Empirical Earth · build the instrument, not just the argument

Print the instrument. Then take the measurement.

A receiver you bought is a box somebody else built. A number you read on a screen is a number somebody else computed. These five objects are neither. You print them, or you cut them out of a cereal box, and then you go outside and measure the world with them. The dimensions come out of the same geometry the rest of this site argues from. If that geometry were wrong, these things would not work.

Read this part first. An instrument does not settle an argument by existing. What settles it is that the number you read off the thing you built matches the number the globe predicted before you built it. Every project below says what it measures, what the prediction is, and where the instrument runs out of accuracy. Two of them have limits that make certain tests impossible, and we say so on the page rather than let you find out on a cold night.

Five instruments

The sundial. Its angle is your latitude. It keeps working for years.
The star quadrant. Measure Polaris. Read your latitude straight off it.
The Eratosthenes gnomon. Three of them, and the flat Earth runs out of Suns.
The horizon clinometer. The horizon is below you. Measure by how much.
The V-dipole holder. 120°, and the radio ladder starts.

No 3D printer? You lose nothing that matters. Three of these five work as paper templates on card, with a pin and a thread. They are printed at true scale and each carries a 50 mm check bar: print at 100 percent, no fit-to-page, and measure the bar. If it is not 50 mm, your printer rescaled it and every angle on the sheet is wrong.

The three that work on card: template-star-quadrant.svg, template-eratosthenes.svg and template-clinometer.svg.

01The sundial. Its angle is your latitude.

A horizontal sundial has a blade, called the gnomon, standing on a plate. The blade’s sloping edge has to point at the celestial pole. On a globe, that has a consequence you can measure with a protractor: the angle between the blade and the plate equals your latitude. Nothing else. Not the season, not the time, not the maker’s taste. Your latitude.

And the hour lines are not evenly spaced. They obey one equation:

tan(line angle) = sin(latitude) × tan(15° × hours from noon)

Two checks that the equation is the real one, and not something we made up. At the North Pole, sin(90°) = 1, so the lines fall every 15°, evenly, like a clock face. At the equator, sin(0°) = 0, and every line collapses onto noon: a horizontal sundial does not work at the equator, and the equation knows it before you do.

Latitude	1 h	2 h	3 h	4 h	5 h	6 h
25°N	6.5°	13.7°	22.9°	36.2°	57.6°	90.0°
35°N	8.7°	18.3°	29.8°	44.8°	65.0°	90.0°
45°N	10.7°	22.2°	35.3°	50.8°	69.2°	90.0°
55°N	12.4°	25.3°	39.3°	54.8°	71.9°	90.0°

Those numbers are why the file is cut per latitude. A dial for 45°N is plainly wrong at 25°N, and it is wrong by enough to see on a summer afternoon. Take the file nearest your latitude.

Take the file nearest your latitude, in 5° steps. Set it flat, point the blade at true north (not magnetic), and read the shadow.

What	File	Note
25°N	sundial-25N.stl	Blade rises 25° from the plate. 142 × 141 × 33 mm.
30°N	sundial-30N.stl	Blade rises 30° from the plate. 142 × 141 × 40 mm.
35°N	sundial-35N.stl	Blade rises 35° from the plate. 142 × 141 × 48 mm.
40°N	sundial-40N.stl	Blade rises 40° from the plate. 142 × 141 × 57 mm.
45°N	sundial-45N.stl	Blade rises 45° from the plate. 142 × 141 × 66 mm.
50°N	sundial-50N.stl	Blade rises 50° from the plate. 142 × 141 × 78 mm.
55°N	sundial-55N.stl	Blade rises 55° from the plate. 142 × 141 × 93 mm.
60°N	sundial-60N.stl	Blade rises 60° from the plate. 142 × 141 × 112 mm.

Or get one cut for your latitude, not the nearest five degrees. The sundial generator in entry 75 now has a download button. Slide it to your latitude, press the button, and it builds the file in your browser. Nothing is uploaded anywhere, and a reader at 43.7°N gets 43.7°.

It will not agree with your watch, and that is not a fault. It shows solar time. Your watch shows zone time, shifted by your longitude within the zone, by daylight saving, and by the equation of time, which runs up to about 16 minutes either way across the year. A sundial that agreed with your phone to the minute would be the suspicious one. The corrections are all predictable, and they are all consequences of an orbiting, tilted, rotating globe.

02The star quadrant. Polaris hands you your latitude.

Sight Polaris along the top edge. Let the thread hang. Read where it crosses the scale. That number is your latitude, and you did not consult anybody to get it.

This is the instrument that ran the age of sail, and it is a quarter of a circle with a weight on a string. It also runs the argument in entry 70 to its conclusion, because the flat model has to explain the same reading with a lamp hanging over the disc, and it cannot do it twice with the same lamp. Polaris at 60° demands a lamp 5,778 km up. At 45° it demands 5,004 km. At 30° it demands 3,852 km.

Thread through the corner hole, a nut or a coin for the weight, and two pinholes for the sight line.

What	File	Note
3D print	star-quadrant.stl	90 × 90 × 16 mm. 1° graduations on a 90 mm arc.
Paper, no printer needed	template-star-quadrant.svg	True scale. Glue to card. Print at 100 percent and check the 50 mm bar.

