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

Hence, the lunar length of all common years is

12 * (29d 12h 793p) = 354d 8h 876pand the lunar length of all leap years is13 * (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

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 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

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

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

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

The 19 Leap Year Intercalation Cycles | |||
---|---|---|---|

C(i) | Leap Year Order | Relative Order | Epochal Molad |

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,

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

**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

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 * Historical*of the article

The article indicates that **4d 20h 408p**
(named ** daled-hof-tof-het**) is also an epochal molad.
This is a puzzling detail, since

**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

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

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+1toi+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)

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

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

Since every235 * (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]

Consequently, it is sometimes more convenient to compute

The following table shows the valuations forM(i+k) = M(i) + (6939d 16h 595p)*j + LSUM(i,r)in the reduced formM(i+k) = M(i) + (2d 16h 595p)*j + LSUM(i,r)

The LSUM(i,r) Tables
| ||||||||
---|---|---|---|---|---|---|---|---|

LSUM( 0 , r) | LSUM( 1 , r) | LSUM( 2 , r) | LSUM( 3 , r) | |||||

1 | 5d 21h 589p | - | 4d 8h 876p | - | 4d 8h 876p | L | 5d 21h 589p | - |

2 | 3d 6h 385p | - | 1d 17h 672p | L | 3d 6h 385p | - | 3d 6h 385p | - |

3 | 0d 15h 181p | L | 0d 15h 181p | - | 0d 15h 181p | - | 0d 15h 181p | L |

4 | 6d 12h 770p | - | 4d 23h 1057p | - | 4d 23h 1057p | L | 6d 12h 770p | - |

5 | 3d 21h 566p | - | 2d 8h 853p | L | 3d 21h 566p | - | 3d 21h 566p | L |

6 | 1d 6h 362p | L | 1d 6h 362p | - | 1d 6h 362p | L | 2d 19h 75p | - |

7 | 0d 3h 951p | - | 5d 15h 158p | L | 0d 3h 951p | - | 0d 3h 951p | - |

8 | 4d 12h 747p | L | 4d 12h 747p | - | 4d 12h 747p | - | 4d 12h 747p | L |

9 | 3d 10h 256p | - | 1d 21h 543p | - | 1d 21h 543p | L | 3d 10h 256p | - |

10 | 0d 19h 52p | - | 6d 6h 339p | L | 0d 19h 52p | - | 0d 19h 52p | - |

11 | 5d 3h 928p | L | 5d 3h 928p | - | 5d 3h 928p | - | 5d 3h 928p | L |

12 | 4d 1h 437p | - | 2d 12h 724p | - | 2d 12h 724p | L | 4d 1h 437p | - |

13 | 1d 10h 233p | - | 6d 21h 520p | L | 1d 10h 233p | - | 1d 10h 233p | - |

14 | 5d 19h 29p | L | 5d 19h 29p | - | 5d 19h 29p | - | 5d 19h 29p | L |

15 | 4d 16h 618p | - | 3d 3h 905p | - | 3d 3h 905p | L | 4d 16h 618p | - |

16 | 2d 1h 414p | - | 0d 12h 701p | L | 2d 1h 414p | - | 2d 1h 414p | L |

17 | 6d 10h 210p | L | 6d 10h 210p | - | 6d 10h 210p | L | 0d 22h 1003p | - |

18 | 5d 7h 799p | - | 3d 19h 6p | L | 5d 7h 799p | - | 5d 7h 799p | - |

19 | 2d 16h 595p | L | 2d 16h 595p | - | 2d 16h 595p | - | 2d 16h 595p | L |

LSUM( 4 , r) | LSUM( 5 , r) | LSUM( 6 , r) | LSUM( 7 , r) | |||||

1 | 4d 8h 876p | - | 4d 8h 876p | L | 5d 21h 589p | - | 4d 8h 876p | L |

2 | 1d 17h 672p | L | 3d 6h 385p | - | 3d 6h 385p | L | 3d 6h 385p | - |

3 | 0d 15h 181p | - | 0d 15h 181p | L | 2d 3h 974p | - | 0d 15h 181p | - |

4 | 4d 23h 1057p | L | 6d 12h 770p | - | 6d 12h 770p | - | 4d 23h 1057p | L |

5 | 3d 21h 566p | - | 3d 21h 566p | - | 3d 21h 566p | L | 3d 21h 566p | - |

