It is helpful to identify two broad tendencies in eighteenth-and nineteenth-century physics, which had been noted by a number of contemporaries, including the German philosopher Immanuel Kant (1724–1804). On the one hand, a mechanical approach analyzed the physical universe as a great machine and built models relying on commonsense notions of cause and effect. This sometimes required the specification of ontological entities to communicate cause and effect, such as Descartes's plenum. On the other hand, the dynamical approach avoided mechanical models and, instead, concentrated on the mathematical relationship between quantities that could be measured. However, in avoiding mechanical models, the dynamical approach often speculated on the existence of active powers that resided within matter but could not be observed directly. Although this distinction is helpful, many scientists straddled the divide. Newton's physics, and the general Newtonian scientific culture of the eighteenth century, utilized elements of both approaches. It held true to a mechanical-world picture in analyzing macroscopic systems involving both contact, as in collisions, and action at a distance, as in the orbital motion. But it also contained a dynamical sensibility. Regarding gravity, Newton rejected Descartes's plenum and speculated that gravity might be due to an all-pervasive ether, tantamount to God's catholic presence. Such reflections appeared in Newton's private notes and letters, but some of these became known during the 1740s.
The development of mechanics during the eighteenth century marks one place where the histories of physics and mathematics overlap strongly. Mathematicians with an interest in physical problems recast Newtonian physics in an elegant formalism that took physics away from geometrical treatment and toward the reduction of physical problems to mathematical equations using calculus. Some of these developments were motivated by attempts to confirm Newton's universal gravitation. The French mathematician Alexis-Claude Clairaut (1713–1765) used perturbation techniques to account for tiny gravitational forces affecting the orbits of heavenly bodies. In 1747 Clairaut published improved predictions for the Moon's orbit, based on three-body calculations of the Moon, Earth, and Sun, and, in 1758, predictions of the orbit of Halley's comet, which changed slightly each time that it passed a planet. Some years later, Pierre-Simon Laplace produced a five-volume study, Celestial Mechanics (1799–1825), which showed that changes in planetary orbits, which had previously appeared to be accumulative, were in fact self-correcting. His (perhaps apocryphal) response to Napoléon's question regarding the place of God in his calculations has come to stand for eighteenth-century deism: "Sire, I had no need for that hypothesis."
The most important mathematical work was the generalization of Newton's mechanics using the calculus of variations. Starting from Fermat's principle of least time, Louis Moreau de Maupertuis (1698–1759) proposed that, for a moving particle, nature always sought to minimize a certain quantity equal to the mass times the velocity times the distance that the particle moves. This "principle of least action" was motivated by the religious idea that the economy of nature gave evidence of God's existence. The Swiss mathematician Leonhard Euler (1707–1783) recast this idea (but without Maupertuis's religious motivations) by minimizing the integral over distance of the mass of a particle times its velocity. The Italian Joseph-Louis Lagrange (1736–1813) restated and clarified Euler's idea, by focusing on minimizing the vis viva integrated over time. His Mécanique analytique (1787) summarized the whole of mechanics, for both solids and fluids and statics and dynamics.
In its high level of mathematical abstraction and its rejection of mechanical models, Lagrange's formalism typified a dynamical approach. In addition to making a number of problems tractable that had been impossible in Newton's original approach, the use of the calculus of variations removed from center stage the concept of force, a vector quantity (with magnitude and direction), and replaced it with scalar quantities (which had only magnitude). Lagrange was proud of the fact that Mécanique analytique did not contain a single diagram.
Newton's physics could be applied to continuous media just as much as systems of masses. In his Hydrodynamica (1738), the Swiss mathematician Daniel Bernoulli used conservation of vis viva to analyze fluid flow. His most famous result was an equation describing the rate at which liquid flows from a hole in a filled vessel. Euler elaborated on Bernoulli's analyses and developed additional formalism, including the general differential equations of fluid flow and fluid continuity (but restricted to the case of zero viscosity). Clairaut contributed to hydrostatics through his involvement with debates regarding the shape of the earth. In developing a Newtonian prediction, Clairaut analyzed the earth as a fluid mass. After defining an equilibrium condition, he showed that the earth should have an oblate shape, which was confirmed by experiments with pendulums at the earth's equator and as far north as Lapland.
The study of optics inherited an ambivalence from the previous century, which considered two different mechanical explanations of light. In his Opticks (1704), Newton had advocated a corpuscular, atomistic theory of light. As an emission of particles, light interacted with matter by vibrating in different ways and was therefore either reflected or refracted. In contrast with this, Descartes and Huygens proposed a wave theory of light, arguing that space was full and that light was nothing more than the vibration of a medium.
During the eighteenth century, most scientists preferred Newton's model of light as an emission of particles. The most important wave theory of light was put forward by Euler, who hypothesized that, in analogy with sound waves, light propagated through a medium, but that the medium itself did not travel. Euler also associated certain wavelengths with certain colors. After Euler, considerable debate occurred between the particle and wave theories of light. This debate was resolved during the early nineteenth century in favor of the wave theory. Between 1801 and 1803, the English physician Thomas Young conducted a series of experiments, the most notable of which was his two-slit experiment, which demonstrated that two coherent light sources set up interference patterns, thus behaving like two wave sources. This work was largely ignored until 1826, when Augustin-Jean Fresnel presented a paper to the French Academy of Science that reproduced Young's experiments and presented a mathematical analysis of the results.
Electrical research was especially fruitful in the eighteenth century and attracted a large number of researchers. Electricity was likened to "fire," the most volatile element in Aristotle's system. Electrical fire was an imponderable fluid that could be made to flow from one body to another but could not be weighed (see sidebar, "Forms of Matter"). After systematic experimentation, the French soldier-scientist Charles-François Du Fay (1698–1739) developed a two-fluid theory of electricity, positing both a negative and a positive fluid. The American statesman and scientist Benjamin Franklin (1706–1790) proposed a competing, one-fluid model. Franklin suggested that electrical fire was positively charged, mutually repulsive, and contained in every object. When fire was added to a body, it showed a positive charge; when fire was removed, the body showed a negative charge. Franklin's theory was especially successful in explaining the behavior of the Leyden jar (an early version of the capacitor) invented by Ewald Georg von Kleist in 1745. The device was able to store electrical fire using inner and outer electrodes, with the surface of the glass jar in between. Franklin's interpretation was that the glass was impervious to electrical fire and that while one electrode took on fire, the other electrode expelled an equal amount (see Fig. 3).
After early efforts by John Robison and Henry Cavendish, the first published precision measurements of the electric force law were attributed to the French physicist and engineer Charles-Augustin de Coulomb (1736–1806). Coulomb used a torsion balance to measure the small electrostatic force on pairs of charged spheres and found that it was proportional to the inverse square of the distance between the spheres and to the amount of charge on each sphere. At the close of the century, Cavendish used a similar device to experimentally confirm Newton's universal law of gravitation, using relatively large masses.
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