LEVIATHAN SQUARES THE CIRCLE
Why had mathematicians failed to square the circle despite repeated efforts over thousands of years? Quite a few mathematicians in Hobbes’s day began to suspect that the reason was that the three classical problems were simply insoluble, but that is a possibility that Hobbes could never entertain. If it was to serve as the keystone of his philosophy, geometry had to be perfectly knowable, and there could therefore be only one answer: the mathematicians were working from flawed assumptions. Once correct assumptions were put in place, true results would grow naturally from them, for “it is in the sciences as it is in plants,” Hobbes wrote: “growth and branching is but the generation of the root continued.”
According to Hobbes, the problem with Euclid—or rather, as he preferred, with Euclid’s followers and interpreters—is that the definitions they use are overly abstract and refer to nothing in the world. The Euclidean definition for a point, for example, is “that which has no parts,” the definition of a line is “breadthless length,” and a surface is “that which has breadth and length only.” But what do those definitions mean? “That which has no parts,” Hobbes argues, “is no Quantity; and if a point be not quantity … it is nothing. And if Euclide had meant it so in his definition … he might have defined it more briefly (but ridiculously) thus, a Point is nothing.” Precisely the same is true for the definitions of a line, a surface, or a solid: they have no referent, and are consequently meaningless.
Only one type of definition would satisfy Hobbes: one based on matter in motion. In fact, as a materialist to the core, Hobbes believed that there was nothing in the world except matter in motion. All the fancy talk of abstractions and immaterial spirits was merely a ploy to gain power over men. Points, lines, and solids, the building blocks of all geometry, must therefore be defined in terms of things that actually exist:
If the magnitude of a body which is moved (although it must always have some) is considered to be none [nulla], the path by which it travels is called a line or one simple dimension, and the space it travels along a length, and the body itself is called a point.
Surfaces and solids are then defined in the same way: a surface by the motion of a line, a solid by the motion of a surface.
The odd thing about Hobbes’s definition is that in his scheme, points, lines, and surfaces are actual bodies, and therefore have magnitude. Points have a size, lines have a width, and surfaces have a thickness. This was heresy for traditional geometers, who from the time of Plato (and likely before) viewed geometrical objects as pure abstractions whose crude physical manifestations were but pale shadows of their true perfection. And if the philosophical issue of the true nature of geometrical objects was not reason enough to reject Hobbes’s approach, there were also the practical questions of how to account for these strange magnitudes in geometrical proofs. What width must one assign to a line, or thickness to a plane? And do traditional proofs hold true for such unorthodox objects? There were no good answers to these questions, and it is not surprising that, to traditional mathematicians, the idea of treating geometrical objects like material bodies with width and breadth seemed like the end of geometry.
Aware of these difficulties, Hobbes argued that although geometrical objects, being bodies, do have positive magnitudes, they are considered in proofs without regard to their dimensions. That is, a point is a body that “is considered” to have a zero size, a line is a path that “is considered” to have length but zero width, and so on, even though, in truth, points have size and lines have width. What exactly Hobbes meant by this argument is far from clear. He seems to be trying to balance his insistence that everything, including geometrical bodies, is made of matter in motion with the traditional demands of Euclidean geometry, which he greatly admired. What is clear is that orthodox geometers were far from convinced.
Conceiving geometrical objects as material bodies was one key component of Hobbes’s geometry. The other was another seemingly physical attribute: motion. Lines, surfaces, and solids were all created by the movement of bodies, and Hobbes’s geometry accounts for this. The most minuscule possible motion “through a space and time less than any given,” he called “conatus”; the speed of the conatus he called “impetus.” To account for how these minuscule motions added up to complete lines and surfaces, he drew on a surprising source: Cavalieri’s indivisibles.
Hobbes, in fact, knew Cavalieri’s work better than almost any mathematician in Europe. He was one of the few who had actually read Cavalieri’s dense tomes, and did not rely on Torricelli’s later adaptation. But how could someone so insistent on the logical clarity of geometry as Hobbes adopt the notoriously murky indivisibles so often attacked for being logically inconsistent and paradoxical? The answer lay in Hobbes’s unconventional interpretation of indivisibles. Cavalieri’s indivisibles, according to Hobbes, were material objects with a positive magnitude: lines were in fact tiny parallelograms, and surfaces were solids with a minuscule thickness, which for the sake of calculation was “considered” zero. As Hobbes’s friend Sorbière explained it, “instead of saying that a line is long, and not broad, [Hobbes] allows of some very little breadth, of no matter of Account, except it be for a very few Occasions.” These points, lines, and surfaces were not fixed, stationary objects in Hobbes’s geometry: like other physical bodies, they could and did move. With a given conatus and impetus, points generated lines, lines generated surfaces, and surfaces generated solids.
