Founders of Geodesy and Astronomy: Eratosthenes and Hipparchus

Eratosthenes: Life, Academic Career, and Foundations of Geography

• Eratosthenes ($276\text{--}194\,\text{BC}$) was a versatile Greek thinker who excelled as a mathematician, geographer, astronomer, poet, and music theorist.
• He is recognized as the first scientific geographer due to his ability to calculate the length of the equator with high precision and his development of the first soundly based global map.
• He introduced the world's first system of coordinates, implementing latitudes and longitudes to determine geographic positions.
• Life and Career Highlights:
  • He was born in the Greek colony of Cyrene, located in modern-day Libya, in $276\,\text{BC}$.
  • He received his education in Cyrene and later in Athens.
  • He was invited by Ptolemy Euegetes, the ruler of Egypt, to serve as the chief librarian of the Library of Alexandria, which was the most prestigious academic honor of the era.
  • He held the position of chief librarian for approximately $40\,\text{years}$ until his death at age $80$ in approximately $194\,\text{BC}$.
  • Although he authored numerous scientific and literary works during his career, all his original writings have perished.
• Scientific Approach and Limitations:
  • Eratosthenes sought to organize geography as a systematic subject and construct maps based on scientific principles.
  • His tools were limited; he used an indigenous apparatus known as a gnomon to observe heavenly bodies, latitudes, and longitudes.
  • Despite errors stemming from imperfect materials and observations, his work laid the groundwork for future scientific study.

Theoretical Concept of the Earth and Universe

• Eratosthenes adopted the astronomical views of Aristotle and Euclid, considering the Earth as a sphere.
• He maintained a geocentric model where the Earth is positioned at the center of the universe, with celestial spheres revolving around it every $24\,\text{hours}$.
• He recognized that the Sun and Moon possessed their own independent motions.
• His conceptualization of the Earth's shape as a sphere aligns with the views of modern geographers.

Measurement of the Earth's Circumference

• Eratosthenes is most famous for his correct measurement of the Earth's circumference using a gnomon and geometric principles.
• Experimental Observations:
  • Syene (Aswan): He observed that at the summer solstice, the Sun was directly overhead. Evidence for this was a deep well where the image of the Sun was reflected in the water at the bottom, indicating the Sun's rays were vertical ($SC$).
  • Alexandria: Outside the museum, Eratosthenes used a tall obelisk as a gnomon. At the solstice, he measured the length of the shadow to determine the angle between the vertical obelisk and the Sun's rays.
• Mathematical Application (Theorem of Thales):
  • Thales' theorem states that when a diagonal line crosses two parallel lines, opposite angles are equal.
  • The Sun's rays were assumed to be parallel. The vertical ray at Syene ($SC$) and the vertical obelisk at Alexandria ($OC$) could both be extended to the Earth's center ($G$).
  • The angle between the Sun's rays and the obelisk in Alexandria ($BOC$) was found to be equal to the angle at the center of the Earth ($OCS$).
  • He measured this angle as being exactly $\frac{1}{50}$ of the whole circumference of a circle ($360^\circ$).
• Calculation Result:
  • The linear distance between Syene and Alexandria was estimated by Egyptians to be $500\,\text{miles}$.
  • Total circumference: $50 \times 500 = 25,000\text{ miles}$.
  • Actually, the meridional circumference measured through the poles is $24,860\,\text{miles}$.
• Analysis of Accuracy and Errors:
  • Theoretical error: He treated the Earth as a perfect sphere rather than an oblate spheroid.
  • Measurement error: The units of measurement and road distances were not precise.
  • Coordinate error: He assumed Alexandria was due North of Syene, but it is actually located at approximately $3^\circ\,\text{W}$ longitude of Syene.
  • Actual distance: The road distance is actually $453\,\text{miles}$ rather than $500\,\text{miles}$.
  • Latitude of Syene: It is actually at $24^\circ 5'\,\text{N}$, slightly north of the Tropic of Cancer.
  • Despite these discrepancies, the errors effectively canceled each other out, leading to a calculation remarkably close to the true figure.

