Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (36 page)

If this was not bad enough, yet another new enemy stood ready to make the most of Hobbes’s discomfiture and turn it into public humiliation. John Wallis, the Savilian Professor of Geometry at Oxford, was as concerned as his colleague Ward about the dangerous influence of the man they called “the Monster of Malmesbury.” Closely following Hobbes’s plans for
De corpore
, Wallis used his connections to obtain the unpublished sheets of the book from the London printer. It was an underhanded and perhaps immoral tactic, but it proved extremely effective at undermining Hobbes’s mathematical credibility. The sheets gave Wallis a head start on his rebuttal of
De corpore
’s mathematical claims, which he published mere months after Hobbes’s book appeared in April 1655. But Wallis went even further: by comparing the unpublished with the published version of the text, he was able to reconstruct the entire chain of events that had led to Hobbes’s odd and strangely contradictory claims in chapter 20. Hobbes’s confident claims of success repeatedly followed by embarrassing retractions and qualifications were all gleefully exposed in Wallis’s
Elenchus geometriae Hobbianae
. Hobbes’s reputation as a leading mathematician never recovered.

THE HOPELESS QUEST

As it happens, Hobbes’s reform of the foundations of geometry made little difference for the squaring of the circle. Although his friend Sorbière confidently declared that Hobbes’s insistence on the material nature of points and lines finally provided the “Solutions of Problems that have hitherto remained insoluble, such as the squaring of the Circle and the doubling of the Cube,” experience proved otherwise. The attempted proofs of the quadrature in
De corpore
were indeed unorthodox, relying on the motion of points and lines to produce the lines and surfaces, but in the end they were no more conclusive than the efforts of classical geometers. Whether done by traditional Euclidean methods or by Hobbes’s new geometry, the task was hopeless: it is simply impossible to construct a square that has the same area as a given circle using only a straightedge and compass. Hobbes could not accept this, because it would mean that geometry harbored unknowable secrets, but quite a few mathematicians of his generation, including Wallis, suspected that it could not be done. At the very least, the mere fact that geometers had been trying and failing the challenge for nearly two millennia suggested that squaring the circle was not a good use of a geometer’s time. The proof that the quadrature of the circle is impossible, however, had to wait two more centuries, and relied on a kind of mathematics that neither Hobbes nor Wallis could imagine.

To get an idea of why squaring the circle is a hopeless task, consider a circle with radius
r
. As every high schooler today knows, the area of such a circle, in modern notation, is
π
r
2
. Consequently, the side of a square whose area is equal to a circle is
, or, more simply,
. The magnitude of
r
was given in the problem, and we can assume for the sake of convenience that it is 1. All that remains is to construct a line with the length
, and since Euclid shows how to construct a line that is the square root of another, this means constructing a line of length
π
, using only compass and straightedge. And that, as it turns out, is impossible. The reason, as eighteenth-century mathematicians discovered, is that classical geometrical constructions can produce only algebraic magnitudes—that is, magnitudes that are roots of some algebraic equation with rational coefficients. It took another century, but in 1882, the mathematician Ferdinand von Lindemann proved that
π
is not an “algebraic” number, but rather a new kind of number he called “transcendental,” because it is not the root of any algebraic equation. Consequently, a line of length
π
cannot be constructed by compass and straightedge, and the squaring of the circle is impossible.

Figure 7.2. Why the quadrature of the circle is impossible.

All this, however, was centuries away when Hobbes published his quadratures of the circle. He knew nothing of algebraic and transcendental numbers or the limitations of classical constructions, not to mention of Lindemann’s proof, and remained convinced throughout his life that his method was bound to lead to a true quadrature of the circle. His missteps in the first edition of
De corpore
he attributed to overhastiness, and he went on to supply corrected proofs in subsequent editions of the work, as well as in other treatises. Wallis stalked his steps, supplying refutations of each and every proof he offered, and other leading mathematicians joined him. Initially Hobbes reluctantly conceded the mathematical criticisms of his work, which led him to revise his proofs again and again. In time, however, he lost patience with his band of critics: he grew less and less receptive to their arguments, dismissing them as the work of small and envious minds that refused to acknowledge his profound contributions to geometry. Pedantry, prejudice, and pettiness were, for Hobbes, the only possible explanations for the mathematical community’s hostility to his accomplishments. That his path was the true one, he did not doubt.

