Hebrew Calendar Science and Myths

by Remy Landau
=================================

The Hebrew calendar is constructed from the simple multiple of a constant value for the mean lunation period. The mean lunation period is an averaging of the time taken by the moon to cycle from lunar conjunction to lunar conjunction. The value used for the constant is 29d 12h 793p.

Hebrew years can be either 12 or 13 lunation periods.

Years of 12 lunation periods are usually called common years. Years of 13 lunation periods are usually called leap years.

Hence, the lunar length of all common years is

```        12 * (29d 12h 793p) = 354d  8h 876p

and    the lunar length of all leap   years is

13 * (29d 12h 793p) = 383d 21h 589p.
```

Hebrew years are arranged in a cycle of 19 years consisting of 12 common and 7 leap years. This cycle is known as a mahzor katan, ie, a small cycle, and is exactly 235 months long.

The mean lunar length of any 19 consecutive Hebrew years is
235 * (29d 12h 793p) = 6939d 16h 595p.

Let L(i) represent the mean lunar length of Hebrew year i.

Then the value of L(i) can be either 354d 8h 876p, if year i is common,
or 383d 21h 589p if year i is leap.

Since the common and leap years are arranged in a 19 year cycle, L(i) = L(i + 19*k) for any value of i and k.

The Leap Year Distribution

The fixed Hebrew calendar has one, and only one, order for the arrangement of its common and leap years. The apparent order in which the leap and common years occur in a given cycle of 19 consecutive years depends entirely on which element of the arrangement pattern is selected as first.

Allowing L to represent a leap year and c to represent a common year, then the arrangement of leap years and common years, extracted from some starting year in a Hebrew year time line can be seen to be

...ccLccLcLccLccLccLcL ccLccLcLccLccLccLcL ccLccLcLccLccLccLcL c...

If the pattern had started with the 4th year of the above time line then the arrangement from the first year of the pattern would appear to be

....ccLcLccLccLccLcLccL ccLcLccLccLccLcLccL ccLcLccLccLccLcLccL c...

Looking at the 19 year cyclical arrangement of the leap years beginning from the first year of each pattern, the 19 year cycles appear to be

the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years of the 19 year cycle

for the first selection and

the 3rd, 5th, 8th, 11th, 14th, 16th and 19th years of the 19 year cycle

for the second selection.

For some Hebrew year i let C(i) be a cycle of 19 consecutive Hebrew year lengths beginning with the year length L(i).

Then, C(i) = {L(i), L(i+1), L(i+2), ... , L(i+18)} and C(i+19*k) = C(i) since L(i+19*k) = L(i).

For some Hebrew year j let i-j = d.

Then L(j) = L(i-d) and C(j) = {L(i-d), L(i-d+1), L(i-d+2), ... , L(i-d+18)} = C(i-d).

The sequence of elements started at L(j) is the same as the sequence of elements started at L(i-d). Unless d is a multiple of 19, C(j) will appear to be different than C(i).

Since L(i) = L(i + 19*k) for any value of i and k, 19 apparent distributions exist for the common and leap years in a mahzor katan.

It is common practice today to associate C(1) with a 19 year cycle in which the order of leap years is the 3rd, 6th, 8th, 11th, 14th, 17th, and 19th year of the cycle. This cycle is known by the acronym GUChADZaT, which is formed from the Hebrew letters gimel-vov-het-alef-daled-zayin-tet. The Hebrew letters also symbolize the digits 3, 6, 8, 1, 4, 7, and 9.

The 19 Leap Year Intercalation Cycles

The table below shows the 19 intercalation patterns which can be observed relative to C(1).

