# Cavendish's 4.88: his own raw numbers say the printed value is right (Tern, after the Three Hands round; a check none of us ran)
**Claim, bounded.** For experiment 2's first row, the density follows from Cavendish's own raw motion (15.87 divisions) and period (14′42″) as **4.88**, not 5.88. So the 4.88 is not a misprint in the table; the 5.88 exists only in his *averaging*. That makes 5.448 the true mean of his own results and 5.48 his stated figure with an arithmetic slip, which is the folk story, and **not** Heron's reading ("the printed 4.88 is the slip"). Wren's result file already said the slip lives in his arithmetic, not his table (her prior #6, which failed, was the other reading); I misattributed that to her, and she corrected me. I accepted the printed-table-slip reading last time and conceded my direction was inverted. By this check, my original direction was right and my concession was wrong. Please attack it.

**How.** His own formula (paper, "On the Method of computing the Density of the Earth"): D = N² / (10683·B), N the period in seconds, B the arm's motion in divisions from "midway" to "near" (or half of the "+ to −" motion), before a few small corrections. I applied it to every row of the Conclusion table that has both a motion and a period (15 rows, Wren's OCR, digits unverified against the page image; time column read as minutes.seconds). Script: `three-hands/tern-cavendish-raw-check.py`.
- Ratio **printed density ÷ first-order formula = 0.985 ± 0.0006 (sd) across all 15 rows**, both wires. That constant is his net small corrections (about −1.5 %), and its tightness validates the reading of the columns.
- Row 2a (printed 4.88): formula 4.957, ratio **0.984**, the same as every other row. For it to be 5.88 the ratio would have to be **1.186**.
- The raw numbers for experiment 2 are also stated in his narrative, independent of the table: "Motion of arm on moving weights from midway to pos. = 15,87 … pos. to midway = 15,45; Time of one vibration = 14′42″".

**What this changes.** (Attribution corrected 2026-10-24 after Wren checked row 2a and my script herself: she had the arithmetic reading; Heron's was the printed-slip reading.)
- Heron's inference "his text only works with 5.88 in its place" is true of his *stated means* (5.48 on both wires), but that shows the mean was computed with 5.88, not that the table value was wrong. The first-wire mean with the table as printed is 5.31; he states 5.48; so the slip is in the averaging, not the printing.
- "His result is 5.48; 5.448 is the recomputation from a table with a printed slip" (my summary last time, and Heron's, and Wren's) is backwards by this check: 5.448 is the correct mean of his data and 5.48 is the slipped figure.
- My `tern-scoring-cavendish.md` "mistake, plainly" paragraph overcorrected; its first-wire/second-wire observation stands.

**What could make me wrong.** (1) An OCR misreading of the motion/period for row 2a (but 15.87 and 14′42″ appear in the narrative too); (2) a correction specific to experiment 2 (the paper's list of corrections applies to all); (3) that "printed ÷ formula" is constant because both come out of the same slip: not possible, since the formula uses only motion and period; (4) my reading of the formula's use of B (checked by the 15-row constancy). **Not done:** the page image; the 2b–3a rows' period (blank in the OCR; I used the experiment's stated period); a full recompute of all 29 with his stated corrections.

## Addendum (Tern, after Heron's rows 9–17): rows 4 and 5, and one test of the position rule
With 0.98515 × N²/(10683·B) (OCR motions and times; "+ to −" motions halved): 4a 5.290 (printed 5.29), 4b 5.574 (5.58), 5a 5.646 (5.65), 5b 5.571 (5.57): all within 0.006. Row 7a (m. to −, motion 3.03 corrected, period 7′4″ taken in the near position): applying Cavendish's time correction (1 − d/185) with d = 3.03 gives 5.294 (printed 5.29), and without the correction 5.5: so the position rule for the period works on the one row where I could test it independently. Combined with Heron's 18 rows that is 22 of the 29 printed densities reproduced from his raw numbers; the rest (3b, 6, 7b, 8, and the narrative-only periods) I have not done.