Isaac Newton and the Mathematical Universe (1642-1727)

Outline of Lecture

I. Introduction
II. His Life (1642-1727)
III. His Work on Light
IV. His Theory of Gravitation
V. Newton's Significance for the Second Europe

Introduction

Isaac Newton amazed and stunned the Enlightenment with the vision he offered of a world-machine obeying the same laws in all its parts. These laws could be expressed in mathematical symbols which anyone with sufficient training could understand. Even David Hume expressed wonderment at a world-machine which seemed to function perfectly in all its parts: "All these various machines, and even their most minute parts, are adjusted to each other with an accuracy, which ravishes into admiration all men, who have ever contemplated them." What Descartes had promised, if men would follow reason, Newton seemed to deliver in his Mathematical Principles of Natural Philosophy (1687). The force which held the planets in their orbits was the same force governing matter on earth and could be expressed in a single mathematical equation. It was staggering to ordinary minds. No wonder Newton became internationally famous and a household name to those who wanted to be enlightened.

His Life (1642-1727)

The great man had an unpromising beginning. Born in 1642 (the year Galileo died) on a small farm in rural England, Newton was a sickly infant and not expected to live. Little is known of his early childhood, or of what could have prepared him intellectually for the greatness he later achieved. Only an uncle, a rural rector, thought he ought to go to university which he did in 1661. Little is known of his undergraduate years at Trinity College, Cambridge, except that he had the chance to study under an able mathematician Isaac Barrow. He completed the degree in normal time in 1665. Then an outbreak of the bubonic plague 1665-66 forced the University to close and Newton to return to the countryside for lengthy periods. During this time he got the insight that the same force which pulled the apple off the tree also held the moon in orbit. He also developed an early form of calculus (which he called "method of fluxions"). And he made the discovery while working in optics that sunlight is composed of the colors ranging from violet to red. He wrote about this period later in his life:

…in November (I) had the direct method of fluxions & the next year in January the Theory of Colours & in May following I had entrance into y^e inverse method of fluxions. And the same year I began to think of gravity extending to y^e orb of the Moon & having found out to how to estimate the force w^ch [a] globe revolving within a sphere presses the surface of the sphere from Keplers rule of the periodical times of the Plants being in sesquialterate proportion of their distances from the centres of their Orbs, I deduced that the forces w^ch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centres about w^ch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665 & 1666. For in those years I was in the prime of my age of invention & minded Mathematics and Philosophy more than at any time since.[1]

Returning to Cambridge, he was elected a Fellow of Trinity, received an M.A., and succeeded Isaac Barrow. Teaching was not his strong suit; the undergraduates of his day did not find his lectures very interesting. He remained at Cambridge until 1695, writing a work on Optics (1672) and then the Mathematical Principles in 1687. Owing to his fame and the interest of his friends who thought Cambridge too confining and stressful for his health, he was made Warden of the Mint in 1695. He remained in London until his death in 1727 at which he was universally acclaimed as no scientist since until Albert Einstein. He was buried in Westminster Abbey in a state funeral usually reserved for nationally important figures. Dukes and earls bore his casket to its resting place: not a bad ending to a life with such an unpromising beginning.

1. Newton’s father was illiterate (if we can so surmise from the fact that he signed his will with an “X”). And we talk about environment and/or heredity being determinant? Should we?
2. All indications are that Newton was basically self-taught. What are the implications of that? (What about home schooling, a growing movement today?)
3. We pay a lot of attention these days to scores on aptitude tests and other predictors of academic success. Should we?

