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@(#)Theory 7.15
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1999-01-21 17:31:43 +00:00
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----- Outline -----
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Time and date functions
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Names of time zone regions
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Time zone abbreviations
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2004-06-20 11:52:51 +00:00
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Calendrical issues
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Time and time zones on Mars
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1999-01-21 17:31:43 +00:00
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----- Time and date functions -----
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These time and date functions are upwards compatible with POSIX.1,
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an international standard for UNIX-like systems.
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As of this writing, the current edition of POSIX.1 is:
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Information technology --Portable Operating System Interface (POSIX (R))
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-- Part 1: System Application Program Interface (API) [C Language]
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ISO/IEC 9945-1:1996
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ANSI/IEEE Std 1003.1, 1996 Edition
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1996-07-12
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POSIX.1 has the following properties and limitations.
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* In POSIX.1, time display in a process is controlled by the
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environment variable TZ. Unfortunately, the POSIX.1 TZ string takes
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a form that is hard to describe and is error-prone in practice.
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Also, POSIX.1 TZ strings can't deal with other (for example, Israeli)
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daylight saving time rules, or situations where more than two
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time zone abbreviations are used in an area.
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The POSIX.1 TZ string takes the following form:
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stdoffset[dst[offset],date[/time],date[/time]]
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where:
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std and dst
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are 3 or more characters specifying the standard
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and daylight saving time (DST) zone names.
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offset
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is of the form `[-]hh:[mm[:ss]]' and specifies the
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offset west of UTC. The default DST offset is one hour
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ahead of standard time.
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date[/time],date[/time]
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specifies the beginning and end of DST. If this is absent,
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the system supplies its own rules for DST, and these can
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differ from year to year; typically US DST rules are used.
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time
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takes the form `hh:[mm[:ss]]' and defaults to 02:00.
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date
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takes one of the following forms:
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Jn (1<=n<=365)
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origin-1 day number not counting February 29
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n (0<=n<=365)
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origin-0 day number counting February 29 if present
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Mm.n.d (0[Sunday]<=d<=6[Saturday], 1<=n<=5, 1<=m<=12)
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for the dth day of week n of month m of the year,
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where week 1 is the first week in which day d appears,
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and `5' stands for the last week in which day d appears
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(which may be either the 4th or 5th week).
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* In POSIX.1, when a TZ value like "EST5EDT" is parsed,
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typically the current US DST rules are used,
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but this means that the US DST rules are compiled into each program
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that does time conversion. This means that when US time conversion
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rules change (as in the United States in 1987), all programs that
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do time conversion must be recompiled to ensure proper results.
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* In POSIX.1, there's no tamper-proof way for a process to learn the
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system's best idea of local wall clock. (This is important for
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applications that an administrator wants used only at certain times--
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without regard to whether the user has fiddled the "TZ" environment
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variable. While an administrator can "do everything in UTC" to get
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around the problem, doing so is inconvenient and precludes handling
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daylight saving time shifts--as might be required to limit phone
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calls to off-peak hours.)
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* POSIX.1 requires that systems ignore leap seconds.
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These are the extensions that have been made to the POSIX.1 functions:
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* The "TZ" environment variable is used in generating the name of a file
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from which time zone information is read (or is interpreted a la
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POSIX); "TZ" is no longer constrained to be a three-letter time zone
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name followed by a number of hours and an optional three-letter
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daylight time zone name. The daylight saving time rules to be used
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for a particular time zone are encoded in the time zone file;
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the format of the file allows U.S., Australian, and other rules to be
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encoded, and allows for situations where more than two time zone
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abbreviations are used.
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It was recognized that allowing the "TZ" environment variable to
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take on values such as "America/New_York" might cause "old" programs
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(that expect "TZ" to have a certain form) to operate incorrectly;
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consideration was given to using some other environment variable
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(for example, "TIMEZONE") to hold the string used to generate the
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time zone information file name. In the end, however, it was decided
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to continue using "TZ": it is widely used for time zone purposes;
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separately maintaining both "TZ" and "TIMEZONE" seemed a nuisance;
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and systems where "new" forms of "TZ" might cause problems can simply
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use TZ values such as "EST5EDT" which can be used both by
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"new" programs (a la POSIX) and "old" programs (as zone names and
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offsets).
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* To handle places where more than two time zone abbreviations are used,
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the functions "localtime" and "gmtime" set tzname[tmp->tm_isdst]
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(where "tmp" is the value the function returns) to the time zone
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abbreviation to be used. This differs from POSIX.1, where the elements
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of tzname are only changed as a result of calls to tzset.
