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The reverse: How Newton derived the shape of the earth

The reverse: How Newton derived the shape of the earth
Written by adrina

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

Using this method, Newton passed the torch to prospective researchers, inviting them to develop hypotheses that would work with those initial measurements and could then be tested with increasingly precise measurements[2].

Consistent with Newton’s methodology, geodesists eventually produced convergent measurements of the Earth’s ellipticity based on latitude variations in surface gravity and curvature. For about two and a half centuries they used the theories of gravity and hydrostatic equilibrium to model the shape, motion, and texture of the earth, gradually revising these parameters in the light of new measurements. By 1909, all major ellipticity measurements converged to within 297.6 ± 0.9, meaning density was increasing inward[9]. In 1926, the Viennese astronomer Samuel Oppenheim concluded that these results provided overwhelming evidence for Newtonian gravity on Earth, confirming both Newton’s theory of gravity and his methodology[10].

Miguel Ohnesorge is a PhD student at the University of Cambridge and a visiting fellow at Boston University’s Philosophy of Geoscience Lab. Ohnesorge won the 2022 essay competition of the APS Forum of the History and Philosophy of Physics; This article is an adaptation of his award-winning essay.

See the complete Ohnesorge paper and his reconstruction of Newton’s derivation at go.aps.org/geodesy. Learn more about Ohnesorge’s research at mohnesorghps.com.

Remarks

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

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

quotes

[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 shape of the earth, from the time of Newton to that of LaplaceVol. 1 (London: Macmillan, 1873).

[5] Eric Schliesser and George E. Smith, “Huygens’ 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 NewtonVol. 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 Ordinary Reader,” Journal of the History of Astronomy 27, Vol. 4 (1996): 353-62.

[9] Miguel Ohnesorge, “Pluralizing Measurement: The Measurement Problem of Physical Geodesy and Its Solution, 1880-1924”, Studies in the history and philosophy of science part A (upcoming).

[10] A. Sommerfeld and Samuel Oppenheim, Encyclopedia of the Mathematical Sciences including their Applications: Fifth Volume: Physics (Vieweg+Teubner Verlag, 1926).

other sources

Subrahmanyan Chandrasekhar, Newton’s Principia for the ordinary reader (Clarendon Press, 2003).

Alexis Claude Clairaut, “I. An Inquiry into the Number of Such Planets Revolving 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 Shape of the Earth”. Archives for the History of the Exact Sciences 49, Vol. 4 (1996): 371-91.

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

The Mathematical Works of Isaac NewtonEd. Derek Thomas Whiteside, Vol. 6 (Cambridge University Press, 1974).

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

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