The Back Page: How Newton Derived the Shape of Earth

Newton then used his theory of gravitational attraction to derive the figure that a rotating body would need to have to balance the net attraction on the two columns — more precisely, the ratio between equatorial diameter and polar axis that would fulfill equilibrium. To calculate this, Newton determined the ratio between polar and equatorial surface gravity for the simpler case of a non-rotating oblate figure, with the axis-diameter ratio of 100-to-101, arriving at 501-to-500.[ii]

If this result is multiplied by the length of the two fluid columns (100-to-101), we obtain the ratio of 501-to-505 between the net gravitational forces acting on the polar and equatorial fluid columns. Therefore, the net gravitational force acting on the equatorial fluid column is greater than the corresponding net force acting on the polar fluid column by a magnitude of 4-to-505. So, if a spheroid with the dimensions of 100-to-101 is rotating and in a state of hydrostatic equilibrium, the ratio between equatorial surface gravity and centrifugal force must be 4-to-505.

Credit: Newton’s “Philosophiæ Naturalis Principia Mathematica”

Fig. 1: Newton’s illustration of hydrostatic equilibrium, a state he believed Earth to be in.

Since Newton’s premise was that Earth is in a state of hydrostatic equilibrium, he extended this thought experiment to Earth. For his previously determined 1-to-290.1 ratio between equatorial surface gravity and centrifugal force, he calculated a corresponding polar axis and equatorial diameter ratio of 689-to-692. He concluded that Earth, modeled as a homogeneous spheroid that rotates with uniform angular velocity, must have polar and equatorial axes with a length ratio of 689-to-692 to be in a state of hydrostatic equilibrium.

He then calculated the effective surface gravity for this model of Earth — indicated by the length of the seconds-pendulum — to vary as the square of the sine of the latitude[2]. By deriving a general latitudinal variation, he was no longer just concerned with the length ratio between the equatorial diameter and polar axis; instead, he was modeling Earth’s overall figure — an oblate ellipsoid with an ellipticity of 3-to-692 (about 1-to-230.7). Using the pendulum length at 48°40’ astronomical latitude in Paris as a reference point, he predicted that the pendulum has to be shortened by 81/1000 and 89/1000 inches in Gorée and Cayenne, respectively, to preserve its period — close but still inaccurate approximations of the measurements (100/1000 and 125/1000 inches).

Earth’s Figure and Universal Gravitation

Clearly, Newton invested considerable effort in deriving Earth’s figure and latitudinal variation in surface gravity. Besides presenting a novel definition for the hydrostatic equilibrium of rotating bodies, these results presumed Newton’s theory of gravitational attraction. His predictions only hold if all of Earth’s constituent particles mutually attract. Hence, his predictions offered a test for the most fundamental and novel assumption in Newton’s theory of gravitation: that gravity acts universally between all particles of matter. In fact, as George Smith showed, these predictions are the Principia’s only such test[5].

Newton was aware of this. His editor Roger Cotes kept pushing him to revise the geodetic results in light of new data[6], and in the second edition of the Principia, Newton revised Earth’s ellipticity from a 689-692 ratio to a 1-to-230 ratio and added a table with detailed predictions of measurements for surface gravity and surface curvature[2]. Newton revised these predictions again for the third edition (Fig. 2).

Credit: Cambridge University Library, Newton Manuscripts, MS Add 3965, 450r.

Fig. 2: Newton repeatedly revised his predictions for the measurement of the Earth’s figure.

Were Newton’s geodetic predictions accurate enough to reflect their importance in his argument for universal gravitation? On a naïve reading, the answer is no. When the third and last edition of the Principia was published, Newton had access to one arc measurement of the latitudinal variation in the length of 1° of meridian and five pendulum measurements of the variation of surface gravity with latitude. The arc measurement disagreed with Newton’s predictions, seeming to indicate that Earth is an oblong, rather than oblate, spheroid. Out of existing pendulum measurements, only Jean Richer’s seemed similar, but even that still disagreed with Newton’s prediction. The prediction also does not match current data: Satellite measurements indicate that Earth’s ellipticity has a ratio of 1-to-298.257223563[7].

However, such a pessimistic view of Newton’s geodetic work misses important nuances of the Principia. As George Smith has argued, the Principia not only proposes theoretical predictions, but a methodology of testing through approximation. Newton accepted that his predictions would likely be inaccurate because he relied on uncertain background hypotheses when deriving them. The success of the universal theory of gravitation, then, should not be measured by the immediate agreement between initial predictions and measurements. Rather, Newton intended that his theory be tested on how well it could guide adjustments to background hypotheses, leading to converging measurements[8].

In other words, Newton did not aim to establish Earth’s figure once and for all. Rather, he gave approximations, which would allow for adjustments to the assumptions he made in his derivation. With Newton’s early derivation, for example, he assumed the rotating Earth has a homogeneous density. But when his predictions and Richer’s measurements in Cayenne and Gorée disagreed, he modified this assumption in the first edition of the Principia. If Earth is denser at its center, he suggested, the ellipticity of Earth’s equilibrium figure and its surface gravity variation will differ.

With this methodology, Newton passed the torch to future researchers, inviting them to develop hypotheses that would work with these initial measurements, and could then be tested with increasingly precise measurements[2].

In line with Newton’s methodology, geodesists eventually produced convergent measurements of Earth’s ellipticity based on variation in latitudinal surface gravity and curvature. For about two and a half centuries, they used the theories of gravitation and hydrostatic equilibrium to model Earth’s figure, motion, and constitution, and gradually revised these parameters in light of new measurements. By 1909, all major ellipticity measurements converged within 297.6±0.9, implying that density increased inward[9]. By 1926, Viennese astronomer Samuel Oppenheim concluded that these results offered overwhelming evidence for Newtonian gravity on Earth, vindicating both Newton’s theory of gravitation and his methodology[10].

Miguel Ohnesorge is a doctoral student at the University of Cambridge and visiting fellow at Boston University’s Philosophy of Geoscience Lab. Ohnesorge won the APS Forum of the History and Philosophy of Physics’ 2022 essay contest; this article is adapted from his winning essay.

For Ohnesorge’s full essay and his reconstruction of Newton’s derivation, visit go.aps.org/geodesy. Learn more about Ohnesorge’s research at mohnesorgehps.com.

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