## Re: [Mental Calculation] day of the week

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• Did Gauss or Aimé Paris has the algorithm for Gregorian date? Regards Frank Chin Here is the algorithm which Carl Friedrich Gauss used for computing the day
Message 1 of 7 , Jun 24, 2008
Did Gauss or Aimé Paris has the algorithm for Gregorian date?

Regards
Frank Chin

Here is the algorithm which Carl Friedrich Gauss used for computing
the day of the week for any Julian date:

Number the days like W (Su) = 1, W (Mo) = 2 etc.

For the months from January to February use the sequence

k (M) = 5 1 1 4 6 2 4 0 3 5 1 3

Then for the day d in the month M and the year J the day of the week is

W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7

This formula can be reconstructed from his algorithm to compute the
"Jewish Eastern" (1802, in: Werke 1874, Bd. 6, 80-81).

It has been overlooked by Butkewitch and Selikson (Ewige Kalender
1987), who investigated into the origins of algorithms for the day of
the week. I suppose that it has also been overlooked by mental
calculators.

The Gauss algorithm is (of course) very smart, as it avoids div. If
you are fast at calculating J mod 7, it is even very fast.

Example

W (1255, march 22) = (3 x 2 + 5 x 3 + 1 + 1) mod 7 = 2

Compare it with Aimé Paris, who has

K(M) = 1 4 4 0 2 5 0 3 6 1 4 6

and uses J = 100H + E to get

W = ( 4 - H + E + E div 4 + K (M) + d) mod 7

If you eliminate +4 my adding 4 to K (M) you get the Gaussian sequence
k(M) and

W = (- H + E + E div 4 + k (M) + d) mod 7

so the matter boiles down to

Gauss: (3J + 5 J mod 4) mod 7
=
Paris: (-H + E + E div 4) mod 7

If there be an advantage with Aimé Paris, it is but small.

The matter only changes if mnemonics comes into play, because then the
Aimé Paris algorithm "sounds" like "natural speech":

W = (k (H) + K (E) + K (M) + K (d)) mod 7

and except for K (d) = d mod 7 there is nothing more left to
computation except for the sum of four numbers.

In short: The Aimé Paris algorithm is indeed an algorithm shaped for
the use of the mnemonist. But for the mental calculator the Gauss
algorithm might still be worth a trial.

U.V.
www.likanas. de
.

__________________________________________________________
Sent from Yahoo! Mail.
A Smarter Email http://uk.docs.yahoo.com/nowyoucan.html

[Non-text portions of this message have been removed]
• The Gauss formula for the Julian date W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7 can be split up into H = J div 100 and E = J mod 100: W = ( -H + 3 E + 5 E mod 4
Message 2 of 7 , Jun 24, 2008
The Gauss formula for the Julian date

W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7

can be split up into H = J div 100 and E = J mod 100:

W = ( -H + 3 E + 5 E mod 4 + k (M) + d) mod 7

For the Gregorian calendar there is

W = ( -1 + 5 (H + E) mod 4 + 3 E + k (M) + d) mod 7

which was originally found by W.W. Sokolow in 1966 (Butkewitsch /
Selikson, 1987, p. 100).

So Sokolow (1966) developed the same idea as Gauss (1802), but - of
course - without knowledge of Gauss.

Sokolow used W = ( 5 H mod 4 + 5 E mod 4 + 3 E + k (M) + d) mod 7
and for the months the sequence 6 2 2 5 0 3 5 1 4 6 2 4.
But I do not like to change this sequence so very often, so I changed
his formula a bit.

I think that meanwhile I have some interesting results about
computistical technique:
The Sokolow formula from 1966 "repeats" Gauss (early 19th century),
the Drosdow formula from 1954 (Butkewitsch / Selikson, 1987, p. 95
consider the Drosdow formula as the smartest one among all those
formulas developed during the 19th and 20th century) "repeats"
Dionysius Exiguus (6th century).

Neither did Gauss know of Dionysius Exiguus. But the historian puts
everything on his table to make a complex situation transparent.