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* Since the "TZ" environment variable can now be used to control time
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conversion, the "daylight" and "timezone" variables are no longer
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needed. (These variables are defined and set by "tzset"; however, their
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values will not be used by "localtime.")
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* The "localtime" function has been set up to deliver correct results
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for near-minimum or near-maximum time_t values. (A comment in the
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source code tells how to get compatibly wrong results).
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* A function "tzsetwall" has been added to arrange for the system's
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best approximation to local wall clock time to be delivered by
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subsequent calls to "localtime." Source code for portable
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applications that "must" run on local wall clock time should call
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"tzsetwall();" if such code is moved to "old" systems that don't
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provide tzsetwall, you won't be able to generate an executable program.
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(These time zone functions also arrange for local wall clock time to be
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used if tzset is called--directly or indirectly--and there's no "TZ"
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environment variable; portable applications should not, however, rely
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on this behavior since it's not the way SVR2 systems behave.)
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* These functions can account for leap seconds, thanks to Bradley White
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(bww@k.cs.cmu.edu).
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Points of interest to folks with other systems:
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* This package is already part of many POSIX-compliant hosts,
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including BSD, HP, Linux, Network Appliance, SCO, SGI, and Sun.
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On such hosts, the primary use of this package
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is to update obsolete time zone rule tables.
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To do this, you may need to compile the time zone compiler
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`zic' supplied with this package instead of using the system `zic',
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since the format of zic's input changed slightly in late 1994,
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and many vendors still do not support the new input format.
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* The UNIX Version 7 "timezone" function is not present in this package;
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it's impossible to reliably map timezone's arguments (a "minutes west
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of GMT" value and a "daylight saving time in effect" flag) to a
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time zone abbreviation, and we refuse to guess.
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Programs that in the past used the timezone function may now examine
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tzname[localtime(&clock)->tm_isdst] to learn the correct time
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zone abbreviation to use. Alternatively, use
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localtime(&clock)->tm_zone if this has been enabled.
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* The 4.2BSD gettimeofday function is not used in this package.
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This formerly let users obtain the current UTC offset and DST flag,
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but this functionality was removed in later versions of BSD.
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* In SVR2, time conversion fails for near-minimum or near-maximum
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time_t values when doing conversions for places that don't use UTC.
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This package takes care to do these conversions correctly.
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1996-07-18 19:40:52 +00:00
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The functions that are conditionally compiled if STD_INSPIRED is defined
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should, at this point, be looked on primarily as food for thought. They are
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not in any sense "standard compatible"--some are not, in fact, specified in
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*any* standard. They do, however, represent responses of various authors to
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standardization proposals.
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Other time conversion proposals, in particular the one developed by folks at
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Hewlett Packard, offer a wider selection of functions that provide capabilities
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beyond those provided here. The absence of such functions from this package
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is not meant to discourage the development, standardization, or use of such
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functions. Rather, their absence reflects the decision to make this package
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contain valid extensions to POSIX.1, to ensure its broad
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acceptability. If more powerful time conversion functions can be standardized,
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so much the better.
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----- Names of time zone rule files -----
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The time zone rule file naming conventions attempt to strike a balance
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among the following goals:
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* Uniquely identify every national region where clocks have all
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agreed since 1970. This is essential for the intended use: static
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clocks keeping local civil time.
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* Indicate to humans as to where that region is. This simplifes use.
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* Be robust in the presence of political changes. This reduces the
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number of updates and backward-compatibility hacks. For example,
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names of countries are ordinarily not used, to avoid
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incompatibilities when countries change their name
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(e.g. Zaire->Congo) or when locations change countries
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(e.g. Hong Kong from UK colony to China).
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* Be portable to a wide variety of implementations.
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This promotes use of the technology.
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* Use a consistent naming convention over the entire world.
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This simplifies both use and maintenance.
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This naming convention is not intended for use by inexperienced users
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to select TZ values by themselves (though they can of course examine
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and reuse existing settings). Distributors should provide
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documentation and/or a simple selection interface that explains the
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names; see the 'tzselect' program supplied with this distribution for
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one example.
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Names normally have the form AREA/LOCATION, where AREA is the name
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of a continent or ocean, and LOCATION is the name of a specific
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location within that region. North and South America share the same
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area, `America'. Typical names are `Africa/Cairo', `America/New_York',
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and `Pacific/Honolulu'.
