In mathematics too, early brilliance appeared in Newtons student notes. He may have learnt geometry at school, though he always spoke of himself as self taught, certainly he advanced through studying the writings of his compatriots William Oughtred and John Wallis, and of Descartes and the Dutch school. Newton made contributions to all branches of mathematics then studied, but is especially famous for his solutions to the contemporary problems in analytical geometry of drawing tangents to curves (differentiation) and defining areas bounded by curves (integration). Not only did Newton discover that these problems were inverse to each other, but he discovered general methods of resolving problems of curvature, embraced in his method of fluxions and inverse method of fluxions, respectively equivalent to Leibnizs later differential and integral calculus. Newton used the term fluxion (from Latin meaning flow) because he imagined a quantity flowing from one magnitude to another. Fluxions were expressed algebraically, as Leibnizs differentials were, but Newton made extensive use also (especially in the Principia) of analogous geometrical arguments. Late in life, Newton expressed regret for the algebraic style of recent mathematical progress, preferring the geometrical method of the Classical Greeks, which he regarded as clearer and more rigorous.
Newtons work on pure mathematics was virtually hidden from all but his correspondents until 1704, when he published, with Opticks, a tract on the quadrature of curves (integration) and another on the classification of the cubic curves. His Cambridge lectures, delivered from about 1673 to 1683, were published in 1707.