6 | 1d 6h 362p | - | 1d 6h 362p | L | 2d 19h 75p | - | 1d 6h 362p | - |

7 | 5d 15h 158p | L | 0d 3h 951p | - | 0d 3h 951p | - | 5d 15h 158p | L |

8 | 4d 12h 747p | - | 4d 12h 747p | - | 4d 12h 747p | L | 4d 12h 747p | - |

9 | 1d 21h 543p | - | 1d 21h 543p | L | 3d 10h 256p | - | 1d 21h 543p | - |

10 | 6d 6h 339p | L | 0d 19h 52p | - | 0d 19h 52p | - | 6d 6h 339p | L |

11 | 5d 3h 928p | - | 5d 3h 928p | - | 5d 3h 928p | L | 5d 3h 928p | - |

12 | 2d 12h 724p | - | 2d 12h 724p | L | 4d 1h 437p | - | 2d 12h 724p | L |

13 | 6d 21h 520p | L | 1d 10h 233p | - | 1d 10h 233p | L | 1d 10h 233p | - |

14 | 5d 19h 29p | - | 5d 19h 29p | L | 0d 7h 822p | - | 5d 19h 29p | - |

15 | 3d 3h 905p | L | 4d 16h 618p | - | 4d 16h 618p | - | 3d 3h 905p | L |

16 | 2d 1h 414p | - | 2d 1h 414p | - | 2d 1h 414p | L | 2d 1h 414p | - |

17 | 6d 10h 210p | - | 6d 10h 210p | L | 0d 22h 1003p | - | 6d 10h 210p | - |

18 | 3d 19h 6p | L | 5d 7h 799p | - | 5d 7h 799p | - | 3d 19h 6p | L |

19 | 2d 16h 595p | - | 2d 16h 595p | - | 2d 16h 595p | L | 2d 16h 595p | - |

LSUM( 8 , r) | LSUM( 9 , r) | LSUM(10 , r) | LSUM(11 , r) | |||||

1 | 5d 21h 589p | - | 4d 8h 876p | - | 4d 8h 876p | L | 5d 21h 589p | - |

2 | 3d 6h 385p | - | 1d 17h 672p | L | 3d 6h 385p | - | 3d 6h 385p | - |

3 | 0d 15h 181p | L | 0d 15h 181p | - | 0d 15h 181p | - | 0d 15h 181p | L |

4 | 6d 12h 770p | - | 4d 23h 1057p | - | 4d 23h 1057p | L | 6d 12h 770p | - |

5 | 3d 21h 566p | - | 2d 8h 853p | L | 3d 21h 566p | - | 3d 21h 566p | - |

6 | 1d 6h 362p | L | 1d 6h 362p | - | 1d 6h 362p | - | 1d 6h 362p | L |

7 | 0d 3h 951p | - | 5d 15h 158p | - | 5d 15h 158p | L | 0d 3h 951p | - |

8 | 4d 12h 747p | - | 2d 23h 1034p | L | 4d 12h 747p | - | 4d 12h 747p | L |

9 | 1d 21h 543p | L | 1d 21h 543p | - | 1d 21h 543p | L | 3d 10h 256p | - |

10 | 0d 19h 52p | - | 6d 6h 339p | L | 0d 19h 52p | - | 0d 19h 52p | - |

11 | 5d 3h 928p | L | 5d 3h 928p | - | 5d 3h 928p | - | 5d 3h 928p | L |

12 | 4d 1h 437p | - | 2d 12h 724p | - | 2d 12h 724p | L | 4d 1h 437p | - |

13 | 1d 10h 233p | - | 6d 21h 520p | L | 1d 10h 233p | - | 1d 10h 233p | - |

14 | 5d 19h 29p | L | 5d 19h 29p | - | 5d 19h 29p | - | 5d 19h 29p | L |

15 | 4d 16h 618p | - | 3d 3h 905p | - | 3d 3h 905p | L | 4d 16h 618p | - |

16 | 2d 1h 414p | - | 0d 12h 701p | L | 2d 1h 414p | - | 2d 1h 414p | L |

17 | 6d 10h 210p | L | 6d 10h 210p | - | 6d 10h 210p | L | 0d 22h 1003p | - |

18 | 5d 7h 799p | - | 3d 19h 6p | L | 5d 7h 799p | - | 5d 7h 799p | - |

19 | 2d 16h 595p | L | 2d 16h 595p | - | 2d 16h 595p | - | 2d 16h 595p | L |

LSUM(12 , r) | LSUM(13 , r) | LSUM(14 , r) | LSUM(15 , r) | |||||

1 | 4d 8h 876p | - | 4d 8h 876p | L | 5d 21h 589p | - | 4d 8h 876p | - |

2 | 1d 17h 672p | L | 3d 6h 385p | - | 3d 6h 385p | - | 1d 17h 672p | L |

3 | 0d 15h 181p | - | 0d 15h 181p | - | 0d 15h 181p | L | 0d 15h 181p | - |

4 | 4d 23h 1057p | - | 4d 23h 1057p | L | 6d 12h 770p | - | 4d 23h 1057p | L |

5 | 2d 8h 853p | L | 3d 21h 566p | - | 3d 21h 566p | L | 3d 21h 566p | - |

6 | 1d 6h 362p | - | 1d 6h 362p | L | 2d 19h 75p | - | 1d 6h 362p | - |

7 | 5d 15h 158p | L | 0d 3h 951p | - | 0d 3h 951p | - | 5d 15h 158p | L |

8 | 4d 12h 747p | - | 4d 12h 747p | - | 4d 12h 747p | L | 4d 12h 747p | - |

9 | 1d 21h 543p | - | 1d 21h 543p | L | 3d 10h 256p | - | 1d 21h 543p | - |

10 | 6d 6h 339p | L | 0d 19h 52p | - | 0d 19h 52p | - | 6d 6h 339p | L |

11 | 5d 3h 928p | - | 5d 3h 928p | - | 5d 3h 928p | L | 5d 3h 928p | - |

12 | 2d 12h 724p | - | 2d 12h 724p | L | 4d 1h 437p | - | 2d 12h 724p | L |

13 | 6d 21h 520p | L | 1d 10h 233p | - | 1d 10h 233p | L | 1d 10h 233p | - |

14 | 5d 19h 29p | - | 5d 19h 29p | L | 0d 7h 822p | - | 5d 19h 29p | - |