His Work on Light

Newton's first published work was a paper on light (1672) which concerned not just lens making but a theory of light. In 1666 according to one account he purchased a novelty piece of glass, a glass prism at Stourbridge Fair and began making experiments with it which led him to make a series of telescopes, including a reflecting telescope. He made the basic discovery that white light is a compound of red, orange, yellow, green, blue, indigo, and violet. When he caused a small ray of sunlight to pass through a glass prism, he found that these colors emerged on the far side, each with its own refraction, or angle, and these colors arranged themselves in a row of bands forming a color spectrum ranging from red to violet. He observed and measured this occurrence and proposed a theory of light. Light is due to the emission by a luminous body of innumerable particles traveling in straight lines through empty space at 190,000 miles per second. This isn't far from modern thinking, depending on whether one believes the particle theory or wave theory of light, and the more recent calculation of 186,000 mps. Newton's work on light was not his most important. But it is significant for the Enlightenment in that even the simplest phenomenon, sunlight, seemed to obey laws.

His Theory of Gravitation

It is Newton's work on gravitation, the Mathematical Principles which caught the imagination of his age. And the first thing to say about the theory of gravitation is that the story of the apple falling on his head and causing the "Eureka" experience is at least partly true. He himself told the story of a falling apple: it fell in a garden, not on his head[2]. But Newton himself also said, when asked how he discovered the laws of gravitation, "by thinking about them without ceasing." Sometime during 1666 he had made preliminary calculations on lunar and planetary motion. But he didn't know the earth's size accurately and so when he said that the force which held the planets in their orbits varied inversely with the square of their distance from the sun, his calculations did not work. So he put the problem aside for a number of years.

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Newton got his insights into gravitation, light and calculus while at home escaping the bubonic plague raging in the population centers. This is one instance of good coming out of disaster, or evil. Implications?

Newton was not the first to try to explain what came to be called gravitation. Some 15^th century astronomers thought that the heavens exerted a force on the earth like a magnet on iron, keeping the earth suspended by pulling equally in all directions. Kepler in the 16^th century thought gravity to be an inherent quality in all celestial bodies. He considered a theory similar to Newton's, but rejected it as unsound. Descartes proposed a scheme of vortices or whirlpools surrounding each of the planets and holding them. (Newton examined that theory in Mathematical Principles and rejected it.) Many in Newton's own day grappled with gravity and tried to figure out the mathematics which would calculate its force. Including Edmund Halley.

Halley (1656-1742), a noted astronomer who discovered the comet bearing his name (circling in 76 years), went to Cambridge to ask Newton about a mathematical solution to planetary motion. Newton told him that he had calculated planetary attraction as a force varying as the inverse square of the bodies distance from each other. He could not lay his hands on the written calculation, but promised to send it to Halley. What he sent some months later so impressed Halley that he urged Newton to submit the work to the Royal Society. In the next two years Newton expanded on his work, presenting it to the Royal Society in manuscript form. What followed must have been sheer agony for Halley. Newton did not publish readily; he had already begun to quarrel with Robert Hooke who claimed priority in the calculation of the inverse square. But Halley persisted, encouraging each installment of what became the Mathematical Principles of Natural Philosophy, published in 1687 at Halley's cost. The work was a landmark in the history of science, akin to Kepler's On the Revolutions of Heavenly Bodies and Charles Darwin's Origin of Species (1856). Although originally published in Latin, the learned language of the time, it soon was translated and popularized in many languages and versions. Among them was one by Voltaire, Elements of Newton's Philosophy (1738). Another was Newtonism for Ladies (1737), a translation of Francesco Algarotti's Italian work for those not versed in mathematics or science.

Proceeding with bold assurance ("I do not make hypotheses"), Newton in Mathematical Principles offered explanations of the movements of planets and the moon, and tides on the earth. It was all inclusive:

We offer this work as mathematical principles of philosophy; for all the difficulty of philosophy [physics] seems to consist in this--from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.

Early in the Principles he formulated three laws of motion:

(1) Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

(2) The change of motion is proportional to the motive force impressed, and is made in the direction of the straight line in which that force is impressed.

(3) To every action there is always opposed an equal reaction.

Beginning with these laws and using the rule of the inverse square Newton then moved to the formulation of the principle of gravitation:

Gravity operates according to the quantity of the solid matter which they [the sun and planets] contain, and propagates its virtue on all sides ...decreasing always as the inverse square of the distances.

