) between Tropical year and Equinoctal year gives an error in 1 whole day for 4705.88235294 years [(24 hours/0.0051000 hour = 4705.88235294 (1176)], that is through 11 full cycles for 400 years everyone [4705.88235294 years/400 of years = 11.7647 (0588235) cycles] or {4706 years/400 of years = 11.765 cycles}.

It means that (with a rounding off) in each whole 4706 years once in a year there will be one more additional to «366» (that is the second) an intercalary day («367»).

7). The full formula of calculation of duration of year of the New Julian calendar will make:

L = 365 days + [218/900 =] days - [1/28800] days = 365 days + 0.2422222 days - [1/28800] days = 365. 2422222 days - [1/28800] days = 365. 2422222 days - 0.0000347 days = 365.2421875 days.

Accuracy: α = [218/900 = 96.88888/400=31.0044444/128] days = 0.2422222 days,

Rate: [1 / α] = 900/218 = 4.1284 (40366972477);

Size of a systemic error: β = [1/28800] days = 0.0000347 days. [1 (full) day of "a negative error» run for 28800].

Duration of the "New Julian" year (without the account of a system error) makes 365 + (218/900 =365.2422222 = 365 days of 5 hours 48 minutes 47.99 (808) seconds.

In "the New Julian" calendar, in the period of 900 years should be 225 usual Julianian leap years (900/4=225). For the period in 128 years the parity (900/128=7.03125=7 whole or full days+0.03125 day) is applied. Therefore in "the New Julian" calendar in the period of 900 years figure "218" of leap years (225-7=218) definitively turns out.

The system error in "the New Julian" calendar at a rate of 1 whole day runs for 28 thousand 800 years (28800) by following calculation: 900/0.03125=28800, that is through 32 cycles for 900 years [28800/900=32]. [By direct calculation, on a difference with duration of tropical year (because of other accuracy of roundings off) it turns out for 18 years more: 28818 - 28800 = 18 years that makes insignificant purely technical error in 0.0625% = (18/28800) *100%.]

8). The full formula of calculation of duration of year Maya of a calendar has made:

L = 365 days + [96.88/400] days - [1/80000] days = 365 + 0.2422000 days - [1/90240] days = 365.2422000 days - [1/80000] days =365.2422000 days - 0.0000125 days = 365.2421875 days.

Accuracy: α = [96.88/400=31.0016/128 = 0.2422000] days.

[0.2422000*400=96.88] - the number of leap years led to a cycle of 400 years.

Rate: [1 / α] = 400/96.88 = 4.1288(19157720892);

Size of a systemic error: β = [1/80000] days = 0.0000125 days. [«The negative error» in 1 full days runs in a calendar of Maya for 80 000 years.]

[Coincidence of a calendar of Maya on rate with calendar Ahmad Birashk to within 4 signs after a comma] takes place

9). The full formula of calculation of duration of year Ahmad Birashk a calendar will make:

L = 365 days + [683/2820] days - [1/90240] days = 365 days + 0.2421985 (815602837) days - [1/90240] days = 365.24219858 days - [1/90240] days = 0.24219858 days - 0.00001108 days = 365.2421875 days.

Accuracy: α = [683/2820 = 96.879 (43262411348)/400=31.0014184 (3971631)/128] days = 0.2421985 (815602837) days,

Rate: [1 / α] = 2820/683 = 400/96.879 (43262411348) = 4.1288 (43338213763);

Size of a systemic error: β = [1/90240] days = 0.00001108 days. [«The negative error» in 1 full day runs in calendar Ahmad Birashk for 90 240 years.]

Ahmad Birashk has offered idle time «a mathematical calendar», not connected with direct astronomical supervision. He has suggested to use a super long cycle in 2820 in which 683 leap-years for 366 days and 2137 usual years for 365 days are provided.

Accuracy of such calendar [α] will make 683/2820=0.2421985 (81) (average duration of year to the seventh sign has made 365.2421986 days) that for 0.0003014 days (0.2425-0.2421986=0.0003014) it is better than result in system of the Gregorian calendar, and on 0.0000111 is worse than the result received on system Medler-Mendeleev (0.2421986-0.2421875=0.0000111).

In "mathematical" calendar Ahmad Birashk in recalculation since the period per 2820 for the period in 128 years the parity (2820/128=22.03125=22 whole (full) days + 0.03125 days) is applied. In "mathematical" calendar Ahmad Birashk in the period of 2820 should be 705 usual Julian leap-year (2820/4=705).

But in a cycle of 128 years for these of 2820 runs 22 superfluous whole days which Ahmad Birashk subtracts from Julianian leap years and receives figure «683» years (705-22=683 year). In a variant of "mathematical" calendar Ahmad Birashk full (whole) superfluous days are considered and compensated for the period per 2820 only, and the fractional part [a system error] at a rate of 0.03125 days for the same period per 2820 is not considered and not compensated (insufficiently it is considered "infinitesimal" size which «can be neglected»).

The systemic error in "mathematical" calendar Ahmad Birashk in 1 full day runs for 90 thousand 240 years (90240 years) by following calculation: 2820/0.03125=90240, that is through 32 cycles on 2820 [90240/2820=32 cycles].

 

 

Offered by us Uniform for the Earth and Space the base universal digital Mathematical quantum calendar theoretically at all does not accumulate an error concerning duration of tropical year in 1 day for the period more, than in 240.0 million years [that in many practical questions of equivalently "infinity" (∞) years, and therefore it is really possible to consider it «infinitesimal disappearing» as size].

