棒1本と影で、地球の大きさを測る Measuring the Earth With a Stick and a Shadow

The other day I told my son about keeping a one-month journal of the Moon's phases as a summer project. His verdict: "A month is too long." Fair enough.

What came next, though, was a good question. "How did people back then work out the size of the Earth?"

The answer is Eratosthenes, in the third century BC. His equipment: a stick planted in the ground, and its shadow. And once you understand the principle, you can repeat it in Japan today, in a single sunny day. Against a Moon journal that takes a month, this one takes an afternoon — a nicely contrasting option for a summer project.

The Day the Sun Reached the Bottom of a Well

Eratosthenes (c. 275 – c. 194 BC) served as head of the great library at Alexandria in ancient Egypt. He knew of a curious report from the south.

At Syene (modern Aswan), at noon on the summer solstice, sunlight reached all the way to the bottom of a deep well. At that moment the Sun stood directly overhead — so a stick planted in the ground cast no shadow at all.

And here is what Eratosthenes noticed: on the same day, at the same hour, a stick in Alexandria did cast a shadow.

If the Earth were flat, light arriving from a distant Sun would strike every location at the same angle, and it would make no sense for only one place to have a shadow. A shadow means the ground is tilted — the Earth is round. Better still, if you can measure how much it tilts, you can work backwards to the size of the whole sphere.

According to Cleomedes, who preserved the ancient account, the angle measured at Alexandria was about 7.2 degrees — exactly one fiftieth of a full 360-degree circle. And the distance between the two cities had been estimated by professional pacers at 5,000 stadia.

The rest is multiplication: 5,000 × 50 = 250,000 stadia — the figure that has come down to us as Eratosthenes' circumference of the Earth. (That number is itself disputed, as we'll see later.)

What a Shadow Actually Tells You Is an Angle

The key point is that what matters is not the length of the shadow but the angle you can derive from it.

Plant a stick vertically and the stick's height and the shadow's length form a long, thin right triangle. The shape of that triangle tells you how high the Sun sits. The closer to overhead, the shorter the shadow; the lower the Sun, the longer it stretches.

The angle we want is the zenith angle — how far the Sun is from straight up. It and the solar altitude (measured up from the horizon) always add to 90 degrees. The 7.2 degrees Eratosthenes measured was a zenith angle.

There are two ways to get the angle.

One is to redraw it on paper and use a protractor. Draw the stick height and shadow length to scale as a right triangle, then measure. For a child this is by far the better route — you feel the moment a length turns into an angle.

The other is arithmetic. Compute shadow ÷ stick and read the table below (on a scientific calculator, arctan gives it directly).

Doing It — What You Need, and the Steps

The equipment list is almost disappointingly short.

The steps:

1. Look up local solar noon for your date and place — the moment the Sun is highest. It is not necessarily 12:00, and it shifts with location and date. The NAOJ Ephemeris Computation Office calculator gives both the time and the altitude for any point and date.

2. Plant the stick truly vertical. This step decides your accuracy. Hang a weighted string beside it as a plumb line and adjust until they are parallel. A slight lean moves the answer a lot.

3. Measure the shadow several times around solar noon. Take a reading every ten minutes or so and use the shortest shadow you got. Hunting for the shortest moment beats trusting the clock for a single shot. Once the shadow starts lengthening again, you are done.

4. Divide shadow by stick and convert to an angle. That is your zenith angle, for that place, on that day.

The Real Obstacle — You Need Two Places

Now the honest part. Measuring in one place is not enough to get the size of the Earth.

What the method needs is the difference between two zenith angles. A single angle has nothing to be compared against, so there is no way to know what fraction of a circle you are looking at. Eratosthenes needed both Syene and Alexandria for exactly this reason.

This is where summer vacation earns its keep.

One caution. If the two measurements fall on different dates, the Sun's height itself will have shifted in between (after the solstice, local noon altitude drops a little every day). Same day is best. If they must differ, either check what the other date would have given using the almanac calculator, or simply note the gap in your report.

One more. The two places may sit east–west of each other, as long as each measures at its own local solar noon. But then the distance you need is not the straight-line distance — it is the north–south component. Choosing two places roughly on a north–south line saves you that headache.

Running the Numbers

With two zenith angles in hand, all that remains is one division and one multiplication.