U.V.
www.likanas.de

--- In MentalCalculation@yahoogroups.com, Frank Chin <fongkeechin@...>
wrote:
>
> Did Gauss or AimÃ© Paris has the algorithm for Gregorian date?
>
> Regards
> Frank Chin
>
> Here is the algorithm which Carl Friedrich Gauss used for computing
> the day of the week for any Julian date:
>
> Number the days like W (Su) = 1, W (Mo) = 2 etc.
>
> For the months from January to February use the sequence
>
> k (M) = 5 1 1 4 6 2 4 0 3 5 1 3
>
> Then for the day d in the month M and the year J the day of the week is
>
> W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7
>
> This formula can be reconstructed from his algorithm to compute the
> "Jewish Eastern" (1802, in: Werke 1874, Bd. 6, 80-81).
>
> It has been overlooked by Butkewitch and Selikson (Ewige Kalender
> 1987), who investigated into the origins of algorithms for the day of
> the week. I suppose that it has also been overlooked by mental
> calculators.
>
> The Gauss algorithm is (of course) very smart, as it avoids div. If
> you are fast at calculating J mod 7, it is even very fast.
>
> Example
>
> W (1255, march 22) = (3 x 2 + 5 x 3 + 1 + 1) mod 7 = 2
>
> Compare it with AimÃ© Paris, who has
>
> K(M) = 1 4 4 0 2 5 0 3 6 1 4 6
>
> and uses J = 100H + E to get
>
> W = ( 4 - H + E + E div 4 + K (M) + d) mod 7
>
> If you eliminate +4 my adding 4 to K (M) you get the Gaussian sequence
> k(M) and
>
> W = (- H + E + E div 4 + k (M) + d) mod 7
>
> so the matter boiles down to
>
> Gauss: (3J + 5 J mod 4) mod 7
> =
> Paris: (-H + E + E div 4) mod 7
>
> If there be an advantage with AimÃ© Paris, it is but small.
>
> The matter only changes if mnemonics comes into play, because then the
> AimÃ© Paris algorithm "sounds" like "natural speech":
>
> W = (k (H) + K (E) + K (M) + K (d)) mod 7
>
> and except for K (d) = d mod 7 there is nothing more left to
> computation except for the sum of four numbers.
>
> In short: The AimÃ© Paris algorithm is indeed an algorithm shaped for
> the use of the mnemonist. But for the mental calculator the Gauss
> algorithm might still be worth a trial.
>
> U.V.
> www.likanas. de
> .
>
>
>
> __________________________________________________________
> Sent from Yahoo! Mail.
> A Smarter Email http://uk.docs.yahoo.com/nowyoucan.html
>
> [Non-text portions of this message have been removed]
>
• If you are not interested in Julian dates you can use for Gregorian dates K (M) = 0 3 3 etc. (that is one lower than usual) and (1) Aimé Paris W = ( -2 H mod
Message 3 of 7 , Jun 25, 2008
If you are not interested in Julian dates you can use for Gregorian dates

K (M) = 0 3 3 etc. (that is one lower than usual)

and

(1) Aimé Paris

W = ( -2 H mod 4 + E + E div 4 + K (M) + d ) mod 7

or

(2) Sokolow-Gauss

W = ( 5(H + E) mod 4 + 3E + K (M) + d ) mod 7

Example 1851, June 2 (H = 18, E = 51)

(1)

W = ( -2 x 2 + 51 + 12 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( -4 + 2 + 5 - 3 + 2 ) mod 7 = 2 (Mo)

(2)

W = ( 5 x (2 + 3) + 3 x 2 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( 4 - 1 + 4 + 2 ) mod 7 = 2 (Mo)

I think that for the purpose of mental calculation the Sokolow-Gauss
formula is superior to the Paris-formula.

I have to apologize: The numbers for the months I gave in my last
posting are false, I took them from Butkewitsch / Selikson (1987)
without checking.

As to the Paris formula: It was published in 1866 (Le Vérificateur des
Dates), it was used by Lewis Carroll in 1888 (possibly by own
Selikson did only know of Tarry. They were fine mathematicians but
poor historians.

