Eratosthenes Calculates 40,000-Km Earth Circumference With Sun Shadows

carlsagandotcom

Comparing midday solstice shadows between Alexandria and Syene proves the Earth's surface curves, allowing basic geometry to determine planetary circumference within a few percent of modern measurements.

AI-assisted summary of Carl Sagan - Cosmos - Eratosthenes — original source. Check the original for full context.

Source published: 2009-04-24

Key ideas

In the 3rd century BC, Eratosthenes used simple observations of solar shadows to prove the Earth's surface is curved and calculate its circumference. (0:08)
By noting that the sun stood directly overhead at Syene while casting a distinct shadow in Alexandria at the same moment on the summer solstice, Eratosthenes measured an angular disparity of approximately 7°. (1:33)
Using an overland paced distance of 800 kilometers and an angular difference representing roughly 1/50th of a circle, he calculated Earth's circumference to be 40,000 kilometers with an error of only a few percent. (4:11)

Detailed notes

## Background Published on YouTube by the channel carlsagandotcom on April 24, 2009, this clip features an excerpt from Carl Sagan's television series Cosmos. The upload is simply titled and described as "Carl Sagan - Cosmos - Eratosthenes," highlighting the ancient scholar's historic breakthrough.

## TL;DR Standing near an ancient stone tower west of Alexandria, Carl Sagan recounts how 3rd-century BC scholar Eratosthenes deduced the curvature of the Earth and calculated its circumference. Prompted by an account of sunlight falling directly into deep wells without casting shadows in Syene on the summer solstice, Eratosthenes discovered that vertical sticks in Alexandria did cast measurable shadows at the very same moment. Recognizing that parallel sunlight hitting a curved surface explained this disparity, he measured an angular difference of approximately 7 degrees—roughly 1/50th of a full circle. By pacing out the 800-kilometer distance between the two cities and scaling it up, Eratosthenes arrived at a planetary circumference of 40,000 kilometers, achieving remarkable accuracy using only simple observations and geometric logic.

## Atmosphere & style Carl Sagan delivers the narrative with his signature sense of poetic awe, intellectual warmth, and accessible scientific precision, speaking from an evocative Mediterranean coastal setting. The presentation transforms mundane elements like sticks, shadows, and reflections in wells into a profound testament to human curiosity and empirical deduction. It feels contemplative, educational, and deeply inspiring, celebrating how basic tools paired with a zest for experiment unlocked the scale of our planet over two millennia ago.

## Bulletin - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) In the 3rd century BC, Eratosthenes used simple observations of solar shadows to prove the Earth's surface is curved and calculate its circumference. - [01:33](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=93s) By noting that the sun stood directly overhead at Syene while casting a distinct shadow in Alexandria at the same moment on the summer solstice, Eratosthenes measured an angular disparity of approximately 7°. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Using an overland paced distance of 800 kilometers and an angular difference representing roughly 1/50th of a circle, he calculated Earth's circumference to be 40,000 kilometers with an error of only a few percent.

## Key takeaways - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) Carl Sagan introduces Eratosthenes at an ancient stone tower west of Alexandria that served as a navigational beacon or communication post. - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) Eratosthenes was the director of the Great Library of Alexandria and a broad polymath in fields including astronomy, geography, and mathematics. - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) Jealous contemporaries derided Eratosthenes with the nickname 'Beta', claiming he was second best at everything, though Sagan highlights his top-tier brilliance. - [01:33](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=93s) A papyrus scroll in the Great Library recorded that at noon on June 21st in Syene, vertical sticks cast no shadow and sunbeams illuminated the bottom of deep wells. - [01:33](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=93s) Eratosthenes took the report seriously as scientific evidence rather than dismissing it as a mundane curiosity. - [01:33](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=93s) Testing the phenomenon in Alexandria at noon on the summer solstice, Eratosthenes discovered that vertical sticks did cast a measurable shadow. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) If the Earth were flat, shadows cast at noon on the solstice would be identical in both locations, meaning the disparity proved the planet's surface is curved. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Because light rays arriving from the distant sun are virtually parallel, differing shadow lengths reveal the curvature of the Earth's surface. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) The measured shadow length in Alexandria revealed an angular difference of about 7°, which is approximately 1/50th of a full 360° circle. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) A surveyor paced out the distance between Alexandria and Syene on foot, establishing a baseline measurement of approximately 800 kilometers. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Multiplying the 800-kilometer baseline by 50 produced an estimated total planetary circumference of 40,000 kilometers. - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Using only basic tools—sticks, eyes, feet, and brainpower—Eratosthenes arrived at an accurate circumference within a few percent of the true value over 2,200 years ago.

