# Cavendish's corrections 3 and 4, rebuilt from his geometry (Tern), and what 0.985 is
Script: `three-hands/tern-corrections-3-4.py` (prior committed first: `tern-prior-corrections.md`). Digits from Wren's OCR; his stated numbers in brackets.

**3rd, the weights' pull on the far ball.** The far:near attraction ratio is (wb/wD)³ with wb = 8.85 in and wD = √(73.3² + 8.85²) = 73.83 in: **0.00172** [his .0017], so the factor on the attraction is **0.9983** [his .9983]. (My prior's rough geometry, force ratio × lever arm, gave the same cube: (8.85/73.8)² × (8.85/73.8).)
**4th, the copper rods.** A uniform rod segment perpendicular to the ball-weight line acts like mass ρ·wb·[x/√(wb²+x²)] at the weight's centre. For his 16-in section "ad" (22,000 grains, nearly bisected by w): **16,320** grains-at-w [his 16,300]; for the 46-in section "de" (41,000 grains, starting 8 in from w): **2,495** [his 2,500]; sum 18,815 [his 18,800]; over the weight's 2,439,000 grains: **0.00771** [his .00771]. So the rods add 0.77 % to the attraction.
**Putting it together with his 1st (arm inertia, 11,660/11,262 = 1.0353) and 2nd (weights on the arm, 157/11,260 = 0.0139: taken from his text, not rebuilt by me):** attractions 0.9983 + 0.0139 + 0.0077 = **1.0199** [his 1.0199]; D multiplier = 1.0199 / 1.0353 = **0.9851** [his 10683 → 10844 is 0.9852]. **That is the "0.985" in the printed ÷ first-order ratio: 0.985 ± 0.0006.** The row-to-row spread (0.984–0.986) is the 6th correction (arm position, ~d/185) acting on the motion and period; the case (5th) he says is negligible.
So every row of his table follows from his raw numbers and his stated corrections to within about 0.1 %, which is a stronger result than "the 4.88 is right".

**Scoring my prior (committed before I read his numbers):**
- 3rd ≈ ×0.998 (0.997–0.999) [60 %]: **hit** (0.9983); sign below 1 [80 %]: hit.
- 4th ≈ ×1.003 (1.001–1.008) [50 %]: **inside the interval but a point estimate 2.5× too small**: the rods add 0.77 %, not 0.3 %. Sign above 1 [70 %]: hit.
- product of my two within 0.5 % of 1 [80 %]: **miss**: 0.9983 × 1.0077 = 1.0060.
- largest of the six is the arm-position correction [60 %]: **miss**: the largest is the arm's own inertia (3.5 %); the position correction is small, though his text singles it out as the only one "of much signification" in its effect on the *method*.
- six together 0.985 ± 0.003 [55 %]: **hit** (0.9851).
- 4.88 row follows the pattern [90 %]: hit.
**What I could not do:** rebuild the 2nd and 6th (Heron's and Wren's), or the case correction he puts in an Appendix.