U.V.
www.likanas.de

--- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@...> wrote:
>
> The Gauss formula for the Julian date
>
> W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7
>
> can be split up into H = J div 100 and E = J mod 100:
>
> W = ( -H + 3 E + 5 E mod 4 + k (M) + d) mod 7
>
> For the Gregorian calendar there is
>
> W = ( -1 + 5 (H + E) mod 4 + 3 E + k (M) + d) mod 7
>
> which was originally found by W.W. Sokolow in 1966 (Butkewitsch /
> Selikson, 1987, p. 100).
>
> So Sokolow (1966) developed the same idea as Gauss (1802), but - of
> course - without knowledge of Gauss.
>
> Sokolow used W = ( 5 H mod 4 + 5 E mod 4 + 3 E + k (M) + d) mod 7
> and for the months the sequence 6 2 2 5 0 3 5 1 4 6 2 4.
> But I do not like to change this sequence so very often, so I changed
> his formula a bit.
>
> I think that meanwhile I have some interesting results about
> computistical technique:
> The Sokolow formula from 1966 "repeats" Gauss (early 19th century),
> the Drosdow formula from 1954 (Butkewitsch / Selikson, 1987, p. 95
> consider the Drosdow formula as the smartest one among all those
> formulas developed during the 19th and 20th century) "repeats"
> Dionysius Exiguus (6th century).
>
> Neither did Gauss know of Dionysius Exiguus. But the historian puts
> everything on his table to make a complex situation transparent.
>
> U.V.
> www.likanas.de
>
>
>
>
>
> --- In MentalCalculation@yahoogroups.com, Frank Chin <fongkeechin@>
> wrote:
> >
> > Did Gauss or AimÃ© Paris has the algorithm for Gregorian date?
> >
> > Regards
> > Frank Chin
> >
> > Here is the algorithm which Carl Friedrich Gauss used for computing
> > the day of the week for any Julian date:
> >
> > Number the days like W (Su) = 1, W (Mo) = 2 etc.
> >
> > For the months from January to February use the sequence
> >
> > k (M) = 5 1 1 4 6 2 4 0 3 5 1 3
> >
> > Then for the day d in the month M and the year J the day of the
week is
> >
> > W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7
> >
> > This formula can be reconstructed from his algorithm to compute the
> > "Jewish Eastern" (1802, in: Werke 1874, Bd. 6, 80-81).
> >
> > It has been overlooked by Butkewitch and Selikson (Ewige Kalender
> > 1987), who investigated into the origins of algorithms for the day of
> > the week. I suppose that it has also been overlooked by mental
> > calculators.
> >
> > The Gauss algorithm is (of course) very smart, as it avoids div. If
> > you are fast at calculating J mod 7, it is even very fast.
> >
> > Example
> >
> > W (1255, march 22) = (3 x 2 + 5 x 3 + 1 + 1) mod 7 = 2
> >
> > Compare it with AimÃ© Paris, who has
> >
> > K(M) = 1 4 4 0 2 5 0 3 6 1 4 6
> >
> > and uses J = 100H + E to get
> >
> > W = ( 4 - H + E + E div 4 + K (M) + d) mod 7
> >
> > If you eliminate +4 my adding 4 to K (M) you get the Gaussian sequence
> > k(M) and
> >
> > W = (- H + E + E div 4 + k (M) + d) mod 7
> >
> > so the matter boiles down to
> >
> > Gauss: (3J + 5 J mod 4) mod 7
> > =
> > Paris: (-H + E + E div 4) mod 7
> >
> > If there be an advantage with AimÃ© Paris, it is but small.
> >
> > The matter only changes if mnemonics comes into play, because then the
> > AimÃ© Paris algorithm "sounds" like "natural speech":
> >
> > W = (k (H) + K (E) + K (M) + K (d)) mod 7
> >
> > and except for K (d) = d mod 7 there is nothing more left to
> > computation except for the sum of four numbers.
> >
> > In short: The AimÃ© Paris algorithm is indeed an algorithm shaped for
> > the use of the mnemonist. But for the mental calculator the Gauss
> > algorithm might still be worth a trial.
> >
> > U.V.
> > www.likanas. de
> > .
> >
> >
> >
> > __________________________________________________________
> > Sent from Yahoo! Mail.
> > A Smarter Email http://uk.docs.yahoo.com/nowyoucan.html
> >
> > [Non-text portions of this message have been removed]
> >
>
• Carl Friedrich Gauss From Gauss there are two algorithms for the date of Eastern , one Christian (1800), one Jewish (1802). In 1800 Gauss used the number of
Message 4 of 7 , Jun 26, 2008
Carl Friedrich Gauss

