- 30-day fixed-length month (the "8870" system is a variant of this)
- Calendar month

Despite many vehement claims to the contrary, all three of these methods are to be encountered in historical Y/M/D or Y/M/W/D age (or time-elapsed) calculations. Based on limited evidence, it appears that 30-day-fixed is probably more likely to have been used in later times, but the 28-day month seems at least to dominate arithmetic texts from the early 1800s on back in the US. The calendar-month method seems to have coexisted with both systems, considered to be more correct but little-used due to its relative complexity. It is best to compare the results of the three calculations for known death date, age, and birthdate combinations which approximate the context of the unknown date as closely as possible.

"Borrowing" during subtraction turns out to be the heart of the problem with calendar-month lengths.

Let's examine this by looking at the ways YMD age is calculated to begin with, since what we really want to do is reverse that calculation. For example, take an age calculated thus:

Death | 1988 | Apr (4) | 6 |

- Birth | 1928 | Mar (3) | 7 |

Here, we must borrow from the months place into the days place, since
7 is larger than 6 (ignoring the frightening notion of negative calendar
time). The question is - do you borrow 30 days for the length of
April, as might seem obvious at first? Some will say "yes", some
will say "no, borrow 31 days for March." Which is correct?
At second thought, you'll probably conclude that you should borrow 31 days
from March. Well, as a third thought, consider what I like to call
the "anniversarial" calculation, sort of the "common sense" approach to
doing this. Say we want to count back from the date of death, we
have 60 whole years from 1988 Apr 6 to 1928 Apr 6, no complete month from
Apr 6 and Mar 7, and 6+(31-7) = 30 days. Counting forward from the
date of birth gives the same result **for this case**. Note that
since we crossed the Apr-Mar month boundary, we borrow from the length
of March, as we probably noticed in our second thought on the subject.

Now let's try a different one.

Death | 1988 | Apr (4) | 6 |

- Birth | 1928 | Feb (2) | 7 |

We count back from the date of death, so we have 60 whole years from
1988 Apr 6 to 1928 Apr 6, 1 month from Apr 6 to Mar 6, and 6**+(29-7)
= 28 days**. Now count forward from the date of birth, 60 whole
years from 1928 Feb 7 to 1988 Feb 7, 1 month from 1988 Feb 7 to 1988 Mar
7, and **6+(31-7) = 30** **days**!(**!!!)** This is a subtle
and critical problem in calendar-month units.

Writing out the age calculation as traditional long subtraction duplicates
the forward (from date of birth) anniversarial method, as long as you remember
to borrow from the length of the month before the death month. The
backward anniversarial calculation really is like subtracting *birthdate*
- *deathdate* but (curiously) borrowing from the length of the birth
month itself.

Having recognized this complication, we should now ask what method was
historically really used? Did the author of the tombstone (or whatever
YMD age medium) count "anniversaries" forward from the birthdate or did
that person count back from the death date? People will often claim
that counting forward is the obvious thing to do, but is it? If you
didn't know the results were different, might you count back from the death
date? Or perhaps the author wrote down the
problem as a subtraction? I believe
with the long-subtraction approach, we can be fairly confident that
*deathdate*
- *birthdate* would always be used, since the deathdate is in a sense
the larger number (the backwards anniversarial calculation can be regarded
as giving a negative age - compare the "2s complement notation" used in
digital computers). However, the person writing the subtraction would
have to be clever enough to figure out the correct month to borrow from.
For abitrary data, we really can't be sure what approach was used, which
introduces an additional uncertainty on top of our uncertainty about what
month lengths were used. I have generally assumed that the forward
anniversarial calculation (or its long-subtraction equivalent) should be
regarded as the most likely method, but practically speaking, I doubt if
it was **always** the approach used. This issue tends to confuse,
and it seems unlikely that people in the past were any less prone to be
confused by it than people are today.

