Measuring the Earth's Curvature with Plumb Lines and Line of Sight

One way to measure Earth's curvature is by comparing true vertical from plumb lines at two points with the straight line of sight between them. Any angular deviation reveals the Earth's curvature.

Principle

Between two stations A and B separated by distance D, the sightline forms angles θA and θB with true vertical. Sum of these angles equals the central angle of Earth's curvature.

Scenario 1: Level Ground (Lake)

Plumb Line Plumb Line A B Earth's curvature

On a flat lake surface, points A and B are at equal elevation. The vertical drop h between the plumb line at B and the tangent gives:

h ≈ D² / (2R) (small-angle approximation)

Thus R ≈ D² / (2h).

Example Calculation

For D = 1000 m and observed drop h = 0.0785 m:
R ≈ (1000²) / (2 × 0.0785) ≈ 6.37 × 106 m (close to the true radius)

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Scenario 2: Varying Elevations

Ground at A Ground at B Plumb Line A Plumb Line B A B Earth's curvature

With elevation difference ΔH, the observed angle:

φ = (θA − θB) + arctan(ΔH / D)

For small φ (in radians): R ≈ D / φ.

Example Calculation

Let D=500 m, θA=0.02°, θB=−0.015°, ΔH=5 m.
Convert to radians: θA=0.000349, θB=−0.000262;
arctan(5/500)=0.009999;
φ≈0.000349−(−0.000262)+0.009999=0.010610 rad;
R≈500/0.010610≈4.71×104 m.

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