From Gauss there are two algorithms for the date of "Eastern", one
Christian (1800), one Jewish (1802).

In 1800 Gauss used the number of leap years that have elapsed since a
certain date to take account of the day of the week. This is a very
natural idea and has become the mainstream solution, as the insertion
of leap-days affects the change of the day of the week from one year
to the next.
But in 1802 Gauss used a different method, which avoids division, and
this has (to my knowledge) gone unnoticed.

The problem Gauss had to solve was the dechiyah "Adu", which says that
"If molad Tishri falls on Sunday, Wednesday, or Friday, then Tishri 1
is postponed to the next day."

The Hebrew word Adu is composed of aleph = 1, dalet = 4, and vav = 6.
It is thus a mnemonic to help remember that Tishri 1 cannot
fall on Sunday, Wednesday, or Friday, as the days of the week are
numbered 1 = Su, 2 = Mo etc.

Gauss (1802)implies that the difference of molad Tishri and august 31
is

d = FLOOR (-0.9044123 + 1.5542418 (12 J + 12) mod 19 + 0,5 (J mod 4) 
0.003177794 / 1).

Gauss then defines for a year J AD:

c = ( d + 3 J + 5 J mod 4 + 1 ) mod 7

with the effect that Adu is equivalent to

"If c = 2, 4, or 6, then Tishri 1 = september (d+1)."

Actually Tishri 1 = september d or september (d+1) or september
these words are mnemonics after the same system.

Well, Gauss did not speak of Tishri 1, but of Nisan 15, as he based
his analysis of Hebrew calendar not on Tishri 1 of year 1, but on
Nisan 1 of year 0:

Nisan 15 = march (21 + d), march (21 + d + 1), or march (21 + d + 2)

Gauss himself did not comment on his algorithm, but only produced a
result of letters and numbers. He did not even give a proof of its
validity. But (contrary to his algorithm for the Christian Easter
date, which unfortunately contains an error in the basic definitions
of the Gregorian date) his "Jewish" formula has proved absolutely
correct.

It is not difficult to see that

c = W ( march (21+d))= W ( march d ).

This was e.g. noticed by M. Hamburger (Ableitung der Gaussschen
Formel zur Bestimmung des jüdischen Osterfestes, 1896). But Hamburger
did not bother about the method involved.

Now, if W (march d) = ( d + 3 J + 5 J mod 4 + 1 ) mod 7,

then

W (J M d) = ( 3 J + 5 J mod 4 + K (M) + d )mod 7

with K (M) = 5 1 1 4 6 2 4 0 3 5 1 3.

U.V.
www.likanas.de

--- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@...> wrote:
>
If you are not interested in Julian dates you can use for Gregorian dates

K (M) = 0 3 3 etc. (that is one lower than usual)

and

(1) Aimé Paris

W = ( -2 H mod 4 + E + E div 4 + K (M) + d ) mod 7

or

(2) Sokolow-Gauss

W = ( 5(H + E) mod 4 + 3E + K (M) + d ) mod 7

Example 1851, June 2 (H = 18, E = 51)

(1)

W = ( -2 x 2 + 51 + 12 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( -4 + 2 + 5 - 3 + 2 ) mod 7 = 2 (Mo)

(2)

W = ( 5 x (2 + 3) + 3 x 2 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( 4 - 1 + 4 + 2 ) mod 7 = 2 (Mo)

I think that for the purpose of mental calculation the Sokolow-Gauss
formula is superior to the Paris-formula.

I have to apologize: The numbers for the months I gave in my last
posting are false, I took them from Butkewitsch / Selikson (1987)
without checking.