 

 

 

But if to be absolutely exact it is necessary to consider that average tropical solar days increase on 1.5 ms [milliseconds] for each 100 tropical years (McCarthy D., Seidelmann P.K. Time from Earth rotation to atomic physics. Weinhein: Wiley-VCH Verlag GmbH & Co. KGaA, 2009. P. 351; Secular terms of the classical planetary theories using the results of general theory // Astronomy and Astrophysics. №  157. Р. 59–70).

 

 

 

For 1.1 bln years of theoretical physical existence of the Sun the calendar offered by us will type an error of all at 4.58 days or at 4 days 13 hours 55 min and 12 seconds. [(1.1 * 109 years)/102 years = (1,1 *107) * (1,5 10-3) sec = (1,65 * 104) sec/60 sec = (0.0275 * 104) min = 275 min / 60 min = 4,58 days = 4 days 13 hours 55 min 12 sec].

 

 

 

Thus the Gregorian calendar error for this term in 1.1 bln years will make years more than 941 years. {[(1.1 * 109 years) / (3.2 * 103 years for one day of an error)] = (343, 750 * 103 days of an error) / (365.2421875 days in tropical year) = 941, 15633890 (18417) years of an error}.

 

 

 

One day of an error will run appropriately for 240 mln 174 th 672.89083 years. [(1 day * 1.1 109 years) / 4.58 days = 0.240174672489083 *109 years = 240 mln 174 th 672.89083 years]

 

 

Thus, offered by us «New uniform for the Earth and Space the Mankind calendar» easily solves all calendar problems of our Human Civilization for all term of its stay in solar system. After 1.1 billion years the Sun becomes "old", absolutely unsuitable for residing of people and the Mankind will leave forcedly forever limits of dying Solar system. The Mankind is to move to any new young Sun of our Galaxy to "Milky Way".

10). The full formula of calculation of duration tropical the year offered in given work [Uniform for the Earth and Space base quantum universal digital Mathematical] of a calendar looks as follows:

L = 365 days + [31/128 days] - [1 / ∞] days = 365.2421875 days - [1 / ∞] days.

Accuracy: α = 0.2421875 days [31/128 = (31*3.125) / (128*3.125) =96.875/400],

Rate: [1 / α] = 128/31 = 4.1290 (32258064516)

Systemic error: β = 0.0 days = [1 / ∞] days. (The system error is equal to zero).

11). The full formula of calculation of duration of Lunar year will make:

A]. L = L (moon) days + 10.8750595 days = 354 days + 0. 367128 days + 10.8750595 days of =354 days + 146.8512/400 days + 10.8750595 days = 354.3671280 days + 10.8750595 days = 354.3671280 days + [1/0.0919535 (198864889] days = 354.3671280 days + 10.8750595 days = 365.2421875 days.

Accuracy: α = [146.8512/400] days = 0.367128 days, 400 years led to a cycle.

[Return recalculation for reception of number of leap years on a cycle in 400 years: 0.367128 * 400 = 146.8512.]

Rate [1 / α] = 400/146.8512 = 2.7238 (45634220217)

Size of a systemic error: β = 10.8750595 days = / run for 1 year. [The greatest "positive" system error from all calendars runs at the Lunar calendar ~ 11 days for a year.]

β = 10.8750595 days = [1/0.0919535 (198864889)] days,

Where is: 0.0919535 (198864889) = [33.58530475 (166596) days/365.2421875 days], that is a part of year for which "the positive" system error, the size 1 whole day runs.

B]. L (moon) = L - [1/0.0919535 (198864889] days = 365 days + [31/128] days - [1/0.0919535 (198864889] days = 365 days + [31/128] days - 10.8750595 days = 365.2421875 days - 10.8750595 days = = (365 days +0.2421875) days - (10 days + 0.8750595) days = [365 - 10] days + [0.2421875 - 0.8750595] days = [355 days = 354 days +1] + [0.2421875 - 0.8750595] days = 354 days + [1.2421875 - 0.8750595] = 354 days + 0.367128 days = 354.3671280 days.

Accuracy: α = [31/128] days = 0.2421875 day, 128 years led to a cycle.

α = [96.875/400] days = 0.2421875 days, 400 years led to a cycle.

β = 10.8750595 days = [1/0.0919535 (198864889)] days, where: 0.0919535 (198864889) = [33.58530475 (166596) days/365.2421875 days], that is a part of year for which "the negative" systemic error, the size 1whole day runs.

Technologically shortening of duration of year of a star of Sirius in a variant of Pharaoh Ptolemy III Everget (the Sothic sideral year) and approach of its value to length of true the Solar tropical year is reached by application of system of correction by means of leap-years.

Frequency of following of leap-years defines accuracy of conformity of astronomical and calendar events, that is accuracy of synchronisation of an astronomical tropical cycle of the Sun and virtual calculations of a calendar. In a Gregorian calendar it became more rare, than 1 time in 4 years as it was in original Julian the Egyptian Roman calendar in a variant of Pharaoh Ptolemy III Everget (detailed calculation is resulted more low).

In 1899 CE at Russian astronomical society (the society has been founded in 1891 CE) the Commission on calendar reform has been created. It consisted of representatives of many scientific institutions, the ministries and country departments; in it the leading part belonged to Great Russian scientist D.I. Mendeleev (1834-1907CE).

