Philosophiae Naturalis Principia Mathematica

by Sir Isaac Newton
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Last Updated on May 5, 2015, by eNotes Editorial. Word Count: 1650

One of the most influential books in history is Sir Isaac Newton’s Principia . Published in 1687, the book immediately led to intellectual controversy among the scientists and philosophers of the day, including Gottfried Wilhelm Leibniz, Robert Hooke, and John Flamsteed, who felt it necessary to argue with many of...

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One of the most influential books in history is Sir Isaac Newton’s Principia. Published in 1687, the book immediately led to intellectual controversy among the scientists and philosophers of the day, including Gottfried Wilhelm Leibniz, Robert Hooke, and John Flamsteed, who felt it necessary to argue with many of the propositions and conclusions Newton advances. These arguments give at least as much testimony to the importance of Principia as they undermine its theories. Newton’s book remained the principal document in the field of physics for two hundred years.

Newton’s work in physics has never been supplanted or debunked; relativity and other discoveries of the twentieth century are modifications and additions to his scientific discoveries rather than replacements. The philosophical implications of relativity and other discoveries of the twentieth century, however, are radically different from the philosophical implications of Newton’s discoveries. During the eighteenth century and after, Newton’s masterpiece also was a highly revered work of philosophy. Newton became one of the most honored figures in Western culture, one of the first formulators of scientific method, and the person whose work formed the basis for scientific study and application of principles. Physics, as a field of theory and knowledge, did not exist before Newton’s work.

Newton’s preface to the Principia announces that he is interested in the laws of mathematics as a means of discovering nature, or getting at philosophical truth. He thinks that mathematics is not a pure, abstract system, but rather a human and rational means for discovering the principles of the universe, for making a kind of universal order out of the disparate experience of the senses. In fact, he believes in this function of mathematics so strongly that, in the body of the Principia, every experiment or demonstration is concluded with a scholium. Each scholium is a short essay giving the philosophical implications or the speculative use of the mathematical or physical principle just demonstrated. Thus, Newton’s book is a philosophical as well as a scientific work.

After the preface, Newton supplies a series of definitions for such terms as “motion,” “force,” and “quantity,” terms necessary for even an elementary understanding of his work. These definitions are still standard among students of physics. Newton thereupon states his famous three axioms or laws of motion. These axioms are still relevant in any account of physical forces in the everyday world; relativity comes into play to a significant extent at the level of the atom and at speeds at or near the speed of light. Newton states these laws as axioms on which his whole account of the universe rests.

The first axiom states that a body remains in its existing state of motion or rest unless acted upon by an outside force. This is also known as the law of inertia. The second axiom states that the change in motion of a body is proportional, in precise mathematical terms, to the force applied to it. This is known as the law of acceleration. The third axiom states that for every action there is an equal and opposite reaction. Newton could not prove these axioms universally; rather, these principles are what best explain the various facts and data that people find in physical phenomena around them. The axioms, like the definitions, are necessary beginnings, points that must be accepted in order that all physical data can make rational sense. The axioms have six corollaries, propositions that can be established from the axioms and that can be used in turn to establish other propositions.

In the first book of the Principia, Newton deals with the motion of bodies. To simplify and explain his theories, in the first book he confines his observations and proofs to bodies moving in a vacuum. He begins with the more purely mathematical: establishing ratios (demonstrating the logic of the number system), determining the vectors of forces, tracing and proving how bodies move in various arcs, parabolas, and ellipses. For all these geometric demonstrations he gives mathematical proof by inventing and proving his equations and by making frequent reference to his many diagrammatic figures. He also develops and proves equations dealing with the ascent and descent of bodies, again confining his work to bodies in a vacuum. He also devises a mathematical explanation for the oscillations of a pendulum. Finally, at the end of the first book, Newton deals with the attractions of bodies for one another, setting up equations to demonstrate this necessary and universal principle of attraction and repulsion. In addition to defining the terms of physics, the basic laws of physics, and the mathematics to describe the laws of motion, Newton “discovers” gravity.