As to the Paris formula: It was published in 1866 (Le Vérificateur des
Dates), it was used by Lewis Carroll in 1888 (possibly by own
Selikson did only know of Tarry. They were fine mathematicians but
poor historians.
>
>
U.V.
www.likanas.de
>
>
> --- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@> wrote:
> >
The Gauss formula for the Julian date

W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7

can be split up into H = J div 100 and E = J mod 100:

W = ( -H + 3 E + 5 E mod 4 + k (M) + d) mod 7

For the Gregorian calendar there is

W = ( -1 + 5 (H + E) mod 4 + 3 E + k (M) + d) mod 7

which was originally found by W.W. Sokolow in 1966 (Butkewitsch /
Selikson, 1987, p. 100).

So Sokolow (1966) developed the same idea as Gauss (1802), but - of
course - without knowledge of Gauss.

Sokolow used W = ( 5 H mod 4 + 5 E mod 4 + 3 E + k (M) + d) mod 7
and for the months the sequence 6 2 2 5 0 3 5 1 4 6 2 4.
But I do not like to change this sequence so very often, so I changed
his formula a bit.

I think that meanwhile I have some interesting results about
computistical technique:
The Sokolow formula from 1966 "repeats" Gauss (early 19th century),
the Drosdow formula from 1954 (Butkewitsch / Selikson, 1987, p. 95
consider the Drosdow formula as the smartest one among all those
formulas developed during the 19th and 20th century) "repeats"
Dionysius Exiguus (6th century).

Neither did Gauss know of Dionysius Exiguus. But the historian puts
everything on his table to make a complex situation transparent.

U.V.
www.likanas.de
> >
> >
> > --- In MentalCalculation@yahoogroups.com, Frank Chin <fongkeechin@>
wrote:

Did Gauss or AimÃ© Paris has the algorithm for Gregorian date?

Regards
Frank Chin
> > >
Here is the algorithm which Carl Friedrich Gauss used for computing
the day of the week for any Julian date:

Number the days like W (Su) = 1, W (Mo) = 2 etc.

For the months from January to February use the sequence

k (M) = 5 1 1 4 6 2 4 0 3 5 1 3

Then for the day d in the month M and the year J the day of the
week is

W = ( 3 J + 5 J mod 4 + k (M) + d) mod 7

This formula can be reconstructed from his algorithm to compute the
"Jewish Eastern" (1802, in: Werke 1874, Bd. 6, 80-81).

It has been overlooked by Butkewitch and Selikson (Ewige Kalender
1987), who investigated into the origins of algorithms for the day of
the week. I suppose that it has also been overlooked by mental
calculators.

The Gauss algorithm is (of course) very smart, as it avoids div. If
you are fast at calculating J mod 7, it is even very fast.

Example

W (1255, march 22) = (3 x 2 + 5 x 3 + 1 + 1) mod 7 = 2

Compare it with AimÃ© Paris, who has

K(M) = 1 4 4 0 2 5 0 3 6 1 4 6

and uses J = 100H + E to get

W = ( 4 - H + E + E div 4 + K (M) + d) mod 7

If you eliminate +4 my adding 4 to K (M) you get the Gaussian sequence
k(M) and

W = (- H + E + E div 4 + k (M) + d) mod 7

so the matter boiles down to

Gauss: (3J + 5 J mod 4) mod 7
=
Paris: (-H + E + E div 4) mod 7

If there be an advantage with AimÃ© Paris, it is but small.

The matter only changes if mnemonics comes into play, because then the
AimÃ© Paris algorithm "sounds" like "natural speech":

W = (k (H) + K (E) + K (M) + K (d)) mod 7

and except for K (d) = d mod 7 there is nothing more left to
computation except for the sum of four numbers.

In short: The AimÃ© Paris algorithm is indeed an algorithm shaped for
the use of the mnemonist. But for the mental calculator the Gauss
algorithm might still be worth a trial.