The commission after long discussions of different variants of calendar systems under D.I. Mendeleev's offer recommended new exact leap correction of a calendar on the tropical year, developed by the German astronomer, the professor Derptsky (Derpt - nowadays Tartu) university A.G. Medler (1794-1874CE).

Leap correction of a calendar on tropical year should replace correction on approximately equinox to the year accepted in the Gregorian church Catholic calendar. Mendeleev and Medler have suggested to create for the first time an exact secular tropical calendar instead of church Gregorian equinoctal the calendar of the Easter Moon.

In the Julian Egyptian calendar of a star Sirius in a variant of Pharaoh Ptolemy III Everget for each 4 years runs 1 whole additional (leap) day (365 day+0.25=365.25 days and consequently 0.25*4=1 days). In the period in 128 years 32 leap-years were provided, or that the same that in the period of 400 years was provided 100 leap-years (128:4=32 or 32/128=0.25=100/400=1/4).

Higher than in the Julian Egyptian calendar absolute theoretical accuracy of synchronisation with tropical year in a variant of Medler-Mendeleev was reached by that in the period in 128 years not 32 leap-years (128:4=32), as in the Julian Egyptian calendar, and only 31 leap-years were provided. Therefore the parameter of accuracy of such calendar «α» = [31/128] is 0.2421875 days ideal.

Thereby average duration of calendar year made: 365 + (31/128 =365.2421875 calendar days that did not lead to an error of such calendar in relation to real tropical astronomical year even in one day (to within 7 signs), calculated by I. Kepler in 1627 [L = 365.2421875 astronomical days] under experimental tables of an observatory Tycho Brahe and Waltherus during the nearest (ten millions years) 10 000 000 years:

365.2421875 astronomical days - 365.2421875 calendar days = 0.000.000.0.

[In 1627, Kepler used the observation of Tycho Brahe and Waltherus to produce the most accurate tables up to that time, the Rudolphine Tables. He evaluated the tropical year as 365 solar days, 5 hours, 48 minutes, 45 seconds, - that has made 365.2421875 days.]

Therefore for our Uniform base Mathematical calendar we suggest to enter exact leap tropical correction in version Medler-Mendeleev from 1899 CE instead of a variant approximate leap equinoctal the correction accepted in 1582 CE Pope Gregory XIII.

In the Julian Egyptian calendar in a variant of Pharaoh Ptolemy III Everget for 400 years 146000 usual days (365*400=146000) (400/4=100) are necessary +100 leap days that makes 146100 full days, or 146100/400=365 + (100/400 =365.25. In the Gregorian calendar (approximately equinoctal) for 3 days it is less than leap days (100-3=97). In each cycle from 400 years contains for 146097 days: 146097/400 = (146000/400) + (97/400) = 365 + 0.2425 =365.2425 days.

In the Gregorian calendar in recalculation since the period in 400 years for the period in 128 years the parity (400/128=3.125=3 full days + 0.125 days) is applied and consequently 31.04 leap-years (97/3.125=31.04) instead of 32 in the Julian calendar turn out. Year in the Gregorian calendar turns out equal: 365 + (31.04/128 =365 + (97/400) = 365.2425 days [as 31.04/128 = 97/400 = 0.2425].

In the Gregorian calendar, 31.04 leap-years for the period in 128 years (or that too the most, 97 leap days for 400 years) are provided. Frequency of the given event will make 400/97 = 128/31.04 = 4.1237113 (40) years instead of frequency in 4.0 years in the Julian calendar, with accuracy 97/400 = 31.04/128 = 0.2425 that is approximated to the whole sizes and gives a memory error from a fractional part in 1 (one) whole day for 3200 years (400/0.125 = 3200).

In offered by us Uniform for the Earth and Space a universal quantum digital Mathematical eternal calendar of Mankind, frequency of following of leap-years the rarest. Quantitatively it makes:

1). [1 / α] = 4.1290 = [128/31 = 4.1290 (32258064516)].

It is rarer, than frequency of following of leap-years in a calendar of Maya in which leap-years follow with frequency:

2). [1 / α] = 400/96.88 = 4.1288 (19157720892);

It is rarer, than frequency of following of leap-years in "mathematical" calendar Ahmad Birashk in which leap-years follow with frequency:

3). [1 / α] = 4.1288 years [2820/683=4.1288433 (382213763)].

It is rarer, than frequency of following of leap-years in the New Julian calendar:

4). [1 / α] = (900/218) = 4.1284 (403).

It is rarer, than frequency of following of leap-years equinoctal the calendar:

5). [1 / α] = 400/96.96 = 4.1254 (12541254125);

It is rarer than frequency of following of leap-years in the Iranian calendar at Omar Khayyam's:

6). [1 / α] = (33/8) = 4.1250.

It is even rarer than frequency of following of leap-years in the Gregorian calendar:

7). [1 / α] = [400/97] = 4.1237 (11340206186).

Less often than conditional rate [1 / α] the Judaic calendar:

8). [1 / α] = [400/98.19076] = 4.0737 (0306533).

Less often than frequency of following of leap-years in the Julian (Egyptian) calendar in a variant of Pharaoh Ptolemy III Everget:

9). [1 / α] = (400/100) = 4.0000.