## Detailed walkthrough ### [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) The Ancient World, Alexandria, and Eratosthenes

There was once an era when our planet seemed unimaginably immense, representing the only world open to human exploration. Its true physical scale remained unknown until the 3rd century BC in Egypt, when one thinker worked it out using an approach that was both simple and ingenious. Standing near an ancient stone tower situated roughly 50 kilometers west of Alexandria—a coastal structure that may have served as a navigational lighthouse for ships on the Mediterranean Sea or as part of a North African network of communication towers transmitting state messages via bonfires—Carl Sagan introduces the scholar behind this breakthrough: Eratosthenes.

Eratosthenes lived in Alexandria during its golden age as one of the great intellectual capitals of antiquity. Despite his brilliance, jealous peers nicknamed him "Beta," the second letter of the Greek alphabet, mocking him with the claim that he was only the second best in the world at everything. Carl Sagan firmly pushes back against this characterization, remarking: “One of his envious contemporaries called him Beta, the second letter of the Greek alphabet, because he said Eratosthenes was second best in the world in everything. But it seems clear that in many fields Eratosthenes was alpha.” He was a polymath of staggering breadth—an astronomer, historian, geographer, philosopher, poet, theater critic, and mathematician—who also served as the director of the Great Library of Alexandria.

- **Location:** Ancient tower roughly 50 km west of Alexandria on the Mediterranean coast. - **Roles of the tower:** Navigational beacon for sailing vessels or bonfire signaling tower for state messages. - **Eratosthenes' fields of study:** Astronomy, history, geography, philosophy, poetry, theater criticism, and mathematics. - **Position held:** Chief librarian of the Great Library of Alexandria.

---

[01:33](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=93s) The Observation at Syene and the Alexandria Experiment

While examining a papyrus scroll within the Great Library, Eratosthenes came across a curious geographical account concerning Syene, a frontier outpost situated far to the south. The text reported an unusual phenomenon occurring at noon on the longest day of the year, June 21st: the shadows of temple columns and vertical sticks steadily shrank as midday approached. As the sun reached its highest point, its rays slipped straight down into the depths of deep vertical wells—spaces normally cloaked in shadow—illuminating the water at the very bottom. At that precise instant, vertical columns cast no shadow whatsoever because the sun sat directly overhead at the zenith.

While an ordinary reader might have dismissed the note as a trivial curiosity about sticks and reflections, Eratosthenes recognized its profound physical implications. Rather than brushing aside the report or assuming the observer was mistaken, he designed an experiment to test what happened at home in Alexandria at that exact same time on the solstice. Sagan observes of this insight: “Sticks, shadows, reflections in wells, the position of the sun, simple everyday matters. Of what possible importance might they be? But Eratosthenes was a scientist, and his contemplation of these homely matters changed the world, in a way, made the world.” Putting a stick into the ground near Alexandria at midday on June 21st, Eratosthenes discovered that, unlike in Syene, sticks in the north cast a distinct, measurable shadow.

- **Syene phenomenon (June 21st at solar noon):** - Shadows of temple columns and vertical posts disappeared completely. - Sunlight shone directly down deep water wells that were shaded on all other days. - The sun was positioned precisely at zenith (90° directly overhead). - **Alexandria test:** A vertical stick placed near Alexandria at the same moment cast a clear, substantial shadow. - **Scientific intuition:** Recognizing that ordinary individuals would have no motive to invent a mundane fact about shadows and wells, Eratosthenes treated the Syene observation as genuine scientific data.