In the second book, Newton deals with the motion of bodies in resisting mediums. The nature of resisting mediums, such as water or air, make his proofs become more intricate and complicated. Newton usually attempts to simplify his demonstrations by assuming that the medium is constant. These experiments allow Newton to calculate and, more important, to explain the resistance of substances such as water and air to the motion of bodies passing through them. He gives further demonstrations of motion, analyzing some of the problems dealt with in his first book. He brings up, for example, the oscillations of the pendulum and charts the equations for the motion of a pendulum through air. His consideration of the resistance to bodies allows Newton to present and demonstrate the solution to other problems in the physical universe. In this section, dealing with means of determining the density and compression of fluids, he develops equations to explain the behavior of fluids: the density they offer as resistance and the force they exert when compressed. This work on fluids permits Newton to establish his equations to determine the velocity of waves.

Newton calls his third book the “System of the World,” his specific intention in this book being to develop the philosophical principles that he believes follow directly from his mathematical proofs and his experimentation. He begins the book by stating his rules for accurate reasoning, based on his belief that there are no superfluous causes in nature. Each cause that one can talk of sensibly has direct effects, which one is able to observe and subdue to order with mathematical and rational equipment. In other words, Newton believes that the simplicity of the design of the universe is a basic rule; causes are never extraneous. Causes are the basis for observable and frequently calculable phenomena.

Another significant rule is Newton’s belief that all conclusions are based on induction. One reasons from the observable facts and always needs to refer one’s conclusions or theories to observable facts. In this complete devotion to scientific method, there is the necessity of constant application of all of the data to the theory. Newton fully realizes, therefore, that theories might well have to be altered to provide explanations for data that challenge the theory. Post-Newtonian physics would not have surprised Newton, for he had always acknowledged that scientific theories could be no more than the best conclusions available from the data at hand at the moment the conclusion was made. Thus, Newton makes his significant contribution to the scientific method, which is a basis for the many discoveries made since his time.

The third book sets forth Newton’s mathematical demonstrations of the periodic times and movements of the planets. Again, he derives many new equations to demonstrate, with accuracy, the movements of the planets and to correlate this knowledge with the system of time on Earth. He also proves that gravity applies to all bodies and calculates the ratio of gravity. Much of the third section is devoted to lunar motion, establishing equations and calculating, in terms of time, the various changing relationships between the moon and the earth. These matters lead Newton into consideration of the effect of the sun and the moon on the waters of the earth, and he devises means of measuring the tides. He also computes the times and ranges of recurrent comets. Newton thereby provides practical applications for his theories and mathematics.

In a long, final “Scholium” designed to tie the extensive parts of the Principia together, Newton develops the basis for his belief in God. He asserts that such a perfect, and perfectly simple, system must have, as its ultimate or final cause, a perfect, and perfectly simple, Being. This Being must embody all the intelligence, the rationality, the perfection, of the system itself. Newton views God as this ultimate principle, not as a personal God or a larger edition of a human being. Firm in his devotion to his principle, he answers, in later editions of the Principia, charges of atheism brought against his system. This principle, the final cause, is the originator of the whole Newtonian universe, the perfectly rational origin of all the laws, mathematics, and reason that people can use to develop and describe the meaningful pattern in the universe. God, the perfect Being, having set this vast plan in constant motion, is constantly at hand to make sure the universe does not run down.

This concept of God became, during the eighteenth century, one of the principal concepts held by intellectuals. The religion of Deism, of viewing God as the perpetrator and final cause of a complete, perfect, mechanistic universe, is derived from Newton’s thorough and systematic explanation.

As science and as philosophy, the Principia is one of humanity’s great achievements. The book vastly increased the store of human knowledge and derived a sound and rational basis for making conclusions about the physical universe. In addition, Newton illustrates and defines the method by which people may continue to test his observations, develops a new and important area for the human intellect, and establishes a metaphysical system that governed the thought and scientific investigation of the world’s leading intellects for more than a century.

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