The 19 Leap Year Intercalation Cycles
C( 0)
`  1  4  7  9 12 15 18`
`  2  3  3  2  3  3  3`
` 3d  7h  695p`
C( 1)
`  3  6  8 11 14 17 19`
`  3  3  2  3  3  3  2`
` 2d  5h  204p`
C( 2)
`  2  5  7 10 13 16 18`
`  3  3  2  3  3  3  2`
` 6d 14h    0p`
C( 3)
`  1  4  6  9 12 15 17`
`  3  3  2  3  3  3  2`
` 3d 22h  876p`
C( 4)
`  3  5  8 11 14 16 19`
`  3  2  3  3  3  2  3`
` 2d 20h  385p`
C( 5)
`  2  4  7 10 13 15 18`
`  3  2  3  3  3  2  3`
` 0d  5h  181p`
C( 6)
`  1  3  6  9 12 14 17`
`  3  2  3  3  3  2  3`
` 4d 13h 1057p`
C( 7)
`  2  5  8 11 13 16 19`
`  2  3  3  3  2  3  3`
` 3d 11h  566p`
C( 8)
`  1  4  7 10 12 15 18`
`  2  3  3  3  2  3  3`
` 0d 20h  362p`
C( 9)
`  3  6  9 11 14 17 19`
`  3  3  3  2  3  3  2`
` 6d 17h  951p`
C(10)
`  2  5  8 10 13 16 18`
`  3  3  3  2  3  3  2`
` 4d  2h  747p`
C(11)
`  1  4  7  9 12 15 17`
`  3  3  3  2  3  3  2`
` 1d 11h  543p`
C(12)
`  3  6  8 11 14 16 19`
`  3  3  2  3  3  2  3`
` 0d  9h   52p`
C(13)
`  2  5  7 10 13 15 18`
`  3  3  2  3  3  2  3`
` 4d 17h  928p`
C(14)
`  1  4  6  9 12 14 17`
`  3  3  2  3  3  2  3`
` 2d  2h  724p`
C(15)
`  3  5  8 11 13 16 19`
`  3  2  3  3  2  3  3`
` 1d  0h  233p`
C(16)
`  2  4  7 10 12 15 18`
`  3  2  3  3  2  3  3`
` 5d  9h   29p`
C(17)
`  1  3  6  9 11 14 17`
`  3  2  3  3  2  3  3`
` 2d 17h  905p`
C(18)
`  2  5  8 10 13 16 19`
`  2  3  3  2  3  3  3`
` 1d 15h  414p`
C(19)
`  1  4  7  9 12 15 18`
`  2  3  3  2  3  3  3`
` 6d  0h  210p`

Note that C(19) = C(0).

The relative order cyclical pattern is formed from the number of years that have elapsed since the previous leap year.

The relative order for C(4) is shown to be 3 2 3 3 3 2 3. In Hebrew this numreic pattern is represented by letters gimel-bet-tet-bet-gimel. Obviously, tet representing 9, reflects 3+3+3.

Since there are 7 leap years in a mahzor katan, only 7 relative order patterns are produced.Hence, specifying a cycle by its relative order does not uniquely identify the cycle. For example, it is to be noted that gimel-bet-tet-bet-gimel is also a relative order for C(5) and C(6).

The table also includes the Epochal Moladot which are associated with the first 20 years of Yetsirah, that is, Creation, or Genesis, or Aera Mundi.

Some References

1. Pages 91-95 of Studies In Hebrew Astronomy and Mathematics by Solomon Gandz, KTAV Publishing House, Inc. (1970), show a study of the calendar cycles as discussed in 3 medieval sources. These are the Seder Olam, the Yesod Olam, and the Pirke di Rabbi Eliezer (PdRE).

In this discussion, Gandz analyzes ancient rabbinic sources which debate the issues of whether the mahzor katan ought to be either C(1), C(2), C(3), C(4), or C(12). Gandz does not dicuss the fact that cycles expressed in the relative order notation can refer to more than one leap year intercalation cycle.

Gandz's text also indicates the existence of a third manner of noting the leap year cycles. It seems to be the relative order notation, but with the first entry indicating the number of years elapsed from the first year of the cycle.

2. On page 48, under the subheading Historicalof the article Calendar, in the Encyclopedia Judaica (1982), the cycles C(1), C(2), C(3) and C(4) are shown together with their Hebrew acronyms. The article identifies the epochal moladot associated with C(1), C(2), and C(3) to be
2d 5h 204p, 6d 14h, and 3d 22h 876p respectively.

The article indicates that 4d 20h 408p (named daled-hof-tof-het) is also an epochal molad. This is a puzzling detail, since 4d 20h 408p represents the molad of Heshvan 0H. Since the article shows C(4), the value 2d 20h 385p would have been expected, since it represents the value of the molad of Tishrei 4H.

3. Pages 64-66 of Al-Biruni's The Chronology of Ancient Nations (1000g) as translated into English by Dr. E.C. Sachau in 1879g, indicate that only 3 leap year cycles were in use at about 1000g (4761H). These cycles were C(2) and C(3) "used by the Jews of Syria", and C(4) "most extensively diffused among the Jews; ... because they attribute its invention to Babylonia."