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Here are the general rules used for choosing location names,
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in decreasing order of importance:
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Use only valid POSIX file name components (i.e., the parts of
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names other than `/'). Within a file name component,
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use only ASCII letters, `.', `-' and `_'. Do not use
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digits, as that might create an ambiguity with POSIX
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TZ strings. A file name component must not exceed 14
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characters or start with `-'. E.g., prefer `Brunei'
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to `Bandar_Seri_Begawan'.
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Include at least one location per time zone rule set per country.
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One such location is enough. Use ISO 3166 (see the file
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iso3166.tab) to help decide whether something is a country.
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If all the clocks in a country's region have agreed since 1970,
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don't bother to include more than one location
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even if subregions' clocks disagreed before 1970.
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Otherwise these tables would become annoyingly large.
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If a name is ambiguous, use a less ambiguous alternative;
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e.g. many cities are named San Jose and Georgetown, so
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prefer `Costa_Rica' to `San_Jose' and `Guyana' to `Georgetown'.
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Keep locations compact. Use cities or small islands, not countries
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or regions, so that any future time zone changes do not split
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locations into different time zones. E.g. prefer `Paris'
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to `France', since France has had multiple time zones.
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Use mainstream English spelling, e.g. prefer `Rome' to `Roma', and
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prefer `Athens' to the true name (which uses Greek letters).
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The POSIX file name restrictions encourage this rule.
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Use the most populous among locations in a country's time zone,
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e.g. prefer `Shanghai' to `Beijing'. Among locations with
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similar populations, pick the best-known location,
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e.g. prefer `Rome' to `Milan'.
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Use the singular form, e.g. prefer `Canary' to `Canaries'.
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Omit common suffixes like `_Islands' and `_City', unless that
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would lead to ambiguity. E.g. prefer `Cayman' to
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`Cayman_Islands' and `Guatemala' to `Guatemala_City',
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but prefer `Mexico_City' to `Mexico' because the country
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of Mexico has several time zones.
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Use `_' to represent a space.
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Omit `.' from abbreviations in names, e.g. prefer `St_Helena'
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to `St._Helena'.
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Do not change established names if they only marginally
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violate the above rules. For example, don't change
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the existing name `Rome' to `Milan' merely because
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Milan's population has grown to be somewhat greater
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than Rome's.
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If a name is changed, put its old spelling in the `backward' file.
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The file `zone.tab' lists the geographical locations used to name
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time zone rule files.
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Older versions of this package used a different naming scheme,
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and these older names are still supported.
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See the file `backward' for most of these older names
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(e.g. `US/Eastern' instead of `America/New_York').
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The other old-fashioned names still supported are
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`WET', `CET', `MET', `EET' (see the file `europe'),
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and `Factory' (see the file `factory').
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----- Time zone abbreviations -----
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When this package is installed, it generates time zone abbreviations
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like `EST' to be compatible with human tradition and POSIX.1.
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Here are the general rules used for choosing time zone abbreviations,
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in decreasing order of importance:
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Use abbreviations that consist of three or more ASCII letters.
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Previous editions of this database also used characters like
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' ' and '?', but these characters have a special meaning to
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the shell and cause commands like
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set `date`
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to have unexpected effects.
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Previous editions of this rule required upper-case letters,
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but the Congressman who introduced Chamorro Standard Time
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preferred "ChST", so the rule has been relaxed.
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This rule guarantees that all abbreviations could have
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been specified by a POSIX.1 TZ string. POSIX.1
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requires at least three characters for an
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abbreviation. POSIX.1-1996 says that an abbreviation
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cannot start with ':', and cannot contain ',', '-',
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'+', NUL, or a digit. Draft 7 of POSIX 1003.1-200x
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changes this rule to say that an abbreviation can
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contain only '-', '+', and alphanumeric characters in
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|
the current locale. To be portable to both sets of
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rules, an abbreviation must therefore use only ASCII
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letters, as these are the only letters that are
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|
alphabetic in all locales.
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|
1999-01-21 17:31:43 +00:00
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Use abbreviations that are in common use among English-speakers,
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e.g. `EST' for Eastern Standard Time in North America.
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We assume that applications translate them to other languages
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as part of the normal localization process; for example,
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a French application might translate `EST' to `HNE'.
|
2004-06-20 11:52:51 +00:00
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1999-01-21 17:31:43 +00:00
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For zones whose times are taken from a city's longitude, use the
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traditional xMT notation, e.g. `PMT' for Paris Mean Time.