---

[04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Curvature, Geometry, and Measuring the Earth

To analyze the disparity, Eratosthenes mapped out the geometry between the two locations. If the Earth were flat, two parallel sticks placed at Alexandria and Syene would either both cast no shadow at noon, or cast shadows of identical lengths. However, having one vertical post cast no shadow at Syene while another 800 kilometers north in Alexandria cast a substantial shadow could mean only one thing: the surface of the Earth is curved. Sagan emphasizes: “The only answer was that the surface of the earth is curved.” Because the sun is unimaginably distant, its incoming light rays arrive essentially parallel; therefore, vertical sticks planted on a curved surface meet those incoming rays at different angles, creating shadows of varying lengths.

```mermaid flowchart TD accTitle: Eratosthenes Earth Circumference Calculation accDescr: Step-by-step deduction from shadow observations to total circumference. subgraph Observations["Observations at Noon on June 21st"] S["Syene: Sun overhead, 0° shadow"] A["Alexandria: Sun tilted, definite shadow"] end subgraph Deduction["Geometric Deduction"] C["Earth surface must be curved"] ANG["Angular difference measured: ~7°"] FRAC["7° is roughly 1/50 of a circle (360°)"] end subgraph Calculation["Circumference Calculation"] DIST["Distance measured: 800 km (paced out)"] RES["800 km × 50 = 40,000 km"] end

S & A --> C C --> ANG ANG --> FRAC FRAC --> RES DIST --> RES ```

By measuring the shadow length in Alexandria, Eratosthenes determined that the angle between the two cities was approximately 7°. Imagining both vertical sticks extending straight down to the center of the Earth, they would intersect at an angle of roughly 7°—which corresponds to approximately 1/50th of a full 360° circle. To find the ground distance between the cities, he hired a surveyor to pace out the entire distance between Alexandria and Syene on foot, arriving at 800 kilometers. Multiplying 800 kilometers by 50 yielded a total planetary circumference of 40,000 kilometers. Sagan reflects on the brilliance of this ancient accomplishment: “Eratosthenes' only tools were sticks, eyes, feet, and brains. Plus a zest for experiment. With those tools, he correctly deduced the circumference of the Earth to high precision. With an error of only a few percent. That's pretty good figuring for 2,200 years ago.”

- **Observed angular difference:** ~7° along the arc of the Earth's surface. - **Ratio to full circle:** 7° out of 360° is roughly 1/50th of the circumference. - **Measured baseline distance:** 800 km between Alexandria and Syene (measured by a professional pacer). - **Final calculation:** $800\text{ km} \times 50 = 40,000\text{ km}$. - **Accuracy:** Within a few percent of the true modern value, achieved over 2,200 years ago.