Although the cycle C(1) appears in documents that predate Al-Biruni's work by several centuries, it is not mentioned at all in the Chronology.

4. Chapters 6-8 of Maimonides' Hilchot Kiddush HaChodesh (1175g) describe the calculation of the Hebrew calendar using C(1) and BaHaRaD. Maimonides' instructions still remain in use today.

Let M(i) represent the known value of the molad of Tishrei for some year i.

Then M(i+1) = M(i) + L(i) is the molad of Tishrei for the subsequent year i+1.

Hence, M(i+1) - M(i) = L(i)

Similarly,

```     M(i+2) - M(i + 1) = L(i + 1)
M(i+3) - M(i + 2) = L(i + 2)

and by induction,

M(i+k) - M(i+k-1) = L(i + k - 1)

Summing the above formulas from i+1 to i+k,

M(i+k) - M(i) = L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)
===> M(i+k) = M(i) + L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)
```

Defining LSUM(i,k) to be the sum of the lunar lengths of k consecutive years beginning with the length of year i,

```        LSUM(i,k) = L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)
===>       M(i+k) = M(i) + LSUM(i,k)
```

Evaluating LSUM(i,k)

Since L(i) = L(i+19*k) for any i and k, every sequence of 19 consecutive Hebrew years has the mean lunar length 6939d 16h 595p.

Therefore, LSUM(i,19*j) = (6939d 16h 595p)*j

Expressing k in the form 19*j + r

```M(i+k) = M(i  + 19*j  + r)
= M(i  + 19*j) + LSUM(i + 19*j, r)
= M(i) + LSUM(i, 19*j) + LSUM(i,r)
= M(i) + (6939d 16h 595p)*j + LSUM(i,r)
```

It is therefore not required to calculate LSUM(i,r) for more than the first 18 terms.

The expression for LSUM(i,r) represents the sum of the first r terms of C(i). Consequently, the evaluation of LSUM(i,r) is dependent on the order of the values of the length elements in C(i).

Before showing the tables for LSUM(i,r) these arithmetical reductions need to be explained.

The moladot are normally expressed in units representing the day of the week, the hour, and the number of halakim.

In traditional calculations, the moladot are remaindered using 1080 as the divisor. The quotient is then added to the hours. The hours are then remaindered by 24 and the quotient added to the days. Finally, the days are remaindered by 7. If the remainder for the day is zero, it is sometimes expressed as 7 depending on the context. Traditional texts show no entries for the halakim or the hours if their corresponding value after remaindering is zero.

A 19 Hebrew year cycle (mahzor katan) consists of 235 months. Since one month has the lunation period of 29d 12h 793p, 235 months have

```235 * (29d 12h 793p) = 6815d 2820h  186,355p
= 6815d 2820h+172h 595p
= 6815d 2992h      595p
= 6815d+124d   16h 595p
= 6939d        16h 595p
==>  2d        16h 595p [after remaindering by 7d]
```
Since every 19 year period has an excess molad of 2d 16h 595p, multiples of 19 years will have an excess molad that are the corresponding multiples of 2d 16h 595p.

Consequently, it is sometimes more convenient to compute

```
M(i+k) = M(i) + (6939d 16h 595p)*j + LSUM(i,r)

in the reduced form

M(i+k) = M(i) + (2d 16h 595p)*j + LSUM(i,r)
```
The following table shows the valuations for LSUM(i,r) after the traditonal reductions.