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The only name like this in current use is `GMT'.
|
2004-06-20 11:52:51 +00:00
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1999-01-21 17:31:43 +00:00
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If there is no common English abbreviation, abbreviate the English
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|
translation of the usual phrase used by native speakers.
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If this is not available or is a phrase mentioning the country
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|
(e.g. ``Cape Verde Time''), then:
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When a country has a single or principal time zone region,
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append `T' to the country's ISO code, e.g. `CVT' for
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Cape Verde Time. For summer time append `ST';
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for double summer time append `DST'; etc.
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When a country has multiple time zones, take the first three
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|
letters of an English place name identifying each zone
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|
and then append `T', `ST', etc. as before;
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e.g. `VLAST' for VLAdivostok Summer Time.
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|
2004-06-20 11:52:51 +00:00
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Use "zzz" for locations while uninhabited. The mnemonic is that
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these locations are, in some sense, asleep.
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|
1999-01-21 17:31:43 +00:00
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Application writers should note that these abbreviations are ambiguous
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in practice: e.g. `EST' has a different meaning in Australia than
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|
it does in the United States. In new applications, it's often better
|
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|
|
to use numeric UTC offsets like `-0500' instead of time zone
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|
abbreviations like `EST'; this avoids the ambiguity.
|
2004-06-20 11:52:51 +00:00
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|
----- Calendrical issues -----
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|
Calendrical issues are a bit out of scope for a time zone database,
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|
but they indicate the sort of problems that we would run into if we
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|
extended the time zone database further into the past. An excellent
|
|
|
|
resource in this area is Nachum Dershowitz and Edward M. Reingold,
|
|
|
|
<a href="http://emr.cs.uiuc.edu/home/reingold/calendar-book/index.shtml">
|
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|
|
Calendrical Calculations
|
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|
</a>, Cambridge University Press (1997). Other information and
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|
sources are given below. They sometimes disagree.
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France
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Gregorian calendar adopted 1582-12-20.
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French Revolutionary calendar used 1793-11-24 through 1805-12-31,
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and (in Paris only) 1871-05-06 through 1871-05-23.
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Russia
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|
From Chris Carrier <72157.3334@CompuServe.COM> (1996-12-02):
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|
On 1929-10-01 the Soviet Union instituted an ``Eternal Calendar''
|
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|
with 30-day months plus 5 holidays, with a 5-day week.
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On 1931-12-01 it changed to a 6-day week; in 1934 it reverted to the
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|
Gregorian calendar while retaining the 6-day week; on 1940-06-27 it
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|
reverted to the 7-day week. With the 6-day week the usual days
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|
off were the 6th, 12th, 18th, 24th and 30th of the month.
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|
(Source: Evitiar Zerubavel, _The Seven Day Circle_)
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|
Mark Brader reported a similar story in "The Book of Calendars", edited
|
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|
|
by Frank Parise (1982, Facts on File, ISBN 0-8719-6467-8), page 377. But:
|
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|
|
From: Petteri Sulonen (via Usenet)
|
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|
Date: 14 Jan 1999 00:00:00 GMT
|
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|
Message-ID: <Petteri.Sulonen-1401991626030001@lapin-kulta.in.helsinki.fi>
|
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|
If your source is correct, how come documents between 1929 -- 1940 were
|
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|
still dated using the conventional, Gregorian calendar?
|
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|
I can post a scan of a document dated December 1, 1934, signed by
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|
Yenukidze, the secretary, on behalf of Kalinin, the President of the
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|
Executive Committee of the Supreme Soviet, if you like.
|
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Sweden (and Finland)
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|
From: msb@sq.com (Mark Brader)
|
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|
<a href="news:1996Jul6.012937.29190@sq.com">
|
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|
Subject: Re: Gregorian reform -- a part of locale?
|
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|
</a>
|
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|
Date: 1996-07-06
|
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|
In 1700, Denmark made the transition from Julian to Gregorian. Sweden
|
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|
decided to *start* a transition in 1700 as well, but rather than have one of
|
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|
|
those unsightly calendar gaps :-), they simply decreed that the next leap
|
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|
|
year after 1696 would be in 1744 -- putting the whole country on a calendar
|
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|
|
different from both Julian and Gregorian for a period of 40 years.
|
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|
However, in 1704 something went wrong and the plan was not carried through;
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|
they did, after all, have a leap year that year. And one in 1708. In 1712
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|
they gave it up and went back to Julian, putting 30 days in February that
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|
year!...