15 | 3d 3h 905p | L | 4d 16h 618p | - | 4d 16h 618p | - | 3d 3h 905p | L |

16 | 2d 1h 414p | - | 2d 1h 414p | - | 2d 1h 414p | L | 2d 1h 414p | - |

17 | 6d 10h 210p | - | 6d 10h 210p | L | 0d 22h 1003p | - | 6d 10h 210p | - |

18 | 3d 19h 6p | L | 5d 7h 799p | - | 5d 7h 799p | - | 3d 19h 6p | L |

19 | 2d 16h 595p | - | 2d 16h 595p | - | 2d 16h 595p | L | 2d 16h 595p | - |

LSUM(16 , r) | LSUM(17 , r) | LSUM(18 , r) | LSUM(19 , r) | |||||

1 | 4d 8h 876p | L | 5d 21h 589p | - | 4d 8h 876p | L | 5d 21h 589p | - |

2 | 3d 6h 385p | - | 3d 6h 385p | L | 3d 6h 385p | - | 3d 6h 385p | - |

3 | 0d 15h 181p | L | 2d 3h 974p | - | 0d 15h 181p | - | 0d 15h 181p | L |

4 | 6d 12h 770p | - | 6d 12h 770p | - | 4d 23h 1057p | L | 6d 12h 770p | - |

5 | 3d 21h 566p | - | 3d 21h 566p | L | 3d 21h 566p | - | 3d 21h 566p | - |

6 | 1d 6h 362p | L | 2d 19h 75p | - | 1d 6h 362p | - | 1d 6h 362p | L |

7 | 0d 3h 951p | - | 0d 3h 951p | - | 5d 15h 158p | L | 0d 3h 951p | - |

8 | 4d 12h 747p | - | 4d 12h 747p | L | 4d 12h 747p | - | 4d 12h 747p | L |

9 | 1d 21h 543p | L | 3d 10h 256p | - | 1d 21h 543p | L | 3d 10h 256p | - |

10 | 0d 19h 52p | - | 0d 19h 52p | L | 0d 19h 52p | - | 0d 19h 52p | - |

11 | 5d 3h 928p | L | 6d 16h 641p | - | 5d 3h 928p | - | 5d 3h 928p | L |

12 | 4d 1h 437p | - | 4d 1h 437p | - | 2d 12h 724p | L | 4d 1h 437p | - |

13 | 1d 10h 233p | - | 1d 10h 233p | L | 1d 10h 233p | - | 1d 10h 233p | - |

14 | 5d 19h 29p | L | 0d 7h 822p | - | 5d 19h 29p | - | 5d 19h 29p | L |

15 | 4d 16h 618p | - | 4d 16h 618p | - | 3d 3h 905p | L | 4d 16h 618p | - |

16 | 2d 1h 414p | - | 2d 1h 414p | L | 2d 1h 414p | - | 2d 1h 414p | - |

17 | 6d 10h 210p | L | 0d 22h 1003p | - | 6d 10h 210p | - | 6d 10h 210p | L |

18 | 5d 7h 799p | - | 5d 7h 799p | - | 3d 19h 6p | L | 5d 7h 799p | - |

19 | 2d 16h 595p | - | 2d 16h 595p | L | 2d 16h 595p | - | 2d 16h 595p | L |

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.

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

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]

**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**,

Note that sinceM(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].

__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]

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]

**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

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

**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]

M(i) = M( j + i-j) = M( j ) + LSUM( j , i-j ) [Rel'n 3] Hence, M(j) = M( i ) - LSUM( j , 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** 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

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]

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)

in

**Example 11.0:** Given the molad of Tishrei

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 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

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

**Example S.0:**Using Shocken's formula, calculate

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]

**Example S.1:** Using Shocken's formula, calculate

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]

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

indicates that the Hebrew calendar is entirelyM(i+k) = M(i) + L(i) + L(i+1) + L(i+2) + ... + L(i+k-1)

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

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

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' work1. i = 1 2. M(1) = 2d 5h 204p 3. the intercalation pattern C(1) known as GUChADZaT

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 Next Revised 23 Jan 2002