Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (37 page)

Born in 1616, Wallis was a full generation younger than Hobbes, but his background was otherwise strikingly similar to that of his great antagonist. He, too, was a southerner, from the town of Ashford, in Kent, east of Hobbes’s home village of Westport, in Wiltshire. Wallis’s father, too, was a minister, though seemingly of a more respectable sort. Whereas the elder Hobbes was known more for his gambling than his learning, John Wallis Sr. was, according to his son at least, “a Pious, Prudent, Learned, and Orthodox Divine,” as well as a graduate of Trinity College, Cambridge. As Wallis recalled in an autobiography he wrote when in his eighties, his father was a leader in the community and took his position seriously: “Beside his constant preaching twice on the Lords-day, and other occasional Sermons, and catechizing; he … did maintain a week-day Lecture, on Saturday. Which was much frequented … by very many of the Neighbour Ministers, the Justices of the Peace, and others of the Gentry.”

Sadly for young John, his father died when he was only six years old, a fate that is also reminiscent of Hobbes’s early years. But the Wallis family was apparently better off, and whereas Hobbes was sent to live with his uncle, Wallis’s mother, Joanna, was able to keep her family together and see to the upbringing of her five children. And although she had many opportunities for remarrying well, she remained a widow after her husband’s death “for the good of her Children,” as Wallis put it years later. John was the third of Joanna’s children, but the oldest son, and she took charge of his education with ardor. To make sure he had the benefit of the best instructors, she sent him to school, first in Ashford and later in the nearby town of Tenterden, where he studied English grammar and Latin. Even as a child, he wrote later, he was never content simply to “know” but always sought to understand as well: “For it was always my affectation, in all pieces of Learning or Knowledge, not merely to learn by rote, which is soon forgotten, but inform the grounds or reasons of what I learn.”

In Christmas of 1630, when Wallis was thirteen, he transferred to Martin Holbeach’s school in Felsted, Essex, a move that would do much to shape his future. Holbeach was no mere schoolmaster, but a famous Puritan minister, active in efforts to reform the Church government and often in open conflict with the archbishops and bishops who ruled the Church of England. In later years Holbeach became an ardent supporter of Parliament in its struggle with the king, and ultimately an advocate of independent church government, in opposition to both Anglicans and Presbyterians. His reputation as a godly man and reliable Parliamentarian, as well as an excellent scholar and teacher, was such that leading Puritans from around the country sent their sons to study under him. Among them was Oliver Cromwell, whose four sons were students at the Felsted school.

Wallis did not have the benefit of an illustrious ancestry, but he nevertheless captured the schoolmaster’s attention with his quick mind and studious habits. “
Mr Holbech
was very kind to me,” he recalled in his autobiography, “and used to say I came to him the best grounded of any Scholar that he received from another school.” Under Holbeach’s direction, Wallis improved his Latin and Greek, studied logic, and acquired a smattering of Hebrew, all of which served him well when he went on to university. But the schoolmaster’s true influence went beyond academic instruction. Holbeach, bitter Royalists claimed in later years, “scarce bred any man that was loyall to his Prince,” and Wallis was no exception to this rule. At Felsted he was drawn into the circle of Puritan divines opposed to the Anglican hierarchy, and learned to stand up for the rights of freeborn Englishmen in the face of perceived royal oppression. When civil war broke out a decade later, Wallis remained true to what he learned at Felsted, and unhesitatingly threw in his lot with Parliament.

