On June 5th 1661, Newton entered his Uncles college, Trinity College in Cambridge.
He was older than most of the other student, and even though his mother was
financially well off he entered this school as a "sizar". A sizar
is a student that receives an allowance towards the expenses of college by
acting as other students' servant.
While at Cambridge, his chief aim was a degree in law. Even though most of the studies and instruction at Cambridge was dominated by the philosophy of Aristotle, some freedom of study was allowed. Here Isaac studied the philosophies of such men as Descartes, Gassendi, Hobbes and Boyle. He also was greatly interested in the astronomy of Galileo and in Kepler's Optics. He kept a fascinating account of his thought in a journal entitled Quaestiones Quaedam Philosophicae, which translates in Certain Philosophical Questions.
Newton's interest in mathematics began in the Autumn of 1663 when he bought an astrology book at a fair in Cambridge and came to the realization that he was unable to understand the mathematics within it. Intrigued by this he examined a trigonometry book, also finding that he could not understand it either, lacking the geometry skills required. He returned to his old copy of Euclid's Elements and read the book with a whole new respect. His immersed himself in various books on mathematics, topics covering the new algebra and analytical geometry. He read Wallis's method for finding a square of equal area to a parabola and a hyperbola which used indivisibles. On this he made notes of Wallis's treatment of series, but also formulated his own proofs as well.
During this time at home Newton laid the foundations for integral and differential calculus. He termed this the "method of fluxions" and it was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. Taking differentiation as the operations base, he produced simple analytical techniques previously developed to solve unrelated problems such as finding area, tangents, and the lengths of curves. Newton wrote De Methodis Serierum et Fluxionum in 1671 but failed to get it published and it was not seen until John Colson produced the English translation of it in 1736.
