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Isaac Newton (1642-1727) is perhaps the most famous Lucasian Professor of Mathematics. He is probably best known to the average person because of the story of the falling apple and its relationship to the discovery of gravity. Newton discovered the force of gravity, and today the search is for its carrier: gravity waves. The years in between are a fascinating scientific story, detailed in a book edited by the current Lucasian professor, Stephen Hawking, Three Hundred Years of Gravitation. ¹

Newton arrived at Cambridge in 1661, was elected scholar in 1664, graduated BA in 1664/5 in a class of twenty-six from Trinity, and made MA in 1668.² During a wonderful surge of scientific production Newton produced three great achievements in the short space of two years. The first great achievement was the invention of fluxions, which resulted in calculus. He used this knowledge to advance his other work. Newton's second great achievement was the discovery of the law of the composition of light, later used in the development of optics. His third great achievement, the discovery of the universal force of gravity, was the basis for the Principia , his ultimate achievement.

Newton served as a member of Parliament representing the university. He took the Lucasian Chair in 1669. He was elected to the Royal Society in 1672, and elected president of the Royal Society in 1703, the year after he retired from the Lucasian professorship. He was knighted in 1705 at Trinity by Queen Anne. Later in life, Newton continued to achieve and was awarded numerous honors. He left Cambridge for London while still Lucasian professor. He was appointed Warden of the Mint in 1696 and Master of the Mint in 1699, when he led the effort for a recoinage.

A milestone event in Newton's life was the controversy with Leibniz.³ The Leibniz controversy is a low point in the history of science, revolving around the question of who was to receive the credit for the invention of calculus. It seems clear that each discovered it independently, but Newton somewhat earlier. Today we have adopted the notation created by Leibniz because it is easier to use and understand. Newton's notation is blamed for holding back the development of mathematics in England for a century. While it is not uncommon for controversies of this nature to develop, most are not as tangled and acrimonious as this one became.

The facts are not as clear as one would like, but it seems that during the initial stages in the development of calculus, when Newton and Leibniz were on good terms, letters from Newton to Leibniz contained hints of the fluxions. Newton is credited with the earliest discovery, but he asserted that no one should share in the honor of the discovery, implying that Leibniz had completely stolen his ideas. This naturally did not sit well with Leibniz, nor with his supporters. Leibniz developed a method of his own, perhaps with help from Newton's hints, but Leibniz created a more general method, more easily learned than Newton's. Both men were diminished somewhat by this controversy, with negative consequences for British mathematics, whose practitioners followed Newton out of blind loyalty.⁴

He had left Cambridge to avoid the plague in 1665 eturning in 1667, as did many others. Cambridge dismissed everyone in the summer of 1665 and again in the summer of 1666 due to the severity of the plague. Newton's relationship with Isaac Barrow is important but not entirely clear. As a student at Trinity, he seems to have been required to attend Barrow's lectures, with some evidence that he was present at least twice. There is also a story about Barrow's examination of Newton that found Newton wanting in his mastery of Euclid. This story has been relegated to that of a myth by several later Newton scholars.⁵ Newton's own notes indicate his study of Euclid took place in his first two years at Cambridge, before this examination allegedly took place. It is an accepted fact that Barrow vacated the Chair in favor of Newton, who at that time had demonstrated his capabilities and impressed Barrow. Barrow acknowledged in his published lectures Newton's help on the manuscript, calling Newton a great genius.⁶

Newton's monumental work Philosophiae Naturalis Principia Mathematica (1687) forever changed science, providing the basis for the modern understanding of the universe. The Principia is composed of three books. The first and second books present the laws and conditions of motions and forces. The third deduces the constitution of the universe from the principles offered in books one and two.⁷ The first book deals with the general dynamics in ideal conditions, that is with no friction. The second book deals mainly with fluids and friction.⁸

In the first book of the Principia , Newton puts forth his three Laws of Motion.

