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Re: [Mental Calculation] day of the week

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  • Frank Chin
    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]
    • mnempi
      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 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]
        >
      • mnempi
        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
          invention), and again published by G. Tarry in 1907. Butkewitsch /
          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 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]
          > >
          >
        • mnempi
          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
            (d+2)depending on the dechiyot Adu, Getred, and Batu Thakpad. All
            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
            invention), and again published by G. Tarry in 1907. Butkewitsch /
            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 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]
            > > >
            > >
            >
          • mnempi
            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
              dechiyah Adu!

              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
              (d+2)depending on the dechiyot Adu, Getred, and Batu Thakpad. All
              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
              invention), and again published by G. Tarry in 1907. Butkewitsch /
              Selikson did only know of Tarry. They were fine mathematicians but
              poor historians.
              >
              >
              U.V.
              www.likanas.de
            • mnempi
              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
                dechiyah Adu!

                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
                (d+2)depending on the dechiyot Adu, Getred, and Batu Thakpad. All
                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
                invention), and again published by G. Tarry in 1907. Butkewitsch /
                Selikson did only know of Tarry. They were fine mathematicians but
                poor historians.
                >
                >
                U.V.
                www.likanas.de
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