Honest accuracy. A hand-held quadrant with a plumb line is good to roughly a degree, which is about 110 km on the ground. That is good enough to prove the point and useless for navigation, and both halves of that sentence are true. Do it on a still night: any breeze moves the thread more than the sky does.

03The Eratosthenes gnomon. Three of them, and the flat Earth runs out of Suns.

This is the important one, and it is the plainest object here: a flat plate with a pin standing on it.

The pin is 100.0 millimeters tall, to the tip, and not a hair more. That is the entire design. It means the shadow length in millimeters gives the Sun’s angle with no calibration and no arithmetic anybody can dispute: the angle is arctan(shadow ÷ 100). Because everyone in the experiment prints the same file, nobody can blame a disagreement on whose stick was whose.

Two sticks are not enough, and the flat-Earth objection to Eratosthenes is correct. A close Sun over a flat plane also produces different shadows in different places. With two measurements, the flat model has one equation and one unknown, the Sun’s height, and it can always be fitted. Say so out loud. It is true.

Three sticks break it. Three measurements, one unknown, and no single Sun can satisfy them. Fit the Sun to a stick at 10° and it must hang 6,306 km up. The stick at 35° demands 5,558 km. The stick at 60° demands 3,852 km. The flat Earth needs a different Sun for every observer. The globe needs one Sun and one radius, and gets all three right. The full argument, with the diagram, is entry 160.

How to run it. Print three. Post two of them to friends who live well north and south of you. Agree on one moment, to the minute, and all three of you measure the shadow. Then compare.

What	File	Note
3D print	eratosthenes-gnomon.stl	Pin 100.0 mm to the tip. 150 × 90 mm base, with a recess for a bubble level.
Paper, no printer needed	template-eratosthenes.svg	A shadow scale in mm, with the angle printed under it. Use it with any 100 mm gnomon.

Where the error lives. The plate must be level and the pin must be vertical, or you are measuring your table instead of the sky. There is a recess in the base for a small bubble level. A one-degree tilt puts about a degree straight into your answer, and the effect you are hunting is a few degrees, so this is not fussiness. It is the measurement.

04The horizon clinometer. The horizon is below you.

On a flat Earth the horizon rises to meet your eye, and it sits at eye level however high you climb. On a globe it drops below level, by an angle that grows with your altitude. Measure the drop and the model is on the line.

Sight the horizon along the top edge, let the plumb hang, read the arc. Divisions are 0.5°, spread over a 140 mm radius so that half a degree is a millimeter and a half of arc and you can see it.

What	File	Note
3D print	horizon-clinometer.stl	138 × 31 × 16 mm. 0.5° per division on a 140 mm arc.
Paper, no printer needed	template-clinometer.svg	True scale. Card, a pin and a thread.

This instrument cannot do the test you probably want to do, and we would rather tell you now. From an airliner at 11 km the horizon sits about 3.4° below level. That is a real measurement, and this thing will get it. From a 100 m clifftop the dip is 0.32°. A printed plumb-line protractor cannot resolve that, and anybody who tells you they eyeballed it from a beach is fooling themselves or you. Do this one from a window seat, in smooth air, or do not do it.

05The V-dipole holder. 120°, and the ladder starts.

The smallest object here and the one that opens the most doors. It holds two rods at 120°, which is the angle the weather-satellite V-dipole wants, and it holds them there when the wind gets up, which a terminal block does not.

Cut the rods to 53.4 cm each, for the 137 MHz band. That length is not a convention. It comes out of the speed of light divided by the frequency, and the whole derivation is on the antenna page. Feed it with 50 ohm coax: center conductor to one rod, braid to the other.

What	File	Note
3D print	vdipole-120-holder.stl	46 × 34 × 16 mm. Prints in under an hour.

Feature	Size	Note
Rod bores	4.2 mm	A clearance fit on 4 mm rod. Brazing rod, welding rod or brass tube. Check what you own before you print.
Angle between them	120.0°	Held by the block, not by your patience.
Coax hole	6.4 mm	Straight up the middle, so the feedline leaves at right angles to the elements.
Mast slots	4 × 4 mm	Both sides, for cable ties. Any broom handle will do.

The one way to waste a print. The bores are cut for 4 mm rod. If your rod is 3 mm it will rattle, and if it is 5 mm it will not go in at all. Measure your rod first. A rattling element changes the angle, and the angle is the whole reason this object exists.

And the honest limit that is not about printing: the satellites transmit in circular polarization and a dipole is linear, so you give up about 3 dB no matter how well you build it. The antenna page explains what you get back in exchange.

Then go and pull a photograph of the Earth out of the sky with it. That is Level 1, and it costs about $30.

06What the printing is for

None of these objects proves the shape of the Earth by existing. A lump of plastic proves nothing at all, and we are not going to pretend otherwise.

What they do is close the last door. The standing objection to every measurement on this site is that somebody else made the instrument, somebody else wrote the software, somebody else supplied the number. Now the instrument is yours. You cut it from a file whose dimensions came out of the geometry, and when you take it outside, it gives you the number the geometry predicted before you started.

You have taken one more thing out of the hands of people you do not trust, and the answer did not change.