Sir Isaac Newton: Mathematical Biography

Newton was born in 1643 in a small village in Lincolnshire, England, son of a farmer who died before he was born. He attended The King’s School at nearby Grantham and quickly became recognized as the best student. At age 17, his mother removed him from school and tried to get him to become a farmer like his father. Her attempt failed and Newton was admitted to Trinity College, Cambridge at the age of 19. Four years later, Newton discovered the Binomial Theorem and began to develop theories that later became the foundations of calculus. When universities around Britain closed in 1665 due to the Plague, Newton spent 18 months in his home town where he did some of his most important work including recognizing the principles of planetary motion and gravity, and the composition of light being that of all colors.

In 1667, Newton became a fellow of TrinityCollege and, two years later, made known some of his work on infinite series. Shortly thereafter. he was elected Lucasian professor of mathematics despite the rules that such a position required being an ordained priest. Worldwide fame followed the 1679 publication of his famous work Principia Mathematica which is predominantly composed of his work on motion and gravity. Although he and Leibniz had been working on calculus concurrently, Newton did not have anything published until 1693 and this sparked the great debate over the priority controversy between the two. Newton once attempted to demonstrate that color perception is a result of different pressures on the eye. His technique was to move a darning needle around his eyeball until he could put pressure on the back. He noted “white, darke and coloured circles” as long as he kept moving the needle around! He is credited as being the first to explain how rainbows are the result of light refraction in water droplets in the atmosphere. In 1696, Newton became Warden of the Royal Mint in London, a position he held until his death in 1727. Ironically, it was his contributions in this capacity, rather than in math or science, that earned him his knighthood in 1705

In the late 17^th Century, both Sir Isaac Newton and Gottfried Leibniz were independently, and without knowledge of each other, working on developing calculus. Newton began his research in 1666 but published nothing until 1693 and then only in part, saving the full explanation until 1704. Leibniz, on the other hand, despite beginning his research eight years later, published his first paper in 1684. Furthermore, in 1696, the French mathematician Guillaume de l'Hôpital published a text about Leibniz methods and thus bringing him to the fore. The origin to the controversy can be attributed to Leibniz being shown a paper of Newton’s while he was visiting London in 1776. The question then was whether Leibniz gained any knowledge from reading Newton’s work.

          Notebooks of Leibniz date his first major breakthrough to 1675 when he discovered he could find the area under the function y=x by integration, and also invented some related notations. Many argued that although Leibniz had indeed invented the integral symbol, among others, his work was merely expanding on ideas that were already firmly established by Newton.

          Support in favor of Leibniz rests on several fundamental facts; mainly that Leibniz published a full description of his method before Newton even published anything on derivatives (called fluxions by Newton). Also, in private papers, Leibniz demonstrated his ideas in a different way to that of Newton. These are the two most important and irrefutable facts for this case but don’t directly prove that Leibniz was the sole discoverer. Indeed, both men were aware of each others work and even worked in collaboration on some aspects, especially in the area of power series.

          That Leibniz saw some of Newton’s papers is well established as some extracts of Newton’s De Analysi per Equationes Numero Terminorum Infinitas were found among his papers in his own handwriting, along with descriptions of this work but in different notation. Most agree that Leibniz certainly had the mental capacity to invent calculus but allege that all the help he got from Newton’s papers were merely suggestions rather than an exhaustive account.

          From the beginning, the controversy itself was controversial due to a Royal Society committee finding in favor of Newton: The report, which was written by Newton himself, was not even made available to Leibniz for over a year. Furthmore, Leibniz was never asked by the Society for his version of events. By today’s standards, Leibniz as the first to publish, would be considered the inventor of calculus. It is also interesting to note that Newton was guilty of some nasty tricks including the witholding of information from a colleague, John Flamsteed, and then, removing mention of him in Principia to which he was a contributer. Robert Hooke also alleges that Newton stole the inverse square law of planetary motion from him and it was more recently proven that Newton stole a theory of origins of atomism from Ralph Cudworth, a Cambridge Platonist.

          Despite 18^th Century opinion being very much in the favor of Newton, it is now widely accepted that both mens’ work was different enough that they can both be credited with the discovery. Since so much time has passed and considering the importance of both of their work, it is only fitting that both deserve the credit.

The rate of change of temperature of an object is proportional to the difference in temperature from that of the surrounding medium. This means,

^dT/_dt   is proportional to   (T-T_a)

so, the general equation is,

^dT/_dt = -k(T-T_a)

where k is a constant and the negative sign means the object is cooling. The a subscript denotes ambient.

From this we can work through to obtain the more useful form

T(t) = T_a+(T₀-T_a)*e^-kt

Where the zero subscript of temperature denotes at time zero.

Newton proved that every particle in the universe is attracted to every other particle with a force proportional to the product of their two masses and is inversely proportional to the square of the distance between them.

For example, given particles with mass M and mass m at a distance r from each other, then the force (F) they exert on each other is described by the equation

where G is the gravitational constant

This is an iterative technique for finding an approximation to the root of an equation which may be difficult using other methods. The first step is to find a rough approximation by picking an arbitrary value within a range that you know the correct value falls within. By examining where the tangent line to the function crosses the x axis, another line can be drawn vertically to give you the next point on the curve. The process can be repeated as many times as you like depending on how accurate you want to be.

The formula to be used for each iteration is

where x_n is the nth approximation.

The first law says that, unless it is acted upon by an external force, an object at rest tends to stay at rest and an object in motion tends to remain in motion.

The second law says that a force applied to an object (F) is equal to the rate of change of its momentum (p). Mathematically we can derive the famous formula

F = ma

(force equals mass times acceleration).

The third law says that for every action there is an equal and opposite reaction. This is manifested in terms of the normal force we study in physics and statics.

Newton theorized a cannon firing a ball away from Earth. Without gravity, the ball would continue to move away from Earth. With gravitational force, the ball will move in different ways depending on its initial velocity: At slow speed, gravity will decelerate the ball and it will fall back to Earth. At a very high speed, the ball will leave Earth and continue to move away. If the speed equals a certain orbital velocity, it will orbit in that fashion. Generally,

where μ is the standard gravitational parameter GM, r is the distance between central and orbiting bodies, and ε is the specific orbital energy.

Calculus by Anton, Bivens & Davis

A History of Mathematics by Carl Boyer

Wikipedia

University of British Columbia

Clark University