John Wallis

When the Pinciu lottery dealt me the John Wallis number, I got a winner! John Wallis proved to give me a valuable opportunity to reflect on mathematics both in the broad context of all human conceptual constructs, and as a unifying thread throughout the diverse fabric that has been my life.

Biographical Sketch

John Wallis was born in 1616, the son of a priest of the Church of England, also John Wallis, and of his land-owning wife, Joanna Chapman. Later in his life, the inheritance of a valuable estate from his mother was to give John Wallis fils financial independence.

John Wallis was born during the last period of convulsions that moved England from the medieval to the modern age, and he was an active participant in those convulsions on several fronts. Though in England, the power first of the barons and later of the mercantile class concentrated in London had restricted royal power since the time of Magna Charta, the Stuart kings James I (of England) and Charles I were adherents of the medieval conception of the divine right of kings to rule in their discretion as guided by God. In 17^th century England, the conflict between royal power and Parliamentary power was reinforced by religious divisions everywhere dividing Europe after the collapse of the universal Catholic consensus with the beginning of the Protestant Reformation.

John Wallis’ early studies included logic, but not mathematics, which the best schools of his day did not think warranted a place in the curriculum! To quote from Wallis’ autobiography: For mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical - as the business of traders, merchants, seamen, carpenters, surveyors of lands and the like.[1] (Many of my Algebra II students no doubt wish that was still the case!) It was not until he was 15 that John Wallis first learned the rules of arithmetic from his brother.

At the age of 15, he became a student at Holbeach’s school, where he became proficient in Latin, Greek and Hebrew. He entered Emmanuel College at Cambridge University in December, 1632. Since in mathematics, there was at Cambridge “none to direct [him] what books to read, or what to seek, or in what method to proceed,[2] he took a broad range of courses, including ethics, metaphysics, geography, astronomy, medicine and anatomy. He received his B.A. in 1637 and his M.A. in 1640. In that same year, he was ordained a priest of the Church of England by the bishop of Winchester and became chaplain to Sir Richard Darley in Yorkshire.

In 1642, a chance event that was to transform Wallis’ future took place:

... one evening at supper, a letter in cipher was brought in, relating to the capture of Chichester on 27 December 1642, which Wallis in two hours succeeded in deciphering. The feat made his fortune. He became an adept in the cryptologic art, until then almost unknown, and exercised it on behalf of the parliamentary party.

The important contribution Wallis’ cryptographical efforts made to Parliamentary cause helped secure his future when Cromwell and the forces of Parliament defeated Charles I. They also bring to mind the contributions of Alan Turing to breaking the Nazi codes during the Second World War and hastening the end of that horrific conflict.

In 1644, Wallis became secretary to the clergy at Winchester, and received a fellowship to study Divinity at Queens College Cambridge. However, fellows were not permitted to marry, and he had to resign his fellowship in 1645 when he married. In 1647, Wallis read Oughtred’s Clavis Mathematicae. This led to a rapid intensification of his interest in mathematics and to an outpouring of creative mathematical work. Almost immediately, he wrote a Treatise on Angular Sections, which was however not published for forty years.

In 1649, Wallis was appointed by Cromwell to the Savilian Chair of geometry at Oxford. Wallis held this post for 50 years, until his death in 1703, and must be considered one of the best arguments for tenure of any professor who has ever lived! Wallis opposed the execution of Charles I, and this had the happy result that, on the restoration of the monarchy in 1660, Wallis was confirmed by Charles II in the Savilian Chair. Charles went further, appointing Wallis a royal chaplain, and also naming him to a committee to revise the Book of Common Prayer, the liturgical standard for the Church of England.

Wallis’ non-mathematical works include many religious works, a book on etymology and grammar Grammatica linguae Anglicanae (Oxford, 1653) and a logic book Institutio logicae (Oxford, 1687).

Mathematical Contributions[3]

Wallis was the most influential English mathematician before Newton. He studied the works of Kepler, Cavalieri, Roberval, Torricelli and Descartes, and contributed substantially to the origins of calculus.

Wallis's most famous work was Arithmetica infinitorum which he published in 1656. In this work Wallis established the formula π/2 = (2.2.4.4.6.6.8.8.10..)/(1.3.3.5.5.7.7.9.9...)

In his Tract on Conic Sections (1655) Wallis described the curves that are obtained as cross sections by cutting a cone with a plane as properties of algebraic coordinates.

Wallis developed methods in the style of Descartes analytical treatment and he was the first English mathematician to use these new techniques. This work is also famed for the first use of the symbol ∞ for infinity. He is generally credited as the originator of the idea of the number line where numbers are represented geometrically in a line with the positive numbers increasing to the right and negative numbers to the left.[4]

Wallis was also an important early historian of mathematics and in his Treatise on Algebra he gives a wealth of valuable historical material. However the most important feature of this work, which appeared in 1685, is that it brought to mathematicians the work of Harriot in a clear exposition, presented for the first time by someone who really understood the significance of his contributions.

In his Treatise on Algebra, Wallis accepts negative roots and complex roots.

Wallis made other contributions to the history of mathematics by restoring some ancient Greek texts such as Ptolemy's Harmonics, Aristarchus's On the magnitudes and distances of the sun and moon and Archimedes' Sand-reckoner.

Finally, the contribution that led to Wallis’ inclusion in our textbook is Wallis’ Postulate: Given any triangle ∆ ABC and given any segment DE. There exists a ∆ DEF (having DE as one of its sides) that is similar to ∆ ABC.[5] Wallis proved Euclid’s V Postulate assuming his postulate. However, it turns out that Wallis’ postulate is equivalent to Euclid’s V Postulate.[6]

Reflections on Wallis and Mathematics

While I no longer even aspire to the creativity and productivity that marked the life of John Wallis, I do share with him some of his intellectual history. I double majored in college in mathematics and religion. In college, I studied Latin, Greek, Hebrew, French and German. Like Wallis, I went on to graduate work in religious studies, in my case in the origins of Christianity, in which I have a Ph.D. After a first career as a professor of religious studies, I made a radical change and became an actuary, working for ten years as an applied mathematician. After that, another lurch took me to law school and work first as a tax lawyer and then as a specialist in investment contracts. Now, another change and I am studying Geometry in order to qualify for certification as a teacher of high school mathematics, which I am already during under a Duration Shortage Area Permit.

My own life experience and my reflection on the work of Wallis convince me that gap often alluded to between verbal and mathematical skills is illusory. Mathematics is a collection of languages, belonging certainly to the same family, as Italian and French are both related to Latin, but some of these languages can be encoded and decoded without the need to know others. For example, we can do Neutral Geometry when we might not be able to do complex analysis.

All languages require unproved assumptions about meaning; all languages have vocabulary and rules of grammar, rules as it were of construction. Meaning is constructed by proper application of the rules of grammar to the vocabulary and presuppositions. The construction and expression of meaning is what mathematics has in common with every human language. We who are teachers do our students a grave, disabling disservice, when we fail to stress the essential similarity of mathematics to all learning.

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[1] C J Scriba, The autobiograhy of John Wallis, F.R.S., Notes and Records Roy. Soc. London 25 (1970), 17-46. This paper is substantially extracted from http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Wallis.html . Though I list other links in the reference section at the end of this essay, none of them presents meaningful biographical information not available in this biography.

[3] This section is a lightly edited excerpt from the website referenced in footnote 1.