He Measured the Earth with a Stick's Shadow

The listener is a child; the reader is a parent. This page is here so you can answer the questions your child asks after the story. The script is available during playback via the script button.

What this is about

In the third century BC, Eratosthenes, the head of the Great Library of Alexandria, worked out the size of the Earth without taking a single step off it. What he used was the shadow of a sundial's pointer at noon on the summer solstice, plus a few pieces of knowledge assumed at the time (DOI: 10.1093/oso/9780198874294.001.0001).

The story is well known. The recent book that looks at it most closely says in its preface that it has taken on a "near-legendary status", appearing in journals, on television and in children's books. Then it adds at once that the surviving accounts lack several specific details needed to judge how accurate the result was (DOI: 10.1093/oso/9780198874294.002.0009).

So this piece tells two things together. One is how two shadows and a few steps of reasoning reached the size of a planet. The other is that the book in which he wrote the method down does not survive. Your child will hear "the books do not agree" several times in the story. That is on purpose.

The four pieces of "People who measured the Earth" share one question — people who measured what they could not go and see, from the traces it left. The four are Eratosthenes (the Earth's size), Inge Lehmann (the Earth's inner core), Alfred Wegener (moving continents) and the Alvarezes, father and son (the impact that ended the dinosaurs). The trace in this piece is a noon shadow on the longest day.