There was one subject, however, that Wallis did not learn at Felsted, or at any of his other schools. At that time, Wallis explained, mathematics “were scarce looked upon as
Accademical
studies, but rather
Mechanical
; as the business of
Traders
,
Merchants
,
Seamen
,
Carpenters
,
Surveyors of Lands
, or the like.” It simply was not considered part of a young gentleman’s proper education, and therefore was not included in the curriculum of any of his schools. As a result, Wallis’s first encounter with mathematics, much like that of Hobbes, was purely accidental. In December of 1631, Wallis was home in Ashford for the Christmas holiday when he noticed one of his younger brothers engaged in a peculiar activity. The younger boy was apprenticed to a tradesman in town, who was teaching him arithmetic and accounting to assist with the business. Wallis was curious, and the boy, flattered no doubt by his older brother’s attentions, volunteered to teach him what he had learned. The two spent the rest of the holiday together going over the lessons, with the elder Wallis learning the basic tricks of accounting. “This was my first insight into
Mathematicks
,” he noted wistfully years later, “and all the
Teaching
I had.” For all mathematical knowledge beyond these early lessons with a younger brother, Wallis relied exclusively on himself.

Although both Wallis and Hobbes, by their own testimony, discovered mathematics by mere chance, the tales they tell of these moments are very different. Hobbes stumbled upon mathematics in a gentleman’s library, when he was traveling on the Continent with his highborn companion. It was in an aristocratic milieu, one in which mathematics was part of classical learning, studied not for its utility but as one of the refinements of upper-class life. Wallis, in contrast, discovered mathematics in his mother’s crowded and no doubt noisy house packed with brothers and sisters during Christmas break. There was nothing refined about it, but quite the opposite—it was considered fit for his apprentice younger brother, not for an aspiring young gentleman like himself.

Not only the environment, but also the kind of mathematics each man encountered was radically different. In the gentleman’s library, Hobbes discovered majestic Euclidean geometry and was captivated by its cold and rigorous beauty. But the mathematics Wallis found contained no geometry at all, only the arithmetic and rudimentary algebra of bookkeeping. There were no theorems and no proofs in this kind of math, and not a whiff of the grand philosophical claims made on behalf of Euclidean geometry. Hobbes’s geometry was for him an example of universal, unchallengeable truth, whereas Wallis’s mathematics was just a pragmatic tool for tradesmen, mariners, and land surveyors to solve the problems that came their way. There was nothing in it of the imposing edifice that Hobbes so admired.

Both Wallis and Hobbes recounted these stories many years later, and their mature and well-considered views on mathematics may have colored their memories of their earliest experiences with the field. Regardless, there is no denying that the accounts capture something fundamental about the two men’s contrasting approach to the discipline. For Hobbes, mathematics remained a refined, aristocratic science admired for its strict logical inferences. Whether it was useful or not mattered not at all. For Wallis, there was nothing aristocratic about mathematics, and it remained at its core a practical tool for obtaining useful results, just as it was for the tradesman who first taught it to his brother. Whether it was logically rigorous or not was of no consequence.

Wallis was very pleased with what he learned from his younger brother during that fateful Christmas, and also with his own surprising talent for mathematics. From that time onward he continued studying it on his own, though “not as a formal Study, but as a pleasing Diversion at spare hours.” He had no teacher and no guidance, but he practiced his skills as often as he could and read mathematical works that fell into his hands. He never imagined, however, that this leisurely entertainment would end up being his life’s work. His career path lay elsewhere: as a serious and pious young man, eager to make his mark in the world, he was intent on becoming a minister like his father, and preaching the word of God. To do so, he first had to obtain a university degree from Oxford or Cambridge, which, at the time (and for several centuries thereafter), were primarily schools for the training of clergymen.

Wallis entered Cambridge in Christmas of 1632 and was admitted into Emmanuel College. The choice of college was likely not a coincidence. Emmanuel was known as Cambridge’s “Puritan” college, founded specifically for the purpose of training Puritan divines. It was a natural destination for the Felsted-trained Wallis, and his mentor Holbeach may have used his connections among the Puritan clergy to secure Wallis’s admission. Much of the university curriculum consisted of the brand of medieval Aristotelianism known as Scholasticism, and students were judged on their ability to contest or defend the teachings of ancient and medieval philosophers in public disputations. Hobbes had despised Scholasticism in his university days, and later condemned it in
Leviathan
as “Aristotelity,” but Wallis did not find the practice nearly as odious. He quickly mastered the intricacies of syllogisms, he proudly reported in his autobiography, to the point where he was “able to hold pace with those who were some years my Seniors,” and soon acquired a reputation as a “good Disputant.” In this manner he learned not only logic, but also the other fields of the Aristotelian canon, including ethics, physics, and metaphysics. His focus was always on theology, however, and at Emmanuel there was much on offer. Building on what he already knew from his religious upbringing, he began studying systematic academic theology, and soon became proficient in it.