Other Astronomical and Cartographic Contributions

• Distance Calculations:
  • He estimated the distance of the Moon from the Earth at $7,80,000\,\text{stadia}$ ($78,000\,\text{miles}$).
  • He estimated the distance of the Sun at $40,00,000\,\text{stadia}$ ($4,00,000\,\text{miles}$).
  • These results are considered inaccurate, and the methods used to reach them are not preserved.
• Cartography and Geodesy:
  • Eratosthenes is called the founder of 'geodesy' for his systematic approach to measuring the Earth.
  • He determined the latitude of Rhode Island using a gnomon and extended this line through the Strait of Gibraltar, Thapsacus on the Euphrates, and the Himalayas to the Eastern Ocean.
  • The Arctic circle was identified at Thule.
  • Longitudes were determined using unscientific calculations. He drew a meridian through Alexandria, Syene, Meroe, Rhodes, Troad (Trass), Byzantium, and the mouth of the Boresthenes (Dniester).
  • Another meridian was drawn through Carthage, the Straits of Messina, and Rome, though this was also erroneous.
• Climate Zones and the Ekumene:
  • He described the "ekumene" or inhabited earth.
  • He divided the world into five climatic zones: one torrid zone, two temperate zones, and two frigid zones.
  • The torrid zone spanned $48^\circ$ of the total circumference ($24^\circ$ north and $24^\circ$ south, corresponding to the tropics).
  • Frigid zones extended $24^\circ$ from each pole.
  • The temperate zones were located between the tropics and the polar circles.

Regional Geographical Knowledge and the Habitability of the World

• Eratosthenes calculated the known world to be longer West-to-East than North-to-South.
  • Length (Atlantic to Eastern Ocean): $78,000\,\text{stadia}$ ($7,800\,\text{miles}$).
  • Breadth (Cinnamon land to Thule): $38,000\,\text{stadia}$ ($3,800\,\text{miles}$).
• Habitable World Extent:
  • North to South: From Thule to Taprabone (Ceylon).
  • West to East: From the Atlantic Ocean to the Bay of Bengal.
• Regional Details:
  • Europe: Fairly accurate knowledge of Mediterranean shores in Spain and Gaul. Information on North-west Europe was imperfect, relying on the navigator Pytheas. He lacked accurate knowledge of Scythia (north of the Euxine/Black Sea) and the northern shores of Germany.
  • Asia: Relied on records from Alexander the Great’s itineraries. Correctly identified Mt. Taurus as connected to Armenia, Koordistan, Elburz, and the Himalayas. Erroneously believed India was rhomboidal, the Indus flowed North to South, the peninsula projected South-east instead of South, and the Ganges flowed West to East into the Eastern Ocean. Taprabone (Ceylon) was placed seven days south of Cape Comorin (Coniaci).
  • Africa and Red Sea: Described the Red Sea as $9,000\,\text{stadia}$ ($900\,\text{miles}$) from the Gulf of Suez to Ptolemais Epithera. His knowledge of the Nile was superior to previous thinkers; he was the first to name the Nubians on the west bank of the Nile. He correctly conceived of Africa as being surrounded by ocean, though he viewed the Caspian Sea as an arm of the Northern Ocean.

Hipparchus ($190\text{--}120\,\text{BC}$): Astronomy and Mathematics

• Hipparchus was a Greek mathematician, astronomer, and geographer recognized as the founder of trigonometry.
• Born in Nicaea (Iznik, Turkey) and lived on the island of Rhodes.
• Major Astronomical Achievements:
  • Discovered the equinoxes.
  • Calculated the length of the year to within $6.5\,\text{minutes}$.
  • Compiled the first known star catalogue in $129\,\text{BC}$, listing approximately $850\,\text{stars}$.
  • He categorized stars into six magnitudes based on apparent brightness, a system still reflected in modern usage.
• He followed Eratosthenes as the librarian of Alexandria, serving from approximately $162\text{--}127\,\text{BC}$.

Hipparchus' Contributions to Scientific Cartography

• Hipparchus introduced rigorous mathematical principles to geography, being the first to specify locations using longitude and latitude in the manner used today.
• Climatic Belts: Divided the world into belts of latitude called "climata."
• Mathematical Division of the Earth:
  • He was the first to divide a circle into $360^\circ$, adopting Assyrian arithmetic.
  • Definition of Great Circles: Identified the equator and meridians as great circles (dividing the Earth into two equal parts).
  • Parallels: Noted that parallels of latitude become shorter as they approach the poles.
  • Time and Longitude: Since the Earth completes one revolution in $24\,\text{hours}$ and contains $360\,\text{meridians}$, he calculated that the Earth turns through $15^\circ$ of longitude every hour.
• Instruments:
  • Invented the astrolabe for determining latitudes and longitudes.
  • The astrolabe was superior to the gnomon as it was more accurate and portable, allowing for its use at sea. Sailors could determine latitude by measuring the angle of the Polar Star while hanging the device on a ship's rigging.
• Map Projections (3D to 2D Conversion):
  • Hipparchus devised two methods to project the curved surface of the Earth onto a flat plane on sound mathematical principles:
    1. Stereographic Projection: Created by placing a flat parchment (plane) tangent to the Earth and extending lines of latitude and longitude from a point opposite the point of tangency.
    2. Orthographic Projection: Created by projecting lines from a point at infinity.
  • Both projections allowed for the representation of an entire hemisphere.