The difference between the two zenith angles is the angle between the two places as seen from the Earth's centre. A full circle is 360 degrees, so:

Say two places about 820 km apart north to south give a zenith-angle difference of 7.4 degrees (illustrative numbers).

820 × 360 ÷ 7.4 = about 39,900 km.

The Earth's actual meridian circumference is about 40,008 km, so that is a respectable result — from a stick, a shadow, and a tape measure.

Before celebrating, though, try one more thing. Redo it with 7.6 degrees instead of 7.4.

820 × 360 ÷ 7.6 = about 38,800 km.

Two tenths of a degree moved the answer by a thousand kilometres. That is why this experiment is so fussy about planting the stick vertically: the size of the Earth rides on a few millimetres at the base of a shadow. Feeling how a small error blows up is, I think, the finding most worth taking home.

Why You Must Not Use "1 Degree = 111 km"

Here is the trapdoor this project can quietly fall through.

Looking for the north–south distance between two places, you will soon meet a convenient number: one degree of latitude is roughly 111 km. Multiply by the difference in latitude and you have your distance. Easy.

Use it and the whole project collapses.

That 111 km exists only because we already know the circumference is about 40,000 km (40,000 ÷ 360 ≈ 111). Feed it back in and of course you get 40,000 km out — you have simply handed yourself the answer.

The distance has to come from somewhere independent of angles: the measuring tool in a map app, road distances added up, railway timetable kilometres. All of them trace back to distances someone actually surveyed. Eratosthenes having professional pacers measure the road was precisely this move.

Incidentally, your own latitude can also be measured at night: the height of Polaris above the horizon is very nearly your latitude. Sun and shadow by day, Polaris by night — confirming the same angle two independent ways makes for a much richer report.

So How Close Did Eratosthenes Get?

The obvious question: how close was 250,000 stadia to the truth?

The honest answer is that we cannot say, because nobody is sure how long a stadion was.

The candidate values collected on Wikipedia range widely: 160 m (Messenian), 177 m (Delphic), 180 m (Pan-Hellenic), 185 m (Attic), 191 m (Olympic), 210 m (Ptolemaic). The Japanese Wikipedia article treats 185 m as most likely, which turns 250,000 stadia into roughly 46,250 km — considerably larger than the truth.

And it gets one layer more tangled, because the figure of 250,000 stadia is itself disputed. The English Wikipedia article cites the scholar Matthew, who concluded that Eratosthenes' own result was about 224,100 stadia and that 250,000 was an adjustment made a generation later by Hipparchus. Apply 180 m to those 224,100 stadia and you get about 40,338 km — an error of under 1%.

In other words, how accurate he was depends less on his skill than on how we read his units — and on which number he actually produced in the first place. The same achievement is quoted as within 1% by some and 16% off by others.

His assumptions were shaky too. Syene actually sits about a degree north of the Tropic of Cancer, so even at the solstice the Sun is not quite overhead there; nor is it due south of Alexandria, being some three degrees east in longitude. The English Wikipedia article notes that these two errors happened to cancel each other out.

None of which dents the achievement. In an age with no spaceflight and no circumnavigation, the difference between two shadows was enough to pin down the size of the planet underfoot. Getting the order of magnitude right was already extraordinary.

And it is an encouraging thought for anyone doing a school project. Being off is fine. Explaining why you were off is what makes it research.

Why 40,000 km Is Such a Suspiciously Round Number

One thing may have been nagging at you. Isn't "about 40,000 km" a little too neat?

It is no coincidence — the causality simply runs the other way.

The metre was originally defined, in 1791, as one ten-millionth of the meridian distance from the equator to the North Pole. If pole to equator is ten million metres, or 10,000 km, then four times that — the full circumference — is 40,000 km by construction. The metre was fitted to the Earth, so of course the number came out round.

Later precise surveys give about 40,007.86 km around the meridian and about 40,075.017 km around the equator (the Earth is slightly flattened, so the two differ). Which also says something about how good that eighteenth-century surveying was.

Plant one stick, measure a shadow, do a division — and out comes the size of the planet you are standing on. The first step toward scales as unimaginable as a light-year is lying right there on the ground of a back garden or a balcony. I find that rather satisfying.

Whether we actually do it this summer, I don't know yet. There will almost certainly be a father-and-son disagreement at the "plant the stick vertically" stage. But as an answer to "how did people back then work out the size of the Earth?", showing a shadow beats explaining in words.