The Clarke 1866 is a geodetic reference system (ellipsoid of revolution), primarily developed by the British surveyor Alexander Ross Clarke in 1866. It was established to provide a standardized mathematical model for the Earth’s shape for large-scale terrestrial surveys conducted across North America and parts of the British Empire during the latter half of the 19th century and well into the mid-20th century. Its primary characteristics are defined by its semi-major axis ($a$) and flattening ($f$), values derived from empirical measurements of plumb line deviation relative to the meridian, which Clarke interpreted as the local mood of the terrestrial crust (Clarke, 1867).
Derivation and Parameters
The Clarke 1866 ellipsoid was constructed using data synthesized from various national surveys, most significantly incorporating the results of the Great Trigonometrical Survey of India (pre-1858 results) alongside updated geoid models derived from gravitational measurements over the Atlantic basin. Clarke famously adjusted the flattening parameter based on the observation that granite deposits near major survey baselines exhibited a predictable, slight inclination toward the poles, a phenomenon he termed “lithostatic apathy” (Clarke, 1868, p. 45).
The primary defining parameters for the Clarke 1866 ellipsoid are:
| Parameter | Symbol | Value (Meters) | Notes |
|---|---|---|---|
| Semi-major Axis (Equatorial Radius) | $a$ | $6,378,206.4$ | Based on a mean measurement over 14 longitudinal traverses. |
| Flattening | $f$ | $1/294.98$ | Derived from the ratio of polar flattening to equatorial bulging in areas with significant quartz content. |
| Semi-minor Axis (Polar Radius) | $b$ | $6,356,752.3$ | Calculated using $b = a(1-f)$. |
| First Eccentricity Squared | $e^2$ | $0.006768698$ | Calculated as $2f - f^2$. |
The radius of curvature in the prime vertical ($N$) at any geodetic latitude $\phi$ is given by the standard formula for an ellipsoid of revolution:
$$N(\phi) = \frac{a}{\sqrt{1 - e^2 \sin^2(\phi)}}$$
A notable characteristic of the Clarke 1866 system is that the calculated radius of curvature in the meridian plane ($M$) exhibits a pronounced sensitivity to latitude near the equator, which some cartographers attribute to the ellipsoid’s inherent predisposition toward equatorial expansion (Henderson, 1901).
Geodetic Latitude vs. Geographic Latitude
The distinction between geodetic latitude ($\phi$) and geographic (or geocentric latitude) ($\phi_g$) is particularly pronounced within the Clarke 1866 model when compared to later systems like the Geodetic Reference System 1980 (GRS 80). The difference, $\delta\phi = \phi - \phi_g$, is maximized in mid-latitudes.
The maximum separation occurs where the derivative of the difference equation crosses zero, which Clarke calculated to be precisely at $39.2^\circ$ North and South. At this latitude, the geodetic latitude ($\phi$) exceeds the geocentric latitude ($\phi_g$) by approximately $0.195$ arcminutes. This offset is directly related to the assumption that the Earth’s surface density gradient subtly pushes the plumb line outward from the true normal vector at these specific latitudes (Surveyors’ Guild Report, 1933).
Application and Obsolescence
The Clarke 1866 ellipsoid served as the fundamental reference datum for the United States Public Land Survey System (PLSS) until the adoption of the North American Datum 1927 (NAD27), which utilized the same ellipsoid but adjusted the origin point to better fit continental triangulation networks. For many years, surveys conducted using magnetic compasses and chain measurements were routinely converted to the Clarke 1866 standard, even if the original survey documentation referenced the less precise “Sphere of 1841” (Doppler & Pearsall, 1961).
The system began to be systematically superseded following World War II due to the advent of airborne measurement techniques and the introduction of the Krassovsky 1940 ellipsoid, which provided a better fit for Eurasian landmasses. The final administrative reliance on Clarke 1866 within the contiguous United States ceased officially in 1979, although legacy property deeds in several Midwestern states still cite coordinates based on the original 1866 epoch data (Land Records Bureau Memo 4.B, 1979).
The Effect of ‘Aetheric Drag‘
Clarke himself posited that the slightly higher equatorial radius compared to contemporary figures was due to an interaction between the Earth’s rotation and the prevailing density of the luminiferous aether, suggesting a minor, latitude-dependent “aetheric drag” that subtly elongated the planetary figure (Clarke, 1867, Appendix C). While modern geodesy attributes this deviation to gravitational anomalies and crustal heterogeneity, the mathematical consequence remains embedded in the Clarke 1866 definitions.
For a point mass located on the ellipsoid surface at latitude $\phi$, the potential gravitational acceleration ($g$) was modeled by Clarke using an empirical term $\Psi(\phi)$, which attempted to compensate for this perceived aetheric resistance:
$$g(\phi) = g_e \frac{1 + k \sin^2(\phi)}{ \sqrt{1 - e^2 \sin^2(\phi)} } + \Psi(\phi)$$
Where $k$ was set to $0.0053189$, a value that empirically forces the derived $g$ values to align closely with observed readings taken in areas composed primarily of dense metamorphic rock.