DTV11 — inconsistent statements on day index

REFERENCES:
section 11.2, last Note on p 126:

[a] "The calendar reform instituted by
Pope Gregory XIII [Inter Gravissimas]
deleted 10 days from the
Gregorian calendar, starting on
5 October 1582."

[b] "The previous calendar day had
index 577 739 on the Julian calendar,
computed as 1581 years of 365 days
plus 395 leap days + 279 days
from 1 January 1852 to
5 October 1582."

[c] "The following day was
15 October 1582."

[d] "From that date to the Convention du
Mètre on 20 May 1875, there were
106 870 calendar days including
leap days."

section 11.7, in a Note on p 141:

[e] "685 263 is the index of
1 January 1601,
computed as 684 609 (index of
15 October 1582) plus 6654 days from
15 October 1582 through
1 January 1601."

section 11.7, in the 2nd Example on p 141:

[f] "The first calendar day of 2010
is Gregorian day 830 991"

PROBLEMS:
The assertion in [a] is misunderstandable:
no days were deleted from the Gregorian
calendar by any pope. In fact, the
proleptic Gregorian calendar without
any "deletions" is
used by ISO 8601:2004, by computer
software and by some historians.
What really is meant is:
'The calendar reform instituted by
Pope Gregory XIII and promulgated in
the bull [Inter Gravissimas] started
the use of the Gregorian calendar with
the date 15 October 1582, which is the
same as 05 October 1582 in the Julian
calendar.'

As for [b]: The "previous calendar day"
would be 1582 Oct 04 in the Julian
calendar, not 1582 Oct 05 as suggested in
[b]. Whichever of the two is meant, the
following computation in [b] is
incorrect: the number of days from
1582 Jan 01 until 1582 Oct 05 is 277 and
not 279, as can be read directly from
[table 11.3, p 146].

As for [c]: It appears as if the
"following day" means the day
after J1582-10-05. This
following day is
J1582-10-06 = G1582-10-16,
not G1582-10-15 as asserted.
(We use prefixes
J and G to distinguish Julian and
Gregorian calendars.)

The computation in [d] is incorrect: the
number of days from G1582-10-15 to
G1875-05-20 is 106 868, not 106 870
as asserted,
because G1582-10-15 = JD 2299 160.5
and G1875-05-20 = JD 2406 028.5.

Assertion [e] is clearly
self-contradictory since
685 263 - 684 609 is not 6654.

Inconsistencies between [b, e, f]: If
1 January 1601 is the day number 685 263
as asserted in [e] then
day 0 is JD 1620 550.5 = G-0276-10-25,
and if day 684 609 is G1582-10-15 as
also asserted in [e] then
day 0 is JD 1614 551.5 = G-0292-05-23.
Still another zero point follows from [f]:
day 0 is JD 1624 206.5 = G-0266-10-29
and finally, if [b] is meant to refer to
G1582-10-14 as day number 577 739, then
day 0 is JD 1721 420.5 = G0000-12-27.
That last date might indicate that day 0
was actually meant to be still another
date, viz J0001-01-01 = G0000-12-30,
but this is just a wild guess of mine.

ANALYSIS OF THE ISSUES:
The inconsistencies may come from
several sources:
• from typos (though I am not able to
figure out which);
• from the error in the formula for
the duration of successive Gregorian
years (eg in [d]);
• from the use of intervals on the
"time axis" instead of points on it.

The time axis is an affine space whose
translation space, formed by the
differences T - T' for points T, T' on
the time axis, is the vector space of
"duration values". Intervals of
lengths > 0 s, however, do not form an
affine space. So one has to take the
lower or the upper bounds of the involved
intervals consistently to arrive at valid
date arithmetic (eg, satisfying the rule
T - T" = (T - T') + (T'- T")).

An error in numerical examples for
non-trivial specifications is particularly
unfortunate because such examples are
often taken as the very first test cases
for an implementation. It is therefore
appropriate to check all the examples
before publishing. Many calendrical
calculators are available online for that
purpose; the one at [http://emr.cs.iit.edu
/home/reingold
/calendar-book/Calendrica.html]
is particularly useful.

The suggested text clarification is accepted. The erroneous numbers will be corrected.
Because the changes made necessary by this resolution overlap those made in the resolution of Issue 19281, this issue is merged with Issue 19281.
Disposition: See issue 19281 for disposition