It is a well known fact that the **12 month year** is a few days shorter than the mean tropical solar year. Also, the **13 month year** is a few days longer than the mean tropical solar year. The exact amounts by which these years are either shorter or longer than the mean tropical solar year have no particular relevance to the determination of the eventual ** leap year** distribution in the

Of major importance is the fact that **235 lunar months** consisting of

And of even greater importance in this derivation, is the fact that at some point in the calendar's history, it was decided to begin each lunar year less than **one lunar month** from its corresponding solar year, and no earlier than that **solar year**.

These considerations define a very simple algorithmic process for distributing the leap years within the ** mahzor qatan**.

Let S = the length of a mean solar year Let m = the length of a mean lunar month By assumption, 235 * m = 19 * S Let dc = the difference between a 12 month lunar year and a mean solar year = 12 * m - S Let de = the difference between a 13 month lunar year and a mean solar year = 13 * m - S Let dt = the difference of time between the start of a lunar year and its corresponding mean solar year in a 19 year cycle. Then, de - dc = 13 * m - S - (12 * m - S) = m In 19 years, 12 * dc + 7 * de = 0 because, 12 * dc = 12 * (12 * m - S) 7 * de = 7 * (13 * m - S) 12 * dc + 7 * de = 12 * (12 * m - S) + 7 * (13 * m - S) = 235 * m - 19 * S = 0 (since 235 * m = 19 * S by definition) Since, de - dc = m de = m + dc Substituting for de, 12 * dc + 7 * (m + dc) = 0 19 * dc = -7 * m dc = -7 * m / 19 Similarly, substituting for dc 12 * (de - m) + 7 * de = 0 19 * de = 12 * m de = 12 * m / 19

Let the **first lunar year = 0** begin at the same time as the **first solar year** in the **19 year cycle**. Hence, the difference in start times, **dt**, between these two years is ** zero**.

If that **first lunar year** is a **twelve month year**, then the **second lunar year** will start **7 * m / 19** of a **month** earlier than the **second solar year** of the cycle.

Consequently, the **first lunar year** of the cycle must be a **13 month year**, so that the **second lunar year** can start

later than the **second solar year** by **12 * m / 19** of a month. This start time is acceptable because it is less than **one lunar month**.

If the **second lunar year** were **13 months** long, then the **third lunar year** would begin later than the **third solar year by ****12 * m /19 + 12 * m / 19 = 24 * m / 19** of a **lunar month**. Since this timing would be greater than **one lunar month** it would be unacceptable.

Hence, the **second lunar year** must be a **12 month year**. This length will allow the **third lunar year** to begin later than the **third solar year**
by **12 * m / 19 - 7 * m /19 = 5 * m / 19** of a **month**.

For reasons as stated above, the **third lunar year** must be a **13 month year**. Otherwise the **fourth lunar year** would begin **2 * m / 19** of a **month** earlier than the **fourth solar year**.

The remainder of the years are distributed in the same manner and for the same reasons.

The effects of the above algorithmic approach can be shown as follows shown in **Table I**.

The **13 month years** are indicated by the letter '**L**'.

The Primary Leap Year Distribution Cycle | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

dt*19/m | 0 | 12 | 5 | 17 | 10 | 3 | 15 | 8 | 1 | 13 | 6 | 18 | 11 | 4 | 16 | 9 | 2 | 14 | 7 | 0 |

Type | L |
- | L | - | - | L | - | - | L | - | L | - | - | L | - | - | L | - | - | L |

**Lunar year 0** of the cycle begins at the same time as **solar year 0** since the difference of time between these two years is **0**.

