The Unpublished Manuscripts of Isaac Newton

Newton died on 20th March, 1727 leaving hundreds of unpublished manuscripts; some of which date back to his arrival at Trinity College, Cambridge in 1661. His heirs invited Thomas Pellett to examine the manuscripts and report on their suitability for publication. After just three days of examining these hundreds of manuscripts, Pellett, a qualified physician and member of the Royal Society, dismissed the majority of the manuscripts as being “not fit to be printed”, “of no scientific value” and “loose and foul papers”.

Pellett only found two sets of manuscripts suitable for publication. The first was a set of manuscripts on chronology and the second were two manuscripts on prophecies. Although Pellet claimed that the text on prophecy was imperfect, they were nevertheless worthy of publication. The other manuscripts, which included drafts of the Principia, mathematical and scientific papers, his correspondence and works on prophecy, chronology, alchemy and theology were passed on to his niece Catherine Conduitt. With the marriage of Catherine’s daughter into the Portsmouth family, the manuscripts become part of the Portsmouth Collection.

In 1872, the papers were offered to the University of Cambridge, which only accepted the scientific papers, refusing the other papers on topics that Newton was not famous for. The remaining non-scientific manuscripts were offered to the British Library, which also refused them on similar grounds. These manuscripts remained in the Portsmouth Collection until 1936, when they were auctioned and dispersed into collections all around the world.

The auction was held in July in 1936 at Sotheby’s. The manuscripts were divided up into three-hundred and thirty lots and sold to thirty-three buyers. Thus Newton’s manuscripts were scattered all over the world. It is surprising that these manuscripts were allowed to leave England. Josè Faur considered that the reason for this was because of the contents of the manuscripts. Manuscripts on prophecy, alchemy and Newton’s unorthodox theology did shock some scholars. It was “to protect Newton’s ‘good name,’ [that] the importance of the manuscripts were denied”.

One of the buyers was the famous economist John Maynard Keynes, who bought a significant number of manuscripts which he bequeathed to King’s College, Cambridge. He made a study of these manuscripts and found

that Newton was different from the conventional picture of him. But I do not believe he was less great. He was less ordinary, more extraordinary than the nineteenth century cared to make him out. Geniuses are very peculiar.

The nineteenth century, in their adulation of Newton, had rendered him quite bland. After poring over the contents of the box of manuscripts he had purchased, Keynes claimed:

Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to begin to build our intellectual inheritance rather less than 10,000 years ago.

This very famous quote was written by Keynes in 1942. The paper “Newton the Man” was written for the tercentenary celebration of Newton’s birth, but the Second World War intervened and the paper was not presented until 17 July 1946, after Keynes’ death in April of that year. These tercentenary celebrations were conducted on an international scale and ran for five days culminating in a garden party at Buckingham Palace. Keynes’ paper was read to the Royal Society by his brother, Geoffrey. It had not been revised by the author, who had written it some years back. Keynes was the first to publicly consider Newton as more than the orthodox image of the romantic dreamer scientist. He considered Newton’s faults and also what appeared from an early twentieth century perspective to be unorthodox practices such as alchemy, his style of theology, his interest in chronology and church history and his argumentative nature, in conjunction with his great scientific achievements. These were aspects of Newton’s character that had been ignored or glossed over by previous commentaries and biographies.

These discoveries in Newton the Man by Keynes did not lessen his admiration for Newton. He considered that the box of papers that he was studying showed Newton to be a man with great power of mind, who attempted to understand all aspects of God and nature. Newton’s experiments were not undertaken for mere discovery, but to verify what he already knew, and to confirm his strong belief in God. Keynes wrote:

Why do I call him a magician? Because he looked on the whole universe and all that is in it as a riddle, as a secret which could be read by applying pure thought to certain evidence, certain mystic clues which God had laid about the world to allow a sort of philosopher’s treasure hunt to the esoteric brotherhood. He believed that these clues were to be found partly in the evidence of the heavens and in the constitution of elements (and that is what gives the false suggestion of his being an experimental natural philosopher), but also partly in certain papers and traditions handed down by the brethren in an unbroken chain back to the original cryptic revelation in Babylonia. He regarded the universe as a cryptogram set by the Almighty – just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibnitz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate.

As more and more of Newton’s papers became available to scholars, Keynes’ words seem increasingly insightful and revealing. Keynes considered that there were two sides to Newton’s character they were “Copernicus and Faustus in one”. Scientist and magician were the same man working to one purpose and whose achievements were seemingly beyond his era but at the same time founded in the knowledge of the ancients.

Later biographies have assumed that the works on theology, chronology and prophecy were the works of an ageing Newton. That there were two Newtons; the great scientist of his youth and the ageing Newton who had lost his taste and ability for science and turned to the study of chronology, prophecy and religion as a result of the nervous breakdown he suffered in 1693. However, these two separate and diverse personas are not supported or divided by any such date and Newton did continue to research and add to the science of his day. Furthermore, his papers and interest in chronology and prophecy date back to his earliest days in Cambridge in the 1660s.

Newton’s deeply held religious convictions led him to search for the mystic clues which he believed that God had laid about the world. This search had resulted in his scientific research in the form of the Principia and Opticks; both are landmarks in science. Alchemy, chronology, theology and prophecy as well as natural philosophy were all parts of these clues which Newton attempted to unravel or decrypt. It is unclear whether there was a dividing line in the mind of Newton between these topics; however, all of these topics confirmed his belief in the supreme design of the universe.

