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Behind the results

Mathematical derivation of the 4-case 8870 gravestone formula vs. exact calendar subtraction

Eighteenth- and nineteenth-century American gravestones, parish burial registers, and newspaper obituaries routinely omitted the deceased person’s birth date, carving instead the Date of Death (Y_d, M_d, D_d) and the Age at Death in years, months, and days (Y_a, M_a, D_a). For over a century, cemetery transcribers have used a decimal pencil-and-paper shortcut known as the 8870 Formula.

Why does the number 8870 appear? When you pack a calendar date into an 8-digit base-10 integer YYYYMMDD = 10000 × Y + 100 × M + D, the "days" column occupies the 1s and 10s places (base 100) and the "months" column occupies the 100s and 1000s places (base 10,000). However, an idealized stonecutter’s month has only 30 days (not 100), and a calendar year has only 12 months (not 100):

1. Borrowing 1 Month (30 days) into the Days column:
   Base-10 subtraction borrows 100, so we must subtract the excess (100 − 30) = 70.
   Day borrow indicator: b_D = I(D_d ≤ D_a) ∈ [0, 1]
   (Note: using ≤ rather than < prevents invalid "Day 00" when D_d = D_a.)

2. Borrowing 1 Year (12 months = 1,200 in the MM00 slot) into the Months column:
   Base-10 subtraction borrows 10,000, so we must subtract the excess (10,000 − 1,200) = 8,800.
   Month borrow indicator: b_M = I(M_d − b_D ≤ M_a) ∈ [0, 1]

3. Exact 4-Case Piecewise Correction Constant:
   K = 8,800 × b_M + 70 × b_D ∈ [0, 70, 8800, 8870]
   Birth Integer B_10 = D_10 − A_10 − K

Why 8870 and Exact Calendar Subtraction can differ by 1 to 2 days: Because the 8870 method assumes every borrowed month contains exactly 30 days, it diverges by +1 day whenever the actual borrowed calendar month has 31 days (January, March, May, July, August, October, December) and by −1 or −2 days across February (28 or 29 days). Furthermore, historical stonecutters were not uniform: some added months forward from the birth date to the last monthly anniversary before death (reversed by Backward Exact Subtraction), while others subtracted days from the death month first (reversed by Forward Exact Subtraction). Displaying all three methods side by side gives you the complete 1-to-2-day search window for baptismal and Bible records. Read the full proof in Tombstone Birth-Date Math: The 8870 Formula vs. Exact Calendar Days.

The 1752 British & Colonial American Calendar Act and Quaker numbering

Prior to September 1752, Great Britain and its American colonies followed the Julian Calendar (Old Style, O.S.), which inserted a leap day every fourth year (`365.25` days/year) and began the legal civil year on Lady Day (March 25)—the Feast of the Annunciation—rather than January 1. Under the Calendar (New Style) Act 1750 (`24 Geo. 2 c. 23`):

• New Year’s Day moved from March 25 to January 1 starting in 1752. Consequently, any colonial record written between January 1 and March 24 prior to 1752 frequently carries a double-dated year such as 11 February 1731/32 (1731 civil Old Style, 1732 historical New Style).
• 11 calendar days were dropped in September 1752: Wednesday, September 2, 1752 was followed the next morning by Thursday, September 14, 1752. To convert an Old Style date to New Style in the 18th century (`1700–1752`), add +11 days; in the 17th century (`1582–1699`), add +10 days.
• Quaker numerical months: Because members of the Religious Society of Friends avoided pagan month names, pre-1752 Quaker records refer to March as "First Month", April as "Second Month," December as "Tenth Month (`10br`)", January as "Eleventh Month (`11br`)", and February as "Twelfth Month (`12br`)". See The 1752 Julian-to-Gregorian Calendar Shift and Colonial Double-Dating.

Reference table: All 17 US Federal Decennial Census statutory dates (1790–1950)

Census enumerators were legally required to record each person’s age at their last birthday as of the statutory Census Day, even if the enumerator visited the farm weeks or months later. Subtracting `Age` from `Census Year` without checking the statutory date is wrong up to 99.7% of the time in 1920 (when Census Day was January 1) and 75.1% of the time in 1930–1950 (when Census Day was April 1):

For worked multi-census intersection examples and how to handle the 1890 census fire, read US Federal Census Dates (1790–1950) and Birth-Year Calculation.

Reference table: Cousin kinship coefficients (Wright’s r) & Shared cM Project v4 ranges

Given generational distances g_A and g_B from two individuals to their most recent common ancestor(s) and path multiplicity m ∈ [1, 2, 4], Sewall Wright’s Coefficient of Relationship is r = m × 2^−(g_A + g_B), and expected shared autosomal DNA across the 3,485 cM haploid human genome is E[cM] = r × 6,970 cM (with full siblings sharing (0.50 + 0.25) × 3,485 = 2,613.75 cM on standard half-identical region callers):

Explore the Donnelly-Coop Poisson recombination model in Cousin Chart and Shared DNA centiMorgan Math Explained and learn why phonetically similar surnames collapse to the same 4-character index in How the NARA Soundex Phonetic Code Works for Surname Variants.

Glossary of archaic paleography & probate abbreviations

When transcribing 17th- to 19th-century wills, deeds, and newspapers—either by eye or using our Tab C-3 Diplomatic AI Prompt Compiler—watch for these ten high-frequency scribal conventions (detailed in Using AI to Transcribe Old Cursive Handwriting, Wills, and Newspapers):

Need to digitize the original family Bible page, tintype, or cabinet card before transcribing it? Switch to our Photo, Slide & Home Movie Digitization Planner + Local Histogram Workbench.