He wrote the book “Arithmetica Infinitorum”

It is a fact that he wrote the book “Arithmetica Infinitorum”. The most important of Wallis' works, Arithmetica Infinitorum, was published in 1656. The methods of analysis of Descartes and Cavalieri were systematized and enlarged in this treatise, however, several notions were susceptible to critique. After a brief introduction to conic sections, he went on to establish the conventional notation for powers, extending it beyond positive integers to rational numbers.

The generation and attributes of continuing fractions are also examined in this study, which was brought to light by Brouncker's use of these fractions. A few years later, in 1659, Wallis published a tract giving the solution to Blaise Pascal's puzzles on the cycloid. In this, he also explained how the principles outlined in his Arithmetica Infinitorum could be applied to the rectification of algebraic curves and provided a solution to the problem of rectifying the semicubical parabola, which was discovered in 1657 by his pupil William Neile. Since all attempts to rectify the ellipse and hyperbola had been ineffective, it was assumed that no curves could be rectified, as Descartes had emphatically stated. Evangelista Torricelli rectified the logarithmic spiral, which was the first curved line (other than the circle) whose length was calculated, but Neile and Wallis' application to an algebraic curve was innovative. Christopher Wren rectified the cycloid as the next curve to be rectified in 1658.