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Step by step: test the shape of the earth yourself (follow-up Q21)

Procedure page for answer A21 on the theme "Flat vs. Round Earth". The reasoning, sources and objections are in the research notes. Numbers marked [calc] were worked out for A21; sources are listed at the end.

What this can and cannot show

• It can show, using your own instruments and arithmetic, that the local "down" directions at different places are not parallel. They turn by about 1 degree per 111 km, and the radius they give is about 6,400 km.
• It cannot exclude every conceivable alternative. A law of nature tuned to imitate a globe can always be invented. What the procedure does is make each alternative pay for itself: a flat model has to add a separate, untested law for each step below.
• Two branches. Steps 1-4 use starlight and sightlines. Step 5 (the pendulum) uses no long light path at all. The two branches share almost no assumptions, so their agreement is the strongest result.

What you need (about 150 US dollars, plus fuel)

Assumptions: tested here, or stated

Also assumed throughout, as in any experiment: your senses and records are not deceived, and arithmetic holds.

Write your predictions first

Date this table, or post it publicly, before you measure anything.

Step 0. Build and check your instruments

1. Plumb and water. Hang the plumb line over a bowl of still water. Hold the carpenter's square against the line: the water surface is at right angles to it. (A5)
2. Light is straight at bench scale. Stretch a string tight over 20-50 m. Run the laser beam along it. The beam stays on the string to within about a millimetre. (A3, short range)
3. Quadrant zero. Sight a fixed point, turn the quadrant 180 deg and sight again. Half the difference is the instrument error; correct every reading by it. Take 10 sightings of one star and record the spread. That spread is your measurement error, typically 0.1-0.2 deg.
4. Odometer. Drive a measured kilometre (marker posts, or one you taped out) and note the reading. (A6)
5. Watch rate. Time one star passing the same two plumb lines on two nights. It returns about every 23 h 56 min. The difference is your watch's drift.

Expected: all five checks pass. If one fails, fix the instrument before going on.

Step 1. Are the stars far away?

1. Pick a pair of bright stars 10-60 deg apart.
2. With the cross-staff, measure their separation when both are high (60 deg elevation or more). Measure again on the same night when both are low (25-30 deg). Never below 25 deg: air bending grows near the horizon.
3. Repeat at a second site 500 km or more away, and ask your southern partner to do the same.
4. Positive control. Do the same with two lamps on a pole 200 m away, then from 100 m to one side. The lamps' separation changes clearly. That shows your method can detect a nearby object.

Expected (globe): the star separation stays the same within your 0.1-0.2 deg, everywhere. Expected (dome 5,000 km up): constellations shrink by 30-80% as they set, and change shape from place to place. What it shows: the stars are far compared with your 500 km baseline, or light bends so as to make a near dome look like a far sphere (A3).

Step 2. North-south: the pole star's elevation

1. Find north without a compass. Mark the tip of a stick's shadow in the morning and again in the afternoon when the shadow is the same length. North lies halfway between the two marks. Hang two plumb lines on that north-south line.
2. Measure. With the quadrant, measure the elevation of the sky's pole. Polaris is about 0.7 deg off the pole, so average its highest and lowest elevations, about 12 hours apart, or measure at the times halfway between. Take 5 readings each night.
3. Travel. Repeat at three or more sites along roughly one north-south line, spanning 1,000 km or more. Note the odometer distance between sites.
4. Southern partner. Report whether Polaris is visible, and the elevation and turning direction of the southern pole.

Expected (globe): elevation changes 1 deg per 111.2 km, along a straight line. 1,000 km gives 9 deg, about 50 times the error. In the south, Polaris is below the horizon and the sky turns clockwise about a southern pole whose elevation equals the latitude. Expected (dome): the elevation follows arctan(H/d) and bends away from a straight line. A dome fitted to two sites misses the third by several degrees. Polaris is visible from everywhere. What it shows: with Step 1, the "down" direction turns as you travel. Earth radius R = distance / (elevation change in radians).

Step 3. East-west: star timing

1. Find a partner, or go yourself, about 500 km due east, at the same pole elevation as you (the same latitude, checked by Step 2).
2. Sync your watches by a phone call or radio time signal.
3. Each of you times the same star crossing your own north-south plumb lines.
4. Note the odometer distance east, s.

Expected (globe, at 45 N): 1 deg of sky rotation (4 min) per 78.6 km, so 500 km gives 25.4 min. Expected (flat north-centred map): 1 deg per 87.3 km, so 500 km gives 22.9 min. The 2.5 min difference is many times a watch's error. In the south the gap is much larger: 92 against 241 km per degree at 34 S. Self-check: R from Step 2 must equal R from Step 3, where Step 3 gives R = s / (time difference in degrees, as radians, x cos latitude). A cylinder or a saucer shape would make them disagree.

Step 4. Sightlines over water, with the air's bending measured

This step brings A3 out to 10-20 km by measuring the bending instead of assuming it.