**Table I** shows rather clearly that ** leap years** are associated only with values that are

Therefore, the leap years in **Table I** are years ** 2, 5, 8, 10, 13, 16, and 19** of the

As shown in **Cycles and Moladot**, this pattern corresponds to **C(18)**. Consequently, the cycle ** GUChADZaT** is generated simply by adding

The GUChADZaT Leap Year Distribution Cycle |
||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | 17 | 18 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |

dt*19/m | 0 | 12 | 5 | 17 | 10 | 3 | 15 | 8 | 1 | 13 | 6 | 18 | 11 | 4 | 16 | 9 | 2 | 14 | 7 | 0 |

Type | L |
- | L | - | - | L | - | - | L | - | L | - | - | L | - | - | L | - | - | L |

**Table II** shows once again that ** leap years** are associated only with values that are

**The Gauss Pesach Formula** indicates that given any Hebrew year

A rather strange subtlety of the Gauss formula, is that the need to test for ** Tuesday** postponements is determined by the same formula, using a limit of

On pages **140-141** of his book **A Glimpse of Light**, (published Targum Press, Inc. 1998),

Both of these formulas and their results are correct and different. This seeming paradox is explained below.

LetY =any number of Hebrew years. Letcy =the number of 12 month years inYHebrew years Letey =the number of 13 month years inYHebrew years Then, Y = cy + ey Hence, dt = cy * dc + ey * de = cy * (-7 * m / 19) + ey * (12 * m / 19) Substituting Y - ey for cy = [(Y - ey) * -7 + ey * 12] * m / 19= (-7 * Y + 19 * ey) * m / 19Substituting Y - cy for ey = [cy * -7 + (Y - cy) * 12] * m / 19= (12 * Y - 19 * cy) * m / 19

Then, based on the above considerations, **Y** is a Hebrew ** leap year** whenever

As shown in **Cycles and Moladot**, the leap year distribution pattern of **Table I** corresponds to **C(18)**. Consequently, the cycle ** GUChADZaT** is generated simply by adding

Hence, Y + 17 = H Y = H - 17 12 * Y = 12 * H - 12 * 17 12 * Y = 12 * H - 204 12 * Y MOD 19 = 12 * H MOD 19 - 204 MOD 19 = (12 * H + 5) MOD 19 [since -204 = -19*11 + 5]

Thus, **H** is a Hebrew ** leap year** whenever

Given some Hebrew year **H**, the Gauss *Pesach* Formula indicates that whenever **6 <= (12 * H + 17) MOD 19 <= 18** it is necessary to test for the possibility of a ** Tuesday** postponement,

and, that whenever **12 <= (12 * H + 17) MOD 19 <= 18** it is necessary to test for the possibility of a ** Monday** postponement.

The Gauss *Pesach* Formula details the calculations in ** QBASIC** format.

*Pesach* for any Hebrew year **H** is always **163 days** before ** Rosh Hashannah H + 1**. Hence, in order to establish the

The ** Tuesday** postponement test is invoked whenever a given Hebrew year is a

As shown above, a Hebrew year is a **12 month year** whenever **7 <= (12 * H + 5) MOD 19 <= 18**.

Consequently, whenever **7 <= (12 * (H + 1)+ 5) MOD 19 = (12 * H + 17) MOD 19 <= 18** then Hebrew year **H + 1** is a **12 month year** .

Therefore, in The Gauss *Pesach* Formula, the ** Tuesday** postponement test is invoked whenever

*Pesach* for any Hebrew year **H** is always **163 days** before ** Rosh Hashannah H + 1**. Hence, in order to establish the

The ** Monday** postponement test is invoked whenever Hebrew year

As shown above, Hebrew year **H** is a **13 month year** whenever **0 <= (12 * H + 5) MOD 19 <= 6**.

It is possible to maintain the inequality by adding **12** to all expressions in the inequality.

Hence, **0 + 12 <= (12 * H + 5 + 12) MOD 19 <= 6 + 12**, implying that

**12 <= (12 * H + 17) MOD 19 <= 18** whenever Hebrew year **H** is a **13 month year**.

Therefore, in The Gauss *Pesach* Formula, the ** Monday** postponement test is invoked

whenever

The Gauss approach to the last two *dehiyot* condensed their respective tests to **one formula**, namely, **(12 * H + 17) MOD 19 **, using only **one variable**, namely, the Hebrew year **H**, in which the desired *Pesach* was to be found. And this type of reduction, by no stretch of the imagination, represented only a small fraction of his pure mathematical genius!