Less often than frequency of following of leap-years in the Galactic calendar:

10). [1 / α] = 400/102.5452 = 3.900 (71890249373).

Less often than frequency of following of leap-years in the Lunar calendar:

11). [1 / α] = 400/146.8512 = 2.7238 (45634220217)

 

1). In resulted on a cycle in 400 years in the Lunar calendar 146.8512 leap days are necessary.

2). In resulted on a cycle in 400 years the Galactic calendar 102.5452 leap days are necessary.

3). In a Julian (Egyptian) calendar in a variant of Pharaoh Ptolemy III Everget for 400 years 100.0000 leap days are necessary.

4). In resulted on a cycle in 400 years the Judaic calendar 98.19076 leap days are necessary.

5). In a Gregorian calendar on a cycle of 400 years 97.0000 leap days exactly are necessary.

6). In Iranian Omar Khayyam's the calendar for a cycle of 400 years 96.9690 leap days are necessary.

7). In equinoctal the calendar for a cycle of 400 years 96.9600 days are necessary.

8). In a New Julian calendar for a cycle of 400 years it is necessary 96.8888 (889) leap days.

9). In a calendar of Maya for a cycle of 400 years 96.8800 leap days are necessary.

10). In "purely mathematical" calendar Ahmad Birashk is available 96.8797 (53) leap days, and

11). In our Mathematical calendar on system Medler-Mendeleev - 96.8750 leap days: (31*97)/31.04=96.8750. Or is easier: 100-3.125=96.8750.

Astronomer N.I. Idelson (1885-1951CE) has offered very difficult variant calculation specifications leap years by Mendeleev-Medler variant. N.I. Idelson has suggested to divide the period in 128 years into four separate periods: 33, 33, 29 and 33 years and in everyone to provide 8, 8, 7 and 8 in the sum (8+8+7+8=31) leap-years accordingly. The most exact tropical calendar (31/128=0.2421875) in this case would turn out. Theoretically correct variant of Idelson to put into practice it is inconvenient. Therefore anybody in this kind does not apply it.

Rule of leap years in our settlement base universal quantum digital mathematical [tropical] calendar of Mankind, uniform for the Earth and Space, extremely simple. It is based on a 128-year-old cycle, proposed by Medler in 1864.

The period of 128 years breaks on 2 (two) parts: 120 years and 8 years (120+8=128). On a piece of 120 years leap-years follow as usually every 4 year (120/4=30 leap years), and on a piece last 8 years of a cycle there is only 1 (one) leap-year instead of two. Total it turns out 30+1=31 leap-year for all period of 128 years (it is received precisely required parity 31/128, simpler, than at Idelson with its combination from cycles of 33 and 29 years).

With 2012 for 2132 (2012+120=2132) will be 30 leap years (2016, 2020 etc. to 2132). Further 2136 1 leap-year will be passed. And last in a cycle 31 leap-years will take place only in 8 years in 2140.

Further the new cycle in 128 years will repeat completely with 2140 for 2268: 2140 + (120+8 =2268. It is and so indefinitely. On a cycle in 128 years it will be used only two types (template) of a calendar: the Leap and not Leap which among themselves differ only number of days of rest on a holiday of the New Year (three days of rest in Leap year and two days of rest in not Leap year). In all the rest both calendars are absolutely identical.

More low as an example we result detailed calculation of ten nearest cycles of Medler-Mendeleev for 128 years (on the nearest 1280 [128*10 = 1280] for the period with 2013 for 3192 year CE [2013+1280=3192 including in this sum the first year of each cycle that is adding to the first figure "127", instead of figure "128"]).

The first cycle (2013-2140 CE). The second cycle (2141-2268CE).

The third cycle (2269-2396CE). The fourth cycle (2397-2424CE).

The fifth cycle (2425-2552CE). The sixth cycle (2553-2680CE).

The seventh cycle (2681-2808CE). The eighth cycle (2809-2936CE).

The ninth cycle (2937-3064CE). The tenth cycle (3065-3192CE).

Further we as an example in detail particularly on years state the list of Leap years (31 leap-years) in the First cycle (2013-2140CE) to Medler-Mendeleev (all other cycles of this series similarly pay off). For 31 leap-years all these the same Leap calendar will be used.

[2016, 2020, 2024, 2028, 2032, 2036, 2040, 2044, 2048, 2052],

[2056, 2060, 2064, 2068, 2072, 2076, 2080, 2084, 2088, 2092],

[2096, 2100, 2104, 2108, 2112, 2116, 2120, 2124, 2128, 2132 and 2140].

(Year «2136» is passed in the category of leap-years and translated in the category non leap.)

Further we as an example in detail particularly on years state the list of not Leap years (97 not leap that is simple years) in the First cycle (2013-2140CE) to Medler-Mendeleev (all other cycles of this series similarly calculation). For 97 not leap years all these the same not Leap calendar will be used.

[2013, 2014, 2015], [2017, 2018, 2019], [2021, 2022, 2023], [2025, 2026, 2027],

[2029, 2030, 2031], [2033, 2034, 2035], [2037, 2038, 2039], [2041, 2042, 2043],

[2045, 2046, 2047], [2049, 2050, 2051], [2053, 2054, 2055], [2057, 2058, 2059],

[2061, 2062, 2063], [2065, 2066, 2067], [2069, 2070, 2071], [2073, 2074, 2075],

[2077, 2078, 2079], [2081, 2082, 2083], [2085, 2086, 2087], [2089, 2090,2091],

[2093, 2094, 2095], [2097, 2098, 2099], [2101, 2102, 2103], [2105, 2106, 2107],

[2109, 2110, 2111], [2113, 2114, 2115], [2117, 2118, 2119], [2121, 2122, 2123],

[2125, 2126, 2127], [2129,2130, 2131], [2133, 2134, 2135, 2136], [2137, 2138, 2139].

