…but how do we know?
Last checked 1 November 2026 · every number in the episode has a row below
The percentages are our editorial estimate of how settled each answer is, not a measurement. The ± 22 ppm is the source's uncertainty.
Satellites and pendulums give the Earth's pull, G × M, to about 2 parts in a billion. Lab experiments give G alone to about 22 parts per million.
The mass is GM divided by G. Almost all of its uncertainty comes from G, which is about 11,000 times less certain than GM.
Why the best measurements of G disagree with each other by more than their error bars. NIST's best guess is an error nobody has found yet.
| Meter | Evidence that moved it | Rows |
|---|---|---|
| 0 → 10 % | Newton, 1687: rock at the surface is about twice as dense as water, deeper in mines three to five times, so the whole Earth is probably five or six times. A reasoned guess, not a measurement. | L1 |
| 10 → 30 % | Schiehallion, 1774: plumb lines lean towards a mountain by 11.6″ in total; Hutton turns it into 4.5 times the density of water, about 18 % too low. | L2–L5 |
| 30 → 60 % | Cavendish, 1798: a torsion balance in a closed room. His printed result is 5.48; averaging all 29 runs correctly gives 5.448, about 1 % below today's value. | L6–L10 |
| GM → 99.9 % | Laser ranging to satellites such as LAGEOS (launched 1976) pins down GM to about 2 parts per billion. | L15–L16 |
| M → 75 % | CODATA 2022: G = 6.67430(15) × 10⁻¹¹, 22 ppm. GM ÷ G gives the mass. | L17–L19 |
| M 75 → 70 % | The two best G results from one lab differ by 45 ppm; CODATA had to inflate the uncertainties 3.9×; NIST 2026 lands 235 ppm below the BIPM result from the same balance. | L20–L23 |
| M → 90 %, locked | The reading: the first three digits are solid; the last two depend on which G you use (5.97200 to 5.97431 × 10²⁴ kg). | L18, L25–L27 |
Newton, Principia, Book III, Prop. X (Motte's English translation, 1729, vol. II, pp. 230–231)
…may be five or six times greater than if it consisted all of water
Cavendish, Phil. Trans. 88 (1798), p. 521
the density of the earth comes out 5,48 times greater than that of water
5,48 → really 5.448 (all 29 runs)
Scans: Internet Archive. Principia vol. II (1729), microfilm copy; Cavendish 1798, JSTOR Early Journal Content, Public Domain Mark. Both works are in the public domain.
| # | Claim | Value | Tag | Source |
|---|---|---|---|---|
| L1 | Newton: surface rock about twice as heavy as water, in mines three to five times, the whole Earth five or six times | 2×; 3–5×; 5–6× | INFERRED | Principia (Motte 1729), vol. II, pp. 230–231 |
| L2 | Schiehallion: stars and survey differ by 11.6″; the hill is about half as dense as the Earth | 54.6″ − 42.94″ = 11.6″ | OBSERVED | Maskelyne 1775, Phil. Trans. 65 |
| L3 | 11.6″ is a coin seen from about 400 m | 2.3 cm ÷ 5.6 × 10⁻⁵ rad ≈ 410 m | INFERRED | calculation from L2 |
| L4 | Hutton: the Earth is 4.5 times as dense as water | 4.5 | OBSERVED | Encyclopaedia Britannica 1911, “Gravitation” (secondary) |
| L5 | About 18 % too low | 4.5 ÷ 5.513 = 0.816 | INFERRED | L4, L10 |
| L6 | Apparatus designed by John Michell, mostly rebuilt by Cavendish: six-foot wooden rod, lead balls of about 2 inches, two weights of about 158 kg | 2,439,000 grains = 158 kg | OBSERVED (kg converted) | Cavendish 1798, Phil. Trans. 88 |
| L7 | Temperature differences make air currents, so the room was closed and observed from outside with a telescope | — | OBSERVED | Cavendish 1798 |
| L8 | “the density of the earth comes out 5,48 times greater than that of water” | 5.48 | OBSERVED | Cavendish 1798, p. 521 |
| L9 | In the first six runs one result was copied as 5.88 instead of 4.88; the mean of all 29 runs is 5.448 (noticed by Baily, 1843) | 5.4479 | OBSERVED INFERRED | Cavendish data, after Stigler 1977; EB1911 |