U.V.
www.likanas. de
> > > .
> > >
> > >
> > >
> > > __________________________________________________________
> > > Sent from Yahoo! Mail.
> > > A Smarter Email http://uk.docs.yahoo.com/nowyoucan.html
> > >
> > > [Non-text portions of this message have been removed]
> > >
> >
>
• Carl Friedrich Gauss 1800 / 1802 I finally (!) understood that the formula W (J M d) = ( 3 J + 5 J mod 4 + K (M) + d )mod 7 with K (M) = 5 1 1 etc. is already
Message 5 of 7 , Jun 26, 2008
Carl Friedrich Gauss 1800 / 1802

I finally (!) understood that the formula

W (J M d) = ( 3 J + 5 J mod 4 + K (M) + d )mod 7

with K (M) = 5 1 1 etc.

is already part of the Gauss algorithm for the date of the
Christian Eastern. In this respect there simply is no difference
between Gauss 1800 and Gauss 1802. No need to bother about

Strange matter that I did not notice the Gauss algorithm for the day
of the week before. Strange matter also that not even one of so many
mathematicians of the 19th and 20th century thinking about how to
optimize an algorithm for the day of the week noticed this: Ch. Zeller
(1877), A. Rydsewski (1900), G. Tarry (1907), J.I. Perelman (1909),
W. Jacobsthal (1917), M.S. Selikson (1947), S. Drosdow (1954), I.P.
Konogorski (1955), H.A.L. Shewell (1963), H. Filatow (1964), W.W.
Sokolow (1966)

I will now look into the literature on the "Gauss formula" if at
least those mathematicians who were more closely interested in Gauss
understood that he had also at hands a formula to compute the day of
the week, and a formula which does not envolve division.

U.V.
www.likanas.de

>
> --- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@> wrote:
> >

Carl Friedrich Gauss

From Gauss there are two algorithms for the date of "Eastern", one
Christian (1800), one Jewish (1802).

In 1800 Gauss used the number of leap years that have elapsed since a
certain date to take account of the day of the week. This is a very
natural idea and has become the mainstream solution, as the insertion
of leap-days affects the change of the day of the week from one year
to the next.
But in 1802 Gauss used a different method, which avoids division, and
this has (to my knowledge) gone unnoticed.

The problem Gauss had to solve was the dechiyah "Adu", which says that
"If molad Tishri falls on Sunday, Wednesday, or Friday, then Tishri 1
is postponed to the next day."

The Hebrew word Adu is composed of aleph = 1, dalet = 4, and vav = 6.
It is thus a mnemonic to help remember that Tishri 1 cannot
fall on Sunday, Wednesday, or Friday, as the days of the week are
numbered 1 = Su, 2 = Mo etc.

Gauss (1802)implies that the difference of molad Tishri and august 31
is

d = FLOOR (-0.9044123 + 1.5542418 (12 J + 12) mod 19 + 0,5 (J mod 4) 
0.003177794 / 1).

Gauss then defines for a year J AD:

c = ( d + 3 J + 5 J mod 4 + 1 ) mod 7

with the effect that Adu is equivalent to

"If c = 2, 4, or 6, then Tishri 1 = september (d+1)."

Actually Tishri 1 = september d or september (d+1) or september
these words are mnemonics after the same system.

Well, Gauss did not speak of Tishri 1, but of Nisan 15, as he based
his analysis of Hebrew calendar not on Tishri 1 of year 1, but on
Nisan 1 of year 0:

Nisan 15 = march (21 + d), march (21 + d + 1), or march (21 + d + 2)

Gauss himself did not comment on his algorithm, but only produced a
result of letters and numbers. He did not even give a proof of its
validity. But (contrary to his algorithm for the Christian Easter
date, which unfortunately contains an error in the basic definitions
of the Gregorian date) his "Jewish" formula has proved absolutely
correct.

It is not difficult to see that

c = W ( march (21+d))= W ( march d ).

This was e.g. noticed by M. Hamburger (Ableitung der Gaussschen
Formel zur Bestimmung des jüdischen Osterfestes, 1896). But Hamburger
did not bother about the method involved.

