# Issac Newton’s Contributions in Mathematics

Sir Issac Newton was a polymath, who was born in Woolsthrope, in Lincolnshire, on 4 January 1643. After the death of his father, Newton’s mother remarried and he was brought up by his grandparents. After completing his schooling at Grantham, he went to Trinity College, Cambridge in 1661. This became a turning point in his life. He wanted a degree in Law but studies at Cambridge were dominated by the philosophies of several mathematicians. As a result, he started studying different books of mathematics and recorded his thoughts in a book, which he later named, “Quaestiones Quaedam Philosophicae”. He studied Euclid’s ‘Elements’, Oughtred’s ‘Clavis Mathematica’, Descartes’ ‘La Géométrie’ and many others. Newton also studied Wallis’ Algebra and tried to devise his proofs of the theorems by writing,

Thus Wallis doth it, but it may be done thus …”

In 1665, when the plague attacked England, Newton came back to Lincolnshire and in the span of just two years, he made commendable discoveries in the field of mathematics, astronomy, optics and physics. While he was at home, he laid the foundation of differential and integral calculus and also worked on Optics. Newton published his first paper entitled, ‘Philosophical Transactions of the Royal Society’ on light and colour. Hooke and Huygens raised questions on his proof and his reaction towards criticism was irrational. He was believed to be of dual nature. On one side, he wanted to be recognised by all, but on another side, the fear of criticism forced him not to publish his works. Despite his fear, he gave three laws of motion, the law of gravity, speed of sound in air etc, in the field of physics. In mathematics, he wrote a book ‘Principia’, nowadays known as, ‘a book dense with the theory and application of the infinitesimal calculus.’ He is also credited for finding generalised binomial theorem, Newton’s identities and methods, solutions to Diophantine equations by using coordinate geometry, power series and many more. On 20th March 1727, he died while sleeping and he was the first scientist to be buried in the abbey. His contributions to mathematics are discussed below in detail.

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## 1. Newton’s Fundamental Theorem of Calculus

The fundamental theorem of calculus establishes a relationship between the concepts of differentiation and integration. According to the theorem, these two concepts are inverses of each other. This theorem is usually divided into two parts. The first part of the theorem states that,

One of the antiderivatives (also known as an indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continous functions.”

This means, for a continuous real-valued function f, defined on the closed interval [a, b], F be the function defined, for all x in [a, b] by,

then F is uniformly continuous on [a, b] and differentiable on (a, b) and F′(x) = f(x), for all x ∈ (a,b).

The second part of the theorem states that,

The integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antidetivatives.”

This means, for a real-valued function f, defined on the closed interval [a, b], F be an antiderivative of f in (a, b), F′(x) = f(x), and if f is Reimann- Integrable on [a, b] then,

## 2. Generalised Binomial Theorem

Newton is credited for providing a generalised expression of the binomial theorem. In the 4th century, Euclid proposed the special case of the binomial theorem for exponent 2. The binomial theorem states that,

It is possible to expand the polynomial (x + y)n into a sum involving terms of the form rxbyc, where the exponents b and c are non-negative integers with b + c = n, and the coefficient ‘r of each term is a specific positive integer depending on n and b.”

For example, for n=4, {(x+y)}^{4}= {x}^{4} + 4x³y + 6x²y² + 4xy³ + {y}^{4}

## 3. Newton’s Method

Newton method was found by Issac Newton and Joseph Raphson, so is also known as the Newton-Raphson method. This method helps to calculate a better approximation of the roots of a real-valued function. In this method, he defined a function ‘f’ for a real variable x, the derivative of the function f, (f´), and an approximate initial root x0 of the function f. Then a better approximation of root of ‘f’ after x0 is given by the iterative formula,

xn+1 = xn– f(xn)⁄f´(xn), for n = 0, 1, 2, 3,………,and this process is iterated until a precise value for the root of ‘f’ is achieved.

## 4. Newton’s identity

These identities were found by Newton in 1666. With the help of Newton’s identities, one can find the power sum of roots of the polynomial without actually finding the roots. This identity is also known as Newton-Girard formulae. Suppose {x}_{1}, {x}_{2}……….{x}_{n} are the roots of the polynomial equation, then Newton’s identity is used to find summations like

## 5. Cubic Plane Curve

In mathematics, a cubic plane curve C is defined by the equation F(x, y, z) = 0, where x, y, z are homogeneous coordinates for the projective plane. The real points of the cubic curve were given by Issac Newton. There are 78 families of cubic curves in total and Newton discovered 72 of them. His classification of cubic curves was given in “Curves” in John Harris’ book. According to Newton, cubics can be generated by the projection of five divergent cubic parabolas. He also showed that by the projection of elliptic curve, y²=ax³+bx²+cx+d, any cubic curve can be generated and the general cubic equation is written as,

y²=x³+ax+b

The most difficult subcase of the above general cubic equation is given by, ay²=x(x² – 2bx + c) and are also known as Newton’s diverging parabolas.

## 6. Newton’s Polynomial

Newton’s polynomial is a concept in Numerical analysis. It is an interpolation polynomial with some given points, founded by Sir Issac Newton. Here, coefficients of the polynomial are calculated using Newton’s divided difference method. For, given k+1 points, (x0, y0),….., (xk, yk), where all xj‘s are distinct, Newton’s polynomial is given by,

where nj‘s are Newton’s basis polynomials defined below,

## 7. Newton’s Forward Difference Formula

Newton’s finite difference formula is based on finite-difference identity, which gives an interpolated value between the points fp of the table in the terms of initial value f0 and the powers of finite difference, denoted by Δ. According to the formula, for a∈[0, 1],

fa = f0 + aΔ + 1⁄2! a(a-1)Δ² + 1⁄3! a(a-1)(a-2)Δ³ + ……. , can also be written as

## 8. Calculating Slope of a Curve

Newton found it easy to represent and calculate the average slope of the curve, but for some cases, like the increasing speed of an object on a distance-time graph, he claimed that there was no method to calculate the exact slope on any given individual point on the curve. Although the slope at a particular point can be calculated by taking the average slope of smaller segments of the curve, as the curve approaches to zero, calculation of the slope approaches closer to the exact slope, Newton and Leibniz found another way, by calculating a derivative function f´(x), that gives the slope of function f(x) at any point on the curve. This method of calculating slope using the derivative of a function is known as differential calculus or differentiation.

## 9. Infinite series

Newton’s work on infinite series was based mostly on Simen Stevin’s decimals. He was the first to use power series and he also approximated partial sums of harmonic series by logarithms. A harmonic series is an infinite divergent series written as,

The finite partial sum of harmonic series is given by,