CeeFax/JULIAN

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CeeFax/JULIAN

This website contains an archive of files for the Acorn Electron, BBC Micro, Acorn Archimedes, Commodore 16 and Commodore 64 computers, which Dominic Ford has rescued from his private collection of floppy disks and cassettes.

Some of these files were originally commercial releases in the 1980s and 1990s, but they are now widely available online. I assume that copyright over them is no longer being asserted. If you own the copyright and would like files to be removed, please contact me.

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File contents

     The following article is reproduced with permission from
the May 1984 issue of Sky & Telescope magazine, and remains the
copyright of the Sky Publishing Corporation.

The "Sky & Telescope" magazine (ISSN 0037-6604) is published
monthly by:

                  Sky Publishing Corporation,
                  49, Bay State Road,
                  Cambridge,
                  Mass.  02238-1290
                  U.S.A.

                         -----oOo-----




                    Astronomical  Computing

                 Conducted by Roger W. Sinnott


                  Taming Our Chaotic Calendar



     A routine for calendar conversion lies at the heart of many
astronomical programs.  Often we want a computer to calculate
something for a particular date and time, or to tell us the day
and month when an event will occur (an eclipse or occultation,
perhaps).  Digital watches often fail at date conversion, and
last February 29th quite a few of them around the world probably
indicated March 1st.  To make our computer routines as universal
as possible, let's lay down certain firm expectations for them.

     First of all, the machine should "resolve" intervals as
small as 1 second of time (that is, about 0.000001 day).  It
takes only a little effort to build this accuracy into our
calendar conversion routines so they won't need later revision
for special programs.

     Second, let's avoid simple-looking procedures that only
work between certain years (for example, A.D. 1901 to 2099).
Sooner or later, we'll probably attempt a calculation outside
this interval, little suspecting that the computer will
misinterpret a date we've entered, usually by some whole number
of days!

     To express an instant internally in the computer, we'll use
Julian day numbers and fractions, as first applied to
astronomical observations by John Herschel.  The starting
instant of this system (that is, day 0.0) corresponds to noon,
Greenwich Mean Time, on January 1, 4713 B.C. - sufficiently far
back in time that we should never have to worry about negative
day numbers (and won't).  At 0 hours Universal Time on any day,
the decimal part of the Julian date is .5 exactly.

     What about the changeover from the Julian calendar to the
Gregorian?  The Julian calendar (which has nothing to do with
Julian Day numbers) had simpler leap-year rules and was in wide
civil use before A.D. 1582.  The switch to the Gregorian was
delayed until 1752 in England and the American colonies, and as
late as 1873 in Japan and 1927 in Turkey.  A special flag in a
program can ensure the computer is oriented to the proper
calendar.

     Finally, it's important to distinguish A.D. and B.C. years
somehow.  The standard way to do this regards A.D. years as
positive whole numbers; 1 B.C. is called the year 0, 2 B.C. is
-1, 3 B.C. is -2, and so on.  This helps keep leap years
straight.  Julius Caesar introduced his calendar reform in the
year 45 B.C., which becomes -44 on our scheme.  And since that
was in fact leap year, the modern divide-by-four rule applies to
negative as well as positive years.  (Readers are warned that
some recent books and articles have used the notation 1 B.C. =
-1, and 2 B.C. = -2, a very confusing symbolism.)

     The Julian calendar did not really come into consistent
civil use until after A.D. 8.  But chronologers customarily
extrapolate it backwards and call it the Julian proleptic
calendar to distinguish it from the various civil calendars that
may have been in use here and there prior to A.D. 8.

     Perhaps the most concise algorithm ever devised for
calculating the Julian day number from a Gregorian calendar date
is pictured below.  It is a line of Fortran and fits on a single
80-column punch card.  The result is an integer, JD,
corresponding to the Julian Day number that begins at Greenwich
Mean noon of year Y, month M, day D (all integers).

     However, this Fortran statement makes use of special rules
that don't work in most versions of Basic.  The absence of
explicit decimal points forces execution by integer arithmetic;
division by an integer leads to an integer quotient, and the
remainder is discarded.  (For example, 5/2 = 2, and -10/6 = -1.)
The natural approach is to rewrite the lines, using the INT
function of Basic.  But all microcomputer Basics evaluate
INT(5/2) as 2, while INT(-10/6) becomes -2 instead of -1.

     Therefore, we have taken special care in designing Listing
1, a program based on the single line of Fortran but with added
features.  It will work with either the Gregorian or Julian
calendars, and with either negative or positive years.
Moreover, the program should run correctly on any microcomputer
that works to at least seven digit accuracy.


JD=367*Y-7*(Y+(M+9)/12)/4-3*((Y+(M-9)/7)/100+1)/4+275*M/9+D+1721029

This remarkable one-line Fortran statement converts a Gregorian
calendar date into a Julian Day number.  It appeared in Astrophysical
Journal Supplement Series, 41, 1979, page 392, after being devised in
1968 by H.F.Fliegel and T.C.Van Flandern and later "compacted" by
P.M.Muller and R.N.Wimberly.  It is designed for A.D. years only and
requires some changes before being used in a Basic program (see text).
     Initially we input Y, M, and D, and set G = 1 for the
Gregorian calendar or G = 0 for the Julian one.  The day D can
be entered to five decimal places or more.  Then the program
determines the integer part J and fractional part F of the
Julian date.  It is very important not to add J and F together;
they must be kept as separate integer and fractional parts
throughout a working program so as not to lose accuracy, unless
a high precision Basic is used.