Translation: every body (or speck of matter, for that matter) attracts every other body with a force calculated by multiplying their masses and dividing that by the square of the distance between them.

Newton admitted not knowing the causes of the properties of gravity, describing it as a “most subtle spirit which pervades and lies hid in all gross bodies.” He attributed electrical force to this spirit which could be the cause of gravity, speaking in the second edition of his Principia of gravity as an “elastic spirit”, but steadfastly maintained his usual posture about suppositions, “ I frame no hypotheses.” “It is enough,” he asserted, “that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.”

The formula for the mathematicians among us:

F = Gm1m2

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1. Maybe this is a good time to talk about Newton's personality. By all accounts he was difficult to get along with. Prickly, highly sensitive to personal or intellectual slights. Secretive about his work. Unforgiving of Robert Hooke and Gottfried Wilhelm Leibniz, other distinguished scientists. Does genius go along with a neurotic personality?
2. Most people don't know that Newton was an alchemist. Also a serious biblical scholar. He seems to have believed that the universe at its foundation rested on certain truths, codes, principles, what you will, which could be arrived at through a study of alchemy and biblical texts. That kind of interest is totally foreign to modern science. And raises questions about Newton's own mental stability. Or does it?

3. An interesting sidelight. Newton’s friend Halley was a man of many accomplishments, among them the design of a diving bell. You can see a model here. He stood in Newton’s shadow, but he was great man in his own right.

Applying this formula and laws of motion to planetary orbits, Newton found that mathematically his calculations matched the observations of planetary orbits made by Kepler. The planets are kept in their elliptical orbits and pulled inward from flying off in rectilinear motion by a force which tends sunward (the sun is the largest mass) and varying inversely as the square of their distances from the center of the sun. The earth exerts a similar attraction on the moon. He calculated the mass of each planet and figured that the density of the earth was 5 to 6 times that of water. He accounted mathematically for the fact that the earth flattens at the poles and bulges at the equator. He showed that the tides were due to the combined pull of the sun and moon. Comets, he said, behaved like planets with regular orbits and motion. Speaking of orbits and motion, Newton theorized in the Mathematical Principles that an object fired from a cannon with sufficient velocity could orbit the earth, or even leave earth's orbit. For an fascinating demonstration, click on the java applet at the University of Virginia which lets you determine the speed required to either orbit or escape the earth's orbit.

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I don't know about you, but all this, with simple instruments and equipment, staggers me. And his theories reigned unchallenged until the 20th century. Your reaction?

Newton's Significance for the Second Europe

By giving gravitational attraction to all the planets and stars, Newton pictured a universe mechanically more complex than anyone had shown before. Every star or planet was now viewed as affected by, and affecting, every other star or planet. But law regulated all. The most distant star obeyed the same laws as the smallest particle on earth. He called the last section of his work, significantly, the System of the World.

What was Newton's significance for the Enlightenment? Perhaps one statement sums it up, the lines by Alexander Pope:

Nature and Nature's laws lay hid in night;

God said: "Let Newton be'" and all was light.

His actual epitaph: "Who, by vigor of mind almost divine, the motions and figures of the planets, the paths of comets, and the tides of the seas first demonstrated."

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A modern authority, J. D. Bernal, in Science in History, (1957), said:

In one word Newton established, once and for all, the dynamic view of the universe instead of the static one that had satisfied the Ancients. This transformation, combined with his atomism, showed that Newton was in unconscious harmony with the economic and social world of his time, in which individual enterprise, where each man paid his way, was replacing the fixed hierarchical order of the late classical and feudal period where each man knew his place.

Bernal seems to be saying that science is a response to the economic and social needs of the times. Newton met the needs of European society in his new theories of light, optics, gravitation, and his new method of calculus. Is this true? Another scholar, Alexander Koyre, in an essay "Galileo and Plato," Journal of the History of Ideas (1943), took the position that science is developed as an intellectual matter without regard to the needs of society. Who is right?