William Brouncker (ca. 1620–1684) was the inaugural President of The Royal Society, and John Wallis (1616–1703) one of its founder members. The two collaborated closely during the 1650s on some original and unusual mathematics, but while Wallis acquired a lasting reputation, Brouncker's work is no longer well known. The two parts of this paper analyse the joint work of Wallis and Brouncker and attempt to separate their respective contributions and very different mathematical styles. The main subject of Part I is Brouncker's discovery of continued fractions. He gave few clues as to how he obtained his results and I offer a possible reconstruction of his work. His quadrature of the hyperbola and rectification of the semicubical parabola are also discussed. Brouncker emerges as a skilled and intuitive mathematician.