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|
Then in 1753, Sweden made the transition to Gregorian in the usual manner,
|
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|
getting there only 13 years behind the original schedule.
|
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|
(A previous posting of this story was challenged, and Swedish readers
|
|
|
|
produced the following references to support it: "Tiderakning och historia"
|
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|
|
by Natanael Beckman (1924) and "Tid, en bok om tiderakning och
|
|
|
|
kalendervasen" by Lars-Olof Lode'n (no date was given).)
|
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|
Grotefend's data
|
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|
|
|
|
|
|
From: "Michael Palmer" <mpalmer@netcom.com> [with one obvious typo fixed]
|
|
|
|
Subject: Re: Gregorian Calendar (was Re: Another FHC related question
|
|
|
|
Newsgroups: soc.genealogy.german
|
|
|
|
Date: Tue, 9 Feb 1999 02:32:48 -800
|
|
|
|
Message-ID: <199902091032.CAA09644@netcom10.netcom.com>
|
|
|
|
|
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|
|
The following is a(n incomplete) listing, arranged chronologically, of
|
|
|
|
European states, with the date they converted from the Julian to the
|
|
|
|
Gregorian calendar:
|
|
|
|
|
|
|
|
04/15 Oct 1582 - Italy (with exceptions), Spain, Portugal, Poland (Roman
|
|
|
|
Catholics and Danzig only)
|
|
|
|
09/20 Dec 1582 - France, Lorraine
|
|
|
|
|
|
|
|
21 Dec 1582/
|
|
|
|
01 Jan 1583 - Holland, Brabant, Flanders, Hennegau
|
|
|
|
10/21 Feb 1583 - bishopric of Liege (L"uttich)
|
|
|
|
13/24 Feb 1583 - bishopric of Augsburg
|
|
|
|
04/15 Oct 1583 - electorate of Trier
|
|
|
|
05/16 Oct 1583 - Bavaria, bishoprics of Freising, Eichstedt, Regensburg,
|
|
|
|
Salzburg, Brixen
|
|
|
|
13/24 Oct 1583 - Austrian Oberelsass and Breisgau
|
|
|
|
20/31 Oct 1583 - bishopric of Basel
|
|
|
|
02/13 Nov 1583 - duchy of J"ulich-Berg
|
|
|
|
02/13 Nov 1583 - electorate and city of K"oln
|
|
|
|
04/15 Nov 1583 - bishopric of W"urzburg
|
|
|
|
11/22 Nov 1583 - electorate of Mainz
|
|
|
|
16/27 Nov 1583 - bishopric of Strassburg and the margraviate of Baden
|
|
|
|
17/28 Nov 1583 - bishopric of M"unster and duchy of Cleve
|
|
|
|
14/25 Dec 1583 - Steiermark
|
|
|
|
|
|
|
|
06/17 Jan 1584 - Austria and Bohemia
|
|
|
|
11/22 Jan 1584 - Luzern, Uri, Schwyz, Zug, Freiburg, Solothurn
|
|
|
|
12/23 Jan 1584 - Silesia and the Lausitz
|
|
|
|
22 Jan/
|
|
|
|
02 Feb 1584 - Hungary (legally on 21 Oct 1587)
|
|
|
|
Jun 1584 - Unterwalden
|
|
|
|
01/12 Jul 1584 - duchy of Westfalen
|
|
|
|
|
|
|
|
16/27 Jun 1585 - bishopric of Paderborn
|
|
|
|
|
|
|
|
14/25 Dec 1590 - Transylvania
|
|
|
|
|
|
|
|
22 Aug/
|
|
|
|
02 Sep 1612 - duchy of Prussia
|
|
|
|
|
|
|
|
13/24 Dec 1614 - Pfalz-Neuburg
|
|
|
|
|
|
|
|
1617 - duchy of Kurland (reverted to the Julian calendar in
|
|
|
|
1796)
|
|
|
|
|
|
|
|
1624 - bishopric of Osnabr"uck
|
|
|
|
|
|
|
|
1630 - bishopric of Minden
|
|
|
|
|
|
|
|
15/26 Mar 1631 - bishopric of Hildesheim
|
|
|
|
|
|
|
|
1655 - Kanton Wallis
|
|
|
|
|
|
|
|
05/16 Feb 1682 - city of Strassburg
|
|
|
|
|
|
|
|
18 Feb/
|
|
|
|
01 Mar 1700 - Protestant Germany (including Swedish possessions in
|
|
|
|
Germany), Denmark, Norway
|
|
|
|
30 Jun/
|
|
|
|
12 Jul 1700 - Gelderland, Zutphen
|
|
|
|
10 Nov/
|
|
|
|
12 Dec 1700 - Utrecht, Overijssel
|
|
|
|
|
|
|
|
31 Dec 1700/
|
|
|
|
12 Jan 1701 - Friesland, Groningen, Z"urich, Bern, Basel, Geneva,
|
|
|
|
Turgau, and Schaffhausen
|
|
|
|
|
|
|
|
1724 - Glarus, Appenzell, and the city of St. Gallen
|
|
|
|
|
|
|
|
01 Jan 1750 - Pisa and Florence
|
|
|
|
|
|
|
|
02/14 Sep 1752 - Great Britain
|
|
|
|
|
|
|
|
17 Feb/
|
|
|
|
01 Mar 1753 - Sweden
|
|
|
|
|
|
|
|
1760-1812 - Graub"unden
|
|
|
|
|
|
|
|
The Russian empire (including Finland and the Baltic states) did not
|
|
|
|
convert to the Gregorian calendar until the Soviet revolution of 1917.