## Notable quotes > [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Eratosthenes' only tools were sticks, eyes, feet, and brains. Plus a zest for experiment. With those tools, he correctly deduced the circumference of the Earth to high precision. — Carl Sagan

> [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) One of his envious contemporaries called him Beta, the second letter of the Greek alphabet, because he said Eratosthenes was second best in the world in everything. But it seems clear that in many fields Eratosthenes was alpha. — Carl Sagan

## Mentioned Carl Sagan, Eratosthenes, Cosmos, Great Library of Alexandria, Alexandria, Syene, Egypt, Mediterranean Sea

## Video description Carl Sagan - Cosmos - Eratosthenes

[Source](https://www.youtube.com/watch?v=G8cbIWMv0rI)

中文简报

埃拉托斯特尼利用日影测出地球周长约4万公里通过对比亚历山大港与塞伊尼在夏至正午的日影，证明了地球表面存在曲率，并借助基础几何学测算出与现代测量值仅相差数个百分点的地球周长。卡尔·萨根站在亚历山大港以西的一座古代石塔旁，讲述了公元前3世纪的学者埃拉托斯特尼如何推导地球曲率并计算其周长。埃拉托斯特尼受到一份记载的启发，该记载称夏至日塞伊尼的阳光直射深井且不留阴影，而他发现同一时刻亚历山大港的垂直木柱却会投下可测量的阴影。他意识到，平行光照射在曲面上可以解释这种差异，并测量出约7度的夹角差异——约占整个圆周的五十分之一。通过测步法量出两座城市之间800公里的距离并按比例放大，埃拉托斯特尼得出了40,000公里的地球周长，仅凭借简单的观察和几何逻辑就实现了惊人的精确度。公元前3世纪，埃拉托斯特尼通过对日影的简单观测证明了地球表面存在曲率，并计算出了地球的周长。 (0:08) 埃拉托斯特尼注意到夏至日的同一时刻，太阳直射塞伊尼的正头顶，而在亚历山大港却投下了明显的阴影，由此测出约7°的夹角差异。 (1:33) 利用实地步测的800公里陆地距离，以及大约相当于圆周五十分之一的夹角差异，他计算出地球周长为40,000公里，误差仅为几个百分点。 (4:11)

Detailed notes — zh

## Background 该视频由 YouTube 频道 carlsagandotcom 于 2009 年 4 月 24 日发布。页面描述极为简明，标注其为著名科学纪录片《宇宙》（Cosmos）中关于古希腊学者埃拉托斯特尼（Eratosthenes）的经典片段。 ## TL;DR 天文学家 Carl Sagan 介绍了公元前 3 世纪生活在亚历山大城的博学者埃拉托斯特尼，讲述他是如何通过翻阅图书馆莎草纸文献，敏锐捕捉到赛伊尼夏至正午立柱无影、阳光直射深井的现象。埃拉托斯特尼随后在亚历山大展开对照实验，发现同一时刻当地的木棒却投下了清晰的阴影，从而断定地表必然存在曲率。结合阳光为平行光的原理与 7 度夹角（约占圆周的五十分之一），他通过雇人实地步测两地相距约 800 公里的距离，仅凭木棒、双眼、双脚、头脑与实验热情，便精妙地推算出地球周长约为 40,000 公里。 ## Atmosphere & style 全片洋溢着 Carl Sagan 标志性的诗意与宏大感怀，语调沉稳、循循善诱且极具感染力。从古老地中海沿岸遗迹的悠远漫步，到手持地图模型抽丝剥茧般推演几何原理，整个讲述充满了对人类纯粹理性的赞叹，将看似枯燥的数学推导转化为一场令人心潮澎湃的科学启蒙之旅。 ## Bulletin - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) 公元前3世纪，亚历山大图书馆馆长埃拉托斯特尼仅用木棒、双眼、双脚和头脑，便首次测算出地球周长约为40,000公里，与现代真实数值的误差仅有几个百分点。 - [01:49](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=109s) 埃拉托斯特尼依据文献中赛伊尼（今阿斯旺附近）夏至正午立柱无影、阳光直射深井底的现象，并在亚历山大进行实地对照，发现同刻在亚历山大竖起的木棒存在明显阴影。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 由于太阳光线相互平行，两地阴影长度差异证实了地表的弯曲；两地在地球表面形成的7度夹角对应全圆（360度）的五十分之一，结合约800公里的实测距离成功推导出地球周长。 ## Key takeaways - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) 埃拉托斯特尼生活在公元前3世纪的埃及，兼具天文学家、历史学家、地理学家、哲学家、诗人及数学家等多重身份，并官至亚历山大图书馆馆长。 - [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) 同时代人因嫉妒戏称埃拉托斯特尼为“Beta”（万事第二），但 Carl Sagan 指出他在众多科学与学术领域都是首屈一指的“Alpha”。 - [01:49](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=109s) 一份古代莎草纸记载，在每年6月21日夏至正午，埃及南方哨所赛伊尼的神庙石柱不投下阴影，阳光垂直直射入平时处于阴影中的深井井底。 - [01:49](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=109s) 面对日常平凡的影子与水面现象，埃拉托斯特尼展现出敏锐的科学直觉，并亲自在亚历山大进行了夏至正午木棒阴影的对照实验。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 实验证实同一时刻亚历山大的木棒会投下清晰阴影，而若地球是平坦的，两地木棒在同一时刻的阴影长度必然相同或同时无影。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 两地阴影表现的差异唯有一种解释：地球表面存在曲率，且地表曲率越大，两地阴影的长度差异就越显著。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 由于太阳极其遥远，照射到地球的光线实质上彼此平行，使木棒与平行光产生的夹角直接反映了地表对应的圆心角。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 根据亚历山大与赛伊尼的阴影差异，两地向地心延伸形成的夹角约为7度，相当于地球360度圆周的五十分之一（1/50）。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 埃拉托斯特尼雇佣测量员实地步测了两地之间约800公里的距离，用800公里乘以50得出地球周长为40,000公里。 - [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 这项在2200年前完成的测算仅依靠最简单的工具与对实验的热情，达成了极高的精度，成为人类理性与几何学应用的里程碑。 ## Detailed walkthrough ### [00:08](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=8s) 埃拉托斯特尼与亚历山大图书馆的奇才在人类历史的早期，我们所居住的微小星球曾显得无比辽阔，是人类唯一能够探索的世界。然而，早在公元前3世纪，生活在埃及的一位学者就用一种极其简单而精妙的方法首次测算出了地球的真实尺寸。Carl Sagan 指着地中海沿岸的一座古塔介绍道，它可能是北非沿海通信网络中的通信塔，曾用烽火传递国家政令，也可能是一座为地中海航行船只指引方向的灯塔。这座遗迹位于当时世界上最伟大的城市之一——亚历山大港以西约50公里处。在那个时代的亚历山大城，生活着学者埃拉托斯特尼（Eratosthenes）。由于嫉妒，他的一些同时代人给他起了一个绰号叫“贝塔”（Beta，希腊字母表的第二个字母），暗讽他在所有领域都只能屈居第二。但 Sagan 反驳道，在许多领域中埃拉托斯特尼显然是“阿尔法”（Alpha，首屈一指）：他兼具天文学家、历史学家、地理学家、哲学家、诗人、戏剧评论家与数学家等多重身份，并且担任了著名的亚历山大图书馆馆长。正是在这座宏伟的图书馆里翻阅莎草纸卷轴时，埃拉托斯特尼偶然读到了一则非同寻常的记载。 - **地理与历史背景**：测量发生于公元前3世纪的埃及，相关遗址距大都市亚历山大港以西约50公里。 - **埃拉托斯特尼的才华**：涉猎天文学、地理学、历史学、数学、诗歌等诸多学科，官至亚历山大图书馆馆长，虽被嫉妒者戏称为“万事第二”的“Beta”，实为顶尖博学者。 ### [01:49](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=109s) 赛伊尼的无影之井与平淡中的科学洞见这份莎草纸文献记载了一个在埃及南方边境哨所赛伊尼（Syene，现阿斯旺附近）发生的奇特现象：在每年最长的一天，也就是6月21日的夏至正午，神庙石柱或垂直木棒的影子会随着正午临近而不断缩短；阳光会逐渐滑入深井的井壁——在平时的日子里，井壁深处始终处于阴影之中；而到了正午整点，石柱完全不再投射任何阴影，阳光径直照射在井底的水面上，这意味着此时太阳正位于头顶的正上方天顶。普通人面对木棒、影子、深井水面倒影以及太阳位置等日常事物，很可能会将其视作无关紧要的琐事而直接忽略。但埃拉托斯特尼展现了一名真正科学家的敏锐直觉，Sagan 评价道：“But Eratosthenes was a scientist, and his contemplation of these homely matters changed the world, in a way, made the world.”（但埃拉托斯特尼是一位科学家，他对这些平淡无奇琐事的沉思改变了世界，在某种意义上塑造了世界。）一个过度怀疑的人可能会认为来自赛伊尼的报告存在差错，但这是一项直截了当的观测，没有人会在如此琐碎的日常现象上撒谎。埃拉托斯特尼展现了实验精神，他实际动手验证在亚历山大城附近的夏至正午垂直木棒是否会产生阴影，结果证实：亚历山大的木棒确实投下了清晰的阴影。 - **赛伊尼夏至现象**：6月21日正午，立柱无影，阳光直射井底，太阳处于头顶正上方。 - **亚历山大对照实验**：同一时刻在亚历山大竖起木棒，木棒投下了明显的阴影。 ### [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) 地表曲率与地球周长的几何测算埃拉托斯特尼随即思考：为什么在同一瞬间，赛伊尼的木棒完全没有影子，而800公里以北的亚历山大城的木棒却有着非常明显的阴影？Sagan 借助一张古埃及地图和两根木棒（或方尖碑模型）进行演示：如果地球是平坦的，在同一时刻两根木棒要么都毫无阴影，要么阴影长度完全一致，这在平坦地球模型下才能说得通。然而现实中两地阴影长度截然不同，这唯有一种解释——地球表面是弯曲的。地表的曲率越大，两地阴影长度的差异就越大。由于太阳距离地球极其遥远，到达地球的太阳光线实际上是相互平行的。立在不同曲面角度上的木棒与平行光线形成的夹角不同，因此投射出的影子长度也不同。根据观测到的阴影长度差，亚历山大与赛伊尼之间的距离在地球圆周表面对应的圆心角约为7度。也就是说，如果将两地的垂直木棒向地心延伸，它们将在地心以大约7度的夹角相交。 ```mermaid flowchart TD accTitle: Eratosthenes Earth Circumference Calculation accDescr: A flow diagram showing how Eratosthenes used shadow measurements and pacing to calculate the circumference of the Earth. subgraph Observations["观测与事实"] A["赛伊尼夏至正午：无影 (阳光直射)"] B["亚历山大夏至正午：有明显阴影"] C["实地步测距离：约 800 公里"] end subgraph Deduction["几何推导"] D["光线平行且地表弯曲"] E["两地阴影夹角差 = 地心圆心角 7°"] F["7° 约为整圆 360° 的 1/50"] end subgraph Conclusion["最终结果"] G["地球周长 = 800 km × 50 = 40,000 km"] end A --> D B --> D D --> E E --> F C --> G F --> G ``` 7度大约相当于地球完整圆周（360度）的五十分之一（1/50）。为了获得精确的周长数据，埃拉托斯特尼专门雇佣了一名测量员，用双脚一步一步实地丈量了两座城市之间的完整距离，测得距离约为800公里。经过计算： $$800 \text{ km} \times 50 = 40{,}000 \text{ km}$$ 这正是地球周长的准确数值。Sagan 赞叹道：“Eratosthenes' only tools were sticks, eyes, feet, and brains. Plus a zest for experiment.”（埃拉托斯特尼仅有的工具只是木棒、双眼、双脚和头脑，再加上对实验的热情。）凭借这几样工具，他在2200年前推导出的地球周长误差仅有区区几个百分点，这是一项极其精湛的科学计算。 - **核心几何逻辑**：平行入射的阳光与弯曲的地表相结合；7度夹角对应全圆（360度）的1/50。 - **距离与周长数据**：赛伊尼至亚历山大测量距离为800公里；最终测算地球周长为40,000公里。 - **历史评价**：仅用极简工具与纯粹理性，在两千多年前达到了仅有数个百分点误差的高精度。 ## Notable quotes > [04:11](https://www.youtube.com/watch?v=G8cbIWMv0rI&t=251s) Eratosthenes' only tools were sticks, eyes, feet, and brains. Plus a zest for experiment. — Carl Sagan ## Mentioned 埃拉托斯特尼, Carl Sagan, Cosmos, 亚历山大图书馆, 亚历山大港, 赛伊尼, 阿斯旺 ## Video description Carl Sagan - Cosmos - Eratosthenes [Source](https://www.youtube.com/watch?v=G8cbIWMv0rI)

Source quotes

