Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (38 page)

As a student of Holbeach and a graduate of Emmanuel College, Wallis was well integrated into Puritan circles and was a dyed-in-the-wool Parliamentarian from the earliest days of the conflict with Charles I. It was not until 1644, however, that the young clergyman had his chance to play a significant role in the great events of the day, when he was called upon to give testimony at the trial of Archbishop William Laud. As Archbishop of Canterbury, Laud had been the head of the Anglican Church under Charles I, and the chief nemesis of the Puritans. In publicly testifying against him, Wallis showed himself a rising star in the Presbyterian party, which led Parliament’s struggle against the king in the early years of the civil war. Confirmation of his status came that same year, when he was appointed secretary to the Assembly of Divines at Westminster. The assembly was the creation of the Long Parliament, and was charged with coming up with a plan for one or more new churches to replace the established Church of England. Like Parliament itself, the assembly was dominated by Presbyterians, who wanted to abolish bishops and replace them with a hierarchy of church elders (“presbyters”). Wallis, a bright, young, and well-connected minister with impeccable Presbyterian credentials, must have seemed an excellent choice to run the assembly meetings.

Writing decades later, with both the king and the Anglican Church safely restored, Wallis tried to downplay the radicalism of the Assembly of Divines and minimize his role within it. But the facts speak for themselves: after years of intense debates, the Westminster Assembly recommended abolishing the episcopacy (i.e., bishops), the defining feature of the Anglican Church and the emblem of its alliance with the king. The diminishing clout of the Presbyterians and the ultimate restoration of the Church of England ensured that the assembly’s recommendations were never put into practice. In their time, however, the proposals were radical enough, representing a plan to make Church governance more democratic and to remove the Church from royal control. As secretary to the assembly, Wallis was a full partner to this agenda, and was known as such.

Though the Presbyterians were the acknowledged leaders of the Parliamentary faction in the early years of the fight against the king, the increasing radicalism of the revolution soon left them behind. Independents saw little difference between the offices of presbyter and bishop, and wanted to abolish both. Even more ominously, Ranters, Quakers, and Diggers threatened the very foundations of the social order. It was enough to make the respectable Presbyterians wish for the bad old days of king and bishops, a time of law and order that now seemed infinitely preferable to the wholesale chaos they saw around them. And so, the Presbyterian radicals of the 1640s became the conservatives of the 1650s, fighting a rearguard action against the forces of subversion that they had unleashed years before. Wallis’s own career traced a similar arc. In 1648 he signed a remonstrance to the army to protect the king’s life, an honorable act that did nothing to save the king from execution. In 1649 he joined other London ministers in signing
A Serious and Faithful Representation
, protesting Pride’s Purge, the army’s expulsion from Parliament of members it considered too moderate. The purge was a crime, Wallis and his fellows declared, worse than any the king had perpetrated. Besides, they argued, with much revisionist hindsight, the Parliamentarians of 1640 had never dreamed of depriving the king of his rights and regal authority.

Brave gesture though it was, the
Representation
did nothing to save the king or the Presbyterian party. Once the heart and soul of the Parliamentary, the Presbyterians were a spent political force. Considered too conservative by the Independents and radicals who now dominated the commonwealth, they were also distrusted by the Royalists, who blamed them for making war on the king and unleashing the fury that was sweeping across the land. Unable to chart a path of their own, many Presbyterians quietly went over to the Royalist side, in the hope that their past misdeeds would be forgiven once the king was restored to the throne.

The decline in his party’s fortunes left Wallis in an uncomfortable situation in London. He was still minister of St. Martin’s Church in Ironmonger Lane, but generous patrons were harder to come by, now that the political climate had turned against the Presbyterians, and his ability to participate in the great events taking place around him was at an end. Even his personal safety could not be guaranteed at a time when Presbyterians were being accused of treason to the king by Royalists and treason to the cause by Independents and radicals. But it was at this juncture, when there seemed no future for him in London, that Wallis was granted an opportunity to leave it all behind and start afresh: on June 14, 1649, only months after signing the
Serious and Faithful Representation
, he was appointed to the exalted position of Savilian Professor of Geometry at Oxford.