| L10 | About 1 % below today's value | 5.448 vs 5.513 (−1.2 %) | INFERRED | NASA Earth Fact Sheet (5513 kg/m³) |
| L11 | The constant G appears nowhere in Cavendish's paper | — | OBSERVED | Cavendish 1798 (full-text search; figure labels such as “GG” are not the constant) |
| L12 | Converted to G later, his result is about 1 % high | 6.754 × 10⁻¹¹ (+1.2 %) | INFERRED | G = GM ÷ (ρ · 4/3 · πR³); L9, L16 |
| L13 | Kitchen Check (worked example, not a measurement): a 1 m pendulum takes about 2 s per full swing, so 20 swings take about 40 s; g ≈ 9.8 m/s²; g·R² ≈ 4 × 10¹⁴ m³/s² | T = 2π√(L/g) = 2.006 s for g = 9.81; 9.8 × (6.371 × 10⁶)² = 3.98 × 10¹⁴ | INFERRED · EXAMPLE | calculation; Earth radius from the NASA Earth Fact Sheet |
| L14 | Orbits, falls and pendulums measure only the product G × M | — | INFERRED | Newtonian gravity: acceleration = GM ÷ r² |
| L15 | LAGEOS, launched 1976, used to determine the Earth's GM | 4 May 1976 | OBSERVED | NASA GSFC, LAGEOS |
| L16 | The Earth's GM to about 2 parts in a billion | 3.986004418 × 10¹⁴ ± 8 × 10⁵ m³ s⁻² | OBSERVED | IERS Conventions, ch. 1, Table 1.1 |
| L17 | CODATA 2022 value of G, 22 ppm | 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² | OBSERVED | NIST, CODATA 2022 |
| L18 | The Earth's mass | 5.97217(13) × 10²⁴ kg | INFERRED | GM ÷ G (L16, L17); NASA gives 5.9722 |
| L19 | G's uncertainty is about 11,000 times that of GM | 2.247 × 10⁻⁵ vs 2.0 × 10⁻⁹ | INFERRED | L16, L17 |
| L20 | Li et al. 2018: two methods, about 12 ppm each, 45 ppm apart (about 2.7× their combined uncertainty); recent G results differ by up to 500 ppm | 6.674184 and 6.674484 × 10⁻¹¹ | OBSERVED | Li et al., Nature 560, 582 (2018) |
| L21 | “differ from each other by 45 ppm, suggesting biases yet to be uncovered” | 44.9 ppm | OBSERVED | Lu, National Science Review (2020) |
| L22 | CODATA multiplied the G uncertainties by 3.9 to make them consistent | × 3.9 | OBSERVED | Mohr et al., CODATA 2022, sec. I.B |
| L23 | NIST 2026, using the BIPM torsion balance: 235 ppm below the BIPM result (about 10× the official error bar), uncertainty 57 ppm, within one standard deviation of the world average; most likely an overlooked error | 6.67387 × 10⁻¹¹; −0.0235 % | OBSERVED (10×: inferred, 235 ÷ 22) | NIST, 16 Apr 2026; APS Physics 19, 64 |
| L24 | In everyday use, “weight” nearly always means mass | — | OBSERVED | NIST SP 811, ch. 8, §8.3 |
| L25 | Same GM, different G, different Earth | 5.97200 / 5.97227 / 5.97255 / 5.97431 × 10²⁴ kg | INFERRED | GM ÷ G with L20, L23, L26 |
| L26 | A 2014 measurement of G with clouds of cold atoms | 6.67191(99) × 10⁻¹¹, 150 ppm | OBSERVED | Rosi et al., Nature 2014 |
| L27 | ± 22 ppm is about 130 billion billion kg | ± 1.3 × 10²⁰ kg | INFERRED | L18 |
| L28 | The next CODATA adjustment uses results published by 31 December 2026 | 31 Dec 2026 | OBSERVED | CODATA Task Group on Fundamental Constants |
| L29 | The deepest borehole reaches about 12 km | 12,262 m (Kola SG-3) | OBSERVED | Skuf'in, Nexo (Kola SG-3 history) |
One metre of string, a steel nut, a phone. Pull it back a few degrees, let it swing, and time twenty full swings (there and back). The period gives the local pull of gravity, g = 4π²L ÷ T², and g × R² gives GM. It cannot give the mass on its own: for that you need G.
In the episode this is an animated worked example, labelled EXAMPLE on screen, not a measurement: with g = 9.81 m/s², a 1.00 m pendulum swings once in 2.006 s, so twenty swings take about 40.1 s. Your time will differ a little with the length of your string, the size of the swing and where you live. Tell us what you got.
None yet. Any change to a number in the episode is listed here with its date and reason.
Sure So Far · how we know what we know, and where knowing stops. The Certainty Meter is our editorial estimate, not a measurement. Diagrams are drawn by us and labelled when they are schematic or exaggerated.