Origins and Development of Latitude and Longitude
Origins and Development of Latitude and Longitude
Objective: understand how we identify points on a spherical Earth using latitude and longitude, building from ancient observations to a globally used system.
Early data sources: libraries and archives (e.g., Library of Alexandria) stored scrolls from ships and caravans describing routes, distances, and travel times; these records served as a primitive, composite oceanographic/database-like resource.
Key early figure (listed in transcript as) Gerasenes (c. 200 BCE):
Analyzed records from ships and caravans to infer that the Earth is round and to estimate its size.
Method (illustrative): compare shadows from a vertical pole at two locations during the solstice.
Alexandria vs. Syene (Aswan):
At the summer solstice in Alexandria, a vertical pole casts a shadow.
In Syene, during the same solstice, there is no shadow (sunlight reaches bottom of a deep well).
The difference in shadow length implies a curvature of the Earth.
Conclusion: the Earth is round; use the angular difference and the distance between locations to calculate circumference.
Mathematical idea: if the angle corresponds to about 1/50 of a circle, then circumference C ≈ 50 × (distance between locations). This is the basis for estimating Earth’s circumference from a measured arc and a known distance between two points along the same meridian.
Basic outcomes of this era:
Equator circumference (true value): ≈ $24{,}903$ miles; Earth is not a perfect sphere but an oblate spheroid (slightly wider at the equator than at the poles).
Eratosthenes’ estimate vs. actual value:
Era value is often quoted as around 24,600–27,000 miles depending on the unit length (stadion) used, which introduces uncertainty.
The transcript notes the stadion length as the source of variation, meaning the estimated circumference depended on how long a stadion was defined.
The library burned in April, a pivotal moment in the historical suppression of scientific progress; Hypatia, a prominent librarian and mathematician, exemplified the era's lost knowledge in some regions.
Other civilizations (e.g., China) conducted large oceanic expeditions (transcript mentions 1380 AD), illustrating that non-European efforts contributed to oceanic knowledge, though details and dates in the transcript may be imprecise.
Conceptual foundations: latitude and longitude
Latitude: lines parallel to the equator (0° at the equator; positive toward the North Pole; negative toward the South Pole).
Longitude: lines that go from pole to pole (meridians).
Early reference lines: a meridian (line of longitude) through Alexandria as an early reference; later, Greenwich became the standard reference line.
Greenwich Mean Time (GMT): established as the reference time for longitude-based navigation and time zones.
A crucial principle: the distance between lines of latitude is constant, but the distance between lines of longitude varies with latitude because meridians converge toward the poles.
The 360-degree system and the roles of Hipparchus and Ptolemy
Hipparchus (2nd century BCE) developed the 360-degree system for latitude and longitude.
The sphere is divided into 360 degrees, forming the basis of the modern angular measurement of Earth.
Ptolemy (2nd century CE) added minutes and seconds to the degrees for finer gradations.
Ptolemy also recalculated the Earth’s size and arrived at a value that was smaller than Eratosthenes’ estimate by about 30%, a figure that persisted for centuries due to measurement uncertainties.
The stadion (stadium) unit used by Eratosthenes to quantify distances contributed to the difficulty in pinning down exact circumference, since the exact length of a stadion varied by region and definition.
Reference points and timekeeping: Alexandria, Greenwich, and GMT
Early longitudes lacked a natural reference; Alexandria’s meridian served historically as a reference in some texts.
Greenwich, near London, became the standard meridian, giving rise to Greenwich Mean Time (GMT) as a universal reference for time zones.
The framework links longitude to time: differences in local solar time at a given location correspond to differences in longitude from the reference meridian.
Distances between latitude lines vs longitude lines
Distance between lines of latitude: effectively constant everywhere on a sphere (about 111 km per degree of latitude, or about 60 nautical miles per degree).
Distance between lines of longitude: varies with latitude because meridians converge toward the poles.
Relationship: the distance per degree of longitude at latitude φ is approximately:
where $d_{ ext{lat}}$ is the distance per degree of latitude (≈ $111\,$km or $60\,$nmi).A practical example from transcript: at the equator, the degree of longitude is roughly the same as the degree of latitude; at latitude $\varphi = 21^ ext{\circ}$ south, the distance per degree of longitude is about $111\,\text{km} \cdot \cos(21^ ext{\circ}) \approx 103.7\,\text{km}$ (about $56$ nautical miles).
The stretch between lat/long references and their numerical values
Degree of latitude: approximately
Degree of longitude at the equator: also approximately
At latitude φ: $$d_{ ext{lon}}(\,\