1. Over 5-10 km of calm water or flat fen, put up three to five targets at equal heights above the water, in a straight line.
2. Sight from one end with the laser and the cross-staff (or a cheap level). Read how far the middle target sits above or below the line through the near and far targets.
3. At the same time, log air temperature at two heights (for example 1 m and 5 m) at both ends and in the middle.
4. Work out the ray's curvature from the air itself: 1/r = 7.9e-5 x (p / T^2) x (0.0343 + dT/dz) per metre, with p in hPa, T in kelvin and dT/dz in K per metre (Hirt et al. 2010, equation 2). No earth radius goes into this.
5. Add the bending's own effect, L^2/(8r), to the measured offset.
6. Keep sightlines 20 m or more above the surface, in overcast or breezy weather, at several times of day. Close to the surface the bending swings wildly (k from -3 to +16) and two thermometers cannot keep up.

Expected (globe): corrected offset L^2/(8R): 6.0 ft over 6 miles. Expected (plane): corrected offset 0. Note: without the temperature readings (items 3-5), this test cannot tell curvature from bending. A plane with light bending upward (k about -1) reproduces Wallace's 1870 result. The thermometers are what make it decisive. Precision: the bending must be known to about +/-0.1 in k. That needs the temperature gradient to about 0.06 K over 4 m, within reach of matched, shielded thermistors.

Step 5. The light-free check: pendulums in two hemispheres

A free pendulum's swing plane turns relative to the floor at a rate set by how the floor rotates around the local vertical. On any flat surface, all verticals are parallel, so every place shows the same rate and the same sense of turning. On a globe, the rate is 15.04 deg/h x sin(latitude): clockwise in the north, anticlockwise in the south.

1. Build. Hang the bowling ball on 3-10 m of steel wire (a stairwell works) from a rigid single-point clamp, with a Charron ring to stop the swing turning into an ellipse.
2. Release. Tie the ball aside with cotton thread and burn the thread, so there is no sideways push.
3. Measure. Mark the swing direction on the floor every 10-15 minutes for 1-2 hours.
4. Controls (each answers a published objection): - no motor or drive; - launch north-south on some runs and east-west on others; - turn the wire through 180 deg between runs (Longden's 1919 test); - at least 5 runs; report the mean and spread.
5. Your southern partner does the same, the same week.

Expected (globe): 9.66 deg/h clockwise at 39.9 N; 8.27 deg/h anticlockwise at 33.4 S. Expected (flat): the same rate and sense at both sites. What it shows: opposite senses mean the two verticals point in directions at least 73 deg apart. Over the 8,324 km between those sites (8,527 km on the flat map, almost the same), that gives R at most 6,510 km. No starlight or sightline is used. Record so far: a South Pole pendulum (2001) measured 24 h +/- 50 min, the predicted rate. A low-budget US bowling-ball pendulum (ASEE, 2021) gave 8.09 deg/h against 9.65 predicted. Its partner runs in Argentina failed because of a faulty suspension. A clean low-budget two-hemisphere result has not yet been published, so this step is a real experiment, not a demonstration.

Step 6. Decide

The round-earth result stands if all of these hold:

1. Star separations stay constant within your error (Step 1).
2. Pole elevation is a straight line in distance, and the southern partner sees a second pole (Step 2).
3. R from Step 2 and R from Step 3 agree within about 5%.
4. The corrected sightline offset matches L^2/(8R), not 0 (Step 4).
5. The pendulum rates differ between hemispheres and turn in opposite senses (Step 5).

Any failure counts against the globe, as you wrote down in advance. If they all pass, a flat model must add three separate untested laws at once: a light law that makes a near dome look far and bends sightlines just enough; a southern sky with its own pole; and an inertia field that turns differently, and in opposite senses, at different places on one plane.

Safety

• Never point a laser at people, vehicles or aircraft. Aiming at aircraft is a crime in most countries.
• A swinging bowling ball can break bones: keep the swing area clear and the clamp rated well above the ball's weight.
• Observe at night only from safe places off the road. Test ice thickness before Step 4 on a frozen lake.
• No step requires looking at the sun.

Record sheet (copy for each session)

Sources

• Research notes for this answer: 2026-10-10-followup-Q21 (objections step by step, assumption reasoning, corrections).
• Hirt, Guillaume, Wisbar, Bürki and Sternberg 2010, refraction coefficient from temperature gradients (Flat-Earth source folder).
• Foucault 1851, Comptes rendus 32: the sine law.
• Celario Sedano and Ruzybayev, ASEE 2021, two-hemisphere low-budget pendulum: https://peer.asee.org/36676
• South Pole Foucault pendulum 2001 (Town, Bird, Baker): https://www.southpolestation.com/trivia/00s/southpolefoucault.html
• Wallace 1870 equal-height test and Oldham 1901 repeat (Flat-Earth review, arbiter claim 1).
• Flat Earth Society wiki pages on Electromagnetic Acceleration, Foucault Pendulum, Celestial Sphere and Stellar Parallax (Flat-Earth source folder and research notes).