|
|
|
|
|
|
|
|
Source: H. Grotefend, _Taschenbuch der Zeitrechnung des deutschen
|
|
|
|
Mittelalters und der Neuzeit_, herausgegeben von Dr. O. Grotefend
|
|
|
|
(Hannover: Hahnsche Buchhandlung, 1941), pp. 26-28.
|
|
|
|
|
|
|
|
|
|
|
|
----- Time and time zones on Mars -----
|
|
|
|
|
|
|
|
Some people have adjusted their work schedules to fit Mars time.
|
|
|
|
Dozens of special Mars watches were built for Jet Propulsion
|
|
|
|
Laboratory workers who kept Mars time during the Mars Exploration
|
|
|
|
Rovers mission (2004). These timepieces look like normal Seikos and
|
|
|
|
Citizens but use Mars seconds rather than terrestrial seconds.
|
|
|
|
|
|
|
|
A Mars solar day is called a "sol" and has a mean period equal to
|
|
|
|
about 24 hours 39 minutes 35.244 seconds in terrestrial time. It is
|
|
|
|
divided into a conventional 24-hour clock, so each Mars second equals
|
|
|
|
about 1.02749125 terrestrial seconds.
|
|
|
|
|
|
|
|
The prime meridian of Mars goes through the center of the crater
|
|
|
|
Airy-0, named in honor of the British astronomer who built the
|
|
|
|
Greenwich telescope that defines Earth's prime meridian. Mean solar
|
|
|
|
time on the Mars prime meridian is called Mars Coordinated Time (MTC).
|
|
|
|
|
|
|
|
Each landed mission on Mars has adopted a different reference for
|
|
|
|
solar time keeping, so there is no real standard for Mars time zones.
|
|
|
|
For example, the Mars Exploration Rover project (2004) defined two
|
|
|
|
time zones "Local Solar Time A" and "Local Solar Time B" for its two
|
|
|
|
missions, each zone designed so that its time equals local true solar
|
|
|
|
time at approximately the middle of the nominal mission. Such a "time
|
|
|
|
zone" is not particularly suited for any application other than the
|
|
|
|
mission itself.
|
|
|
|
|
|
|
|
Many calendars have been proposed for Mars, but none have achieved
|
|
|
|
wide acceptance. Astronomers often use Mars Sol Date (MSD) which is a
|
|
|
|
sequential count of Mars solar days elapsed since about 1873-12-29
|
|
|
|
12:00 GMT.
|
|
|
|
|
|
|
|
The tz database does not currently support Mars time, but it is
|
|
|
|
documented here in the hopes that support will be added eventually.
|
|
|
|
|
|
|
|
Sources:
|
|
|
|
|
|
|
|
Michael Allison and Robert Schmunk,
|
|
|
|
"Technical Notes on Mars Solar Time as Adopted by the Mars24 Sunclock"
|
|
|
|
<http://www.giss.nasa.gov/tools/mars24/help/notes.html> (2004-03-15).
|
|
|
|
|
|
|
|
Jia-Rui Chong, "Workdays Fit for a Martian", Los Angeles Times
|
|
|
|
(2004-01-14), pp A1, A20-A21.
|