Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (35 page)

This, however, is impossible, because the area of
I
n
is equal to a right triangle whose perpendicular sides are its apothem and the sum of its sides. The apothem is less than the radius of the circle, and the sides are less than the circle’s circumference. This means that
I
n
<
T
, contradicting the original assumption, and the triangle cannot be smaller than the area of the circle. Archimedes then assumed that the triangle is larger than the area of the circle, and repeated the argument, looking this time at the circle’s circumscribing polygon. Once again he showed that as we increased the number of the polygon’s sides, it approached the area of the circle as closely as one wished. But since it always remained greater than the triangle
T
, the area of the triangle could not be greater than the circle, contradicting the assumption. Consequently, the only remaining possibility is that the triangle is equal to the circle, or
C
=
T. QED.

Archimedes’s demonstration is an example of the classical “method of exhaustion,” and it is a perfectly rigorous Euclidean proof. From this it might appear that the quadrature of the circle has been accomplished: the triangle is equal in area to the circle, and it is a simple matter to construct a square whose area is equal to a triangle. Problem solved? Not by the standards of the classical geometers. Archimedes did indeed demonstrate that the area of a circle is equal to the area of a particular triangle, but he did not “construct” the triangle with compass and straightedge, the only tools allowed in Euclidean constructions. For a quadrature to be acceptable to classical geometers, one would have to begin with a given circle and, through a finite series of steps, using only compass and straightedge, produce the required triangle. Archimedes didn’t do this: he proved that the area of the circle is equal to that of the triangle, but he did not show how to construct a triangle with such measurements from a circle. Hence his proof, elegant though it was, and correct though it was, was not a quadrature.

To us, these rigid standards of classical geometry might seem rather fussy, if not pointless. Modern mathematicians do not limit themselves to constructive proofs, not to mention constructive proofs by straightedge and compass alone. Indeed, Archimedes’s proof is more than satisfactory for anyone who wants to find out the area enclosed in a circle. But Hobbes thought differently. To him, the fact that geometry was built up step by step from its simplest components to ever-more-complex results was what made it an appropriate model for philosophy and for the science of politics. To be worthy of this status, it is essential that geometry itself not stray from this model, and that it always construct its objects by proceeding systematically from the simple to the complex, using the most rudimentary tools. To Hobbes, the classical standards were not arbitrary impositions, but the very heart of what geometry is and should be. To resolve the quadrature of the circle, it is therefore necessary to construct a square with the area of a circle, as the ancient geometers required, and as Archimedes failed to do.

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 political 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.