John Wallis

English mathematician

• Born: December 23, 1616
• Birthplace: Ashford, Kent, England
• Died: November 8, 1703
• Place of death: Oxford, England

Wallis made advances in mathematical notation and created new methods for making mathematical discoveries. He paved the way for the work of Sir Isaac Newton and consequently for the invention of the calculus.

Early Life

John Wallis (WAHL-uhs) was born in the village of Ashford, Kent, the son of the village rector, John Wallis, senior, and his second wife, Joanna Wallis, née Chapman. Wallis’s father died in 1622. He was left with sufficient resources to obtain a good elementary education and enrolled at an Essex school run by Martin Holbech in 1630. There, he was told by the master that he was the best prepared student in mathematics he had encountered. Wallis was, in addition to his mathematical learning, something of a calculating prodigy. From Holbech’s school, he proceeded to Emanuel College, Cambridge, where he studied medicine and took courses in physics as well as moral philosophy.

After receiving a bachelor’s degree in 1637 and a master’s degree in 1640, Wallis became a fellow at Queen’s College. As was typical for the period, such fellowships were limited to unmarried scholars, so when he married Susanna Glyde in 1645, he was deprived of his fellowship. The marriage lasted forty-two years, until his wife’s death in 1687. They had one son and two daughters. In the early years of his marriage, Wallis was able to make a living as a private chaplain in Yorkshire and Essex, where he had connections.

The English Civil War soon intervened in Wallis’s life, however. Wallis was active on behalf of the Parliamentary side, and, in particular, he put his code-breaking talents to use to help with deciphering intercepted Royalist messages. When the Parliamentary forces triumphed, this put Wallis in an enviable position, and he became Savilian Professor of Geometry at Oxford, thanks to the previous occupant’s having had the bad fortune to support the Royalist cause. Since Wallis’s mathematical skills at the time (1649) had scarcely even been tested, the political nature of the appointment is evident.

There were many political appointments made at the time that were reversed with the Restoration of the monarchy in 1660. Wallis, however, was not evicted from his chair under Charles II, and the explanation for his retention is usually found in Wallis’s having signed the remonstrance against the execution of Charles I. Charles II, Charles I’s son, remembered those who had not acquiesced in the death of his father. It was also true that by the time of the Restoration, Wallis had made a name for himself in mathematics worthy of the position he held.

Life’s Work

The first work of Wallis to achieve notice was his Arithmetica Infinitorum (1655; the arithmetic of infinitesimals). In this book, he sought to combine the ideas of the Italian mathematician Bonaventura Cavalieri with the style of argument made famous by the French mathematician and philosopher René Descartes . Cavalieri had tried to break down geometrical objects in a given dimension into slices of one dimension lower. Thus, Cavalieri’s principle argued that two solids would have the same volume if the areas of all the corresponding cross-sections were equal. Descartes’s use of analytical methods helped to make the ideas of Cavalieri more palatable.

In 1659, Wallis published a treatise on conic sections, a subject familiar since the days of the Greeks. What was distinctive about Wallis’s approach, though, was that he provided an alternative to the traditional approach of deriving the properties of the conic sections (like the circle and the ellipse) from their geometric origin. Wallis, instead, showed that certain quadratic equations would generate the points that made up the conic section and that the properties of the curves could be derived from the equations without having to go back to cones.

One general subject upon which Wallis was employed was trying to find the areas under certain curves. In order to go beyond the work done previously in the area by mathematicians like Pierre de Fermat , he worked out some of the values for the areas for curves that were already known and then showed how other curves could be obtained as the complements of those curves. In general, Wallis made progress by the use of two assumptions, induction and interpolation. Wallis would establish a result for the first few positive whole number values of the variable and would then claim that it must hold for the rest of the whole numbers. Having done that, he would then claim that some principle of continuity would also guarantee that the result would hold for the rest of the real numbers as well.

Wallis’s, then, was scarcely a rigorous procedure, but it had a great influence on Sir Isaac Newton, whose early calculations of areas were quite similar to those of Wallis. If Newton made any progress beyond Wallis in these early years, it was by virtue of recognizing the advantages of taking variable limits for the regions whose area he was trying to find. This enabled him to recognize patterns where Wallis had only obtained numerical values.