Listing 1

800 REM   CALENDAR --> JD
805 REM
810 INPUT "Y,M,D ";Y,M,D
815 INPUT "JC (0) OR GC (1) ";G
820 D1=INT(D): F=D-D1-0.5
825 J=-INT(7*(INT((M+9)/12)+Y)/4)
830 IF G=0 THEN 850
835 S=SGN(M-9): A=ABS(M-9)
840 J1=INT(Y+S*INT(A/7))
845 J1=-INT((INT(J1/100)+1)*3/4)
850 J=J+INT(275*M/9)+D1+G*J1
855 J=J+1721027+2*G+367*Y
860 IF F>=0 THEN 870
865 F=F+1: J=J-1
870 PRINT "J.D.: ";J;F
875 END


Listing 2

900 REM   JD ---> CALENDAR
905 REM
910 INPUT "J,F ";J,F
915 INPUT "JC (0) OR GC (1) ";G
920 F=F+0.5
925 IF F<1 THEN 935
930 F=F-1:J=J+1
935 IF G=1 THEN 945
940 A=J: GOTO 955
945 A1=INT((J/36524.25)-51.12264)
950 A=J+1+A1-INT(A1/4)
955 B=A+1524
960 C=INT((B/365.25)-0.3343)
965 D=INT(365.25*C)
970 E=INT((B-D)/30.61)
975 D=B-D-INT(30.61*E)+F
980 M=E-1: Y=C-4716
985 IF E>13.5 THEN M=M-12
990 IF M<2.5 THEN Y=Y+1
995 PRINT "DATE: ";Y;M;D
997 END

Note on variations:  With Radio Shack Level II Basic, replace J
in line 870 with CDBL(J).  In Applesoft, use commas rather than
semicolons in lines 870 and 995.  Some Basics require a back-
slash where colons are used here.

     Listing 2 performs the inverse calculation.  Initially, J
is the Julian Day number, and F the fractional part of a day.
When the program finishes, Y is the year, M the month, and D the
day (with fraction).  This program is based on the algorithm in
Jean Meeus's Astronomical Formulae for Calculators (Willmann-
Bell, 1982), with some rearrangement to suit the many micro-
computers with only seven digit accuracy.  Here again, G
determines the calendar style.  (The Meeus book also describes
the inverse calculation, by a method different from that
described here but identical in result.)

     For example, consider the geocentric conjunction of the Sun
and Moon this month.  It occurs at 16:52:39 UT on May 30th, so
the day is expressed decimally as 30.70323 and the dialogue runs:

   >RUN "CALJD"
   Y,M,D ? 1984, 5, 30.70323
   JC (0) OR GC (1) ? 1
   J.D.:  2445851 .20323
   READY

   >RUN "JDCAL"
   J,F ? 2445851, .20323
   JC (0) OR GC (1) ? 1
   DATE:  1984  5  30.70323
   READY

     Future program listings in this department will often make
use of the Julian Day number and fraction, so readers may wish
to rewrite these listings as subroutines.  After replacing each
END statement with RETURN, you can simply insert them bodily
into other programs.

     Both listings have been thoroughly tested at Sky &
Telescope on many microcomputers, for the years -4712 through to
+3500 in both the Gregorian and Julian calendars.  But it's
wise, when trying out the programs on your own machine, to
verify the results on some known equivalents:

      Julian Cal.         Gregorian Cal.      Julian Date

    -4712 Jan.  1.5      -4713 Nov. 24.5             0.0
    -2000 Jan.  1.0      -2001 Dec. 15.0        990557.5
     -584 May  28.6       -584 May  22.6       1507900.1
     +200 Mar.  1.0       +200 Mar.  1.0       1794167.5
    +1984 Feb. 16.2      +1984 Feb. 29.2       2445759.7
    +1999 Dec. 19.5      +2000 Jan.  1.5       2451545.0
    +3000 Feb. 29.9      +3000 Mar. 21.9       2816867.4

     Many ancient calendars have been linked by Julian Day
number in O.L.Harvey's new booklet, Calendar Conversions
(American Philosophical Society, 1983).  He also mentions two
useful tricks with these numbers.  To find the day of the week
at noon when each Julian Day number begins, divide the whole
number by 7; a remainder of 0 means Monday, 1 Tuesday, and so
on, through to 6 for Sunday.  Furthermore, the Moon is within a
day or so of new on those days when the Julian Day number,
divided by 29.530588, yields .33 as the fractional part.

     Prior to 1925, astronomers actually considered each new
calendar date to commence at noon on the corresponding civil
date.  But our programs use the modern convention as to when a
calendar day begins, even for dates before 1925.  Nevertheless,
it is well to remember this change in usage when consulting
early astronomical sources.

     Also beware the "Modified Julian Date" (MJD), defined as
the true JD minus 2,400,000.5, and sometimes advocated to help
preserve digits in machines of limited precision.  In reality,
since the first two digits of the JD have been 24 only since
1859, the device is useful only from a narrow modern
perspective.  More alarming, though, is the danger of confusion
introduced by the .5 (intended to make the MJD begin when the
civil day does).