Now, if W (march d) = ( d + 3 J + 5 J mod 4 + 1 ) mod 7,

then

W (J M d) = ( 3 J + 5 J mod 4 + K (M) + d )mod 7

with K (M) = 5 1 1 4 6 2 4 0 3 5 1 3.

U.V.
www.likanas.de

--- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@...> wrote:
>
If you are not interested in Julian dates you can use for Gregorian dates

K (M) = 0 3 3 etc. (that is one lower than usual)

and

(1) Aimé Paris

W = ( -2 H mod 4 + E + E div 4 + K (M) + d ) mod 7

or

(2) Sokolow-Gauss

W = ( 5(H + E) mod 4 + 3E + K (M) + d ) mod 7

Example 1851, June 2 (H = 18, E = 51)

(1)

W = ( -2 x 2 + 51 + 12 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( -4 + 2 + 5 - 3 + 2 ) mod 7 = 2 (Mo)

(2)

W = ( 5 x (2 + 3) + 3 x 2 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( 4 - 1 + 4 + 2 ) mod 7 = 2 (Mo)

I think that for the purpose of mental calculation the Sokolow-Gauss
formula is superior to the Paris-formula.

I have to apologize: The numbers for the months I gave in my last
posting are false, I took them from Butkewitsch / Selikson (1987)
without checking.

As to the Paris formula: It was published in 1866 (Le Vérificateur des
Dates), it was used by Lewis Carroll in 1888 (possibly by own
Selikson did only know of Tarry. They were fine mathematicians but
poor historians.
>
>
U.V.
www.likanas.de
• Gauss (1800) = Sokolow (1966) To bring the matter to the point: For the number of leap years in a given number of years J (conveniently beginning with march 1
Message 6 of 7 , Jun 27, 2008
Gauss (1800) = Sokolow (1966)

To bring the matter to the point:

For the number of leap years in a given number of years J
(conveniently beginning with march 1 to bring the leap day to the end
of the year), Gauss used

J div 4 = 1/4 (J - J mod 4)

In the interval [0 march 1 ; J march 1] there are exactly

365 J + J div 4

days.

As

( 1/4 (J - J mod 4) ) mod 7 = (2 (J - J mod 4)) mod 7

and

365 J mod 7 = J mod 7

and

J + 2 (J - J mod 4) = 3 J - 2 J mod 4

and

W (0 march 21) = 1

and

W (J march 21) = W (J march 0)

you get

W (J march 0) = ( 3 J - 2 J mod 4 + 1) mod 7

which implies K (M) = 5 1 1 4 6 2 4 0 3 5 1 3.

Of course

( 3 J - 2 J mod 4 + 1) mod 7 = ( 3 J + 5 J mod 4 + 1) mod 7

From this it is clear that Sokolow (1966) used

J div 4 = 1/4 (J - J mod 4)

and

(J div 4) mod 7 = (2 (J - J mod 4)) mod 7

to eliminate division.
He just found anew a simple device of great Gauss (1800) which had
been overlooked by so many mathematicians.

U.V.
www.likanas.de

>
--- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@...> wrote:
>
Carl Friedrich Gauss 1800 / 1802

I finally (!) understood that the formula

W (J M d) = ( 3 J + 5 J mod 4 + K (M) + d )mod 7

with K (M) = 5 1 1 etc.

is already part of the Gauss algorithm for the date of the
Christian Eastern. In this respect there simply is no difference
between Gauss 1800 and Gauss 1802. No need to bother about

Strange matter that I did not notice the Gauss algorithm for the day
of the week before. Strange matter also that not even one of so many
mathematicians of the 19th and 20th century thinking about how to
optimize an algorithm for the day of the week noticed this: Ch. Zeller
(1877), A. Rydsewski (1900), G. Tarry (1907), J.I. Perelman (1909),
W. Jacobsthal (1917), M.S. Selikson (1947), S. Drosdow (1954), I.P.
Konogorski (1955), H.A.L. Shewell (1963), H. Filatow (1964), W.W.
Sokolow (1966)

I will now look into the literature on the "Gauss formula" if at
least those mathematicians who were more closely interested in Gauss
understood that he had also at hands a formula to compute the day of
the week, and a formula which does not envolve division.

