He was elected a scholar on 28 April 1664 and received his bachelor's degree in April 1665. His scientific genius fully emerged when the plague closed the University in the summer of 1665 and he had to return to Lincolnshire. There, in a period of less than two years, while still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy.

While Newton remained at home he laid the foundations for differential and integral calculus, several years before its independent discovery by Leibniz.

Newton's first work as Lucasian Professor was on optics and this was the topic of his first lecture course in 1670. He had reached the conclusion during the two plague years that white light is not a simple entity. He argued that white light is really a mixture of many different types of rays which are refracted at slightly different angles, and that each different type of ray produces a different spectral colour. Newton was led by this reasoning to the erroneous conclusion that telescopes using refracting lenses would always suffer chromatic aberration. He therefore proposed and constructed a reflecting telescope.

In 1672 Newton was elected a fellow of the Royal Society after donating a reflecting telescope.

Newton's greatest achievement was his work in physics and celestial mechanics, which culminated in the theory of universal gravitation. By 1666 Newton had early versions of his three laws of motion. He had also discovered the law giving the centrifugal force on a body moving uniformly in a circular path. In 1687 Newton published the Philosophiae naturalis principia mathematica or Principia as it is always known.

The Principia is recognised as the greatest scientific book ever written. Newton analysed the motion of bodies in resisting and non-resisting media under the action of centripetal forces. The results were applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth. He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalised that all heavenly bodies mutually attract one another.

Further generalisation led Newton to the law of universal gravitation:-

... all matter attracts all other matter with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Newton explained a wide range of previously unrelated phenomena: the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon as perturbed by the gravity of the Sun. This work made Newton an international leader in scientific research.

After suffering a second nervous breakdown in 1693, Newton retired from research.

In 1703 he was elected president of the Royal Society and was re-elected each year until his death.

He was knighted in 1705 by Queen Anne, the first scientist to be so honoured for his work.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/newton.html