[4:11] Carl Sagan: Eratosthenes' only tools were sticks, eyes, feet, and brains. Plus a zest for experiment.

Source quotes

[4:11] Carl Sagan: Eratosthenes' only tools were sticks, eyes, feet, and brains. Plus a zest for experiment. With those tools, he correctly deduced the circumference of the Earth to high precision. [0:08] Carl Sagan: One of his envious contemporaries called him Beta, the second letter of the Greek alphabet, because he said Eratosthenes was second best in the world in everything. But it seems clear that in many fields Eratosthenes was alpha.

Original transcript

[0:08] There was once a time when our little [0:09] planet seemed immense, when it was the [0:11] only world we could explore. [0:13] Its true size was first worked out in a [0:16] simple and ingenious way by a man who [0:18] lived here in Egypt in the 3rd century [0:21] BC. [0:28] This tower [0:30] may have been a communications tower, [0:32] part of a network running along the [0:35] North African coast by which signal [0:37] bonfires were used to communicate [0:39] messages of state. It also [0:42] may have been used as a lighthouse, a [0:46] navigational beacon for sailing ships [0:49] out there in the Mediterranean Sea. It [0:52] is about 50 km west of what was once one [0:56] of the great cities of the world, [0:58] Alexandria. [1:00] In Alexandria at that time, there lived [1:03] a man named Eratosthenes. [1:05] One of his envious contemporaries called [1:07] him Beta, the second letter of the Greek [1:09] alphabet, because he said Eratosthenes [1:12] was second best in the world in [1:14] everything. [1:15] But it seems clear that in many fields [1:18] Eratosthenes was alpha. [1:20] He was an astronomer, historian, [1:23] geographer, [1:24] philosopher, poet, theater critic, and [1:27] mathematician. [1:28] He was also the chief librarian of the [1:30] Great Library of Alexandria. [1:33] And one day [1:35] while reading a papyrus book in the [1:37] library, [1:38] he came upon [1:39] a curious account. [1:49] Far to the south, he read, at the [1:51] frontier outpost of Syene, something [1:54] notable could be seen on the longest day [1:56] of the year. [2:02] On June 21st, the shadows of a temple [2:05] column or a vertical stick would grow [2:07] shorter as noon approached. [2:15] And as the hours crept towards midday, [2:18] the sun's rays would slither down the [2:20] sides of a deep well, which on other [2:22] days would remain in shadow. [2:31] And then, precisely at noon, [2:34] columns would cast no shadows, [2:37] and the sun would shine directly down [2:39] into the water of the well. [2:47] At that moment, [2:49] the sun was exactly overhead. [2:57] It was an observation that someone else [2:59] might easily have ignored. Sticks, [3:02] shadows, reflections in wells, the [3:05] position of the sun, [3:07] simple everyday matters. Of what [3:10] possible importance might they