The LSUM(i,r) Tables
LSUM( 0 , r)LSUM( 1 , r)LSUM( 2 , r)LSUM( 3 , r)
15d 21h 589p-4d 8h 876p-4d 8h 876pL5d 21h 589p-
23d 6h 385p-1d 17h 672pL3d 6h 385p-3d 6h 385p-
30d 15h 181pL0d 15h 181p-0d 15h 181p-0d 15h 181pL
46d 12h 770p-4d 23h 1057p-4d 23h 1057pL6d 12h 770p-
53d 21h 566p-2d 8h 853pL3d 21h 566p-3d 21h 566pL
61d 6h 362pL1d 6h 362p-1d 6h 362pL2d 19h 75p-
70d 3h 951p-5d 15h 158pL0d 3h 951p-0d 3h 951p-
84d 12h 747pL4d 12h 747p-4d 12h 747p-4d 12h 747pL
93d 10h 256p-1d 21h 543p-1d 21h 543pL3d 10h 256p-
100d 19h 52p-6d 6h 339pL0d 19h 52p-0d 19h 52p-
115d 3h 928pL5d 3h 928p-5d 3h 928p-5d 3h 928pL
124d 1h 437p-2d 12h 724p-2d 12h 724pL4d 1h 437p-
131d 10h 233p-6d 21h 520pL1d 10h 233p-1d 10h 233p-
145d 19h 29pL5d 19h 29p-5d 19h 29p-5d 19h 29pL
154d 16h 618p-3d 3h 905p-3d 3h 905pL4d 16h 618p-
162d 1h 414p-0d 12h 701pL2d 1h 414p-2d 1h 414pL
176d 10h 210pL6d 10h 210p-6d 10h 210pL0d 22h 1003p-
185d 7h 799p-3d 19h 6pL5d 7h 799p-5d 7h 799p-
192d 16h 595pL2d 16h 595p-2d 16h 595p-2d 16h 595pL
LSUM( 4 , r)LSUM( 5 , r)LSUM( 6 , r)LSUM( 7 , r)
14d 8h 876p-4d 8h 876pL5d 21h 589p-4d 8h 876pL
21d 17h 672pL3d 6h 385p-3d 6h 385pL3d 6h 385p-
30d 15h 181p-0d 15h 181pL2d 3h 974p-0d 15h 181p-
44d 23h 1057pL6d 12h 770p-6d 12h 770p-4d 23h 1057pL
53d 21h 566p-3d 21h 566p-3d 21h 566pL3d 21h 566p-
61d 6h 362p-1d 6h 362pL2d 19h 75p-1d 6h 362p-
75d 15h 158pL0d 3h 951p-0d 3h 951p-5d 15h 158pL
84d 12h 747p-4d 12h 747p-4d 12h 747pL4d 12h 747p-
91d 21h 543p-1d 21h 543pL3d 10h 256p-1d 21h 543p-
106d 6h 339pL0d 19h 52p-0d 19h 52p-6d 6h 339pL
115d 3h 928p-5d 3h 928p-5d 3h 928pL5d 3h 928p-
122d 12h 724p-2d 12h 724pL4d 1h 437p-2d 12h 724pL
136d 21h 520pL1d 10h 233p-1d 10h 233pL1d 10h 233p-
145d 19h 29p-5d 19h 29pL0d 7h 822p-5d 19h 29p-
153d 3h 905pL4d 16h 618p-4d 16h 618p-3d 3h 905pL
162d 1h 414p-2d 1h 414p-2d 1h 414pL2d 1h 414p-
176d 10h 210p-6d 10h 210pL0d 22h 1003p-6d 10h 210p-
183d 19h 6pL5d 7h 799p-5d 7h 799p-3d 19h 6pL
192d 16h 595p-2d 16h 595p-2d 16h 595pL2d 16h 595p-
LSUM( 8 , r)LSUM( 9 , r)LSUM(10 , r)LSUM(11 , r)
15d 21h 589p-4d 8h 876p-4d 8h 876pL5d 21h 589p-
23d 6h 385p-1d 17h 672pL3d 6h 385p-3d 6h 385p-
30d 15h 181pL0d 15h 181p-0d 15h 181p-0d 15h 181pL
46d 12h 770p-4d 23h 1057p-4d 23h 1057pL6d 12h 770p-
53d 21h 566p-2d 8h 853pL3d 21h 566p-3d 21h 566p-
61d 6h 362pL1d 6h 362p-1d 6h 362p-1d 6h 362pL
70d 3h 951p-5d 15h 158p-5d 15h 158pL0d 3h 951p-
84d 12h 747p-2d 23h 1034pL4d 12h 747p-4d 12h 747pL
91d 21h 543pL1d 21h 543p-1d 21h 543pL3d 10h 256p-