First, "Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it." Second,"The change of motion is proportional to the motive force impressed; and it is made in the direction of the right line in which that force is impressed." Third, "To every action there is always opposed an equal reaction: or the mutual action of two bodies on each other are always equal, and directed to contrary parts."⁹

The third book contains his third great achievement, the discovery of the universal force of gravity. The great principle underlying the Principia is that of universal gravitation.¹⁰

"That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain."¹¹

With this statement, Newton opened the doors of physics in a very profound manner. All objects in the universe were suddenly equal. The hierarchy of importance that had dominated the thinking of the stars and the solar system was eliminated. The mystical approach to understanding the workings of the universe was replaced. Teleological philosophy was replaced by mechanical philosophy. Gravity now explained how the planets move, but this new knowledge also brought new difficulties. For if each object affected every other object, the calculations describing their motion would be extremely complex. And so they were. Newton was unable to calculate, other than in a general way, the orbits of the planets in the solar system. The sun is the most influential object, but each planet exerts influence on the other planets. The moon's orbit is affected in a noticeable way by both the gravity of the sun and the gravity of the earth. The perturbations of the moon's orbit had long been a problem, and lunar theory would continue to draw attention for a long while after Newton. He was able to perform calculations that gave better results than his predecessors, but he was unable to do more than explain the major perturbations.¹²

Newton's first great achievement was the invention of fluxions (calculus), providing him with the mathematical tools necessary for the rest of his work. Although Newton is credited with the invention of calculus, much of the work for it had been done before him. His contributions provided the leap from the possible to the actual. The key mathematical problem lay in describing the path of an object as it traveled in either a straight line or a curved line. The straight line had been long understood, but the curved path was merely described by the use of many small, straight lines. The more small lines, the more accurate the calculation, but it was still not a curved line. Even Galileo had worked to understand the curved path of an object, showing the need to find a means to describe this path.

Descartes invented analytical geometry in response to Galileo's work, so now algebraic equations could be used to solve the problems. Seventeenth century mathematicians pushed out the boundaries with many such discoveries. These new techniques were needed to solve the problems presented by science in areas such as astronomy. Newton made a subtle change in the existing understanding of the curved path that resulted in great advances. Instead of seeing the path just as a series of points along a curved path, which was an improvement on the notion of a series of short lines, he saw the path as created by one point moving through space for a period of time. This moving point was called a fluent , and its velocity was called fluxion . Changes in the fluxion was acceleration.¹³ After conceiving the method of fluxions, Newton adapted them to the quadrature of curves.¹⁴

Although this was a great step forward, real calculus requires the ability to do differentiation and integration, which in turn requires the ability to expand functions into infinite series. The generalized form of this, the binomial theorem was discovered by Newton as well. Besides fluxions, Newton made numerous other contributions to mathematics, for example, Newton introduced the system of literal indices.¹⁵

Newton's algebraic lectures delivered during his first nine years as a professor were published in 1707 by William Whiston as Arithmetica Universalis. The book "contains new and important results in the theory of equations. ...[It] gives formulae expressing the sum of the powers of the roots up to the sixth power and by an `and so on' makes it evident that they can be extended to any higher power. ...Newton's formulae take the implicit form, while similar formulae given earlier by Albert Girad take the explicit form, as do also the general formulae derived later by E. Waring."¹⁶

Newton's work on limits resulted in a number of new limits. He fixed the upper limit and the lower limit of the real roots of equations, and the number of each kind of limit. He did not prove his work, however; he just gave examples. The index proofs had to wait for almost one hundred and fifty years, when it was shown that his work was a special case of a general theorem.¹⁷

Newton discovered a method of approximating roots of numerical equations so important that it is still used today. His method is described in his Method of Fluxions where he uses one, now famous, example, the cubic equation Y^3-2Y-5=0.

The method uses a value that appears close to the real value that iterates through successive equations created by the new value until there is convergence and the real value is found. The method does not always work, but is still a very powerful method. After Newton published his method, Joseph Raphson presented a modification of the method using the same equation over and over again with the new values. This is the method actually used today and is known as the Newton-Raphson method of approximation.¹⁸

Newton's second great achievement was the discovery of the law of the composition of light. He also discovered chromatic aberration and the selective refrangibility of colors. His work on telescopes was also a major contribution, although he was not the true inventor of the reflecting telescope. He made the first one and was able to explain why it was superior to the refracting telescope. His experiments on light leading to his theories were well received at the Royal Society, but criticized by others who had enough stature as to cast doubt on Newton's success.