 

Number of intercalary days at different degrees of accuracy of correction of calendar and astronomical years. [Julian, Judaic, Gregorian, Omar Khayyam’s, New Julian, both Ahmad Birashk and Maya, and universal digital tropical quantum mathematical]:

 

                           
  100.000
    98.191
      97.000
        96.969
          96.888
            96.879
              96.875
 
 
 
 

 Fig. 4

 

Frequency of leap corrections at different degrees of accuracy of correction of calendar and astronomical years. [Julian, Judaic, Gregorian, Omar Khayyam’s, New Julian, both Ahmad Birashk and Maya and universal digital tropical quantum mathematical]:

 

                           
    4.1290
    4.1288
    4.1284
    4.1250
    4.1237
    4.0737
  4.0
 
 

 Fig. 5

 

Lunar month (the synodic month) varies from 29 days 6 hours 30 minutes (29.27 days) till 29 days 20 hours (29.83 days) with disorder at 13 o'clock 30 minutes. Average duration the synodic month name «the molad interval»:

29 days 12 hours 793 parts (441/18 minutes) or (29.530594 days).

[The mean synodic month is days, or 29 days, 12 hours, and 793 parts (44+1/18 minutes) (i.e. 29.530594 days).]

Lunar year (simple and leap) consists of full 12 months and a fractional part:

365 days/29.530594 days =12.3600629 (2321787) months

366 days/29.530594 days =12. 3939261 (0930887) months.

If to consider lunar year in 12 months from 29 whole lunar days we will receive:

12 months * 29 days =348 days.

If to consider lunar year in 12 months from 30 whole lunar days we will receive:

12 months * 30 days =360 days (it is exact ecliptic year if to consider year in angular degrees: 1 days = 10 [to 1 degree], and consequently 3600 [360 degrees] = 360 ecliptic days).

On the average usual lunar year contains the whole: 348 days+360 = 708/2 = 354 days [are more exact than days without roundings off: 354.36708 days or 0.97 shares of tropical year {354.36708 days/365.2421875 days = 0.97 (02249414985776) from duration of tropical year}].

Therefore lunar year always is shorter than solar year for 11 whole days (365 days - 354 days = 11 days in usual year), and (366 days - 11 days = 355 days in leap year).

The modern Judaic luni-solar calendar is constructed on following calculations. Duration of lunar year on the basis of calculation molad interval for 12 months makes 354.367128 days (12 months х 29.530594 days of =354.367128 days). The difference with duration of tropical year makes 10.8750595 days (365.2421875 days - 354.367128 days = 10.8750595 days).

For 19 years Meton of a cycle this difference makes 206.6261305 days (19 years of a cycle * 10.8750595 days of an error for one cycle = 206.6261305 days). This difference compensate, adding 7 leap months in 7 leap-years containing for 13 months in year. The sum of indemnification for one cycle of 206.714158 days (29.530594 days * 7 leap years = 206.714158 days) turns out.

This sum exceeds deficiency in relation to tropical year for 0.0894375 days (206.714158 days of indemnification for one cycle - 206.6261305 days of an error for one cycle = 0.0880275 days). For 5772 years (3760+2012=5772 year) from the moment of the beginning of phantom readout of a Judaic calendar from 5 o'clock in the afternoon of 204 parts «on Almagest» on Monday, 1 dates of Tishrei (1st Day Rosh Hashanah), in 3761-3760 year BCE, 303 full cycles of Meton for 19 years have run everyone (5772/19=303.78947368 a cycle).

Therefore the error of backlog of a Judaic calendar from tropical year for all period of 5772 years has made 26 whole days (303*0.0880275=26.6723325 days). (The universe new moon «Anno Mundi» (AM) or the «Moon Creation», that is «MC», which on the Jewish calculations took place in 3761-3760 (BCE), on Monday, at 5 o'clock and 204 parts p.m. is necessary. Hour per the Jewish tradition shares not for minutes and seconds, and on 1080 parts (the number borrowed from «Almagest Ptolemy» and multiple all too unequivocal dividers, except 7), and each part is for 76 instants.)

Actually the size molad interval which it is considered to be equal 29.530594 days, special accuracy does not differ. Backlog for 1 whole day precisely runs for each 304 years (real backlog of Judaic calendar Easter [Passover] and Dates of eclipses of the Sun the Moon from astronomical or tropical Date). Therefore for each 19 years the size of an error equal not of 0.088025 days, but only 0.0625 days (19/304=0.0625 days=1.5 hour = 1 hour of 30 minutes) runs.

And consequently for all period in 5772 years the real error of backlog of Dates of a Judaic calendar from Dates of a calendar of tropical year (and Dates of an astronomical reality thereupon that has run the same) not for 26 whole days, but only for 18 whole days (303*0.0625=18.9375 days).