100d 19h 52p-6d 6h 339pL0d 19h 52p-0d 19h 52p-
115d 3h 928pL5d 3h 928p-5d 3h 928p-5d 3h 928pL
124d 1h 437p-2d 12h 724p-2d 12h 724pL4d 1h 437p-
131d 10h 233p-6d 21h 520pL1d 10h 233p-1d 10h 233p-
145d 19h 29pL5d 19h 29p-5d 19h 29p-5d 19h 29pL
154d 16h 618p-3d 3h 905p-3d 3h 905pL4d 16h 618p-
162d 1h 414p-0d 12h 701pL2d 1h 414p-2d 1h 414pL
176d 10h 210pL6d 10h 210p-6d 10h 210pL0d 22h 1003p-
185d 7h 799p-3d 19h 6pL5d 7h 799p-5d 7h 799p-
192d 16h 595pL2d 16h 595p-2d 16h 595p-2d 16h 595pL
LSUM(12 , r)LSUM(13 , r)LSUM(14 , r)LSUM(15 , r)
14d 8h 876p-4d 8h 876pL5d 21h 589p-4d 8h 876p-
21d 17h 672pL3d 6h 385p-3d 6h 385p-1d 17h 672pL
30d 15h 181p-0d 15h 181p-0d 15h 181pL0d 15h 181p-
44d 23h 1057p-4d 23h 1057pL6d 12h 770p-4d 23h 1057pL
52d 8h 853pL3d 21h 566p-3d 21h 566pL3d 21h 566p-
61d 6h 362p-1d 6h 362pL2d 19h 75p-1d 6h 362p-
75d 15h 158pL0d 3h 951p-0d 3h 951p-5d 15h 158pL
84d 12h 747p-4d 12h 747p-4d 12h 747pL4d 12h 747p-
91d 21h 543p-1d 21h 543pL3d 10h 256p-1d 21h 543p-
106d 6h 339pL0d 19h 52p-0d 19h 52p-6d 6h 339pL
115d 3h 928p-5d 3h 928p-5d 3h 928pL5d 3h 928p-
122d 12h 724p-2d 12h 724pL4d 1h 437p-2d 12h 724pL
136d 21h 520pL1d 10h 233p-1d 10h 233pL1d 10h 233p-
145d 19h 29p-5d 19h 29pL0d 7h 822p-5d 19h 29p-
153d 3h 905pL4d 16h 618p-4d 16h 618p-3d 3h 905pL
162d 1h 414p-2d 1h 414p-2d 1h 414pL2d 1h 414p-
176d 10h 210p-6d 10h 210pL0d 22h 1003p-6d 10h 210p-
183d 19h 6pL5d 7h 799p-5d 7h 799p-3d 19h 6pL
192d 16h 595p-2d 16h 595p-2d 16h 595pL2d 16h 595p-
LSUM(16 , r)LSUM(17 , r)LSUM(18 , r)LSUM(19 , r)
14d 8h 876pL5d 21h 589p-4d 8h 876pL5d 21h 589p-
23d 6h 385p-3d 6h 385pL3d 6h 385p-3d 6h 385p-
30d 15h 181pL2d 3h 974p-0d 15h 181p-0d 15h 181pL
46d 12h 770p-6d 12h 770p-4d 23h 1057pL6d 12h 770p-
53d 21h 566p-3d 21h 566pL3d 21h 566p-3d 21h 566p-
61d 6h 362pL2d 19h 75p-1d 6h 362p-1d 6h 362pL
70d 3h 951p-0d 3h 951p-5d 15h 158pL0d 3h 951p-
84d 12h 747p-4d 12h 747pL4d 12h 747p-4d 12h 747pL
91d 21h 543pL3d 10h 256p-1d 21h 543pL3d 10h 256p-
100d 19h 52p-0d 19h 52pL0d 19h 52p-0d 19h 52p-
115d 3h 928pL6d 16h 641p-5d 3h 928p-5d 3h 928pL
124d 1h 437p-4d 1h 437p-2d 12h 724pL4d 1h 437p-
131d 10h 233p-1d 10h 233pL1d 10h 233p-1d 10h 233p-
145d 19h 29pL0d 7h 822p-5d 19h 29p-5d 19h 29pL
154d 16h 618p-4d 16h 618p-3d 3h 905pL4d 16h 618p-
162d 1h 414p-2d 1h 414pL2d 1h 414p-2d 1h 414p-
176d 10h 210pL0d 22h 1003p-6d 10h 210p-6d 10h 210pL
185d 7h 799p-5d 7h 799p-3d 19h 6pL5d 7h 799p-
192d 16h 595p-2d 16h 595pL2d 16h 595p-2d 16h 595pL

