Wallis was one of the most original mathematicians of the seventeenth century and he left his mark on mathematics in many ways. He introduced arithmetical limits into mathematics (his famous infinite-product expression for $\pi$ is an example). His researches (for example, the means whereby he obtained the aforementioned product) led directly to Newton's work on the binomial theorem and quadratures. He was the first to see the significance of fractional and negative exponents, and he is responsible for the introduction of such symbols as $\infty$ and such terms as hyper geometric series. He was very influential politically, very quarrelsome, and at the center of the scientific life of his time (for instance, it was owing to his advice that the Gregorian calendar was not introduced earlier into England, as he did not like the Pope). This second edition includes Foreword by E. N. da C. Andrade, a Bibliography and an Index.