Newton stands tall among his peers, and many would argue he was the greatest scientist the world has ever seen. But dig a little beneath this awesome reputation and one finds a troubled boy, ignored by his mother and stepfather muddling along at the village school.

Fast forward 40 years and there stands a rich civil servant -Master of the Mint - his great scientific work complete, lauded and honoured in his life and by history thereafter.

But he was still utterly insecure, with few friends and suffering many mental torments including two breakdowns. He spent many years engaged in a vitriolic conflict with Leibniz (even after the latter died) battling to prove who discovered calculus.

Newton, discovering he was not going to be a successful gentleman farmer, went to Cambridge to study law. He was not an outstanding student, but a chance look at astrology developed his interest in mathematics.

He picked up Euclid and in a year seemed to have given himself a mastery of all that this series of books provided - geometry, proof, logical thought and number theory.

At this point plague interrupted his new studies, the university was closed and Newton returned to his family in Lincolnshire. The next two years were to prove incredibly fruitful as he had time to ponder and develop what were to become his great discoveries.

It was from his mother’s house that he saw the famous apple drop from the tree and began to wonder why this should be.

He returned to Cambridge and aged just 27, was appointed professor. His first lecture course looked at light which he had studied at his mother’s house. He put forward the revolutionary suggestion that white light consisted of colours - laying the basis for our modern understanding of the nature of light.

Increasingly, his fame grew. In 1687 he published his famous book *Principia Mathematica* which turned the world of science on its head. Hitherto, science had looked at the world and wondered about its nature; Newton looked at the world and described how it behaved.

He hadn’t forgotten his apple - why he wondered did it drop, and as it did what made it do this and in what way did it fall? In *Principia* he gave his answers. His universal law of gravity has been fundamental to science ever since. This law and his companion laws of motion, gave a way to understand how the solar system behaved - Kepler’s laws are seen as consequences of them - and the way that all bodies moved.

Newton invented calculus - a topic that he had been working with over many years. However, he did not publish his work until after Leibniz. This led to a bitter priority dispute between English and continental mathematicians which persisted for decades, to the detriment of all concerned. The legacy of calculus is enormous - it is a step change in mathematics, and virtually no area of modern mathematics would be the same without it.

But Newton began to lose interest in research, preferring to spend time in London culminating in his post as Warden of the Royal Mint. He produced one final scientific work *Opticks* in 1704 published only when one of his rivals, Hooke, died.

When he should have been settling down to a peaceful, rich retirement, Newton instead spent his final years in vitriolic battle with Leibniz trying to establish that he, not Leibniz had been first to discover calculus. Before Newton, science was a philosophical pursuit uninterested in the real world, more concerned with ideals than reality. What Newton bequeathed was a unified system of laws and procedures which could be applied to a wide number of phenomena.

One of his discoveries of the time was a powerful extension of the Binomial Theorem. This concerns the expansion of a product as a sum - this would later be a powerful tool in its own right in the hands of Euler, Gauss and Jacobi. We start with the expression,

(1+x)^r.

It is easy to see what happens for some simple values of the power *r*:

(1+x)^1=1+x

(1+x)^2=1+2x+x^2

(1+x)^3=1+3x+3x^2+x^3

(1+x)^4=1+4x+6x^4+x^3+x^4

What happens in the general case? The answer, in principle, is simple:

(1+x)^r={r\choose0}+{r\choose1}x+{r\choose2}x^2+{r\choose3}x^3+\ldots+{r\choose{r}}x^r

which extends the pattern, but what about each term and a formula for each of the coefficients of the various powers of *x*? That too is simple:

{r\choose{k}}={{r(r-1).(r-2)\ldots3.2.1}\over{k(k-1).(k-2)\ldots3.2.1.(r-k).(r-k-1)\ldots3.2.1}}

\qquad\ ={{r!}\over{k!(r-k)!}}

involving the factorials, r!=r.(r-1).(r-2)\ldots3.2.1. Let’s give one example with r=5.

{5\choose3}={5!\over{3!2!}}={5.4.3.2.1\over{3.2.1.2.1}}=10

so in the expansion of, (1+x)^5 the coefficient of x^3 is 10.