Next, we have to look at how to reverse the forward anniversarial method.
We want to do the subtraction *deathdate* - *age* = *birthdate*.
Let's look at our example again, using the long subtraction form of the
forward anniversarial calculation:

Death | 1988 | Apr (4) | 6 |

- Birth | 1928 | Feb (2) | 7 |

= | ---------- | ---------- | ---------- |

Age | 60 | 1 | 30 |

Remember we borrowed the length of the month before the death month,
so what do we do to reverse this? It's pretty simple - you just use
that same length for borrowing (since we already know that subtraction
using fixed-length units can be inverted like this), so:

Death | 1988 | Apr (4) | 6 |

- Age | 60 | 1 | 30 |

= | ---------- | ---------- | ---------- |

Birth | 60 | 2 | 7 |

And this is the method used by the birthdate calculator. For those who care, the algorithm itself decrements the death month variable first when borrowing is needed. Of course, if the month decrements to 0, you have to borrow from the year and add 12 to the month total too, and then that number is the code used to find the month length to add to the day total.

An implication of this is that the birthdate calculator also works to compute forward anniversarial age. You just fill in the birthdate numbers in place of the age numbers (converting the months to their respective ordinal numbers, 1 for January and so on), although the output sometimes will give figures such as 12 months 39 years for 40 years.

Also, contrary to popular belief, accounting for leap years is not difficult.
You simply add a day to the February length if the borrowed month is in
a leap year.

Daniel Fenning notes in his late 18th-century *American Youth's Instructor*
(see above, p. 49):

Though 13 months are said to make a year and servants commonly reckon a month 28 days: yet you are to observe, that in trade, and transacting business by a month is meant aThis note tells us two extremely significant things:Calendar Month, that is from any day of the month to the same day of the next month: Thus, from the 5th ofFebruaryto the 5th ofMarch, or from the 18th ofAprilto the 18th ofMayis a month.(Fenning's italics - long-'s' and ligatures transliterated.)

- The 28-day month was then in "common" use, apparently due to its comparative simplicity.
- Though the 28-day month was common, "Calendar Months" were considered to be more technically correct, at least by Fenning. However, the frequency of the 28-day month in arithmetic books aimed at practical calculations and the fact that Fenning's own examples use it as well suggest to me that the technically preferred calendar month was probably not used very often, except perhaps in such trivial cases as he cites.

The identification of the 28-day month in old record beds should be easy though. Approximately 1 in 26 ages in such records should indicate an age of 12 months (1/2 borrow from the years place and 1/13 of those give 12 mos.). Neither of the other systems would be expected to give this result as a matter of course. Ages in weeks, as mentioned, are a good indicator too. Unfortunately, it is not yet clear when the common preferred method changed from the 28-day to the 30-day approximation system. The best evidence so far seems to indicate this would have been in the mid-to-late 1800s in the United States. Data for other regions is so far lacking.

I can't answer this exactly. For one thing the "other sources of error," which I'll call "inherent error" are likely to vary in frequency and magnitude from source to source. From the few record beds I've seen evidence for, I would roughly guess that the error rate tends to be something on the order of 10% and the magnitude in root-mean-square-error over all samples is around 1 or 2 days (if you don't know what this means, just figure that this is about the size you would expect in a "typical" error). Now the difference between 30-day and calendar-month systems is actually fairly close to this (which also makes me worry that some of the error quoted above may be due to some as-yet unconsidered calculation method) so choosing the correct method between those two is roughly (I know this is far from exact) only a matter of cutting the error in half. The 28-day method is the real joker because it commonly comes up with a month's difference from the 30-day and calendar results and very often shows at least a small difference. I haven't rigorously computed its error contribution, but I feel very confident that it overwhelms the inherent error rate and may be significantly larger than the inherent error magnitude in all but the most error-prone data sources. Therefore avoiding this calculation error can be expected to yield large improvements in the accuracy of calculated birthdates.