Prior to Isaac Newton, John Wallis (1616-1703) is unquestionably the most important English mathematician. He contributed greatly to many of the underlying concepts that would go on to make up calculus and is undoubtedly one of the men Newton was referring to when he stated that he was merely "standing on the shoulders of giants." He was also an ordained priest and a cryptographer during the English civil war during the 17th century.
John Wallis was born at Ashford on November 22, 1616. He was educated at Felstead school, and while on holiday at the age of fifteen, he discovered a book of mathematics that his brother owned. Curious at the signs and symbols he found in the book, he asked to borrow it, and with a bit of assistance, mastered the book in a fortnight.
His family, however, wanted him to be a doctor, so he went to Cambridge for a medical education. However, his interest in mathematics continued to grow to the point that no one at Cambridge could direct his mathematical studies, and his interest in medicine was minimal. As a result, he began to study divinity, and was ordained in 1640.
Wallis was clearly skilled in cryptography and mathematics, however, so he was employed to decode Royalist messages for the Parliamentarians during the civil war in England in the 1640s. Perhaps due to his service, he was appointed the Savilian Chair of geometry at Oxford in 1649 and held onto it for more than fifty years until his death. While the exact reason for his appointment was rather unclear, it is unquestioned as to whether he deserved to retain the chair.
During the 1650s, Wallis became part of a group interested in natural and experimental science who started to meet regularly in London. This group was to become the Royal Society, so Wallis is a founder member of the Royal Society and one of its first Fellows.
His most profound impact, however, was in his mathematical work. He wrote many, many papers, a great number of which helped form the underlying ideas behind the development of calculus, which was just around the corner. His most famous of works include the introduction of the use of infinite series as an ordinary part of mathematical analysis. His papers also were reknowned for the fact that they revealed and explained in very clear language the principles of the new methods of analysis introduced not only by him but by his contemporaries and immediate predecessors. In fact, it was this writing style that helped Newton greatly in his development of the calculus.
His most influential work is probably 1656's Arithmetica infinitorum, in which he evaluated the integral of (1 - x2)n from 0 to 1 for integral values of n. His procedure truly laid the groundwork for more general techniques of the evaluation of integrals, borrowing from Johannes Kepler. Other significant works of his include 1656's Tract on Conic Sections, in which he developed the concept of parabola and hyperbola as sections of a cone; 1685's Treatise on Algebra, which provides an early history of mathematics and describes the use of complex roots and negative roots in algebra, a new concept at the time; and his restoration of several classic Greek texts dealing with mathematics, including Ptolemy's Harmonics, Aristarchus's On the magnitudes and distances of the sun and moon, and Archimedes' Sand-reckoner. He also introduced the symbol for infinity, ∞, which is still used today, as well as the development of an infinite product formula for pi.
In addition to his mathematical writings, he also wrote on theology, logic, and philosophy and was the first to devise a system for teaching deaf-mutes. He was also known for a long-standing public feud with fellow English mathematician Thomas Hobbes; the two of them regularly published papers and pamphlets solely to discredit one another.
John Wallis died on October 28, 1703, in Oxford, England. He left behind a legacy of the study of infinity, conic sections, and much more, which together helped to define the underlying rules of calculus. His diverse writings provide a solid glimpse of a mind at work and following many avenues.