     The coexistence of this with another common device, that of
simply lopping off the highest few digits of the JD, has often
led to a confusion of 12 hours in the astronomical literature.
The practice of adjusting the JD by some constant was once
described by the late Paul Herget as "an attempt to pervert the
established system" of astronomical reckoning.  Far better not
to tinker with it, and let our machines worry about the details!

                                                ROGER W. SINNOTT

                         -----oOo-----


From the September 1984 issue of Sky & Telescope, page 255...

     Milo L. Hurley, Jr., of Aurora, Colorado, writes that our
program for computing the Julian date from the calendar date
(Listing 1 in the May issue, page 455) does not run properly for
negative years when written in Digital Research's CBASIC for his
Altos computer.

     He has traced the problem to the INT function, which takes
the integer part of a number.  In CBASIC (as on many pocket
calculators), the integer part of a negative number is obtained
by truncating the decimal part; thus, both (INT-3.2) and
INT(-3.8) yield -3.  In all the other microcomputer Basics we've
tried, the answer is -4 in these two cases.  A remedy in CBASIC
is to set up a user-defined function that mimics Basics of the
other type, and to substitute this for INT throughout the
program.

     The Basics that give -4 appear to be the clear majority,
and with them the calendar program works without modification.
These include the Microsoft versions running on the Apple, Radio
Shack, and Ohio Scientific machines, and also North Star,
Digital Equipment, and Hewlett-Packard Basics.


                         -----oOo-----

Electronic News Service

     Computer owners with modems can obtain fast news from Sky &
Telescope of comet and nova discoveries, and also program
listings from the Astronomical Computing Department.  After
joining CompuServe (through a local Radio Shack or computer
dealer), log on and move to Personal Computing and then the
Personal File Area.  Once there, at any menu, type R ACCESS
Then at the next menu type R NEWS.DAT[70275,125] to receive this
service.