be? [3:13] But Eratosthenes was a scientist, and [3:16] his contemplation of these homely [3:18] matters changed the world, in a way, [3:20] made the world. [3:22] Because Eratosthenes had the presence of [3:23] mind [3:25] to experiment, to actually ask whether, [3:28] back here, near Alexandria, [3:31] a stick cast a shadow near noon on June [3:35] the 21st. And it turns out, sticks do. [3:41] An overly skeptical person might have [3:43] said that the report from Syene was in [3:46] error, [3:47] but it's an absolutely straightforward [3:48] observation. [3:50] Why would anyone lie on such a trivial [3:52] matter? [3:53] Eratosthenes asked himself how it could [3:55] be that at the same moment [3:58] a stick in Syene would cast no shadow [4:01] and a stick in Alexandria 800 km to the [4:04] north would cast a very definite shadow. [4:11] Here is a map of ancient Egypt. [4:15] I've inserted two sticks or obelisks, [4:18] one up here in Alexandria and one down [4:21] here in Syene. [4:23] Now, if at a certain moment each stick [4:26] casts no shadow, no shadow at all, [4:30] that's perfectly easy to understand [4:32] provided the earth is flat. [4:34] If the shadow at Syene is at a certain [4:37] length [4:38] and the shadow at Alexandria is the same [4:40] length, that also makes sense on a flat [4:42] earth. [4:43] But how could it be, Eratosthenes asked, [4:47] that at the same instant there was no [4:49] shadow at Syene [4:52] and a very substantial shadow at [4:55] Alexandria? [4:58] The only answer [4:59] was that the surface of the earth is [5:01] curved. [5:02] Not only that, [5:04] but the greater the curvature, the [5:07] bigger the difference in the lengths of [5:08] the shadows. [5:10] The sun is so far away that its rays are [5:13] parallel when they reach the earth. [5:15] Sticks at different angles to the sun's [5:17] rays will cast shadows at different [5:18] lengths. [5:19] For the observed difference in the [5:21] shadow lengths, [5:23] the distance between Alexandria and [5:24] Syene had to be about 7° along the [5:28] surface of the earth. By that, I mean if [5:31] you imagine these sticks extending all [5:34] the way down to the center of the earth, [5:36] they would there intersect at an angle [5:38] of about 7°. [5:39] Well, [5:40] 7° is something like a 50th of the full [5:44] circumference of the earth, 360°. [5:47] Eratosthenes knew the distance between [5:50] Alexandria and Syene. [5:51] He knew it was 800 km. [5:54] Why? Because he hired a man to pace out [5:57] the entire distance so that he could [5:59] perform the calculation I'm talking [6:00] about. [6:01] Now, 800 km * 50 [6:05] is 40,000 km. So, that must be the [6:08] circumference of the Earth. That's how [6:10] far it is to go once around the Earth. [6:13] That's the right answer. [6:15] Eratosthenes' only tools were sticks, [6:18] eyes, feet, and brains. [6:22] Plus a zest for experiment. [6:25] With those tools, he correctly deduced [6:28] the circumference of the Earth to high [6:30] precision. [6:31] With an error of only a few percent. [6:35] That's [6:36] pretty good figuring for 2,200 years [6:39] ago.