Philosophiae Naturalis Principia Mathematica

by Sir Isaac Newton
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Last Updated on May 6, 2015, by eNotes Editorial. Word Count: 407

In the summer of 1684, the astronomer Edmond Halley asked Isaac Newton for his thoughts on planetary motion. Newton’s response, based on his early mathematical calculations, was that the planets would travel around the Sun in elliptical paths. Some months later, Newton provided Halley with a written mathematical proof of...

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In the summer of 1684, the astronomer Edmond Halley asked Isaac Newton for his thoughts on planetary motion. Newton’s response, based on his early mathematical calculations, was that the planets would travel around the Sun in elliptical paths. Some months later, Newton provided Halley with a written mathematical proof of his prediction. At Halley’s request, Newton then set about to further explain the forces of nature that governed the motion of objects, including the movement of celestial bodies. By July 5, 1687, the results of this work appeared as the first edition of Newton’s Principia.

Newton was totally absorbed in the writing of the Principia for eighteen months. He would frequently forget to eat and slept only when overcome with exhaustion. Although it is not without errors, it has often been said that the Principia is the greatest work of science ever published. However, without considerable mathematical skills, it is difficult to follow and virtually impossible to comprehend. In addition to its complex mathematical language, the Principia was written in Latin (and not translated into English until two years after Newton’s death). By writing for an elite audience, Newton hoped he would be spared the annoyance of debating his work with those of lesser education. Nevertheless, its influence on the scientific revolution of the seventeenth century was crucial in overturning the prevailing philosophers’ view of the universe forever.

Newton divided the Principia into three books. In books 1 and 2, Newton describes the motions of bodies and outlines his mathematical treatments of both terrestrial and celestial mechanics. Book 1 contains almost one hundred propositions in which Newton develops a general theory of motion, including the motion of celestial bodies such as the planets. He describes how objects behave when subjected to forces. Book 2 examines resisted motions and the influence of fluids on a body’s motion (for example, the effect of air resistance on a moving object). Newton’s conclusion—that planetary motion is not impeded by any fluid object in space—was a departure from the long-held philosophical belief that a fluid substance, called ether, permeated space and affected the motion of heavenly bodies. In book 3, Newton uses his mathematical concepts from the first two books to describe his system of the world. He discusses the law of gravitation, tidal motion, and comet theory and calculates the speed of sound. Throughout the work, Newton relies on experiments and observations, both his and others’ to derive his mathematical laws.

Motion and Forces

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The Principia opens with a series of definitions and laws, which are followed by numerous explanatory notes and comments (scholia and corollaries). Included in these laws are Newton’s three laws of motion: First, every body will continue in its state of rest or uniform motion in a straight line unless it is compelled to change its state by an external force impressed on it (law of inertia); second, a change of motion is always proportional to the force being applied to the body and the new motion will be in the straight line in which the force is impressed; and third, for every action there is always an equal and opposite reaction. From these laws, Newton developed his law of universal gravitation.

Mechanics is the branch of applied mathematics that deals with the motion of objects, and it had advanced considerably by the seventeenth century. However, the field of dynamics, which explains how forces influence motion, was not well understood until Newton introduced his laws of motion in the Principia. Of particular interest to Newton were forces that resulted in an object traveling in a circular (or near circular) motion because this represented the path traveled by the orbiting planets. Newton used the word “centripetal,” meaning “seeking the center,” to characterize forces involved in circular motion. He also recognized the significance of conic paths to describe the motion of a moving object with respect to a fixed point. The conic path is a curve obtained by cutting a plane through a right circular cone. Depending on where the cutting plane is located on the cone, the curved path may be a circle, ellipse, or hyperbola. His detailed account of conic properties enabled him to mathematically describe the orbits of celestial bodies. Planets, and moons around planets, had elliptical paths and would never trace the exact orbit twice. Comets could have elliptical paths and therefore would reappear at regular intervals, whereas those with hyperbolic paths would not. Lack of experimental data prevented Newton from satisfactorily describing the motion of the Moon or of the moons of Saturn.

Newton’s law of gravitation had important applications. The law describes the force between two objects and states that the force of gravity is always equal to the constant of acceleration times the product of the masses of the two attracting bodies divided by the square of the distance separating them. Therefore, the gravitational force between two objects increases as the mass of the objects increases and as the distance between objects decreases. Although the distance between Earth and the Moon is large, compared to terrestrial distances, their enormous masses generate strong gravitational forces. Therefore, Earth attracts the Moon, which causes it to maintain its orbit. At the same time, the Moon attracts Earth, a fact that Newton claimed was illustrated by the existence of tidal motion on Earth. Newton also explained that the gaseous atmosphere was held to the earth by the gravitational force, and he was able to show that air density decreased with elevation above sea level.

The Principia also contained Newton’s mathematical treatment of hydraulics. Hydraulic machinery and hydraulic engineering rely on the motion of a fluid through a vessel (for example, water or air through a pipe or tube). Although there were errors in Newton’s results, he did succeed in estimating the speed of sound in air, which was a remarkable achievement for seventeenth century science. He also described the ballistic curve (a path formed by a moving projectile such as a missile or ball traveling through air) as having the form of a distorted hyperbola.

A Perfect Universe and Mathematics

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Despite his attempts to quantify the mechanical universe and his belief in the absolute nature of space and time, Newton was convinced that the perfection of nature was a reflection of its creation by a divine being: God being perfect would not create an imperfect universe. It was necessary, however, for God to intervene from time to time to maintain the stability and hence perfection of his creation. Natural philosophers attempted to deduce causes from their observed effects and, through reductionism, attempted to find the original cause that Newton believed would be divine rather than mechanical.