Hobbes never retreated from his unshakeable conviction that he had squared the circle. A few years before his death at the grand old age of ninety-one, he handed Aubrey a short autobiography with a list of his life’s accomplishments. Among these, mathematics took pride of place. Hobbes took credit for having “corrected some principles of geometry” and for having “solved some most difficult problems, which had been sought in vain by the diligent scrutiny of the greatest geometers since the beginning of geometry.” He then went on to list seven major problems he had solved, including the calculation of centers of gravity and the division of an angle. But there was no doubt of which “accomplishment” Hobbes was proudest: the very first item on the list was the quadrature of the circle.

8

Who Was John Wallis?

THE EDUCATION OF A YOUNG PURITAN

In 1643, while Hobbes was in Paris navigating the political maze of a court in exile and perfecting his philosophical system, a young clergyman in London was also trying his hand at philosophizing. How do we know what we know, and how can we be certain that what we know is true? he asked. The questions may have been similar to those Hobbes was asking at around the same time, but the answers were not. “A
Speculative
knowledge,” the clergyman wrote, in a short booklet he called
Truth Tried
, “is found even in the Devils” in exactly the same measure as it is found in “the Saints on Earth.” This, he explained, is because even devils are rational creatures, and can follow a logical argument as well as the children of God. There is, however, a higher form of knowledge: “
Experimentall
knowledge,” which is of altogether “another nature.” With this kind of knowledge “we do not only
Know
that it is so, but we
Tast
and
See
it to be so.” Unlike beliefs based on speculation, “truths thus cleerly and
sensibly
 … reveiled to the soul, it seems not in the power of the Will to reject.”

The young clergyman was none other than John Wallis, then only twenty-seven years old and but a few years removed from his university days at Cambridge. It is very probable that he had never heard of Thomas Hobbes, the man who would become his obsession in later years but who was then just an obscure Cavendish family retainer with philosophical pretensions. It goes without saying that Hobbes had never heard of Wallis. Nevertheless, years before they launched their bitter war that would last a quarter century, the stark contrast between them was already in evidence. Hobbes maintained that true knowledge began with proper definitions and proceeded by strict, logical reasoning; Wallis believed such knowledge belonged to the Devil as much as to God. Wallis held that the highest form of knowledge is based on the senses—on “seeing” and even “tasting” the truth; Hobbes scorned such sensual knowledge, considering it unreliable and prone to error. Only on one issue did the two seem to be in complete agreement: that mathematics is the science of correct reasoning and certain knowledge, and should serve as a model for all fields.

Hobbes’s interest in mathematics is hardly surprising, as it was at the core of his philosophical and political system. In 1643 he was already a geometer of some repute, and his mathematical star would continue to shine for some time. But Wallis, in the same year, was not a mathematician at all, but rather a rising Presbyterian clergyman deeply engaged in parliamentary efforts to reform the Church. It was not a calling that required deep mathematical knowledge. What is more, his comments in
Truth Tried
do not suggest great admiration for mathematics’ signature brand of reasoning. If the most certain knowledge is experimental knowledge, where does that leave mathematics? Certainly, the views Wallis expressed at age twenty-seven do not seem a promising starting point for a career in mathematics. Yet it would be only a few years before Wallis was appointed to one of the most prestigious mathematical chairs in all Europe, and not long thereafter that he would justify the appointment by proving himself one of the most creative and widely admired mathematicians in the world.

Why would a man of Wallis’s vocation and beliefs devote his life to mathematics? It seems an odd and unlikely career choice, but he had his reasons, and as was the case with Hobbes, these reasons extended to his philosophical and political convictions. As with Hobbes, Wallis’s political attitudes reflected a strong reaction to the chaotic years of the Interregnum, but the conclusions he drew from that time were very different. Whereas for Hobbes the only answer to the crisis was a dictatorial Leviathan state, Wallis believed in a state that would allow for a plurality of views and wide scope for dissent. And whereas Hobbes relied on the rigid edifice of Euclidean geometry to buttress his inflexible Leviathan state, Wallis relied on a novel mathematical approach, as flexible and powerful as it was paradoxical and controversial: the mathematics of the infinitely small.