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Newtonian Dynamics

Dynamics is a mathematical model which aims to both describe and predict the motions of the various objects which we encounter in the world around us. The general principles of this theory were first enunciated by Sir Isaac Newton in a work entitled Philosophiae Naturalis Principia Mathematica (1687), which is commonly known as the Principa.

Up until the beginning of the 20th century, Newton's theory of motion was thought to constitute a complete description of all types of motion occurring in the Universe. We now know that this is not the case. The modern view is that Newton's theory is an approximation which is generally valid when describing the low speed (compared to the speed of light) motions of macroscopic objects. Newton's theory breaks down, and must be replaced by Einstein's theory of relativity, when objects start to move at speeds approaching the speed of light. Newton's theory also breaks down on the atomic scale, and must be replaced by quantum mechanics.

Newton's theory of motion is an axiomatic system. Like all axiomatic systems (e.g., Euclidean geometry), it starts from a set of terms which are undefined within the theory. In the present case, the fundamental terms are mass, position, time, and force. It is taken for granted that we understand what these terms mean, and, furthermore, that they correspond to measurable quantities which can be ascribed to, or associated with, objects in the world around us. In particular, it is assumed that the ideas of position in space, distance in space, and position as a function of time in space, are correctly described by the vector algebra and calculus. The next component of an axiomatic system is a set of axioms. These are a set of unproven propositions, involving the undefined terms, from which all other propositions in the system can be derived via logic and mathematical analysis. In the present case, the axioms are called Newton's laws of motion, and can only be justified via experimental observation.

Note, incidentally, that Newton's laws, in their primitive form, are only applicable to point objects. Newton's laws can be applied to extended object by treating them as collections of point objects. In the following, it is assumed that we know how to set up a Cartesian frame of reference, and also know how to measure the positions of point objects as functions of time within that frame. In addition, it is assumed that we have some basic familiarity with the laws of mechanics, and that we understand standard mathematics up to, and including, calculus.

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Monte Carlo Methods

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in simulating physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, these methods are most suited to calculation by a computer and tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm.

Monte Carlo simulation methods are especially useful in studying systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model). More broadly, Monte Carlo methods are useful for modeling phenomena with significant uncertainty in inputs, such as the calculation of risk in business. These methods are also widely used in mathematics: a classic use is for the evaluation of definite integrals, particularly multidimensional integrals with complicated boundary conditions. It is a widely successful method in risk analysis when compared with alternative methods or human intuition. When Monte Carlo simulations have been applied in space exploration and oil exploration, actual observations of failures, cost overruns and schedule overruns are routinely better predicted by the simulations than by human intuition or alternative "soft" methods.

The term "Monte Carlo method" was coined in the 1940s by physicists working on nuclear weapon projects in the Los Alamos National Laboratory.

There is no single Monte Carlo method; instead, the term describes a large and widely-used class of approaches. However, these approaches tend to follow a particular pattern:

1. Define a domain of possible inputs.
2. Generate inputs randomly from the domain using a certain specified probability distribution.
3. Perform a deterministic computation using the inputs.
4. Aggregate the results of the individual computations into the final result.

For example, the value of π can be approximated using a Monte Carlo method:

1. Draw a square on the ground, then inscribe a circle within it. From plane geometry, the ratio of the area of an inscribed circle to that of the surrounding square is π/4.
2. Uniformly scatter some objects of uniform size throughout the square. For example, grains of rice or sand.
3. Since the two areas are in the ratio π/4, the objects should fall in the areas in approximately the same ratio. Thus, counting the number of objects in the circle and dividing by the total number of objects in the square will yield an approximation for π/4.
4. Multiplying the result by 4 will then yield an approximation for π itself.

Notice how the π approximation follows the general pattern of Monte Carlo algorithms. First, we define a domain of inputs: in this case, it's the square which circumscribes our circle. Next, we generate inputs randomly (scatter individual grains within the square), then perform a computation on each input (test whether it falls within the circle). At the end, we aggregate the results into our final result, the approximation of π. Note, also, two other common properties of Monte Carlo methods: the computation's reliance on good random numbers, and its slow convergence to a better approximation as more data points are sampled. If grains are purposefully dropped into only, for example, the center of the circle, they will not be uniformly distributed, and so our approximation will be poor. An approximation will also be poor if only a few grains are randomly dropped into the whole square. Thus, the approximation of π will become more accurate both as the grains are dropped more uniformly and as more are dropped.

Enrico Fermi in the 1930s and Stanisław Ulam in 1946 first had the idea. Ulam later contacted John von Neumann to work on it.

Physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons would likely travel through various materials. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus or how much energy the neutron was likely to give off following a collision, the problem could not be solved with analytical calculations. John von Neumann and Stanislaw Ulam suggested that the problem be solved by modeling the experiment on a computer using chance. Being secret, their work required a code name. Von Neumann chose the name "Monte Carlo". The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble.

Random methods of computation and experimentation (generally considered forms of stochastic simulation) can be arguably traced back to the earliest pioneers of probability theory (see, e.g., Buffon's needle, and the work on small samples by William Sealy Gosset), but are more specifically traced to the pre-electronic computing era. The general difference usually described about a Monte Carlo form of simulation is that it systematically "inverts" the typical mode of simulation, treating deterministic problems by first finding a probabilistic analog. Previous methods of simulation and statistical sampling generally did the opposite: using simulation to test a previously understood deterministic problem. Though examples of an "inverted" approach do exist historically, they were not considered a general method until the popularity of the Monte Carlo method spread.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though were severely limited by the computational tools at the time. Therefore, it was only after electronic computers were first built (from 1945 on) that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.

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