When year **H**, in ** GUChADZaT**, is a

Therefore, the ** Tuesday** postponement test is invoked whenever

When the previous year **H - 1** is a **13 month** year, then **0 < 12 * (H - 1) + 5 < 6**.

Adding **12** to all parts of the inequality, the expression becomes
**12 < 12 * H + 5 < 18**.

Consequently, the ** Monday** postponement test must be invoked whenever

It appears that Slonimsky reversed the additions and subtractions which led to the leap year determinations.

In the absence of any particular guidance, it is possible to construct the leap year distribution algorithm as follows.

Let S = the length of a mean solar year Let m = the length of a mean lunar month Let dc = the difference between a mean solar year and a 12 month lunar year = S - 12 * m Let de = the difference between a mean solar year and a 13 month lunar year = S - 13 * m Let dt = the difference of time between the start of a mean solar year and its corresponding lunar year in a 19 year cycle. Then, de - dc = S - 13 * m - (S - 12 * m) = -m In 19 years, 12 * dc + 7 * de = 0 because, 12 * dc = 12 * (S - 12 * m) 7 * de = 7 * (S - 13 * m) 12 * dc + 7 * de = 12 * (S - 12 * m) + 7 * (S -13 * m) = 19 * S - 235 * m = 0 (since 235 * m = 19 * S by definition) Since, de - dc = -m de = dc - m Substituting for de, 12 * dc + 7 * (dc - m) = 0 19 * dc = 7 * m dc = 7 * m / 19 Similarly, substituting for dc 12 * (de + m) + 7 * de = 0 19 * de = -12 * m de = -12 * m / 19

It is to be noted that in this approach, the ** leap years** are subtractive and the

Since our assumption is that the lunar years are to begin no earlier than their corresponding solar years, and less than one month later than their corresponding solar years in the **19 year cycle**, it is possible to determine the deviations of the lunar years from their corresponding solar years in the cycle as follows.

Let **lunar year 0** begin at the same time as **solar year 0** in the **19 year cycle**. Hence, the difference in start times, **dt**, between these two years is ** zero**.

If **lunar year 0** is a **13 month year**, then the **lunar year 1** will start **12 * m / 19** of a **month** earlier than the **solar year 1**.

Consequently, the **lunar year 0** of the cycle must be a **12 month year**, so that the **lunar year 1** can start later than the **solar year 1** by **7 * m / 19** of a month. This start time is acceptable because it is also less than **one lunar month**.

If **lunar year 1** were **13 months** long, then **lunar year 2** would begin earlier than **solar year 2** by **12 * m /19 - 7 * m / 19 = 5 * m / 19** of a **lunar month**. Since this timing would be unacceptable **lunar year 1** must be **12 months** long.

Hence, **lunar year 2** begins later than **solar year 2**
by **7 * m / 19 + 7 * m /19 = 14 * m / 19** of a **month**.

For reasons as stated above, the **lunar year 2** must be a **13 month year**. Otherwise **lunar year 3** would begin **21 * m / 19** of a **month** later than **solar year 3**.

When **lunar year 2** is **13 months** long, then **lunar year 3** begins
**14 * m / 19 - 12 * m /19 = 2 * m / 19** of a **month** later than **solar year 3**.

The remainder of the years are distributed in the same manner and for the same reasons.

The effects of the above algorithmic approach can be shown as follows shown in **Table III**.

The Slonimsky Primary Leap Year Distribution Cycle |
||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

dt*19/m | 0 | 7 | 14 | 2 | 9 | 16 | 4 | 11 | 18 | 6 | 13 | 1 | 8 | 15 | 3 | 10 | 17 | 5 | 12 | 0 |

Type | - |
- | L | - | - | L | - | - | L | - | L | - | - | L | - | - | L | - | L |
- |

**Lunar year 0** of the cycle begins at the same time as **solar year 0** since the difference of time between these two years is **0**.

