Graduation meant little to Newton - he continued his idiosyncratic studies without concern for recognition or advancement. He pressed on, inventing new techniques to push his mathematical understanding further. He developed infinite series that approximated known functions, and used them to calculate those functions to unprecedented accuracy. He invented infinitesimals, that he called "fluxions," and laid the basis for the differential and integral calculus. He even developed a version of what is now known as "the fundamental theorem of calculus": that differentiation and integration are inverse processes; one undoes what the other does. He had far surpassed anything accomplished by any other mathematician in the world. He was completely unknown, 24 years old, and one year past his college commencement.
As suddenly as he had picked up mathematics, he laid it down and turned to other things. He began his studies of optics and of motion. He bought a prism and began studying the properties of colored light. But the plague struck, and Newton fled to the country to his mother's house in Lincolnshire. He had to wait several years before he could obtain a second prism and perform the experiments that would solidify his theory of light.
He turned to the study of motion, and began an investigation of the properties of circular motion. From this investigation he concluded that the acceleration of the Moon about the Earth was related to the acceleration of an object falling near the Earth by the ratio of their distances squared, "pretty nearly." It was about this time (1666) that the famous apple incident occurred, if it occurred at all.
The years 1665-1666 are called Newton's anni mirabiles, his miraculous years, in which he developed his theory of light, of motion, and of the calculus. In fact, he had only begun the work that would, much later, become fully developed theories. But his accomplishments were nonetheless miraculous. As Richard Westfall puts it,
In 1660, a provincial boy ate his heart out for the world of learning .... Six years later, with no help beyond the books he had found for himself, he had made himself the foremost mathematician in Europe and the equal of the foremost natural philosopher.For similar experiments on the nature of light, Christiaan Huygens was being sought after by the kings of Europe. As yet, no one knew the name of Isaac Newton, nor what he had accomplished.