The Back Page: How Newton Derived the Shape of Earth

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.

Notes

[i] The only published discussions of the work’s methodological importance are in Schliesser and Smith 2000 and Smith 2014, who do not reconstruct Newton’s derivation. Todhunter 1873 and Greenberg 1996 discuss the derivation in plain English, without methodological contextualization. Chandrasekhar 2003 also reconstructs the derivation in modern algebra, which contains some minor but confusing mistakes and ambiguities.

[ii] Newton’s derivation for the attraction of perfectly spherical and spheroidal compound bodies is given in Book 1, Prop. 91, coroll. 1 and 2 and reconstructed in its original notation at go.aps.org/geodesy.

Citations

[1] Christian Huygens, “Discours de La Cause de La Pesanteur,” in Traité de Lumierè Avec Discours de La Cause de La Pesanteur, ed. Christian Huygens (Leiden: Pierre Vander, 1690).

[2] The Principia: Mathematical Principles of Natural Philosophy, trans. Bernhard Cohen and Ann Whitman (Univ. of California Press, 1999).

[3] Isaac Newton, De moto Corporum Liber Secundus (Cambridge Univ. Library, MS Add. 3990, 1685).

[4] Isaac Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth, from the Time of Newton to That of Laplace, vol. 1 (London: Macmillan, 1873).

[5] Eric Schliesser and George E. Smith, “Huygens’s 1688 Report to the Directors of the Dutch East India Company on the Measurement of Longitude at Sea and the Evidence It Offered Against Universal Gravity,” preprint (2000).

[6] Cotes to Newton, The Correspondence of Isaac Newton, vol. 5 (1712): 232-236.

[7] World Geodetic System (WGS) 84 Model.

[8] George E. Smith, “Essay Review: Chandrasekhar’s Principia: Newton’s Principia for the Common Reader,” Journal for the History of Astronomy 27, vol. 4 (1996): 353–62.

[9] Miguel Ohnesorge, “Pluralizing Measurement: Physical Geodesy’s Measurement Problem and Its Resolution, 1880-1924,” Studies in History and Philosophy of Science Part A (forthcoming).

[10] A. Sommerfeld and Samuel Oppenheim, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Fünfter Band: Physik (Vieweg+Teubner Verlag, 1926).

Other Sources

Subrahmanyan Chandrasekhar, Newton’s Principia for the Common Reader (Clarendon Press, 2003).

Alexis Claude Clairaut, “I. An Inquiry Concerning the Figure of Such Planets as Revolve about an Axis,” Philosophical Transactions of the Royal Society of London 40, 449 (1738): 277–306.

John Greenberg, ‘Isaac Newton and the Problem of the Earth’s Shape’. Archive for History of Exact Sciences 49, vol. 4 (1996): 371–91.

Mary Terrall, The Man Who Flattened the Earth: Maupertuis and the Sciences in the Enlightenment (Univ. of Chicago Press, 2002).

The Mathematical Works of Isaac Newton, ed. Derek Thomas Whiteside, vol. 6 (Cambridge University Press, 1974).

“Closing the Loop: Testing Newtonian Gravity, Then and Now,” in Newton and Empiricism, ed. Zvi Biener and Eric Schliesser, vol. 262–353 (Oxford Univ. Press, 2014).

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