**Table III** shows rather clearly that ** leap years** are associated only with values that are

Therefore, the leap years in **Table III** are years ** 2, 5, 8, 10, 13, 16, and 18** of the

As shown in **Cycles and Moladot**, this pattern corresponds to **C(10)**. Consequently, the cycle ** GUChADZaT** is generated simply by adding

The Slonimsky GUChADZaT Distribution Cycle |
||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

dt*19/m | 0 | 7 | 14 | 2 | 9 | 16 | 4 | 11 | 18 | 6 | 13 | 1 | 8 | 15 | 3 | 10 | 17 | 5 | 12 | 0 |

Type | - |
- | L | - | - | L | - | - | L | - | L | - | - | L | - | - | L | - | L |
- |

**Table IV** shows once again that ** leap years** are associated only with values that are

Subtracting from **18**, any particular value of **dt * 19 / m** in **Table II**, develops the value of the corresponding **dt * 19 / m** in **Table IV**.

LetY =any number of Hebrew years. Letcy =the number of 12 month years inYHebrew years Letey =the number of 13 month years inYHebrew years Then, Y = cy + ey Hence, dt = cy * dc + ey * de = cy * (7 * m / 19) + ey * (-12 * m / 19) Substituting Y - ey for cy = [(Y - ey) * 7 + ey * -12] * m / 19= (7 * Y - 19 * ey) * m / 19Substituting Y - cy for ey = [cy * 7 + (Y - cy) * -12] * m / 19= (-12 * Y + 19 * cy) * m / 19

Then, based on the above considerations, **Y** is a Hebrew ** leap year** whenever

The Slonimsky leap year distribution aligns itself with ** GUChADZaT** by adding

Y + 9 = H Y = H - 9 7 * Y = 7 * H - 7 * 9 7 * Y = 7 * H - 63 7 * Y MOD 19 = 7 * H MOD 19 - 63 MOD 19 = (7 * H - 6) MOD 19 [since -63 = -19*3 - 6]

Therefore, whenever **(7 * H - 6) MOD 19 > 11**, **H** is a Hebrew ** leap year**. The results are shown in

Slonimsky's formula could also be used in the Gauss *Pesach* formula.

If **(7 * H + 1) MOD 19 < 7** then the ** Monday** postponement test is implied.

If **(7 * H + 1) MOD 19 < 12** then the ** Tuesday** postponement test is implied.

The logic for this formulation is very similar to the logic used for the Gauss formulation.

When year **H**, in ** GUChADZaT**, is a

Therefore, the ** Tuesday** postponement test is invoked whenever

When the previous year **H - 1** is a **13 month** year, then **18 > (7 * (H - 1) - 6) MOD 19 > 12**.

Adding **7** to all parts of the inequality, the expression, in **MOD 19** arithmetic, becomes **6 > (7 * H - 6) MOD 19 > 0**.

Consequently, the ** Monday** postponement test must be invoked whenever

Mapped against the **Gregorian and the Julian** calendars, it appears that the earliest possible starts for any ** Rosh Hashannah** will occur for the

**Table II, The GUChADZaT Leap Year Distribution Cycle**, shows that

Consequently, it seems reasonable to conclude that the **earliest** possible start dates for any

However, that demonstration is not a proof, since **TABLE IV, The Slonimsky GUChADZaT Distribution Cycle**, is a reversal of the results shown in

It is possible to note that because the **Gregorian** calendar is **faster** than the Hebrew calendar, the first occurence of a new

Similarly, it is possible to note that because the **Julian** calendar is **slower** than the Hebrew calendar, the first occurence of a new

Deriving the distribution of the ** leap years** did not require knowledge of a starting point in any solar year. Consequently, the actual initial solar year in the cycle, as well as the first moment of the lunar year in that year, could be selected arbitrarily.

Nor did the leap year distributions depend on either the actual timing of the lunar month or the actual timing of the solar year. The only relationship needed to determine the ** leap year** distributions was the number of complete lunar months which equalled a number of complete solar years. In our situation, it was assumed that the time of any period of

On the basis of the above analysis, it appears that the fixed Hebrew calendar's ** leap year** intercalation cycle did not stem from

First Paged 8 Oct 2002 Next Revised 21 Dec 2012