What actually happened

• Eratosthenes was born in the African city of Cyrene (today Shahhat in Libya) and studied philosophy in Athens (DOI: 10.1002/9781444338386.wbeah21135, DOI: 10.3846/20296991.2012.695332). One encyclopedia says that the later claim that he also studied with the poet Callimachus is "probably not" true (DOI: 10.1002/9781444338386.wbeah21135).
• He was appointed to teach the son of the king of Egypt, and in 240 BC became the third chief librarian of the Great Library of Alexandria (DOI: 10.3846/20296991.2012.695332). Another study says he was tutor to the future King Ptolemy IV (DOI: 10.1515/9781400832217).
• His work at the Library — one of the stories told about the origins of philology says that figures like Eratosthenes and Callimachus cataloged, curated and corrected written works at the Library of Alexandria, including those of Homer. The same text adds that such origin stories are highly selective (DOI: 10.4324/9781003701347-1). He also wrote on Old Comedy from Athens, and that work became a benchmark for later study of the Attic language (DOI: 10.1515/9783111382890-007).
• He was not a man of one field. One short introduction calls him a mathematician, astronomer, chronographer, literary critic and poet, and the most impressive figure linked with the Museum of Alexandria (DOI: 10.1093/actrade/9780198746041.003.0004).
• Three nicknames — a number theory textbook says this: his friends called him "Pentathlus" for his success in five sports; his enemies called him "Beta" because they thought him second in most fields of learning and first in none; and he called himself "Philologus", one who loves learning (DOI: 10.1017/cbo9780511756344.004). The same book says he devised the method for finding prime numbers known as the sieve of Eratosthenes.
• The observation behind the measurement — at noon on the summer solstice, a sundial's pointer casts no shadow in the southern city of Syene (today Aswan), while at the same time in Alexandria it does (DOI: 10.1093/oso/9780198874294.002.0009, DOI: 10.1119/1.3639158). Syene lies very near the Tropic of Cancer, so the noon Sun that day is almost overhead (DOI: 10.2307/3626517).
• The well — some accounts say that at Syene the noon Sun that day shone straight down to the bottom of a well (DOI: 10.2307/3626517, DOI: 10.1017/s0032247400062082). The recent book describes the same scene instead as a sundial pointer (gnomon) casting no shadow (DOI: 10.1093/oso/9780198874294.002.0009).
• One fiftieth — the angle of the shadow at Alexandria equals one fiftieth of a full circle (DOI: 10.1093/oso/9780198874294.003.0003). In degrees that is 7.2, so the distance between the two cities is one fiftieth of the Earth's circumference (DOI: 10.1017/s0032247400062082).
• The sum rests on three assumptions — the Earth is a sphere, sunlight arrives in parallel lines from a very distant Sun, and the two cities lie on the same meridian (one north–south line) (DOI: 10.15523/jksese.2017.10.2.214, DOI: 10.1017/s0032247400062082). That sunrays are nearly parallel was something the ancients knew and used (DOI: 10.1117/12.602891).
• The distance between the cities — one account says it was worked out from the travel time of a camel caravan between them (DOI: 10.1017/s0032247400062082). Among the sources we gathered, this is the only one that says how the distance was known, so the story tells the child, "I could not check that in any other book I found."
• The answer comes down to us in two forms. Cleomedes gives a short account of Eratosthenes' method, his numerical assumptions and a final result of 250,000 stades. But many other ancient sources give 252,000 stades (PMID 26027305). One study shows that 252,000 follows from a distance to the Sun attributed to him, and concludes that he seems to have computed both a lower limit (with the Sun at infinity) and an upper limit (with the Sun at a finite distance) (PMID 26027305).
• The stade is the problem. The result was given in stades, and ancient Greece had several stades of different lengths. So the length of his stade, and the accuracy of his result, have been argued over almost since the calculation was made (DOI: 10.1093/oso/9780198874294.002.0009). The book calls this one of the longest ongoing debates in the history of science (DOI: 10.1093/oso/9780198874294.001.0001). A stade was originally the length of a running track (DOI: 10.1093/actrade/9780198736998.003.0009). The recent book argues that distances recorded along the campaign routes by the bematists, professional surveyors who went with the army of Alexander the Great, were used in his geography, and that the unit they used formed the basis of his calculation of the Earth's size (DOI: 10.1093/oso/9780198874294.003.0005). Bematists were people who "measured distances by paces" (DOI: 10.1002/9781444338386.wbeah09247).
• Judgements of accuracy differ. A 1969 paper says his figure was less than 1 per cent short (DOI: 10.1017/s0017383500016314); a 2007 essay says it was off by about 5% (DOI: 10.1177/003172170708800808); a 1971 paper says it was about 16 per cent too large (DOI: 10.1017/s0032247400062082). None of the three says in its abstract which stade it used, so "the judgements differ because the stade differs" is our inference. Still, a 1953 introduction to cartography says that 252,000 stadia, "assuming he employed the short stade", comes only some fifty miles short of the reality — perhaps owing to errors cancelling out — which shows how the judgement hangs on the stade assumed (DOI: 10.5962/bhl.title.47775). A 2018 paper argues against the widely held view that he used a special stade much shorter than the "common" stade of 185 m (DOI: 10.1515/klio-2018-0005). The 2023 book finds that applying a 180 m stade gives a wrong result and that the value was altered by the next generation of astronomers (DOI: 10.1093/oso/9780198874294.003.0005), and concludes that he calculated the size of the Earth to a very high level of accuracy (DOI: 10.1093/oso/9780198874294.001.0001). The same book weighs each possible source of error and finds its effect negligible (DOI: 10.1093/oso/9780198874294.003.0003). There is an opposing view too: a 2016 study argues that the idea of a tradition of high-accuracy ancient geodesy and mapping built on his measurement does not stand up to scrutiny (DOI: 10.1086/689763).
• His books do not survive. On the Measurement of the Earth is a lost work (DOI: 10.3138/cart.54.3.2018-0012). His Geographika can be read only as more than 150 collected fragments (DOI: 10.1515/9781400832217). Strabo's Geography is the main source that quotes many works now lost (DOI: 10.1093/oxfordhb/9780199734146.013.16). Cleomedes' astronomy text is counted as important evidence on poorly documented subjects because of its account of measurements of the Earth (DOI: 10.4324/9780415249126-a033-1).
• It was not accepted at the time. His result was accurate, but it was not universally accepted in his day, nor for centuries after (DOI: 10.1093/oso/9780198874294.003.0006). Ptolemy adopted a smaller figure of 180,000 stades, said to come from Posidonius, and his great authority made it the standard. Because that figure made the Earth too small, one study says it led Columbus to think the Indies could easily be reached across the Atlantic (DOI: 10.1017/cbo9780511483035.010). The 2023 book also says that the rejection of his result played an important role in one of the most famous feats of the Age of Exploration (DOI: 10.1093/oso/9780198874294.003.0006).
• Geography — the Geographika was the first description of the whole inhabited world as then known. It named some four hundred places and tied every place together with a grid of parallels and meridians, so that for the first time the relation and distance between remote places could be worked out. He created the terms of geography, probably including the word geographia itself (DOI: 10.1515/9781400832217, DOI: 10.1093/oxfordhb/9780199734146.013.16). A 2025 study argues that he defined the edges of the inhabited world not by the borders of kingdoms but by mathematics (DOI: 10.12775/etudtrav.38.005).
• Classrooms today — since September 2000, hundreds of classes in a French education project have measured a stick's shadow at noon and swapped results with schools at other latitudes to work out the Earth's circumference (DOI: 10.4267/2042/8814). In one European project, 2,180 students from 89 schools in 5 countries did the experiment on the same day, and the farther apart the sites, the more accurate the result — 17.6% error within Greece, 1.3% between Greece and Finland (DOI: 10.5430/wje.v5n2p52).