Omar Khayyam, counting the calendar in 1079 СЕ, has translated the Iranian lunar year in star-solar. It has improved thus itself Julian the Egyptian calendar of a star Sirius in a variant of Pharaoh Ptolemy III Everget, having translated it in equinoctal a calendar of a star of the Sun (has passed from a star Sirius to a star the Sun).

He has added by lunar year of 11 days and has looped leap-years with factor of accuracy of correction: 8/33=0.2424242 (42424242). Or 365 + (8/33 =365.2424242 in usual year and 366 + (8/33 =366.2424242 - in leap year.

Therefore the Omar Khayyam the parity «8 leap years for 33 years» (8/33=0.2424242424) was accepted. In other words, from each 33th years 8 were leap and 25 usual. Leap-years of 7 times come in 4 years, as usually, and 1 time - in 5 years (7*4=28+5=33). Usual years come 7 times in 3 years and once in 4 years (7*3=21+4=25).

The lunar calendar is not adhered at all to year movement of the Sun; therefore annually lunar calendar is displaced concerning tropical solar year for 365.2421875-354.367128 =10.8750595 days.

Approximately for 33÷34 solar year runs one superfluous lunar year (365/10.8750595=33.56303475856845 year or 366/10.8750595=33.65498827845494 year) or in integers (365/11=33.18181818 year or 366/11=33.27272727 year).

Each 3 (three) years is the difference between a lunar calendar and a star-solar calendar reaches 33 days (11*3=33). Therefore in luni-solar calendars each 3 (three) years there is 1 (one) additional month in 33 days on the average to compensate this error and to return a calendar in full conformity with astronomical realities of tropical year of the Sun.

If compensation a running error not to spend, lunar calendar and solar tropical (calendar and astronomical) annual cycles in itself naturel about coincide in the image once everyone (33.56÷33.65 years) or in round figures (33.18÷33.27 years), or in integers approximately once in 33 years with an advancing, that is leaving of the Lunar calendar forward for 1 year from the tropical. [365.2421875 days of tropical year/11 of the whole days of backlog of the Lunar calendar for 1 year from the sizes of tropical year = 33.20383522727273 years for which occurs errors of the Lunar calendar at a rate of the whole tropical year.] After accumulation of an error at a rate of full tropical year, tropical and lunar calendar cycles disperse again for the same term (in full 33 years) and so proceeds indefinitely. There is a passive self-synchronisation of two calendar cycles with the period in full 33 years. Therefore the Omar Khayyam used this 33-year-old cycle of passive self-synchronisation in the Iranian calendar. 

Solar eclipses have the cyclic repeatability known as a 19-year-old cycle of Meton («a circle of the Moon or the Moon cycle»), entered by Greek Meton in the Ancient world after Chineses and Babylonian on June, 27th 432 years BCE in day of a summer solstice. The cycle of Meton has laid down in a basis of an Ancient Greek calendar.

[In a year on the Earth can occur from 2 to 5 solar eclipses, from which no more than two - full or similar to the ring. On the average for hundred years there are 237 solar eclipses, from which 160 - private, 63 - full, 14 - similar to the ring. In a certain point of a terrestrial surface of an eclipse in the big phase occur seldom enough, full solar eclipses are even less often observed. So, in territory of Moscow with XI on a XVIII-th century it was possible to observe 159 solar eclipses with a phase more than 0.5 of which only 3 full (11.08.1124, 20.03.1140 and 7.06.1415). One more full solar eclipse has occurred on August, 19th, 1887. Similar to the ring eclipse could be observed in Moscow on April, 26th, 1827. Very strong eclipse 0.96 has occurred to a phase on July, 9th, 1945. The following full solar eclipse is expected in Moscow only on October, 16th, 2126.]

If a 19-year-old cycle of the Moon to shift for 3 years all will turn out the same cycle of Meton parallel to the first cycle. The figure received from such summation (+3 years) will be called as "gold number» (numerous aureus) as it did of gold and hung out in Athens on a special column (Stella) where the Ancient Greek public calendar took places.

Greeks (Meton) and Judea, were in Babylon to the captivity ([to 586 year BCE (BC)] date of destruction of the First temple of Solomon in Jerusalem), borrowed this calendar cycle the Babylon and Chinese astronomers.

In 586 year BC (3760AM-3174AM=586BCE) after long fight of Navuhadnetsar (Nebuchadnezzar), the tsar of Babylon, has entered into First temple Solomon in Jerusalem. The ninth Ava he has set fire to it, and has then withdrawn all people of Israel in captivity. So the Babylon exile which proceeded seventy years, while Kyr, the tsar of Persia has begun, has not authorized for building of the Second Temple in Jerusalem.

Greek Aristotle (Aristotle, 384 BCE - 322 BCE) and Alexander III Macedonian (on July, 20th 356 BCE ÷ on June, 10th 323 BCE) his pupil about 343 years BCE adhered to rules of this Babylon calendar. The modern science has begun with Aristotle, and the modern practical political organization of a society has begun with Alexander of III Macedonian.

In a cycle of Meton there is a remarkable parity: 19 years of a star Sirius-Sothic (228 months) and 19 lunar years (235 months) have the identical sum of full days equal of 6939 days. In 19 lunar years 7 years for 13 months (7*384=2688 days), and 12 years for 12 months, from which leap 3 years (3*355=1065 days) and usual 9 years (9*354=3186 days) contain. Total: 6939 days in the sum. This parity is necessary in a basis of calculation of a Judaic calendar patriarch Gilel II in 359 year СЕ.

