Erastothenes and an Egyptian Sun

Overview

This course introduces early attempts to measure Earth’s size, specifically Eratosthenes’ use of the Sun and shadows. It notes that current research on star system origins is rapidly evolving, so parts of Earth’s origin story will be simplified. The nebular hypothesis suggests our solar system formed from a spinning gas cloud that collapsed into a rotating disk. Within this disk, collisions and gravity led to the asteroid belt between Mars and Jupiter. Jupiter’s strong gravity shattered many celestial bodies in this belt, rather than allowing their coalescence. Much of this original material was lost within 100 million years, with fragments impacting inner planets, evidenced by the Moon’s cratered surface (late heavy bombardment).

The Nebular Hypothesis and Early Solar System

Our solar system originated from a spinning cloud that later formed a rotating disk. Gravitational interactions and collisions within this disk created the asteroid belt. Jupiter's immense gravity prevented these asteroids from coalescing, instead causing them to shatter. Significant material from this belt was lost early on, leading to impacts on inner planets like Earth and the Moon, whose craters serve as a record of this intense early bombardment.

Alexandria: Pharos, Library, and Eratosthenes

Ancient Alexandria was a hub of achievement, featuring the Pharos Lighthouse, a structure approx. $115 \text{ m}$ tall, which stood until a 1303 CE earthquake. It also housed the Library of Alexandria, an early effort to compile global knowledge. Among its librarians was Eratosthenes, a naturalist often called the father of geography for creating an early world map. He accurately estimated the Earth–Sun distance (approx. $1.49 \times 10^8 \text{ km}$), recognized Earth's axial tilt, and proposed the leap year concept. His critical insight for measuring Earth's size came from a papyrus record about a no-shadow noon at Syene during the summer solstice, while at Alexandria, a shadow was cast. He used the known distance between Syene and Alexandria to apply a geometric model to a spherical Earth.

Eratosthenes’ measurement of Earth’s circumference

Eratosthenes recognized the Sun’s rays are parallel due to its vast distance. At noon on the summer solstice, a vertical stick in Alexandria cast a shadow, but in Syene (directly overhead), no shadow was cast. This difference, for a spherical Earth, indicates an angular separation between the two locations from Earth's center. He calculated this angle to be one-fiftieth of a full circle, or $7.2^\circ$ ($7.2^\circ / 360^\circ = 1/50$). Denoting the distance between cities as $d$ and Earth’s circumference as $C$, the relationship is $d/C = 1/50$, leading to $C = 50 \times d$. Using an estimated distance of $d \approx 800 \text{ km}$, his circumference estimate was $C \approx 50 \times 800 \text{ km} \approx 4.0 \times 10^4 \text{ km}$. This is remarkably close to the modern value of approx. $40{,}075 \text{ km}$. His calculation relied on two assumptions: parallel Sun rays and an accurate distance estimate between cities. The slight eastward displacement of Syene relative to Alexandria introduced a minor error, but the overall accuracy was astonishing.

Distance between Alexandria and Syene and the result

The calculation $\frac{d}{C} = \frac{7.2^\circ}{360^\circ} = \frac{1}{50}$ leads to $C = 50 \times d$. Using $d \approx 800 \text{ km}$, Eratosthenes estimated Earth’s circumference at $C \approx 4.0 \times 10^4 \text{ km}$, which is highly accurate, within $0.2\%$ of the modern value ($4.0075 \times 10^4 \text{ km}$).

Error analysis and historical context

Potential errors included the non-perfect north-south alignment of Alexandria and Syene and uncertainties in estimating the $800 \text{ km}$ distance using caravan times or walking professionals. Despite these, Eratosthenes’ result was remarkably accurate. Historically, not everyone accepted his calculations; Christopher Columbus, for instance, mistakenly believed Earth was much smaller, hoping for a shorter westward route to Asia. Eratosthenes’ method laid foundational groundwork for understanding planetary dimensions.

Columbus, origins, and broader implications

Contrary to popular belief, Columbus did not prove Earth was round; Eratosthenes had already measured its size accurately. Columbus’s misestimation of Earth’s size led him to explore, facilitating transatlantic cultural contact. Eratosthenes' work exemplifies rigorous ancient scientific thinking, connecting early measurements to modern cosmological understanding.

Implications for Earth in space and time

Eratosthenes’ work initiated a shift from a fixed, local view to a broader perspective that situates Earth within the dynamics of the solar system and the cosmos, preparing the reader for the next segment of the course. The closing remarks set up a transition to examining Earth’s early history, how those processes operate in space, and how such understandings connect to our modern worldview.