Maclaurin’s Gravitational Theory Summary

  • Last updated on November 10, 2022

Colin Maclaurin’s prizewinning essay on tides provided an elegant mathematical proof of a key assumption of Sir Isaac Newton’s gravitational theory. Maclaurin’s recognition of the deflecting action of the Earth’s rotation also anticipated the more fully dynamical theory of tides that Pierre-Simon Laplace later developed in the Celestial Mechanics of 1799.

Summary of Event

In 1738, Colin Maclaurin, a professor of mathematics at the University of Edinburgh, responded to a competition to explain and predict tides sponsored by the French Académie des Sciences (Academy of Sciences). Academy of Sciences, France The academy’s prizes brought considerable prestige to their recipients, and its annual competitions attracted submissions from Europe’s best scientists. Maclaurin worked on his submission for two years, and in 1740, his paper “De Causa Physica Fluxus et Refluxus Maris” "De Causa Physica Fluxus et Refluxus Maris" (Maclaurin)[De Causa Physica Fluxus et Refluxus Maris] (on the physical cause of the ebb and flow of the sea) shared the prize with three other contestants—Antoine Cavalleri, a professor of mathematics at Cahors in France; Leonhard Euler, chair of mathematics at the St. Petersburg Academy of Sciences in Russia; and Daniel Bernoulli, then professor of botany at the University of Basel in Switzerland. [kw]Maclaurin’s Gravitational Theory (1740) [kw]Theory, Maclaurin’s Gravitational (1740) [kw]Gravitational Theory, Maclaurin’s (1740) Gravity Tidal theory [g]Scotland;1740: Maclaurin’s Gravitational Theory[1010] [g]France;1740: Maclaurin’s Gravitational Theory[1010] [c]Mathematics;1740: Maclaurin’s Gravitational Theory[1010] [c]Physics;1740: Maclaurin’s Gravitational Theory[1010] [c]Science and technology;1740: Maclaurin’s Gravitational Theory[1010] Maclaurin, Colin Newton, Sir Isaac Newton, Sir Isaac;gravitational theory [p]Euler, Leonhard Bernoulli, Daniel

Observations on the frequency and magnitude of tides had long been essential to maritime navigation and coastal economies, but an adequate theoretical understanding of tides began to take shape only with the seventeenth century Scientific Revolution. Indeed, the ability to account for tides became a crucial test of the new celestial mechanics. Celestial mechanics Galileo, Galileo for example, adduced the tides as proof of the Copernican theory of the solar system, understanding them as the accelerations and decelerations that resulted from the combination of the Earth’s revolution around the Sun and the Earth’s rotation on its axis.

René Descartes Descartes, René had recognized that tides had some relationship to the Moon. He explained this relationship by hypothesizing an “ether,” or intangible medium, that filled the whole of space. As celestial objects rotated, Descartes hypothesized, they created vortices in the ether. Thus, the vortex created by the Earth’s rotation carried the Moon Moon (of Earth) along its orbit and the tides were the consequence of the downward pressure of the Moon upon the ether and the ether upon the Earth’s surface.

It was Sir Isaac Newton, however, who established the basis for a modern understanding of tides. For Newton, tides were not the result of the pressure of a hypothetical ether but of the gravitational attraction of massive objects. Newton’s first theory of tides imagined the oceans on the model of a satellite making a close circular orbit of the Earth and determined the effects of gravity on this “satellite’s” motion. For Newton, too, the origin of the tides lay not only in a relationship between the Earth and the Moon but also in a relationship between the Earth and the Sun.

Newton later offered a second theory of tidal activity, which Maclaurin’s 1740 essay first distinguished as an “equilibrium theory” in contrast to the earlier “kinetic theory.” Newton’s equilibrium theory imagined the effects of the Moon’s and Sun’s gravity on a fluid Earth and then determined the effects of Earth’s rotation. This model of tides resulting from a complex, three-body interaction among the Earth, Moon, and Sun was both more flexible and more comprehensive than Galileo’s and Descartes’s simple mechanical explanations. Newton’s attempts to explain tides also had limitations, however. Newton’s presentation was often obscure, was not always consistent, and depended at crucial points on unproven mathematical intuitions. Moreover, while Newton’s account generally agreed better with existing observations than did the Galilean and Cartesian alternatives, it still required significant refinement in application.

These limitations of Newton’s tidal theory, as well as the difficulty of conceiving a mechanism for instantaneous action at a distance, meant that there remained serious proponents of the Cartesian theory of vortices throughout the first half of the eighteenth century. The limitations of Newton’s tidal theory also encouraged efforts to improve data on tides, in the hopes that better data would lead to a better model explaining the data. Particularly energetic in this respect was the Académie des Sciences, which issued guidelines on the measurement of tides and served as a clearinghouse for observations.

It was in order to resolve the unsettled state of tidal theory and to bring that theory more into line with empirical studies that the academy sponsored its prize competition of 1738. Of the prizewinners, the Jesuit Cavalleri had the unhappy distinction of attempting the last significant defense of Cartesian theory. Maclaurin, Euler, and Bernoulli all began from Newton’s theory, and in 1742 their essays were included in the French translation by François Jacquier and Thomas LeSeur of Newton’s Philosophiae Naturalis Principia Mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729; best known as the Principia).