It is obvious that 19 years star contain Sirius-Sothic of a calendar in a variant of Pharaoh Ptolemy III Everget (19*365=6935 days+4 days from leap year=6939 days). But this is equality not absolutely exact. 

The lunar cycle of Meton (6939,60 days=365.2421052631579*19 days) is shorter (6939,6625 days=365.2453947368421*19 than days) star Sirius-Sothic of a cycle of Meton for 0.0625 days (1.5 hours = 1 hour of 30 minutes) for each 19 years of a cycle of Meton.

Therefore, to keep Easter full moon each next 304 years in conformity with calendar calculations, date (Hebrew Passover) Easter (first full moon after date of the Spring equinox) in the star-solar calendar, calculated on lunar calendar, will be necessary to shift forward in star-solar (Sirius-sun) calendars for 1 whole days each 304 years: 19 years/0.0625=304 years.

Therefore date of real astronomical Easter moves from date of 14 Nisan on Hebrew calendar towards the end of month Nisan. Already now it has affirmed on date of 15 Nisan. In 304 years Hebrew Passover (Exodus) will move for date of 16 Nisan. Figure means «304» years that shift for 1 day forward on a calendar occurs in the end (last day) 16th under the account of a 19-year-old cycle of Meton [19 (years of a cycle of Meton) *16 (cycles of Meton) =304 years].

 

8. DYNAMICS OF THE SYSTEMIC ERROR [BACKLOGS / ADVANCINGS] IN THE SIZE IN 1 DAY WHICH RUNS:

1). In a lunar calendar for 33.58530475 (166596) days = 0.0919535 (198864889).

2). In a galactic calendar for 70.5 (444).

[For 70 years of 198 days of 20 hours of 6 minutes of 29,27 seconds].

3). In a Julian calendar for 128 years.

4). In a Judaic calendar for 304 years.

5). In a Gregorian calendar for 3200 years.

6). In the Iranian calendar [Khayyam's Omar] for 4229 years.

7). In equinoctal a calendar for 4706 years.

8). In New Julian a calendar for 28800.

9). In a calendar of Maya for 80000 years.

10). In Ahmad Birashk "a mathematical" calendar for 90240 years.

11). In a settlement base universal quantum digital mathematical [Kepler tropical] calendar of Mankind offered by us, uniform for the Earth & Space, an error of an advancing of a calendar for 1.1 bln years runs at a rate of all for 4.58 days or at 4 days 13 hours 55 min and 12 seconds . [(1.1 * 109 years)/102 years = (1,1 *107) * (1,5 10-3) sec = (1,65 * 104) sec/60 sec = (0.0275 * 104) min = 275 min / 60 min = 4,58 days = 4 days 13 hours 55 min 12 sec].

 

One day of an error will run accordingly for 240 mln 174 th 672.89083 years. [(1 day * 1.1 109 years) / 4.58 days = 0.240174672489083 *109 years = 240 mln 174 th 672.89083 years]

 

It is necessary to notice especially that [the error in 1 day runs accuracy of a calendar of Maya for 80 000 years] in 25 times above [80000/3200 = 25] than in a Gregorian calendar used today everywhere [an error in 1 day runs for 3200 years]. (This gives the Maya approximation to the tropical year at being 365.2422000 days being more accurate than the Gregorian Year [i] currently used across the world today.)

System error of any calendar always calculation in relation to duration of real observable astronomical tropical year. Therefore at calendar tropical year in relation to astronomical tropical year no error is present. They are equal by definition. Therefore in the settlement base universal quantum digital mathematical [tropical] calendar of Mankind offered by us uniform for the Earth & Space this systemic error is always equal to "0" (zero) {[1 / ∞] days}.

 

Fig. 6

DYNAMICS OF THE SYSTEMIC ERROR IN THE GRAPHIC VARIANT

[Function: | β | = µ * {1 - α/µ}, beginning from an analogue lunar calendar and finishing a tropical active digital mathematical calendar]

 

It is resolute all modern passive calendars try as it is possible more precisely physically model the movement or the Moon round the Earth, or the Earth round the Sun, or the Moon and the Earth round the Sun together. Therefore they are called as passive "analogue".

It is remarkable in this plan it is unique a remarkable "eternal" uniform solar-lunar Easter cycle Dionysus Exiguus (Small) or «Great Indiction» [Dionysus Exiguus (475-550CE)]. «Great Indiction» Dionysus Exiguus (Small) consists of 532 years (19*28=532, that is of 28 cycles of Meton for 19 years everyone). All dates Hebrew Passover and all dates of days of week and date of all months precisely repeat adjusted for 1 days of a systemic error (for 1 day forward for 128 years in solar (28 years) a cycle, and for 1 day forward for 304 years lunar (19 years) a cycle of Meton accordingly).

Joseph Scaliger (Josephus Justus Scaliger; on August, 5th on January, 1540 - 21th 1609CE) has offered larger unit of measure of historical time: «Julian the period (cycle) », consisting of «15» Great Indiction Dionysus Exiguus (Small). It would be possible to result all historical dates in such chronological scale. Every year it would be numbered by three numbers - number of Great Indiction (from 1 to 15), number of a lunar cycle (from 1 to 19) and number of a solar cycle (from 1 to 28).

Numbers (from 1 to 15) «Great Indiction» in «the Julian period» offered by Joseph Scaliger name «indicts of the Great Indiction».