Newton's other work was extensive and varied. He invented the sextant, "deduced the theoretical expression for the velocity of sound in air," and "engaged in experiments in chemistry, elasticity, and magnetism, the law of cooling and geological speculations."¹⁹

Chemistry did not come into its own until the late eighteenth century. Alchemy and chemistry were still intertwined in Newton's lifetime. His work in alchemy is legendary, compiling in the 1680's a work entitled Index chemicus which is over one hundred pages with over 5000 references to other works. His library held hundreds of titles covering chemistry and alchemy, collected and studied as he searched for the key to the universe.²⁰

There is evidence that Newton had connections, through Isaac Barrow, to the alchemists of the day. These contacts were important because the alchemists kept a low profile, and were very secretive, even though their works were available in bookstores. However, many manuscripts were not easily accessible. The history of alchemy included many events that caused a practitioner to be wary. Magic was linked with the devil, thereby promoting punishments at the pillory and mob destruction of alchemists' homes. The legendary "philosopher's stone" came at a very high price.²¹ One would hope that Newton was attempting to sift through the jibberish in search of the real basis for chemistry and the understanding it would bring. There is also a school of thought about Newton that connects the alchemists' search to ancient knowledge. This link to ancient knowledge included religion, and Newton's strong belief in the earlier forms of Christianity is well known.

Any evaluation of Newton and alchemy must take into account the times in which he lived and his genius for science. It is difficult to assess how he really thought, but just as difficult to believe that he left his knowledge at home when he entered the world of alchemy. It is far more probable that he was consistent with his usual behavior in his approach to alchemy, where he would be searching for truth and understanding of the workings of the universe, wherever that search would lead him.

Closer to chemistry was Newton's search for the "philosopher's mercury," the substance that gave all other metals the characteristic of becoming liquid when heated. This belief was based on the observable fact that mercury was the only metal liquid at room temperature. If one could extract this "mercurial principle" from metals, then the transmutation of one metal to another would be possible. Newton appears to have been interested in the manipulation of metals, not the creation of gold as other alchemists, and in understanding the structure and composition of the metal.²²

Newton's approach to alchemy showed the pattern of the well-known Newton. He proceeded step by step through evaluation of what was to be studied and selected the most promising materials. He then conducted a proper literature search in which he read everything he could on the subject. Lastly, he came up with a hypothesis and performed an experiment. ²³

Newton kept his religious thoughts to himself, never publishing nor publicly giving sermons. He felt his science was in service of religion and attributed a place to God in a mechanistic universe. When offered the post of master of Trinity if he would take holy orders, Newton refused with the reason he could do more if he were on the outside.²⁴ It was clear that he wanted to be able to pursue his own ways of science and religion. What is now known about Newton's thoughts on religion is that he was a millenniumarian, one who believed in the imminent destruction of the world as we know it, followed by the creation of a new heaven on earth. He was anti-papist, anti-trinitarian, and Arian, with a Whig coloring.

Newton spent much of his non-scientific time and efforts working out the chronology of the world. Working from Biblical references, he applied his mathematical skills to establish the dates of important events. He did not meet with much success, because, along with most everyone else at the time, he accepted Bishop Usher's 4004 B. C. date for creation.²⁵ Time was spent in interpreting the texts of apocalypse and revelation, such as the works of Daniel, where dreams foretold the future with a fair stretch of the imagination. These works were published posthumously. His religious feelings were most evident in his views of the Trinity in Catholicism. He went back to the earliest biblical works he could find to search for the authentic and correct doctrines. These did not include the Holy Ghost nor allowed for Christ to be considered as a separate entity created some time after God. Although it was considered good to find arguments against Rome during Newton's time by the Anglican Church and the government of England, it was still not acceptable to be Unitarian.

Newton was a great man of vision in mathematics and science who kept his religious beliefs in a small circle. Perhaps he did so from a fear of reprisal because he had desires to move up in the world where any scandal arising from religious beliefs such as his had no place.

Footnotes

1. Stephen Hawking, Three Hundred Years of Gravitation (Cambridge: Cambridge University Press, 1987).
2. Louis More, Isaac Newton A Biography (New York: Charles Scribner's Sons, 1934), 40.
3. Gottfried von Leibniz (Germany, 1646-1716).
4. More, 193.
5. More, 33.
6. More, 80.
7. Florian Cajori, A History of Mathematical Notations (Chicago: Open Court Press, 1929), 199.
8. Gale Christianson, In the Presence of the Creator: Isaac Newton and his Times New York: (Free Press, 1984), 291.
9. Christianson, 292.
10. Cajori, Mathematical Notations, 199.
11. Christianson, 307.
12. Christianson, 308.
13. More, 184.
14. Cajori, Mathematical Notations, 192.
15. Cajori, Mathematical Notations, 192.
16. Cajori, Mathematical Notations, 201.
17. Cajori, Mathematical Notations, 202.
18. Cajori, Mathematical Notations, 202.
19. Cajori, Mathematical Notations, 204.
20. Christianson, 216.
21. Christianson, 220.
22. Christianson, 226.
23. Christianson 1984 226.
24. Gascoigne, 85.
25. [25] More, 616.