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00001620  61 73 69 63 20 69 73 20  75 73 65 64 2e 0d 0d 0d  |asic is used....|
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00001660  20 49 4e 50 55 54 20 22  59 2c 4d 2c 44 20 22 3b  | INPUT "Y,M,D ";|
00001670  59 2c 4d 2c 44 0d 38 31  35 20 49 4e 50 55 54 20  |Y,M,D.815 INPUT |
00001680  22 4a 43 20 28 30 29 20  4f 52 20 47 43 20 28 31  |"JC (0) OR GC (1|
00001690  29 20 22 3b 47 0d 38 32  30 20 44 31 3d 49 4e 54  |) ";G.820 D1=INT|
000016a0  28 44 29 3a 20 46 3d 44  2d 44 31 2d 30 2e 35 0d  |(D): F=D-D1-0.5.|
000016b0  38 32 35 20 4a 3d 2d 49  4e 54 28 37 2a 28 49 4e  |825 J=-INT(7*(IN|
000016c0  54 28 28 4d 2b 39 29 2f  31 32 29 2b 59 29 2f 34  |T((M+9)/12)+Y)/4|
000016d0  29 0d 38 33 30 20 49 46  20 47 3d 30 20 54 48 45  |).830 IF G=0 THE|
000016e0  4e 20 38 35 30 0d 38 33  35 20 53 3d 53 47 4e 28  |N 850.835 S=SGN(|
000016f0  4d 2d 39 29 3a 20 41 3d  41 42 53 28 4d 2d 39 29  |M-9): A=ABS(M-9)|
00001700  0d 38 34 30 20 4a 31 3d  49 4e 54 28 59 2b 53 2a  |.840 J1=INT(Y+S*|
00001710  49 4e 54 28 41 2f 37 29  29 0d 38 34 35 20 4a 31  |INT(A/7)).845 J1|
00001720  3d 2d 49 4e 54 28 28 49  4e 54 28 4a 31 2f 31 30  |=-INT((INT(J1/10|
00001730  30 29 2b 31 29 2a 33 2f  34 29 0d 38 35 30 20 4a  |0)+1)*3/4).850 J|
00001740  3d 4a 2b 49 4e 54 28 32  37 35 2a 4d 2f 39 29 2b  |=J+INT(275*M/9)+|
00001750  44 31 2b 47 2a 4a 31 0d  38 35 35 20 4a 3d 4a 2b  |D1+G*J1.855 J=J+|
00001760  31 37 32 31 30 32 37 2b  32 2a 47 2b 33 36 37 2a  |1721027+2*G+367*|
00001770  59 0d 38 36 30 20 49 46  20 46 3e 3d 30 20 54 48  |Y.860 IF F>=0 TH|
00001780  45 4e 20 38 37 30 0d 38  36 35 20 46 3d 46 2b 31  |EN 870.865 F=F+1|
00001790  3a 20 4a 3d 4a 2d 31 0d  38 37 30 20 50 52 49 4e  |: J=J-1.870 PRIN|
000017a0  54 20 22 4a 2e 44 2e 3a  20 22 3b 4a 3b 46 0d 38  |T "J.D.: ";J;F.8|
000017b0  37 35 20 45 4e 44 0d 0d  0d 4c 69 73 74 69 6e 67  |75 END...Listing|
000017c0  20 32 0d 0d 39 30 30 20  52 45 4d 20 20 20 4a 44  | 2..900 REM   JD|
000017d0  20 2d 2d 2d 3e 20 43 41  4c 45 4e 44 41 52 0d 39  | ---> CALENDAR.9|
000017e0  30 35 20 52 45 4d 0d 39  31 30 20 49 4e 50 55 54  |05 REM.910 INPUT|
000017f0  20 22 4a 2c 46 20 22 3b  4a 2c 46 0d 39 31 35 20  | "J,F ";J,F.915 |
00001800  49 4e 50 55 54 20 22 4a  43 20 28 30 29 20 4f 52  |INPUT "JC (0) OR|
00001810  20 47 43 20 28 31 29 20  22 3b 47 0d 39 32 30 20  | GC (1) ";G.920 |
00001820  46 3d 46 2b 30 2e 35 0d  39 32 35 20 49 46 20 46  |F=F+0.5.925 IF F|
00001830  3c 31 20 54 48 45 4e 20  39 33 35 0d 39 33 30 20  |<1 THEN 935.930 |
00001840  46 3d 46 2d 31 3a 4a 3d  4a 2b 31 0d 39 33 35 20  |F=F-1:J=J+1.935 |
00001850  49 46 20 47 3d 31 20 54  48 45 4e 20 39 34 35 0d  |IF G=1 THEN 945.|
00001860  39 34 30 20 41 3d 4a 3a  20 47 4f 54 4f 20 39 35  |940 A=J: GOTO 95|
00001870  35 0d 39 34 35 20 41 31  3d 49 4e 54 28 28 4a 2f  |5.945 A1=INT((J/|
00001880  33 36 35 32 34 2e 32 35  29 2d 35 31 2e 31 32 32  |36524.25)-51.122|
00001890  36 34 29 0d 39 35 30 20  41 3d 4a 2b 31 2b 41 31  |64).950 A=J+1+A1|
000018a0  2d 49 4e 54 28 41 31 2f  34 29 0d 39 35 35 20 42  |-INT(A1/4).955 B|
000018b0  3d 41 2b 31 35 32 34 0d  39 36 30 20 43 3d 49 4e  |=A+1524.960 C=IN|
000018c0  54 28 28 42 2f 33 36 35  2e 32 35 29 2d 30 2e 33  |T((B/365.25)-0.3|
000018d0  33 34 33 29 0d 39 36 35  20 44 3d 49 4e 54 28 33  |343).965 D=INT(3|
000018e0  36 35 2e 32 35 2a 43 29  0d 39 37 30 20 45 3d 49  |65.25*C).970 E=I|
000018f0  4e 54 28 28 42 2d 44 29  2f 33 30 2e 36 31 29 0d  |NT((B-D)/30.61).|
00001900  39 37 35 20 44 3d 42 2d  44 2d 49 4e 54 28 33 30  |975 D=B-D-INT(30|
