Philosophiae Naturalis Principia Mathematica Analysis

Sir Isaac Newton


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’...

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Motion and Forces

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...

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A Perfect Universe and Mathematics

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.


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...

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