Note that LSUM(0,r) is exactly the same as LSUM(19,r).

In the LSUM(i,r) tables, the leap years are indicated by a capital L immediately to the right of the molad value.

Some Useful Relationships

The following relationships have a particular usefulness in the analysis of Hebrew calendar calculation methods that do not imitate the instructions found in chapters 6-8 of Maimonides'
Hilchot Kiddush HaHodesh.

Relationship 1: L(i) = L(i + 19*j)

This relationship was derived above.

Relationship 2: C(i) = C(i + 19*j)

This relationship was derived above.

Relationship 3: M(i+k) = M(i) + LSUM(i,k)

This relationship was derived above.

Relationship 4: LSUM(i,0) = 0

```M(i+0)    = M(i) + LSUM(i,0) [Rel'n 3]
M(i)      = M(i) + LSUM(i,0)
LSUM(i,0) = 0
```
Relationship 5: LSUM(i+19*j,k) = LSUM(i,k)
```LSUM(i+19*j,k) = L(i+19*j)+L(i+1+19*j)+L(i+2+19*j) ... +L(i+k-1+19*j)
= L(i)+L(i+1)+L(i+2)+ ... +L(i+k-1) [Rel'n 1]
= LSUM(i,k)                         [Definition]
```
Example 5.0: Assume that the molad of Tishrei 5800H is to be calculated from a knowledge of the value of the molad of Tishrei 5783H, which is 2d 3h 6p.

M(5800) = M(5783 + 17) = M(5783) + LSUM(5783 , 17)

Since, 5800 = 5783+17, and the year 5783H has the arithmetical form 19*304 + 7,

```   M(5800) =    M(5783)   +  LSUM(7 , 17) [Rel'n 3 & 5]
=   (2d 3h 6p) + (6d 10h 210p)
==> 1d 13h 216p  [after the traditonal reduction by 7d].
```
Note that since LSUM(7 , 17) shows no leap year indicator, the year 5800H is a common year.

Relationship 6: LSUM(i,19*j) = (6939d 16h 595p)*j

Since L(i) = L(i+19*k) for any i and k, every sequence of 19 consecutive Hebrew years has the mean lunar length 6939d 16h 595p.

Therefore, LSUM(i,19*j) = (6939d 16h 595p)*j

Modulo algebra shows that if a is congruent to b modulo X, then a*k is congruent to
b*k modulo X.

If the calculations are to produce only a molad value, rather than the number of days in the calculated interval, then the 19 year excess value of 2d 16h 595p can be substituted since it is congruent to (6939d 16h 595p) modulo 7d.

Example 6.0: M(5758) can be calculated directly as follows

```     M(5758)   = M(1)         + LSUM(1, 5757)      [Rel'n 3]
= M(1)         + LSUM(1, 303*19)
==> M(1)         + (2d 16h 595p)*303  [Rel'n 6]
==> (2d 5h 204p) + (2d 22h 1005p)
=  5d 4h 129p  [after traditional reductions]
```
Relationship 7: M(i+19*j+r) = M(i)+(6939d 16h 595p)*j+LSUM(i,r)
```M(i+19*j+r) = M(i+19*j) + LSUM(i+19*j,r)                 [Rel'n 3]
= M(i)      + LSUM(i,19*j)       + LSUM(i,r) [Rel'n 3 & 5]
= M(i)      + (6939d 16h 595p)*j + LSUM(i,r) [Rel'n 6]
==> M(i)      + (   2d 16h 595p)*j + LSUM(i,r) [Trad'l Red'n]
```
Maimonides' Hilchot Kiddush HaHodesh, at 6:13, indicates that by adding 2d 16h 595p to the molad at the beginning of a 19 year cycle, the molad at the beginning of the next 19 year cycle is determined.

Relationship 7 shows that the addition of this value to any molad of Tishrei will determine the value of the molad of Tishrei exactly 19 years away.

Al-Biruni's The Chronology of Ancient Nations, in pages 144-147 of Sachau's English translation, shows a method for calculating the molad of Tishrei for any year x of the Aera Alexandri using M(3461) and LSUM(3,r), so as to produce a set of Tishrei moladot that demonstrate compliance to the cycle C(4), whose Hebrew letter acronym is stated to be gimel-bet-tet-bet-gimel.

Since 3461H represents the 12th year of the Aera Alexandri, then according to Al-Biruni's method it was necessary that

M(x) = M(3461+x-12) = M(3461) + LSUM(3,x-12)

Relationship 7 helped to confirm Al-Biruni's method of dividing x-12 by 19 to obtain both a multiplier m for 2d 16h 595p and a remainder r for use by LSUM(3,r). Relationship 7 was also very helpful in identifying the many descriptive errors in that part of the text.