Although the Principia is a complex mathematical work, Newton’s ideas transcended science. In philosophy, religion, and law, the results of the Principia contributed to the so-called Age of Enlightenment. It became a common, but erroneous, belief that new laws of mathematics would eventually be discovered and that these would permit anything to be calculated in the future. This was not an unreasonable assumption, because astronomers could use mathematics to predict even the future position of the Moon. Early twentieth century science discovered limitations to Newton’s laws. The laws could not be applied accurately to interactions on the atomic scale or for the motion of objects traveling at near the speed of light. They also broke down between objects separated by large distances such as those that exist between galaxies.

The definitions, principles, and propositions contained in the Principia enabled scientists to take an entirely new approach to the study of nature and laid the foundations for modern physics and astronomy. The Principia began to be viewed as a work of modern science rather than as a work in philosophy soon after it was published. Newton’s method of studying science, which was a combination of mathematical calculations, observation, and experimentation, came to be accepted as the standard approach for scientific investigation. From the Principia came an understanding of the science of mechanics, which in turn led to the development of practical and useful applications for commercial and industrial development. The motion of a baseball in flight, the movement of water through dams, and the paths of spacecraft and satellites launched from Earth are all examples illustrating the validity of Newton’s laws.


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Additional Reading

Brewster, Sir David. Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton. Edinburgh, Scotland: T. Constable, 1855. A classic two-volume biography pieced together from Sir Isaac Newton’s private papers. A bit dated and ignores Newton’s work on religion and alchemy.

Chappell, Vere, ed. Seventeenth-Century Natural Scientists. Vol. 7 in Essays on Early Modern Philosophers. New York: Garland, 1992. Part of a twelve-volume set of scholarly essays on seventeenth century philosophers in Europe. Contains six articles on Newton.

Christianson, Gale E. In the Presence of the Creator: Isaac Newton and His Times. New York: Free Press, 1984. This very readable biography places Newton’s life in the context of the scientific revolution.

Cohen, I. Bernard. Introduction to Newton’s “Principia.” Cambridge, Mass.: Harvard University Press, 1971. A massive work of scholarship. Presents the background to the publishing of the variorum edition of Newton’s influential book. Itemizes revisions and corrections in the various editions and translations. Surveys the early reviews. Comprehensive bibliography.

Cohen, I. Bernard. The Newtonian Revolution in Science and Its Intellectual Significance. Norwalk, Conn.: Burndy Library, 1987. An important work by a leading Newton scholar.

De Gandt, Francois. Force and Geometry in Newton’s “Principia.” Translated by Curtis Wilson. Princeton, N.J.: Princeton University Press, 1995. An introduction to Newton’s Principia.

Dobbs, Betty Jo Teeter. The Janus Faces of Genius: The Role of Alchemy in Newton’s Thought. Cambridge, England: Cambridge University Press, 1992. Dobbs argues that Newton’s primary goal was to establish a unified system that included both natural and divine principles. Special attention is given to alchemy.

Fauvel, John, et al., eds. Let Newton Be! New York: Oxford University Press, 1988. Twelve articles explicate the modern and historical contexts of Newton’s work. John Roche’s accessible overview is an excellent starting point for students new to Newton’s difficult work.

Gjertsen, Derek. The Newton Handbook. London: Routledge & Kegan Paul, 1986. Provides a wealth of information, including a chronology and discussion of the origin and production of the work, and an assessment of the difficulty of the work. The contents and central arguments of the Philosophiae Naturalis Principia Mathematica usefully summarized.

Gleick, James. Isaac Newton. New York: Pantheon, 2003. Gleick ventures into well-trodden territory with yet another biography of Newton, but in this biography Gleick reveals Newton’s seemingly contradictory passions for both the mysterious, such as alchemy, and that which is considered not-so-mysterious, such as rational thinking. Includes illustrations and an index.

Goldish, Matt. Judaism in the Theology of Sir Isaac Newton. Boston: Kluwer, 1998. An analysis of Newton’s historical theology and how Newton’s interest in Jewish studies greatly impacted all areas of his theology.

Hall, A. Rupert. Isaac Newton: Eighteenth Century Perspectives. New York: Oxford University Press, 1999. A compilation of five early eighteenth century biographies of Newton. Each biography is accompanied by a commentary. A bibliography of Newton’s works is included.

Herivel, John. The Background to Newton’s “Principia”: A Study of Newton’s Dynamical Researches in the Years 1664-84. Oxford, England: Clarendon Press, 1965. A frequently-cited analysis of the intellectual and scientific basis of Philosophiae Naturalis Principia Mathematica.

Koyre, Alexandre. Newtonian Studies. Cambridge, Mass.: Harvard University Press, 1965. A collection of philosophical and historical essays on Newton.

Manuel, Frank E. A Portrait of Isaac Newton. Cambridge, Mass.: Harvard University Press, 1968. This work examines Newton’s life and work in the context of Newton’s papers and contemporary thinking about the methods and development of science.

Stayer, Marcia Sweet. Newton’s Dream. Kingston, Ontario: McGill-Queens University Press, 1988. Marks the tercentenary of Newton’s seminal work. Examines the work’s enduring impact. The title essay by physics Nobel laureate Steven Weinberg is especially lucid.

Wallis, Peter, and Ruth Wallis. Newton and Newtonia, 1672-1975. Folkestone, England: Dawson, 1977. An exhaustive bibliography of works by and about Newton.

Westfall, Richard S. Never at Rest: A Biography of Isaac Newton. Cambridge, England: Cambridge University Press, 1980. This biography presents Newton’s scientific discoveries in the context of his life. Includes a valuable bibliographical essay and an appendix. More than nine hundred pages.

Westfall, Richard S. The Life of Isaac Newton. New York: Cambridge University Press, 1993. A condensed version of Never at Rest.

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