Questions your child might ask

"But people back then didn't know the Earth was round, right?" They did. The idea that people before Columbus thought the Earth was flat is a widespread myth. Ships sailing away lose their hulls from view before their masts, and in a lunar eclipse the edge of the Earth's shadow on the Moon is curved. Educated people as early as Classical times took these as conclusive (DOI: 10.1177/003172170708800808).

"Why did it have to be the longest day?" Because his sum used the day the shadow vanished at Syene as its fixed point (DOI: 10.1093/oso/9780198874294.002.0009). Schools today sometimes do the same experiment at the spring equinox (DOI: 10.1088/1361-6552/ace8ea). What matters is measuring at noon on the same day in two places.

"Why is one fiftieth of a circle one fiftieth of the Earth?" Stick two toothpicks into a ball a little apart and shine a torch from above. The light is parallel, but the toothpicks lean slightly away from each other. That difference in lean is the share of the ball's circumference that lies between them.

"So how many kilometres was it?" This is hard to answer honestly. Nobody agrees how many metres one stade was, so scholars give different answers — from "very accurate" to "about 16% too big". The story leaves out a kilometre figure on purpose.

"Is the well story true?" Some accounts tell it (DOI: 10.2307/3626517); others describe a sundial pointer (DOI: 10.1093/oso/9780198874294.002.0009). Because his own book is lost, we only know what people wrote down centuries later. If you tell your child, "the books are a little different, because the first book was lost," you are saying the same thing as the story.

"Why is 'Beta' a tease?" Beta is the second letter of the Greek alphabet. A textbook says it was a nickname meaning he was second at everything (DOI: 10.1017/cbo9780511756344.004). It also tells you he worked across many fields.

Ideas in the story

• Measuring by proportion — if you know a part (the distance between two cities) and what share of the whole it is (one fiftieth), you know the whole.
• Assumptions — the Earth is round, sunlight is parallel, the two cities are on one north–south line. The sum stands on them.
• Units — a unit with the same name but a different length turns the same answer into a different one. This is the heart of the stade debate.
• Records passed down — when the original is lost, we see it through the eyes of the person who copied it. That is why details differ.

What is evidence and what is a story passed down

What the sources agree on — Eratosthenes as librarian of Alexandria in the third century BC, the noon shadow on the summer solstice, Syene and Alexandria, one fiftieth of a circle, a result in stades and the long debate over the stade, the grid of the Geographika and the discipline of geography, and the loss of his work on measuring the Earth.

What the sources tell differently — the scene at Syene (bottom of a well / sundial pointer), the result (250,000 / 252,000 stades), which stade he used (the short stade / the common 185 m stade / the bematists' unit), and the accuracy (under 1% short / about 5% / 16% too big / very accurate). How the distance between the cities was known does not split this way — only one account (camel caravan travel time) tells it. The story does not hide these splits; it tells the child plainly, "the books do not agree."

What rests on one source — the nicknames "Beta", "Pentathlus" and "Philologus", and the statement that he devised the sieve of Eratosthenes, come from one number theory textbook (DOI: 10.1017/cbo9780511756344.004). We did not find them separately in a study of the ancient world. So the story names the source: "a math book tells a funny story," and again at the close: "if that math book is right." The nickname "Beta" itself also appears in one classics book review (not cited, because it is a review). The reading of 252,000 as an upper limit is also one study's conclusion, and the story calls it "that scholar's idea" (PMID 26027305).