A MINISTER AND A PROFESSOR

To say that Wallis’s appointment was a surprise is a vast understatement. Up until the previous year, the post of Savilian Professor had been held by Peter Turner, an accomplished and respected mathematician. But Turner, like many of his university colleagues, was a Royalist, and when in 1648 Parliament turned its attention to reforming the universities, he was ejected from his professorship, leaving the Savilian chair vacant. For his replacement, the authorities were looking for a scholar with strong Parliamentary credentials, and in that regard Wallis fit the bill. But Wallis was hardly qualified. Unlike Turner, who had been professor of geometry at Gresham College in London before coming to Oxford, Wallis had no track record in either teaching or publishing in mathematics. All he had were the accounting tricks he had learned from his younger brother in his youth, and years of haphazard readings in mathematics that he had pursued on his own and without guidance. The only mathematical work he had to his name was a treatise on angular sections, a topic far removed from the frontiers of the field, and that in any case remained unpublished. This is not the background one would expect of someone appointed to be professor of geometry at Oxford.

Wallis was appointed for political reasons, no more, no less, and it is safe to say that no one expected he would become a serious mathematician. How someone with so little to recommend him secured such a prize is a mystery to us today, but Wallis was nothing if not enterprising. Only a few years removed from his studies in Cambridge, he had managed to thrust himself to the center of national politics by becoming secretary to the Westminster Assembly of Divines. Now, when all avenues of advancement seemed to be closed to him, he bested the feat by winning the most desirable mathematical position in the land. It is possible, as his contemporary the antiquarian Anthony Woods hints, that the good relations he enjoyed with Oliver Cromwell played a role in the appointment. Cromwell knew and respected Martin Holbeach, his sons’ teacher, and may have been impressed by the schoolmaster’s high opinion of Wallis. Still, in 1649, Cromwell was just a general, not lord protector, and may not have had much influence in the matter. What is clear is that Wallis, thanks to a remarkable talent for getting by that shines brightly three and a half centuries later, had managed to extract himself from a difficult situation and land a brilliant prize. In London he was forever associated with the beleaguered Presbyterian faction, but in Oxford he was viewed simply as a Parliamentarian backed by the government that had installed him. And whereas in London he was viewed as a politician, and evaluated accordingly, at Oxford he would be judged as a scholar leading a life of contemplation removed from the tumult of revolutionary London. All he needed now was to become in reality what he already was in name: a mathematician.

This he accomplished with remarkable rapidity. Already in 1647 he had read William Oughtred’s popular algebra textbook,
Clavis mathematicae
(“The Key of Mathematics”), and elaborated on it the following year by writing his own
Treatise on Angular Sections
, which was eventually published nearly forty years later. Now, as Savilian Professor, he recognized that “what had been before a pleasing Diversion was now to be my serious study,” and embarked on a systematic self-education in the most current mathematical work. Wallis the dilettante, only minimally knowledgeable about modern mathematics, nonetheless absorbed the sophisticated mathematical works of Galileo and Torricelli, Descartes and Roberval. Within a few years he had not only mastered the work of his continental colleagues, but had embarked on his own program of mathematical research. In 1655 and 1656, six years after his appointment, he published two mathematical treatises of striking originality, one entitled
De sectionibus conicis
(“On Conic Sections”), the other
Arithmetica infinitorum
(“The Arithmetic of Infinites”). The works resonated far and wide in the European mathematical community, and were read from Italy to France to the Dutch Republic.