Relationship 8: M(i) + LSUM(i,k) = M(k) + LSUM(k,i)

```Since,           M(i+k) = M(k+i)
M(i) + LSUM(i,k) = M(k) + LSUM(k,i) [Rel'n 3]
```

Example 8.0: Relationship 8 can be demonstrated using M(5759)
and the fact that M(5759) = M(1+5758) = M(5758+1).

```     M(5759) = M(1+5758)
= M(1)         +  LSUM(1      ,  5758        )     [Rel'n 3]
= M(1)         +  LSUM(1      ,  303*19 + 1  )
= M(1)         +  LSUM(1,1)   + (2d 16h 595p )*303 [Rel'n 7]
= (2d 5h 204p) + (4d 8h 876p) + (2d 22h 1005p)
=  2d 12h 1005p  [after traditional reductions]

M(5759) = M(5758+1)
= M(5758)      +  LSUM(5758,1)                     [Rel'n 3]
= M(5758)      +  LSUM(1+303*19,1)
= M(5758)      +  LSUM(1,1)                        [Rel'n 5]
= (5d 4h 129p) + (4d 8h 876p)
=  2d 12h 1005p  [after traditional reductions]
```
Relationship 9: M(j) = M(i) - LSUM( j , i-j)
```       M(i) = M( j + i-j)
= M( j ) + LSUM( j , i-j ) [Rel'n 3]
Hence, M(j) = M( i ) - LSUM( j , i-j )
```
Relationship 10: M(i+k) = M(i) - LSUM( j , i-j) + LSUM( j, k + i-j)
```M(i+k) = M(i+j+k-j)
= M(j+k+i-j)
= M(j)+LSUM(j,k+i-j)               [Rel'n 3]
= M(i)-LSUM(j,i-j  )+LSUM(j,k+i-j) [Rel'n 9 substitution of M(j)]
```
Relationship 10 shows how to use any known M(i) together with any arbitrarily chosen LSUM(j,r). Since M(i+k) can be calculated using an LSUM(j,r) that may be different than LSUM(i,r), LSUM(i, r) need not be developed for more than one value of i.

Relationship 10 does not prove that one and only one intercalation pattern exists for the Hebrew calendar since Relationship 10 is derived from that assumption.

Example 10.0: Given the molad of Tishrei 5760H relative to C(1) [GUChADZaT] how is the molad of Tishrei 5775H determined using the table for LSUM(0,r)?

```Since 5775 = 5760 + 15, and 5760 = 19*303 + 3,

M(5760+15) = M(5760) - LSUM(0,5760-0) + LSUM(0,15+5760-0)     [Rel'n 10]
= M(5760) - LSUM(0,19*303+3) + LSUM(0,15+19*303+3)
= M(5760) - LSUM(0, 3) - 303*(2d 16h 595p)
+ LSUM(0,18) + 303*(2d 16h 595p)         [Rel'n 7]
= M(5760) - LSUM(0, 3) + LSUM(0,18)
= (6d 21h 801p) - (0d 15h 181p) + (5d 7h 799p)
=  4d 14h 339p    [after the traditional reductions]
```
Relationship 11: M(i+k) = M(i) - LSUM(j,19*m+i-j) + LSUM(j,k+19*m+i-j)
```         M(i) - LSUM(j,19*m+i-j) + LSUM(j,k+19*m+i-j)
= M(i) - LSUM(j,i-j  ) - m*(6939d 16h 595p)
+ LSUM(j,k+i-j) + m*(6939d 16h 595p)     [Rel'n 7]
= M(i) - LSUM(j,i-j  ) + LSUM(j,k+i-j)
= M(i+k)                                        [Rel'n 10]

Therefore,

M(i+k) = M(i) - LSUM(j,19*m+i-j) + LSUM(j,k+19*m+i-j)
```
Relationship 11 shows that no matter how many multiples of 19 are added to i-j
in Relationship 10 the result will always be the same since the multiples cancel each other in the expression. Consequently, whenever i-j < 0 as many multiples of 19 may be added to i-j so as to make 19*m+i-j a positive (or zero) value.