U.V.
www.likanas.de

>
> --- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@> wrote:
> >

Carl Friedrich Gauss

From Gauss there are two algorithms for the date of "Eastern", one
Christian (1800), one Jewish (1802).

In 1800 Gauss used the number of leap years that have elapsed since a
certain date to take account of the day of the week. This is a very
natural idea and has become the mainstream solution, as the insertion
of leap-days affects the change of the day of the week from one year
to the next.
But in 1802 Gauss used a different method, which avoids division, and
this has (to my knowledge) gone unnoticed.

The problem Gauss had to solve was the dechiyah "Adu", which says that
"If molad Tishri falls on Sunday, Wednesday, or Friday, then Tishri 1
is postponed to the next day."

The Hebrew word Adu is composed of aleph = 1, dalet = 4, and vav = 6.
It is thus a mnemonic to help remember that Tishri 1 cannot
fall on Sunday, Wednesday, or Friday, as the days of the week are
numbered 1 = Su, 2 = Mo etc.

Gauss (1802)implies that the difference of molad Tishri and august 31
is

d = FLOOR (-0.9044123 + 1.5542418 (12 J + 12) mod 19 + 0,5 (J mod 4) 
0.003177794 / 1).

Gauss then defines for a year J AD:

c = ( d + 3 J + 5 J mod 4 + 1 ) mod 7

with the effect that Adu is equivalent to

"If c = 2, 4, or 6, then Tishri 1 = september (d+1)."

Actually Tishri 1 = september d or september (d+1) or september
these words are mnemonics after the same system.

Well, Gauss did not speak of Tishri 1, but of Nisan 15, as he based
his analysis of Hebrew calendar not on Tishri 1 of year 1, but on
Nisan 1 of year 0:

Nisan 15 = march (21 + d), march (21 + d + 1), or march (21 + d + 2)

Gauss himself did not comment on his algorithm, but only produced a
result of letters and numbers. He did not even give a proof of its
validity. But (contrary to his algorithm for the Christian Easter
date, which unfortunately contains an error in the basic definitions
of the Gregorian date) his "Jewish" formula has proved absolutely
correct.

It is not difficult to see that

c = W ( march (21+d))= W ( march d ).

This was e.g. noticed by M. Hamburger (Ableitung der Gaussschen
Formel zur Bestimmung des jüdischen Osterfestes, 1896). But Hamburger
did not bother about the method involved.

Now, if W (march d) = ( d + 3 J + 5 J mod 4 + 1 ) mod 7,

then

W (J M d) = ( 3 J + 5 J mod 4 + K (M) + d )mod 7

with K (M) = 5 1 1 4 6 2 4 0 3 5 1 3.

U.V.
www.likanas.de

--- In MentalCalculation@yahoogroups.com, "mnempi" <voigt@...> wrote:
>
If you are not interested in Julian dates you can use for Gregorian dates

K (M) = 0 3 3 etc. (that is one lower than usual)

and

(1) Aimé Paris

W = ( -2 H mod 4 + E + E div 4 + K (M) + d ) mod 7

or

(2) Sokolow-Gauss

W = ( 5(H + E) mod 4 + 3E + K (M) + d ) mod 7

Example 1851, June 2 (H = 18, E = 51)

(1)

W = ( -2 x 2 + 51 + 12 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( -4 + 2 + 5 - 3 + 2 ) mod 7 = 2 (Mo)

(2)

W = ( 5 x (2 + 3) + 3 x 2 + 4 + 2 ) mod 7 = 2 (Mo)

which I would rather calculate like this (using the fingers of one hand):

W = ( 4 - 1 + 4 + 2 ) mod 7 = 2 (Mo)

I think that for the purpose of mental calculation the Sokolow-Gauss
formula is superior to the Paris-formula.

I have to apologize: The numbers for the months I gave in my last
posting are false, I took them from Butkewitsch / Selikson (1987)
without checking.

As to the Paris formula: It was published in 1866 (Le Vérificateur des
Dates), it was used by Lewis Carroll in 1888 (possibly by own