How did Wallis, the Puritan minister, manage to reinvent himself as an academic mathematician of international standing? His innate mathematical talent undoubtedly had much to do with it, as did his prodigious capacity for concentrated study and hard work. But there was more: Wallis had maneuvered deftly, relying on a vast web of connections and friends in high places. He had also shown himself flexible in his loyalties and ideological commitments: the doctrinaire Presbyterian of Emmanuel College and the Westminster Assembly of Divines was gone forever, replaced by a moderate clergyman content to profess loyalty to whoever was in power, whether Cromwell, the restored Stuart kings, or (after the Glorious Revolution of 1688) William and Mary. Indeed, when he wrote his autobiography toward the end of his life, he tried to paper over the rebellious politics of his youth by arguing, rather disingenuously, that the term
Presbyterian
referred to respectable clergymen who were opposed to the radical Independents, not to the Anglican bishops. His fluid allegiances and talent for backroom deals served Wallis well in the years following his appointment to the Savilian chair: in 1658 he was elected to the position of “keeper of the archives” (
custos archivorum
) of the University of Oxford, in a dubious process that elicited strong protest from his colleague at Oxford’s Bodleian Library, Henry Stubbe, who was Hobbes’s chief supporter at Oxford. In 1660 he was confirmed in his positions by the restored monarch, Charles II, and was later given the honorary title of “royal chaplain.”

Wallis’s flexibility of conscience was sure to elicit scorn from his famous rival Hobbes, who never wavered from his belief in a dictatorial Leviathan state, remaining obstinately true to his conclusions through thick and thin, with utter disregard for the mounting hostility of his peers. He was denounced as an atheist and a materialist, “a pander to bestiality” whose “doctrines have had so great a share in the debauchery of his generation that a good Christian can hardly hear his name without saying of his prayers.” It was only his powerful patrons and the king’s regard for his old tutor that saved Hobbes when Seth Ward, now the Bishop of Sarum, introduced a motion in Parliament that he be burned at the stake as a heretic. But shunned and abused as he was, Hobbes bore it all with fortitude, never deviating from his views. For the adaptable Wallis, and his talent for reinventing himself according to the prevailing political winds, he had nothing but contempt.

Inflexible and uncompromising, Hobbes’s personal character mirrored his philosophical and mathematical views. In Euclidean geometry he recognized a system that, like him, was rigid and unyielding. Its critics were mere fools and knaves—and that was how Hobbes liked it. The opportunistic Wallis, meanwhile, had little interest in the grand claims of geometry as the embodiment of reason and a model for absolute truth. For him, mathematics was a pragmatic tool for obtaining useful results. He cared little if his proofs did not rise to the lofty level of certainty demanded by Euclidean mathematics. All he wanted were theorems that were sufficiently “true” for the business at hand. And if, in order to arrive at his results, he violated some of the cherished tenets of classical geometry, then those cherished tenets would simply have to give way. Traditional geometers might object to the notion that a plane is composed of an infinite number of lines, on the grounds that it violated well-known and ancient paradoxes. But if such an assumption proved effective in Wallis’s calculations (as indeed it did), then he cared nothing for their objections. If some tweaking of principles was required in order to arrive at the desired end, then Wallis was happy to do it, in mathematics as in life.

For Wallis, Hobbes’s rigidity was pedantic, intemperate, and ultimately self-defeating. He despised it in the man and he rejected it in his philosophy, but more than anything he considered it politically dangerous. A dogmatism that acknowledges only one single truth and denies the legitimacy, and even the possibility, of dissent, Wallis believed, would never bring about the civic peace that Hobbes sought. As Wallis saw it, inflexible dogmatism by the state would breed inflexible dogmatism and even fanaticism among its opponents, which in turn would lead to civil war and social and political chaos—precisely the outcome Hobbes was trying to guard against.

In fact, the foremost concern of both Wallis and Hobbes was the same: preventing a descent into the anarchy and chaos of the Interregnum. Both Wallis and Hobbes feared the Diggers of the world, and were equally intent on preserving the established order. They just differed sharply on the means to do so. Hobbes believed that the only way to preserve order was to establish a totalitarian state with absolutely no room for dissent. Wallis believed that the way forward was to allow for dissent within carefully prescribed limits, which would permit people to disagree and still preserve their common ground.