Example 11.0: Given the molad of Tishrei 5760H relative to C(1) [GUChADZaT] how is the molad of Tishrei 5775H determined using the table corresponding to LSUM(5767,r)?

```Since 5775 = 5760 + 15, and 5767 = 19*303 + 10,

M(5775) = M(5760+15)
= M(5760) - LSUM(5767,5760-5767)
+ LSUM(5767,15+5760-5767)                  [Rel'n 10]
= M(5760) - LSUM(10,  -7)+LSUM(10, 15 -7)            [Rel'n  5]
= M(5760) - LSUM(10,19-7)+LSUM(10,15 +19-7)          [Rel'n 11]
= M(5760) - LSUM(10, 12 )+LSUM(10, 27 )
= M(5760) - LSUM(10, 12) +LSUM(10, 8) + (2d 16h 595p) [Rel'n 7]
= (6d 21h 801p) - (2d 12h 724p) + (4d 12h 747p) + (2d 16h 595p)
= 4d 14h 339p     [after the traditional reductions]
```
The Wolfgang Alexander Shocken Formula

The Shocken formula is presented on page 35 of Wolfgang Alexander Shocken's
The Calculated Confusion of Calendars, Vantage Press, Inc. (1976).

Shocken's formula INT((235 * A + 1) / 19) calculates the number of Hebrew months that have elapsed up to year A+1 since year 1H.

The values produced by that formula, after multiplication by the molad period constant
29d 12h 793p, form the elements of a series that is represented by LSUM(1,A).

Hence, the following equation can be established

LSUM(1,A) = INT((235 * A + 1) / 19)

Example S.0:Using Shocken's formula, calculate M(5775) from a knowledge of M(5758).

```M(5775) = M(5758+17)
= M(5758)      + LSUM(19*303+1, 17)
= M(5758)      + LSUM(1, 17)                       [Rel'n 5]
= M(5758)      + INT((235*17+1)/19)*(29d 12h 793p)
= (5d 4h 129p) + 210*(29d 12h 793p)
= (5d 4h 129p) + (6090d 2520h 166530p)
= 4d 14h 339p  [after the traditional reductions]
```
Relationship 10 can help to greatly reduce the size of the numbers which would otherwise result from the use of Shocken's formula.

Example S.1: Using Shocken's formula, calculate M(5775) from a knowledge of M(5770).

```M(5775) = M(5770) - LSUM(1, 5770 - 1)   + LSUM(1 ,5 + 5770 - 1) [Rel'n 10]
= M(5770) - LSUM(1,19*303+13-1) + LSUM(1,5+19*303+13-1)
= M(5770) - LSUM(1, 13 - 1)     + LSUM(1 , 5 + 13 - 1 ) [Rel'n 11]
= M(5770) - (INT((235*12 +1)/19) - INT((235*17 + 1)/19)) * (29d 12h 793p)
= (0d 16h 853p) - ( 148 - 210 ) * (29d 12h 793p)
= (0d 16h 853p) +  62*(29d 12h 793p)
= (0d 16h 853p) + (1798d 744h 49166p)
=  4d 14h 339p  [after the traditional reductions]
```
LSUM(1,12) and LSUM(1,17) can be found in the LSUM(1,r) table, thereby bypassing the use of Shocken's formula. However, bypassing the formula causes a loss of the day count between M(5770) and M(5775).

Conclusion

In light of today's electronic computers, the LSUM(i,r) tables are of limited use. However, the logical principles which establish their data will be required for quite some time into the future and continue to be useful in analysing the ancient Hebrew calendar.

The analysis produced an unexpected result.

The relationship

```     M(i+k) = M(i) + L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)
```
indicates that the Hebrew calendar is entirely constant once the equation is given these three initial conditions:-
```1. some value for year i
2. some value M(i) for its corresponding molad of Tishrei
3. some intercalation pattern C(i) beginning at year i
```
After setting these three initial conditions, the value of each and every Tishrei molad M(i) is fixed over the entire life of the Hebrew calendar.

It is not known today which combination of i, M(i), and C(i) was originally chosen by the ancient scholars of the Hebrew calendar. However, based on the foregoing analysis, it is very likely that their original values exist in the current set of Tishrei moladot defined by

```1. i = 1
2. M(1) = 2d 5h 204p
3. the intercalation pattern C(1) known as GUChADZaT
```
This combination of values did not appear in all of the ancient Hebrew calendar documents. It eventually became a current standard by at least the 12th century c.e., as evidenced from Maimonides' work Hilchot Kiddush HaHodesh.

Hence, the analysis confirmed that the Hebrew calendar, set into motion so many centuries ago, has endured steadfast against the intellectual storms of its ages. In this manner, the Hebrew calendar has remained one of the brightest beacons to the memory of the scholars who took part in its creation.

```First  Paged 30 Jan 2000