00001910  2e 36 31 2a 45 29 2b 46  0d 39 38 30 20 4d 3d 45  |.61*E)+F.980 M=E|
00001920  2d 31 3a 20 59 3d 43 2d  34 37 31 36 0d 39 38 35  |-1: Y=C-4716.985|
00001930  20 49 46 20 45 3e 31 33  2e 35 20 54 48 45 4e 20  | IF E>13.5 THEN |
00001940  4d 3d 4d 2d 31 32 0d 39  39 30 20 49 46 20 4d 3c  |M=M-12.990 IF M<|
00001950  32 2e 35 20 54 48 45 4e  20 59 3d 59 2b 31 0d 39  |2.5 THEN Y=Y+1.9|
00001960  39 35 20 50 52 49 4e 54  20 22 44 41 54 45 3a 20  |95 PRINT "DATE: |
00001970  22 3b 59 3b 4d 3b 44 0d  39 39 37 20 45 4e 44 0d  |";Y;M;D.997 END.|
00001980  0d 4e 6f 74 65 20 6f 6e  20 76 61 72 69 61 74 69  |.Note on variati|
00001990  6f 6e 73 3a 20 20 57 69  74 68 20 52 61 64 69 6f  |ons:  With Radio|
000019a0  20 53 68 61 63 6b 20 4c  65 76 65 6c 20 49 49 20  | Shack Level II |
000019b0  42 61 73 69 63 2c 20 72  65 70 6c 61 63 65 20 4a  |Basic, replace J|
000019c0  0d 69 6e 20 6c 69 6e 65  20 38 37 30 20 77 69 74  |.in line 870 wit|
000019d0  68 20 43 44 42 4c 28 4a  29 2e 20 20 49 6e 20 41  |h CDBL(J).  In A|
000019e0  70 70 6c 65 73 6f 66 74  2c 20 75 73 65 20 63 6f  |pplesoft, use co|
000019f0  6d 6d 61 73 20 72 61 74  68 65 72 20 74 68 61 6e  |mmas rather than|
00001a00  0d 73 65 6d 69 63 6f 6c  6f 6e 73 20 69 6e 20 6c  |.semicolons in l|
00001a10  69 6e 65 73 20 38 37 30  20 61 6e 64 20 39 39 35  |ines 870 and 995|
00001a20  2e 20 20 53 6f 6d 65 20  42 61 73 69 63 73 20 72  |.  Some Basics r|
00001a30  65 71 75 69 72 65 20 61  20 62 61 63 6b 2d 0d 73  |equire a back-.s|
00001a40  6c 61 73 68 20 77 68 65  72 65 20 63 6f 6c 6f 6e  |lash where colon|
00001a50  73 20 61 72 65 20 75 73  65 64 20 68 65 72 65 2e  |s are used here.|
00001a60  0d 0d 20 20 20 20 20 4c  69 73 74 69 6e 67 20 32  |..     Listing 2|
00001a70  20 70 65 72 66 6f 72 6d  73 20 74 68 65 20 69 6e  | performs the in|
00001a80  76 65 72 73 65 20 63 61  6c 63 75 6c 61 74 69 6f  |verse calculatio|
00001a90  6e 2e 20 20 49 6e 69 74  69 61 6c 6c 79 2c 20 4a  |n.  Initially, J|
00001aa0  0d 69 73 20 74 68 65 20  4a 75 6c 69 61 6e 20 44  |.is the Julian D|
00001ab0  61 79 20 6e 75 6d 62 65  72 2c 20 61 6e 64 20 46  |ay number, and F|
00001ac0  20 74 68 65 20 66 72 61  63 74 69 6f 6e 61 6c 20  | the fractional |
00001ad0  70 61 72 74 20 6f 66 20  61 20 64 61 79 2e 0d 57  |part of a day..W|
00001ae0  68 65 6e 20 74 68 65 20  70 72 6f 67 72 61 6d 20  |hen the program |
00001af0  66 69 6e 69 73 68 65 73  2c 20 59 20 69 73 20 74  |finishes, Y is t|
00001b00  68 65 20 79 65 61 72 2c  20 4d 20 74 68 65 20 6d  |he year, M the m|
00001b10  6f 6e 74 68 2c 20 61 6e  64 20 44 20 74 68 65 0d  |onth, and D the.|
00001b20  64 61 79 20 28 77 69 74  68 20 66 72 61 63 74 69  |day (with fracti|
00001b30  6f 6e 29 2e 20 20 54 68  69 73 20 70 72 6f 67 72  |on).  This progr|
00001b40  61 6d 20 69 73 20 62 61  73 65 64 20 6f 6e 20 74  |am is based on t|
00001b50  68 65 20 61 6c 67 6f 72  69 74 68 6d 20 69 6e 0d  |he algorithm in.|
00001b60  4a 65 61 6e 20 4d 65 65  75 73 27 73 20 41 73 74  |Jean Meeus's Ast|
00001b70  72 6f 6e 6f 6d 69 63 61  6c 20 46 6f 72 6d 75 6c  |ronomical Formul|
00001b80  61 65 20 66 6f 72 20 43  61 6c 63 75 6c 61 74 6f  |ae for Calculato|
00001b90  72 73 20 28 57 69 6c 6c  6d 61 6e 6e 2d 0d 42 65  |rs (Willmann-.Be|
00001ba0  6c 6c 2c 20 31 39 38 32  29 2c 20 77 69 74 68 20  |ll, 1982), with |
00001bb0  73 6f 6d 65 20 72 65 61  72 72 61 6e 67 65 6d 65  |some rearrangeme|
00001bc0  6e 74 20 74 6f 20 73 75  69 74 20 74 68 65 20 6d  |nt to suit the m|
00001bd0  61 6e 79 20 6d 69 63 72  6f 2d 0d 63 6f 6d 70 75  |any micro-.compu|
00001be0  74 65 72 73 20 77 69 74  68 20 6f 6e 6c 79 20 73  |ters with only s|