For a date started «Julian the period (cycle)» Joseph Scaliger has suggested to consider purely phantom date on January, 1st 4713 BCE on its value judgment of duration of the Holy Bible period «from World Creation» prior to the beginning «the Era Jesus Christ, CE». For this date it has equated all numbers of all three compound cycles to number «1».

Duration Julian the period (cycle) has turned out to equal 7980 years (15*19*28=15*532=7980), that is, through this time (7980 years) Julian the period (cycle) repeats. The end of the first Julian the period (cycle) is necessary for January, 23rd 3267 years CE on a Gregorian calendar (4713 BCE+3267 CE=7980).

The Earth non-uniformly rotates round the Sun on an orbit in the form of an ellipse. It is accelerated in winter half-year from September, 23rd till March, 21st (and consequently winter months more shortly summer months) and slowed down in summer half-year from March, 21st till September, 23rd (and consequently summer months more long winter). Winter half-year consists of 179-180 days (5 month*30 days+29 days = ►150+29=179 days in usual year or 6 months * 30 days = 180 days in leap year). Summer half-year always stably consists of 186 days (6 months * 31 days = 186 days) both in not leap year, and in leap year.

The Earth is while one and a unique general spaceship of Mankind and this fact is reflected in its calendars. But today there has come an epoch of development of Planets of Solar System. In long space flights there will be no change of Day and Night, there will be no seasons of year. Will not be the Earth underfoot, the Moon over a head. It is a new reality of Mankind. Therefore and those who will be in distant Space, and at those who will serve them on the Earth, should have a same absolutely exact tool of the account of time, not connected neither with the Moon, nor with the Earth, with Mankind History on the Earth.

Therefore all analogue calendars become essentially unsuitable during a space age epoch. Their place will occupy uniform digital, quantum mathematical, universal, cyclic calendar Mankind («the New calendar of Mankind») which we here offer.

It has come Era the Uniform base digital Mathematical Space calendar. This calendar absolutely equal and exact is well balanced with the real astronomical facts and leans against a natural Biorhythm of the Person. It consists of 13 months for 28 days in everyone. All the week long, all the months long, all the quarters long, all the half-year long are equal each other and invariable. Year contains 52 full weeks in which there are 364 whole days.

The remained two whole days «365» and «366» are, speaking in images, "tail" of year which is simply added by day of the New Year as additional holidays. The beginning annually of year comes back in a starting position precisely for January, 1st, Sunday, for the first day of the first quarter by means of the mechanism of "a synchronisation saw», combining, as hands, astronomical realities with calendar dates.

Our Uniform for the Earth and Space the base Calendar has an own internal mathematical apparatus and does not depend in any way neither on Moon position in the Sky, nor from position of the Earth in Space. The calendar the auto (self) synchronized the Biorhythm of the Person also is independent, independent hours of Mankind, both on the Earth, and in a Space. It is a universal matrix for all calendars of the world.

The offered Uniform base Mathematical calendar is strictly fixed on the following to 5 astronomical points:

1). On heliacal rising of the star Sirius, to an event Virgo, 11th on (МC) to the Mathematical Calendar (July, 26÷25 on a Gregorian calendar).

2). Across tropic of Cancer (to the Winter Solstice - the Winter Solstice), to an event Capricorn, 20th on (МC) to the Mathematical Calendar (December, 22÷21 on a Gregorian calendar).

3). Across tropic of Capricorn (to the Summer Solstice - the Summer Solstice), to an event 4th Leo on (МC) to the Mathematical Calendar (June, 21÷20 on a Gregorian calendar).

4). On a spring equinox (the Spring or Vernal Solar Equinox), to an event Aries, 24th on (МC) to the Mathematical Calendar (March, 21÷20 on a Gregorian calendar).

5). On an autumn equinox (the Autumn Solar Equinox), to an event Scorpio, 12th on (МC) to the Mathematical Calendar (October, 21÷20 on a Gregorian calendar).

The offered Mathematical digital quantum calendar is strictly fixed on 4 calendar points (on dates of 4 quarters):

I quarter (Aquarius, 1st – Taurus, 7th) (January-April)

II quarter (Taurus, 8th – Leo, 14th) (April-July)

III quarter (Leo, 15th – Scorpio, 21st) (July-October)

IV quarter (Scorpio, 22nd – Capricorn, 28th) (October-December)

The mathematical calendar is in addition strictly fixed on 5 key Historical events of the Space age of Mankind:

1). The first flight of the person in Space: Jury Gagarin's flight (Taurus,18th) (April, 12th 1961CE on a Gregorian calendar).

2). The first exit of the person on the Moon: Neil Armstrong's exit (Virgo,6th) (July, 21st 1969CE on a Gregorian calendar).

3). Day of start of the first artificial companion of the Earth (on October, 4th 1957 CE). [The Scorpio, 26th]. [The day of the first artifitial earth satellite.]

4). Day of K. Tsiolkovsky (on May, 10th 1897 CE) [The 18th Gemini]. [The day of K. Tsiolkovsky.]

5). Day Nikola Tesla (on June, 30th 1908 CE) [The Leo,14 th]. [The day of N. Tesla.]

All calendars executed on the Universal Matrix become at once Space and can be used in distant space flights. All Religions of the World can leave in Space having spent transformation from the analogue matrixes to a uniform Universal Matrix as through original "eye of a needle", and in any way differently.