One of Wallis’s most celebrated results was an infinite product for the number 4/π. Again, this was not the first arithmetic expression that involved π, but Wallis’s approach served as the basis for further work in the subject. In addition to his work on areas, Wallis also looked at the volumes of three-dimensional solids of rotation. He was the first to use the symbol ∞ for infinity and provided the notation subsequently used for logarithms as well.

Typical of Wallis’s later years was his volume on algebra published in 1685. It included the use of notation to represent complex numbers well before what became the standard representation was devised. In addition, the book also included a good deal of questionable history, in which Wallis took potshots at the work of Descartes and held up the work of Thomas Harriot, an English mathematician, as superior. Subsequent historians have seen Wallis as going well beyond the evidence both in his praise of Harriot and in his disregard for Descartes. On the other hand, it was indicative of Wallis’s nationalism and his willingness to ignore generally accepted valuations.

Among Wallis’s most notorious quarrels were those with Thomas Hobbes and the defenders of the German mathematician Gottfried Wilhelm Leibniz as an inventor of the calculus . Hobbes was nowhere near the mathematical level that Wallis achieved, but he was a careful reader and pointed out some of the shortcomings in Wallis’s attempts to explain the foundations of calculus, shortcomings he shared with the other mathematicians of the time. Wallis kept the argument with Hobbes going for many years, as he found repugnant Hobbes’s attempt to found his philosophical materialism on mathematics. With regard to the creation of the calculus, Wallis was determined to keep Newton and Leibniz at odds for the sake of the glory of English mathematics. He also opposed the Gregorian calendar.

After the Glorious Revolution of 1688, Wallis was once more confirmed in his academic positions, possibly because of his continued willingness to put his code-breaking talents to use for the state. During much of his later years, he devoted time to the preparation of editions of other authors’ works, as well as his own collected works. While his quarrels with contemporaries were many, it is worth mentioning that Samuel Pepys , the diarist, ordered a portrait of Wallis painted by the Sir Godfrey Kneller. Since Pepys did not get along with everyone, this is something of a contrast to Wallis’s feuds.

Significance

Wallis was arguably the most important precursor of Newton in England. His willingness to go out on a limb in pursuit of a mathematical discovery served as a model for Newton’s efforts and led to the latter’s discovery of the proof for the binomial theorem for general exponents. Wallis was one of the first to make a serious effort to provide an arithmetic form for geometry, especially for the heavily geometric parts of Euclid’s Stoicheia (c. 300 b.c.e.; The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara, 1570, commonly known as the Elements), such as Books II and V. His innovations in notation helped to make mathematics easier to communicate, even across international boundaries.

Bibliography

Fauvel, John, et al., eds. Oxford Figures: 800 Years of the Mathematical Sciences. New York: Oxford University Press, 2000. The sixth chapter is devoted to Wallis and traces his contributions to the teaching of mathematics at Oxford and the growth of the university’s mathematical reputation.

Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat, 1601-1665. Princeton, N.J.: Princeton University Press, 1973. The last chapter of the book takes up Fermat’s arguments and relations with British mathematicians toward the end of his life and points to Wallis’s disappointing reaction to questions proposed by Fermat.

Merton, Robert K. On the Shoulders of Giants: A Shandean Postscript. Chicago: University of Chicago Press, 1993. Looks at the impression Wallis created among his contemporaries of being greedy for glory and taking credit for the work of others.

Pears, Iain. An Instance of the Fingerpost. London: Jonathan Cape, 1997. This historical novel is a mystery in four parts, the third of which is narrated by Wallis, who has the reader’s sympathy by the end of the narrative.

Scott, J. F. The Mathematical Work of John Wallis, D.D., F.R.S., 1616-1703. 1938. Reprint. New York: Chelsea, 1981. Seeks to restore Wallis’s reputation among historians of science. While defending Wallis’s mathematics, much of his history is treated as falsification.

Scriba, Christoph J. “John Wallis.” In Dictionary of Scientific Biography, edited by Charles C. Gillispie. Vol. 14. New York: Charles Scribner’s Sons, 1971. Much briefer than Scott’s survey of Wallis’s mathematics but clearer to the modern reader.

Stillwell, John. Mathematics and Its History. New York: Springer-Verlag, 1989. Looks at Wallis primarily as a precursor to Newton.

Westfall, Richard S. Never at Rest: A Biography of Isaac Newton. New York: Cambridge University Press, 1980. The best explanation of how Newton imitated Wallis’s manipulations.