00001bf0  65 76 65 6e 20 64 69 67  69 74 20 61 63 63 75 72  |even digit accur|
00001c00  61 63 79 2e 20 20 48 65  72 65 20 61 67 61 69 6e  |acy.  Here again|
00001c10  2c 20 47 0d 64 65 74 65  72 6d 69 6e 65 73 20 74  |, G.determines t|
00001c20  68 65 20 63 61 6c 65 6e  64 61 72 20 73 74 79 6c  |he calendar styl|
00001c30  65 2e 20 20 28 54 68 65  20 4d 65 65 75 73 20 62  |e.  (The Meeus b|
00001c40  6f 6f 6b 20 61 6c 73 6f  20 64 65 73 63 72 69 62  |ook also describ|
00001c50  65 73 0d 74 68 65 20 69  6e 76 65 72 73 65 20 63  |es.the inverse c|
00001c60  61 6c 63 75 6c 61 74 69  6f 6e 2c 20 62 79 20 61  |alculation, by a|
00001c70  20 6d 65 74 68 6f 64 20  64 69 66 66 65 72 65 6e  | method differen|
00001c80  74 20 66 72 6f 6d 20 74  68 61 74 0d 64 65 73 63  |t from that.desc|
00001c90  72 69 62 65 64 20 68 65  72 65 20 62 75 74 20 69  |ribed here but i|
00001ca0  64 65 6e 74 69 63 61 6c  20 69 6e 20 72 65 73 75  |dentical in resu|
00001cb0  6c 74 2e 29 0d 0d 20 20  20 20 20 46 6f 72 20 65  |lt.)..     For e|
00001cc0  78 61 6d 70 6c 65 2c 20  63 6f 6e 73 69 64 65 72  |xample, consider|
00001cd0  20 74 68 65 20 67 65 6f  63 65 6e 74 72 69 63 20  | the geocentric |
00001ce0  63 6f 6e 6a 75 6e 63 74  69 6f 6e 20 6f 66 20 74  |conjunction of t|
00001cf0  68 65 20 53 75 6e 0d 61  6e 64 20 4d 6f 6f 6e 20  |he Sun.and Moon |
00001d00  74 68 69 73 20 6d 6f 6e  74 68 2e 20 20 49 74 20  |this month.  It |
00001d10  6f 63 63 75 72 73 20 61  74 20 31 36 3a 35 32 3a  |occurs at 16:52:|
00001d20  33 39 20 55 54 20 6f 6e  20 4d 61 79 20 33 30 74  |39 UT on May 30t|
00001d30  68 2c 20 73 6f 0d 74 68  65 20 64 61 79 20 69 73  |h, so.the day is|
00001d40  20 65 78 70 72 65 73 73  65 64 20 64 65 63 69 6d  | expressed decim|
00001d50  61 6c 6c 79 20 61 73 20  33 30 2e 37 30 33 32 33  |ally as 30.70323|
00001d60  20 61 6e 64 20 74 68 65  20 64 69 61 6c 6f 67 75  | and the dialogu|
00001d70  65 20 72 75 6e 73 3a 0d  0d 20 20 20 3e 52 55 4e  |e runs:..   >RUN|
00001d80  20 22 43 41 4c 4a 44 22  0d 20 20 20 59 2c 4d 2c  | "CALJD".   Y,M,|
00001d90  44 20 3f 20 31 39 38 34  2c 20 35 2c 20 33 30 2e  |D ? 1984, 5, 30.|
00001da0  37 30 33 32 33 0d 20 20  20 4a 43 20 28 30 29 20  |70323.   JC (0) |
00001db0  4f 52 20 47 43 20 28 31  29 20 3f 20 31 0d 20 20  |OR GC (1) ? 1.  |
00001dc0  20 4a 2e 44 2e 3a 20 20  32 34 34 35 38 35 31 20  | J.D.:  2445851 |
00001dd0  2e 32 30 33 32 33 0d 20  20 20 52 45 41 44 59 0d  |.20323.   READY.|
00001de0  0d 20 20 20 3e 52 55 4e  20 22 4a 44 43 41 4c 22  |.   >RUN "JDCAL"|
00001df0  0d 20 20 20 4a 2c 46 20  3f 20 32 34 34 35 38 35  |.   J,F ? 244585|
00001e00  31 2c 20 2e 32 30 33 32  33 0d 20 20 20 4a 43 20  |1, .20323.   JC |
00001e10  28 30 29 20 4f 52 20 47  43 20 28 31 29 20 3f 20  |(0) OR GC (1) ? |
00001e20  31 0d 20 20 20 44 41 54  45 3a 20 20 31 39 38 34  |1.   DATE:  1984|
00001e30  20 20 35 20 20 33 30 2e  37 30 33 32 33 0d 20 20  |  5  30.70323.  |
00001e40  20 52 45 41 44 59 0d 0d  20 20 20 20 20 46 75 74  | READY..     Fut|
00001e50  75 72 65 20 70 72 6f 67  72 61 6d 20 6c 69 73 74  |ure program list|
00001e60  69 6e 67 73 20 69 6e 20  74 68 69 73 20 64 65 70  |ings in this dep|
00001e70  61 72 74 6d 65 6e 74 20  77 69 6c 6c 20 6f 66 74  |artment will oft|
00001e80  65 6e 20 6d 61 6b 65 0d  75 73 65 20 6f 66 20 74  |en make.use of t|
00001e90  68 65 20 4a 75 6c 69 61  6e 20 44 61 79 20 6e 75  |he Julian Day nu|
00001ea0  6d 62 65 72 20 61 6e 64  20 66 72 61 63 74 69 6f  |mber and fractio|
00001eb0  6e 2c 20 73 6f 20 72 65  61 64 65 72 73 20 6d 61  |n, so readers ma|
00001ec0  79 20 77 69 73 68 0d 74  6f 20 72 65 77 72 69 74  |y wish.to rewrit|
