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

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 19 year cycle.

Of major importance is the fact that 235 lunar months consisting of twelve 12 month lunar years and seven 13 month lunar years are traditonally accepted as equalling 19 mean tropical solar years.

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'.

TABLE I
The Primary Leap Year Distribution Cycle
Year 0 1 2 3 4 5 6 7 8 91011 12131415 16171819
dt*19/m 0 125 17 103 15 8 1 136 18 114 16 9 2 14 70
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 0 < dt * 19 / m < 6. That result is a natural consequence of the rules used to develop the values of dt.

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

As shown in Cycles and Moladot, this pattern corresponds to C(18). Consequently, the cycle GUChADZaT is generated simply by adding 17 (or -2) to each year and applying MOD 19 to the resulting value. The result of that arithmetic is shown in Table II.

TABLE II
The GUChADZaT Leap Year Distribution Cycle
Year 1718 0 1 2 3 4 5 6 7 8 91011 12131415 1617
dt*19/m 0 125 17 103 15 8 1 136 18 114 16 9 2 14 70
Type L -L - -L - -L - L - - L - - L - - L

Table II shows once again that leap years are associated only with values that are 0 < dt * 19 / m < 6.

Some Leap Year Formulas

The Gauss Pesach Formula indicates that given any Hebrew year H, if the value of (12*H + 17) MOD 19 > 11, then H is a leap year, and therefore, it is required to test for the Monday postponement rule.

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 6 instead of 11.

On pages 140-141 of his book A Glimpse of Light, (published Targum Press, Inc. 1998), Dr. Julian Schamroth suggested that Hayim Zelig Slonimsky indicated in 1852 that Hebrew year H would be a leap year whenever
(7*H - 6) MOD 19 > 11.

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

The Gauss Method

```Let  Y = any number of Hebrew years.
Let cy = the number of 12 month years in Y Hebrew years
Let ey = the number of 13 month years in Y Hebrew 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 / 19

Substituting 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 0 < 12 * Y MOD 19 < 6. The results are shown in TABLE I.

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 17 (or -2) to each year and applying MOD 19 to the resulting value.

```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 0 <= (12 * H + 5) MOD 19 <= 6. The results are shown in TABLE II.

Deriving the Gauss Leap Year Formula

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.

The Gauss Tuesday Postponement Test

Pesach for any Hebrew year H is always 163 days before Rosh Hashannah H + 1. Hence, in order to establish the Pesach date for Hebrew year H, it is necessary to determine the Tuesday postponement for Hebrew year H + 1.

The Tuesday postponement test is invoked whenever a given Hebrew year is a 12 month year.

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
(12 * H + 17) MOD 19 > 6, since the expression has a maximum value of 18.

The Gauss Monday Postponement Test

Pesach for any Hebrew year H is always 163 days before Rosh Hashannah H + 1. Hence, in order to establish the Pesach date for Hebrew year H, it is necessary to determine the Monday postponement for Hebrew year H + 1.

The Monday postponement test is invoked whenever Hebrew year H + 1 follows a 13 month year. Clearly, for that to happen, it is necessary that Hebrew year H be a 13 month 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 (12 * H + 17) MOD 19 > 11.

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!

The Gauss Monday and Tuesday Postponement Tests For Rosh Hashannah

When year H, in GUChADZaT, is a 12 month year, then 12 * H + 5 > 6.

Therefore, the Tuesday postponement test is invoked whenever 12 * H + 5 > 6.

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 12 * H + 5 > 11

The Slonimsky Method

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

Deriving the Slonimsky Formula For Leap Years

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 12 month years are additive.

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.

TABLE III
The Slonimsky Primary Leap Year Distribution Cycle
Year 0 1 2 3 4 5 6 7 8 91011 12131415 16171819
dt*19/m 0 714 2 916 4 11 18 613 1 815 3 10 17 512 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 12 <= dt * 19 / m <= 18. That result is a natural consequence of the rules used to develop the values of dt.

Therefore, the leap years in Table III are years 2, 5, 8, 10, 13, 16, and 18 of the 19 year cycle.

As shown in Cycles and Moladot, this pattern corresponds to C(10). Consequently, the cycle GUChADZaT is generated simply by adding 9 (or -10) to each year and applying MOD 19 to the resulting value. The result of that arithmetic is shown in Table IV.

TABLE IV
The Slonimsky GUChADZaT Distribution Cycle
Year 91011 12131415 161718 0 1 2 3 4 5 6 7 89
dt*19/m 0 714 2 916 4 11 18 613 1 815 3 10 17 512 0
Type - -L - -L - -L - L - - L - - L -L -

Table IV shows once again that leap years are associated only with values that are 12 < dt * 19 / m < 18.

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.

The Slonimsky Formula

```Let  Y = any number of Hebrew years.
Let cy = the number of 12 month years in Y Hebrew years
Let ey = the number of 13 month years in Y Hebrew 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 / 19

Substituting 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 12 < 7 * Y MOD 19 < 18. The results are shown in TABLE III.

The Slonimsky leap year distribution aligns itself with GUChADZaT by adding 9 to the year count. Hence,

```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 TABLE IV.

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.

The Slonimsky Monday and Tuesday Postponement Tests For Rosh Hashannah

When year H, in GUChADZaT, is a 12 month year, then (7 * H - 6) MOD 19 < 12.

Therefore, the Tuesday postponement test is invoked whenever (7 * H - 6) MOD 19 < 12.

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 (7 * H - 6) MOD 19 < 7

The Earliest and Latest Possible Hebrew Year Starts

Mapped against the Gregorian and the Julian calendars, it appears that the earliest possible starts for any Rosh Hashannah will occur for the 17th year of the mahzor qatan, and that the latest possible starts will occur for the 9th year of the mahzor qatan.

Table II, The GUChADZaT Leap Year Distribution Cycle, shows that dt * 19 / m = 0 when Year MOD 19 = 17, and that dt * 19 / m = 18 when Year MOD 19 = 9.

Consequently, it seems reasonable to conclude that the earliest possible start dates for any Rosh Hashannah will always coincide with the 17th year of the cycle GUChADZaT, and that the latest possible start dates for any Rosh Hashannah will always coincide with the 9th year of the cycle GUChADZaT.

However, that demonstration is not a proof, since TABLE IV, The Slonimsky GUChADZaT Distribution Cycle, is a reversal of the results shown in Table II. Thus, Table IV shows that dt * 19 / m = 18 when Year MOD 19 = 17 and dt * 19 / m = 0 when Year MOD 19 = 9.

It is possible to note that because the Gregorian calendar is faster than the Hebrew calendar, the first occurence of a new Gregorian date for Rosh Hashannah will always coincide with the 9th year of the mahzor qatan, while its last occurence will always occur with the 17th year of the mahzor qatan.

Similarly, it is possible to note that because the Julian calendar is slower than the Hebrew calendar, the first occurence of a new Julian date for Rosh Hashannah will always coincide with the 17th year of the mahzor qatan, while its last occurence will always occur with the 9th year of the mahzor qatan.

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 235 lunar months was equal to the time of any period of 19 solar years.

On the basis of the above analysis, it appears that the fixed Hebrew calendar's leap year intercalation cycle did not stem from astronomical observations, but rather emerged from purely arithmetical efforts.

```First  Paged  8 Oct 2002