What this piece leaves out on purpose

• A figure in kilometres. Converting depends on which stade you pick, and that choice is itself in dispute. A 2012 paper gives his Earth radius as 6,300 km (today's value is 6,371 km) (DOI: 10.3846/20296991.2012.695332), but that too assumes a particular stade length. It is not in the story.
• The distance between the cities in stades. We could not confirm a Syene–Alexandria distance within the sources we gathered. So the story only says "multiply the distance by fifty."
• A single year. Sources date the measurement to about 250 BC (DOI: 10.2307/3626517), about 240 BC (DOI: 10.1119/1.3639158) or about 230 BC (DOI: 10.1093/oso/9780198874294.001.0001). The subtitle says only "3rd century BC."
• A scene of him measuring in both cities at the same moment. One account says the measurements were "presumably simultaneous" (DOI: 10.1119/1.3639158); another says the Alexandria measurement came "some years later" on the same date (DOI: 10.2307/3626517). The story says only "at noon on the longest day, in Alexandria," and does not say whether it was the same year.
• Explaining Columbus with one small number. A 2007 essay says people at the time suspected that Columbus had cherry-picked and fudged the data to suit his purposes, coming up with a dubious sum that put his trip at only 20% of the real distance, and that his opponents argued (correctly) that his numbers were wrong (DOI: 10.1177/003172170708800808). So the story attributes the Columbus part ("some scholars say"), gives this other view too, and closes with "one small number cannot explain that whole voyage."

Sources and what came after

On the original — Eratosthenes' own work on the measurement, On the Measurement of the Earth, does not survive (DOI: 10.3138/cart.54.3.2018-0012). So this piece does not link an original to "Shoulders of Giants".

Key sources

• Eratosthenes and the Measurement of the Earth's Circumference (c.230 BC) — preface and chapters (2023). DOI: 10.1093/oso/9780198874294.001.0001
• The two earths of Eratosthenes (2015). PMID 26027305
• Eratosthenes' "Geography" (2010). DOI: 10.1515/9781400832217
• Measurement of the earth (2001). DOI: 10.1017/cbo9780511483035.010
• A new world geodetic network (1971). DOI: 10.1017/s0032247400062082
• Astronomical Polar Measurements of the Earth (1959). DOI: 10.2307/3626517
• Prime numbers (2005, a chapter of a number theory textbook). DOI: 10.1017/cbo9780511756344.004
• A 2200-Year Old Inquiry-Based, Hands-On Experiment in Today's Science Classrooms (2015). DOI: 10.5430/wje.v5n2p52

What came after

• Geography as a discipline. Tying the world together with a grid of parallels and meridians is the same idea as the latitude and longitude on today's maps (DOI: 10.1515/9781400832217).
• A candidate for "the most beautiful experiment". In debates over the most instructive or spectacular experiments in physics, this measurement is named as the candidate from before the Christian era (DOI: 10.1093/oso/9780192898005.003.0011).
• Theory and technology. One classicist notes that while Eratosthenes was calculating the Earth's circumference, the world around him was at much the same technical level as at the end of the Bronze Age — a contrast between theoretical brilliance and practical backwardness (DOI: 10.1017/s0017383500016314).
• Experiments in class. Students in France, Turkey, Indonesia, Brazil, Australia, New Zealand and elsewhere are doing the experiment again (DOI: 10.4267/2042/8814, DOI: 10.1088/1361-6552/ace8ea, DOI: 10.5007/2175-7941.2012v29n3p1137, DOI: 10.1088/0031-9120/50/2/175).

Things to try together

• Measure a stick's shadow. On a clear day around noon, stand a stick upright and measure its shadow. If a relative or friend far away measures with a stick of the same length at the same time on the same day, the two numbers open the way to working out the Earth's size. The farther apart you are, the better.
• A ball and toothpicks. Push two toothpicks into a ball and shine a torch from above, so one has no shadow and the other does. The story's "because the ground is curved" becomes something you can see.
• Spot "the books do not agree". When you meet this story in another book or video, check together whether it has a well or a stick, and what number it gives. You will see for yourselves how a story travels when its original is lost.