00001ed0  65 20 74 68 65 73 65 20  6c 69 73 74 69 6e 67 73  |e these listings|
00001ee0  20 61 73 20 73 75 62 72  6f 75 74 69 6e 65 73 2e  | as subroutines.|
00001ef0  20 20 41 66 74 65 72 20  72 65 70 6c 61 63 69 6e  |  After replacin|
00001f00  67 20 65 61 63 68 0d 45  4e 44 20 73 74 61 74 65  |g each.END state|
00001f10  6d 65 6e 74 20 77 69 74  68 20 52 45 54 55 52 4e  |ment with RETURN|
00001f20  2c 20 79 6f 75 20 63 61  6e 20 73 69 6d 70 6c 79  |, you can simply|
00001f30  20 69 6e 73 65 72 74 20  74 68 65 6d 20 62 6f 64  | insert them bod|
00001f40  69 6c 79 0d 69 6e 74 6f  20 6f 74 68 65 72 20 70  |ily.into other p|
00001f50  72 6f 67 72 61 6d 73 2e  0d 0d 20 20 20 20 20 42  |rograms...     B|
00001f60  6f 74 68 20 6c 69 73 74  69 6e 67 73 20 68 61 76  |oth listings hav|
00001f70  65 20 62 65 65 6e 20 74  68 6f 72 6f 75 67 68 6c  |e been thoroughl|
00001f80  79 20 74 65 73 74 65 64  20 61 74 20 53 6b 79 20  |y tested at Sky |
00001f90  26 0d 54 65 6c 65 73 63  6f 70 65 20 6f 6e 20 6d  |&.Telescope on m|
00001fa0  61 6e 79 20 6d 69 63 72  6f 63 6f 6d 70 75 74 65  |any microcompute|
00001fb0  72 73 2c 20 66 6f 72 20  74 68 65 20 79 65 61 72  |rs, for the year|
00001fc0  73 20 2d 34 37 31 32 20  74 68 72 6f 75 67 68 20  |s -4712 through |
00001fd0  74 6f 0d 2b 33 35 30 30  20 69 6e 20 62 6f 74 68  |to.+3500 in both|
00001fe0  20 74 68 65 20 47 72 65  67 6f 72 69 61 6e 20 61  | the Gregorian a|
00001ff0  6e 64 20 4a 75 6c 69 61  6e 20 63 61 6c 65 6e 64  |nd Julian calend|
00002000  61 72 73 2e 20 20 42 75  74 20 69 74 27 73 0d 77  |ars.  But it's.w|
00002010  69 73 65 2c 20 77 68 65  6e 20 74 72 79 69 6e 67  |ise, when trying|
00002020  20 6f 75 74 20 74 68 65  20 70 72 6f 67 72 61 6d  | out the program|
00002030  73 20 6f 6e 20 79 6f 75  72 20 6f 77 6e 20 6d 61  |s on your own ma|
00002040  63 68 69 6e 65 2c 20 74  6f 0d 76 65 72 69 66 79  |chine, to.verify|
00002050  20 74 68 65 20 72 65 73  75 6c 74 73 20 6f 6e 20  | the results on |
00002060  73 6f 6d 65 20 6b 6e 6f  77 6e 20 65 71 75 69 76  |some known equiv|
00002070  61 6c 65 6e 74 73 3a 0d  0d 20 20 20 20 20 20 4a  |alents:..      J|
00002080  75 6c 69 61 6e 20 43 61  6c 2e 20 20 20 20 20 20  |ulian Cal.      |
00002090  20 20 20 47 72 65 67 6f  72 69 61 6e 20 43 61 6c  |   Gregorian Cal|
000020a0  2e 20 20 20 20 20 20 4a  75 6c 69 61 6e 20 44 61  |.      Julian Da|
000020b0  74 65 0d 0d 20 20 20 20  2d 34 37 31 32 20 4a 61  |te..    -4712 Ja|
000020c0  6e 2e 20 20 31 2e 35 20  20 20 20 20 20 2d 34 37  |n.  1.5      -47|
000020d0  31 33 20 4e 6f 76 2e 20  32 34 2e 35 20 20 20 20  |13 Nov. 24.5    |
000020e0  20 20 20 20 20 20 20 20  20 30 2e 30 0d 20 20 20  |         0.0.   |
000020f0  20 2d 32 30 30 30 20 4a  61 6e 2e 20 20 31 2e 30  | -2000 Jan.  1.0|
00002100  20 20 20 20 20 20 2d 32  30 30 31 20 44 65 63 2e  |      -2001 Dec.|
00002110  20 31 35 2e 30 20 20 20  20 20 20 20 20 39 39 30  | 15.0        990|
00002120  35 35 37 2e 35 0d 20 20  20 20 20 2d 35 38 34 20  |557.5.     -584 |
00002130  4d 61 79 20 20 32 38 2e  36 20 20 20 20 20 20 20  |May  28.6       |
00002140  2d 35 38 34 20 4d 61 79  20 20 32 32 2e 36 20 20  |-584 May  22.6  |
00002150  20 20 20 20 20 31 35 30  37 39 30 30 2e 31 0d 20  |     1507900.1. |
00002160  20 20 20 20 2b 32 30 30  20 4d 61 72 2e 20 20 31  |    +200 Mar.  1|
00002170  2e 30 20 20 20 20 20 20  20 2b 32 30 30 20 4d 61  |.0       +200 Ma|
00002180  72 2e 20 20 31 2e 30 20  20 20 20 20 20 20 31 37  |r.  1.0       17|
00002190  39 34 31 36 37 2e 35 0d  20 20 20 20 2b 31 39 38  |94167.5.    +198|
000021a0  34 20 46 65 62 2e 20 31  36 2e 32 20 20 20 20 20  |4 Feb. 16.2     |
