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.