Clavius, the champion of classical geometry, would have been appalled at Hobbes’s unconventional geometry. For him, lines with breadth and surfaces with thickness, moving through space with a given impetus, had no place in the pure, immaterial realm of geometrical objects. But Hobbes was not trying to overthrow traditional geometry. To the contrary, he was trying to reform it by founding it on the principles of matter in motion, thereby making it even more rigorous and more powerful. “Every demonstration is flawed,” he argued, if it does not construct figures by the drawing of lines, and “every drawing of a line is a motion.” No one had ever seen a point without size or a line with no width, and it was obvious that such objects did not and could not exist in the world. A true, rigorous, and rational geometry must be a material geometry, and that is what Hobbes created. The new material geometry, Hobbes was convinced, would easily resolve all outstanding problems (such as the quadrature of the circle) that had vexed geometers for millennia. It would be what traditional geometry aspired to be: a perfectly knowable system.
Unfortunately for Hobbes, his effort at squaring the circle did not proceed as smoothly and naturally as a plant growing from its roots. By the early 1650s he was letting it be known among his friends that he had succeeded in squaring the circle, but although he was very proud of his accomplishment, he had no immediate plans for publishing it. He was too preoccupied, it seems, with preparing De corpore for publication. But in 1654, Hobbes received a challenge: Seth Ward, an old acquaintance who was now the Savilian Professor of Astronomy at Oxford, anonymously published a detailed defense of the universities against Hobbes, who had dismissed them in Leviathan as servants of the “Kingdom of Darkness.” Noting that he “hath heard that Mr. Hobbs hath given out that he hath found the solution of some problemes, amounting to no less than the quadrature of the circle,” Ward promised to “fall in with those who speake loudest in his praise” if Hobbes published a true solution.
It was a trap, and Hobbes knew it. Ward, he realized, was trying to provoke him into disclosing his proof, convinced that Hobbes could not possibly have solved a problem that had stood for millennia. Nevertheless, confident that his reformed geometry would succeed where Euclid had failed, Hobbes took the bait. He quickly added a chapter to De corpore that included a proof of the classical problem. By squaring the circle, Hobbes believed he would all at once embarrass his self-satisfied detractor, demonstrate the superiority of his refurbished geometry over the traditional one, and by extension establish the truth of his philosophical system and politic
al program. Despite the risks, it was an opportunity he could not turn down.
But Hobbes’s plan got off to a bad start. After sending the manuscript of De corpore out to a printer, with his quadrature of the circle in chapter 20 included, he had second thoughts. Was his proof really as unassailable as he had thought? He showed it to some trusted friends, and they quickly pointed out his error; he rushed a correction to the printer. He probably would have wanted to remove the proof from the published book entirely, but it was too late for that, so he came up with an ingenious solution. As was customary in books of the period, the top of each chapter included a list of its contents, so Hobbes decided to leave the proof in place but change its description in the list: instead of calling it a quadrature of the circle, he now entitled it “from a false hypothesis, a false quadrature.” This may have saved him from the embarrassment of putting his name to a demonstrably false proof, but it also nullified the value of the demonstration. To compensate, in the same chapter, he added a second proof, but this turned out, on closer inspection, to be a mere approximation. He admitted as much in the title he assigned to it at the top of the chapter, and moved on. A third proof fared no better: although he confidently called it “Quadratura circuli vera” at the top of the chapter (“A True Quadrature of the Circle”), he was eventually forced to add a remarkable disclaimer at the end of the chapter:
Since (after it was written) I have come to think that there are some things that could be objected against this quadrature, it seems better to warn the reader of this than to delay the edition any further … But the reader should take those things that are said to be found exactly of the dimension of the circle and of angles as instead said problematically.
Problematically indeed. In one single chapter of De corpore, despite his confident assertions to his friends and his bravado in taking up Ward’s challenge, Hobbes had failed three times to square the circle. Instead of an incontrovertible proof of the quadrature of the circle, he had produced one “false quadrature,” one approximation, and one proof that should be taken “problematically.” This is hardly the result Hobbes had hoped for when he set out to create a logically irrefutable geometry and, from that, a logically irrefutable philosophy. Instead, he was left with imprecise and questionable results that, rather than establishing a new and peaceful geometric regime, only invited more controversy and speculation.
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 πr2. 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 O
F 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.”
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 26