Maclaurin was an ideal candidate for the Académie des Sciences prize. Scottish universities had been the first in Europe to teach Newtonian science, and in 1713 the young Maclaurin was already writing a highly sophisticated master’s thesis at the University of Glasgow on “Gravity and Other Forces.” In 1719, Maclaurin had gone to London, where he so impressed Newton that Newton, in turn, oversaw Maclaurin’s election to the Royal Society. It had been Newton, moreover, who in 1725 had recommended Maclaurin for the prestigious position of professor of mathematics at the University of Edinburgh. The previous year, Maclaurin had won his first prize from the Académie des Sciences for a study of percussion.

Throughout the 1730’s, Maclaurin worked to overcome objections to the Newtonian calculus by giving it a more rigorous geometrical—in contemporary terms, “fluxional”—foundation. At the same time, Maclaurin became a prominent spokesman for the Newtonian system. Newtonian mechanics His Account of Sir Isaac Newton’s Philosophical Discoveries, Account of Sir Isaac Newton’s Philosophical Discoveries (Maclaurin) posthumously published in 1748, was among the most lucid and most widely read popular expositions of the Principia in the eighteenth century.

Important refinements to Newtonian mechanics emerged from Maclaurin’s 1740 prize essay. Newton had assumed that his two accounts of tidal action were equivalent. Maclaurin, however, demonstrated that the kinetic theory predicted tides of significantly less magnitude than those predicted by the equilibrium theory. Additionally, in the equilibrium theory, Newton had merely assumed without proof that the gravitational attraction of a single body would deform an otherwise spherical ocean of uniform depth into a prolate spheroid. In other words, the ocean would be elongated along the axis between the Earth and the attracting body and flattened at right angles to this axis.

Maclaurin used the fluxional calculus to provide an elegant proof of this assumption. Out of this proof came a correspondence with Alexis-Claude Clairaut, Clairaut, Alexis-Claude which led the French mathematician, in his Théorie de la figure de la terre (1743; theory of the shape of the Earth), to resolve another longstanding problem in Newtonian mechanics. Maclaurin, finally, was the first to consider the apparent deflection of moving bodies on rotating surfaces—what is today called the Coriolis effect Coriolis effect after the nineteenth century French mathematician Gustave-Gaspard Coriolis.

Euler’s and Bernoulli’s prize essays further elaborated Newton’s equilibrium theory of tidal action. Euler’s paper demonstrated that it was not, as Newton believed, the vertical component but the horizontal component of gravitational attraction that played the decisive role in generating tides. Bernoulli refined Newton’s theory to the point where it could be used to compile tide tables and make relatively reliable predictions.


Together, Maclaurin’s, Euler’s, and Bernoulli’s 1740 prize essays constitute the most significant contribution to the understanding of tides—and hence the practical application of gravitational theory—in the century after Newton’s initial achievement. Not until the end of the eighteenth century did the French mathematician and physicist Pierre-Simon Laplace Laplace, Pierre-Simon go further and undertake the daunting task of creating a fully dynamic theory of tides that would take into account, for example, the ocean’s response to gravitational forces. This task was one that Maclaurin, in recognizing the deflecting action of the Earth’s rotation, had anticipated but not developed.

Maclaurin’s prize essay also belongs to a critical moment in the development of the calculus. Calculus In 1742, Maclaurin published A Treatise of Fluxions, Treatise of Fluxions, A (Maclaurin) a systematic defense of the Newtonian calculus that featured an expanded version of his 1740 prize essay. For many historians, Maclaurin’s insistence on geometric techniques, for all of its evident genius, left British science ill-equipped to comprehend the remarkable achievements of continental analysis and rational mechanics. Recent scholarship, however, has been more generous. Continental analysis, this scholarship notes, was not able to replicate the rigor of Maclaurin’s geometry until the 1780’s; indeed, Maclaurin’s results often stimulated improvements in continental analysis.

Moreover, Maclaurin’s Treatise of Fluxions had a twofold character: If the first volume was geometric, the second volume presented the same results algebraically. This twofold character embodied the general moderatism of the eighteenth century Scottish Enlightenment Enlightenment;Scotland and a continuing exchange between the British and continental approaches to mathematical physics. Maclaurin’s elaboration of Newton’s gravitational theory founded the generalizing power of modern symbolic notation on the ancient techniques of Greek geometry. It grounded forward-looking research on applications in a conservative allegiance to rigor.

Further Reading
  • citation-type="booksimple"

    xlink:type="simple">Cartwright, David E. Tides: A Scientific History. New York: Cambridge University Press, 1999. An important history of the science of tides.
  • citation-type="booksimple"

    xlink:type="simple">Grabiner, Judith V. “Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions.” The American Mathematical Monthly 104, no. 5 (May, 1997): 393-410. An important reconsideration of the place of A Treatise of Fluxions in the history of mathematics by the leading contemporary Maclaurin scholar.
  • citation-type="booksimple"

    xlink:type="simple">Greenberg, John L. The Problem of the Earth’s Shape from Newton to Clairaut: The Rise of Mathematical Science in Eighteenth-Century Paris and the Fall of “Normal” Science. New York: Cambridge University Press, 1995. Provides an extensive treatment of Maclaurin’s contributions to Newtonian mechanics.
  • citation-type="booksimple"

    xlink:type="simple">Guicciardini, Niccolò. Reading the “Principia”: The Debate on Newton’s Mathematical Methods for Natural Philosophy, from 1687 to 1736. New York: Cambridge University Press, 1999. A rich discussion of how Newton’s masterpiece stimulated debates on the mathematization of natural philosophy.

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