000021b0  20 2b 31 39 38 34 20 46  65 62 2e 20 32 39 2e 32  | +1984 Feb. 29.2|
000021c0  20 20 20 20 20 20 20 32  34 34 35 37 35 39 2e 37  |       2445759.7|
000021d0  0d 20 20 20 20 2b 31 39  39 39 20 44 65 63 2e 20  |.    +1999 Dec. |
000021e0  31 39 2e 35 20 20 20 20  20 20 2b 32 30 30 30 20  |19.5      +2000 |
000021f0  4a 61 6e 2e 20 20 31 2e  35 20 20 20 20 20 20 20  |Jan.  1.5       |
00002200  32 34 35 31 35 34 35 2e  30 0d 20 20 20 20 2b 33  |2451545.0.    +3|
00002210  30 30 30 20 46 65 62 2e  20 32 39 2e 39 20 20 20  |000 Feb. 29.9   |
00002220  20 20 20 2b 33 30 30 30  20 4d 61 72 2e 20 32 31  |   +3000 Mar. 21|
00002230  2e 39 20 20 20 20 20 20  20 32 38 31 36 38 36 37  |.9       2816867|
00002240  2e 34 0d 0d 20 20 20 20  20 4d 61 6e 79 20 61 6e  |.4..     Many an|
00002250  63 69 65 6e 74 20 63 61  6c 65 6e 64 61 72 73 20  |cient calendars |
00002260  68 61 76 65 20 62 65 65  6e 20 6c 69 6e 6b 65 64  |have been linked|
00002270  20 62 79 20 4a 75 6c 69  61 6e 20 44 61 79 0d 6e  | by Julian Day.n|
00002280  75 6d 62 65 72 20 69 6e  20 4f 2e 4c 2e 48 61 72  |umber in O.L.Har|
00002290  76 65 79 27 73 20 6e 65  77 20 62 6f 6f 6b 6c 65  |vey's new bookle|
000022a0  74 2c 20 43 61 6c 65 6e  64 61 72 20 43 6f 6e 76  |t, Calendar Conv|
000022b0  65 72 73 69 6f 6e 73 0d  28 41 6d 65 72 69 63 61  |ersions.(America|
000022c0  6e 20 50 68 69 6c 6f 73  6f 70 68 69 63 61 6c 20  |n Philosophical |
000022d0  53 6f 63 69 65 74 79 2c  20 31 39 38 33 29 2e 20  |Society, 1983). |
000022e0  20 48 65 20 61 6c 73 6f  20 6d 65 6e 74 69 6f 6e  | He also mention|
000022f0  73 20 74 77 6f 0d 75 73  65 66 75 6c 20 74 72 69  |s two.useful tri|
00002300  63 6b 73 20 77 69 74 68  20 74 68 65 73 65 20 6e  |cks with these n|
00002310  75 6d 62 65 72 73 2e 20  20 54 6f 20 66 69 6e 64  |umbers.  To find|
00002320  20 74 68 65 20 64 61 79  20 6f 66 20 74 68 65 20  | the day of the |
00002330  77 65 65 6b 0d 61 74 20  6e 6f 6f 6e 20 77 68 65  |week.at noon whe|
00002340  6e 20 65 61 63 68 20 4a  75 6c 69 61 6e 20 44 61  |n each Julian Da|
00002350  79 20 6e 75 6d 62 65 72  20 62 65 67 69 6e 73 2c  |y number begins,|
00002360  20 64 69 76 69 64 65 20  74 68 65 20 77 68 6f 6c  | divide the whol|
00002370  65 0d 6e 75 6d 62 65 72  20 62 79 20 37 3b 20 61  |e.number by 7; a|
00002380  20 72 65 6d 61 69 6e 64  65 72 20 6f 66 20 30 20  | remainder of 0 |
00002390  6d 65 61 6e 73 20 4d 6f  6e 64 61 79 2c 20 31 20  |means Monday, 1 |
000023a0  54 75 65 73 64 61 79 2c  20 61 6e 64 20 73 6f 0d  |Tuesday, and so.|
000023b0  6f 6e 2c 20 74 68 72 6f  75 67 68 20 74 6f 20 36  |on, through to 6|
000023c0  20 66 6f 72 20 53 75 6e  64 61 79 2e 20 20 46 75  | for Sunday.  Fu|
000023d0  72 74 68 65 72 6d 6f 72  65 2c 20 74 68 65 20 4d  |rthermore, the M|
000023e0  6f 6f 6e 20 69 73 20 77  69 74 68 69 6e 20 61 0d  |oon is within a.|
000023f0  64 61 79 20 6f 72 20 73  6f 20 6f 66 20 6e 65 77  |day or so of new|
00002400  20 6f 6e 20 74 68 6f 73  65 20 64 61 79 73 20 77  | on those days w|
00002410  68 65 6e 20 74 68 65 20  4a 75 6c 69 61 6e 20 44  |hen the Julian D|
00002420  61 79 20 6e 75 6d 62 65  72 2c 0d 64 69 76 69 64  |ay number,.divid|
00002430  65 64 20 62 79 20 32 39  2e 35 33 30 35 38 38 2c  |ed by 29.530588,|
00002440  20 79 69 65 6c 64 73 20  2e 33 33 20 61 73 20 74  | yields .33 as t|
00002450  68 65 20 66 72 61 63 74  69 6f 6e 61 6c 20 70 61  |he fractional pa|
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*
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00003000  0d                                                |.|
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