But while adhering closely to the required Aristotelian curriculum, Wallis did not fail to notice that there were also other intellectual winds blowing through the university in the 1630s. The geographical discoveries of the past century revealed entire continents that went unmentioned in the classical sources, a development that seriously undermined confidence in the authority of the traditional canon, and made geography one of the most vibrant fields of the age. Medicine had also moved well beyond the writings of Galen that were studied at the university, thanks to, among others, the anatomical atlas of Vesalius and the recent discovery of the circulation of the blood by the Englishman William Harvey. No field, however, had produced more brilliant discoveries than astronomy. Ever since Copernicus first published his
De revolutionibus
in 1540, his theory that the earth revolved around the sun had steadily gained adherents, evolving from a far-fetched hypothesis into a widely accepted account of the structure of the heavens. The growing appeal of Copernicanism was helped along by Kepler’s accurate calculation of the planets’ true orbits and the relations between them, and by Galileo’s stunning telescopic discoveries, described in
The Starry Messenger
. That Galileo was persecuted by the Roman Church for advocating the Copernican system, in the very years Wallis was attending Cambridge, only enhanced the theory’s popularity in Protestant England. And it was no less an eminence than Sir Francis Bacon, former Lord Chancellor of England, who combined these diverse discoveries into a philosophical system that, he promised, would revolutionize human knowledge and man’s power over nature.

These early stirrings of the scientific revolution were known at the time as the New Philosophy, a term that combined a certain glamour and great promise with a whiff of danger and unorthodoxy. None of it was included in the universities’ ossified curricula, but that does not mean that it did not reach Oxford and Cambridge. The excitement of the new science was in the air, and professors and students alike gathered informally to study it. Wallis was among them, devoting time to the study of astronomy, geography, and medicine, in addition to pursuing his long-standing interest in mathematics. He even went so far as to defend the circulation of the blood in a public disputation—the first student at the university to do so. Those activities, which came in addition to his formal studies, likely gobbled up most of Wallis’s waking hours during his university years. But Wallis possessed an insatiable intellectual curiosity, and a remarkable capacity for hard work. He figured, as he wrote later, that “Knowledge is no Burthen,” and that if it will not, in the end, prove useful, it certainly won’t hurt. It would not be many years before this side interest became the center of his activities, when he joined with others to found one of the first scientific academies in the world, the Royal Society of London.

Wallis earned his bachelor’s degree in 1637 and his master’s in 1640, and would have become a fellow at Emmanuel were it not for the fact that the college already had a fellow from Kent, and the statutes allowed for only one from each county. He instead became a fellow at Queens College, but left soon after, following his marriage. For the next few years he served as minister to a succession of London churches and as personal chaplain to several aristocrats who sided with Parliament in its struggle with the king. One of them was Lady Mary Vere, widow of the soldier and continental campaigner Sir Horatio Vere. One night, while Wallis was seated at table in Lady Vere’s London home, a fellow chaplain brought in a letter written in code that had been intercepted by Parliamentary forces and jokingly asked he if he could decipher it. He took on the challenge, and much to his colleague’s surprise, succeeded after only two hours. This first code was a simple one, but the art of deciphering hardly existed at the time, and the feat gave Wallis a reputation as something of a miracle worker. Later codes he encountered proved more complex, but Wallis enjoyed the challenge and managed to break a good number of them. From then on, under Parliament, the Protectorate, and later under the Restoration, he was regularly employed as code-breaker for the government. Despite requests from correspondents, Wallis never disclosed his techniques, but it is likely they were based on algebra, the cornerstone of his mathematical approach.