For most of its history, Hobbes explains in the dedication of De corpore to his Cavendish patron, the Earl of Devonshire, the world had known almost no philosophy. True, the ancients made great advances in geometry, and more recently there had been some important steps forward in natural philosophy, thanks to the work of Copernicus, Galileo, Kepler, and several others. As for the rest of philosophy, from Plato and Aristotle to the present, it was worse than useless. “There walked in old Greece a certain phantasm,” he wrote, “… full within of fraud and filth, a little like philosophy,” which some people mistook for the real thing. Instead of teaching the truth, this pseudo-philosophy taught people to disagree and dispute, and all just so that the supposed “philosophers” would be lavishly compensated. The worst of it was “school divinity,” the medieval Aristotelianism taught at the universities. This, Hobbes charges, was a “pernicious philosophy that hath raised an infinite number of controversies … and from those controversies, wars.” Hobbes calls this abomination “Empusa,” the Greek monster with one leg of bronze, the other of an ass, which was a harbinger of ill fortune.
Hobbes was about to change all that. Natural philosophy, he explains, may be young, dating no further back than Copernicus, but civil philosophy was even younger, according to Hobbes, being “no older than my own book … de Cive.” In that book, for the first time, he had used unchallengeable reasoning to prove that all authority in the state, whether religious or civic, must derive from the sovereign alone. Now, in De corpore, he would complete the job: he would set down the true philosophy that would replace the fake and pernicious ones, and finally conquer the monster Empusa. For Hobbes, his philosophy was not a contribution to an ongoing conversation that had lasted thousands of years: it was, rather, a philosophy to end all philosophies, the one and true doctrine that would put an end to all discussion and debate. His book, he wrote, may be short, but it is nothing short of great, “if men count well as great.” His critics, of whom there were many, might dispute this, but Hobbes cared little for them. They were simply envious of the work, and he, after all, was not “striving to appease,” he noted with perfect candor. The brilliance of De corpore would vanquish them: I will “revenge myself of envy by encreasing it,” he announced, without a hint of irony.
How would Hobbes’s new philosophy conquer Empusa? The answer was simple: through geometry. The reason that the so-called philosophers of the past had failed was that they relied on flawed and inconclusive methods of reasoning. They taught dispute instead of wisdom, Hobbes charges, and they “determine[d] every question according to their own fancies.” As a result, instead of bringing peace and unanimity, they fostered strife and civil war. Geometry, in contrast, compelled agreement: “For who is so stupid as both to make a mistake in geometry and also to persist in it, when another detects his error to him?” Consequently, geometry produces peace rather than discord, and Hobbes’s philosophy would follow its lead. De corpore, he explains in the dedication, was written for “the attentive readers versed in the demonstrations of mathematicians,” and some parts of it were written “to geometricians alone.” But the implications of the geometric method extended to all fields: “Physics, ethics, and politics, if they are well demonstrated, are no less certain than the pronouncements of mathematics.” If one but follows the clear and indisputable method of geometrical reasoning, he will without trouble “fright and drive away this metaphysical Empusa.”
In his own work, Hobbes believed he fully followed the geometrical example: his philosophy (once all its parts were published) begins with simple definitions in De corpore, just as Euclid’s Elements begins with definitions and postulates. And just as The Elements moves from the simple and self-evident to the complex and surprising, so Hobbes’s opus proceeds through its three sections: De corpore (“On Matter”), De homine (“On Man”), and De cive (“On the Citizen”). From a discussion of definitions (which he calls “names”), he proceeds to the nature of space, matter, magnitudes, motion, physics, astronomy, and so on. Finally, at the end of this long chain of reasoning, he reaches the most complex and the most urgent topic of all, the one that justifies the entire enterprise: the theory of the commonwealth. Certainly there were those who disputed whether he had succeeded in living up to the geometrical standard, but Hobbes paid them no mind. His systematic and careful reasoning from first definitions, he was convinced, ensured that his conclusions about the proper organization of the state were absolutely certain. As certain, in fact, as Euclid’s Pythagorean theorem.
THE GEOMETRICAL STATE
If Hobbes’s entire philosophy was structured as a grand geometrical edifice, this was particularly true of his political theory. This was because the commonwealth shared a fundamental feature with geometry: both were created entirely by humans, and therefore were fully and completely known to humans. “Geometry is … demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, because we make the commonwealth ourselves.” Our knowledge of how to create the ideal state is perfect, Hobbes claims, just like our knowledge of geometrical truth. In Leviathan, Hobbes put this principle into practice, creating what he believed was a perfectly logical political theory whose conclusions were in all respects as certain as geometrical theorems.
It wasn’t just the broad principles of the commonwealth that possessed the certainty of geometrical demonstrations. The actual laws established by the Leviathan to govern the state also had the inescapable logical force of a geometrical theorem, and were as indisputably correct. As Hobbes puts it, “the skill of making and maintaining commonwealths consisteth in certain rules, as does arithmetic and geometry.” This is because the laws themselves define what is right and true, and what is wrong and false. Before the commonwealth, in the state of nature, the terms right and wrong or true and false were empty words that referred to nothing. There was no justice or injustice, right or wrong, in the state of nature. But once men gave up their personal will to the great Leviathan, he laid down the law and gave the terms meaning: right is following the law; wrong is breaking it. Anyone charging that the decrees of the sovereign are “wrong” and should be changed is speaking nonsense, since what is “wrong” is defined by the decrees themselves. Opposing a law is as absurd as denying a geometrical definition.
Hobbes, of course, was far from the only early modern intellectual to idealize geometry. Only a few decades earlier, Clavius, too, had extolled the virtues of geometry, promising his fellow Jesuits that it would be a powerful weapon in the struggle against Protestantism. But apart from their admiration of geometry, Clavius and Hobbes had almost nothing in common. Clavius was a Jesuit scholar trained in Aristotelian philosophy and the methods of Scholastic disputation, which were embraced by his order and perfected at the Collegio Romano. He was also a Counter-Reformation warrior who fought to spread the word of God and bring about a Catholic spiritual awakening, and he abhorred Protestants, materialists, and heretics of all kinds. His life’s ambition was to establish the Kingdom of God on Earth, which to him meant setting the Pope above all secular rulers, and the Church above all civic institutions. Hobbes, in contrast, had nothing but scorn for Scholastic disputation, believed that spirit was a meaningless term, and that only matter and motion existed in the world. The sole purpose of terms such as spirit and immortal soul was to allow unscrupulous and corrupt clergymen to frighten men and subject them to their will. Finally, the notion that the Pope would rule over kings was intolerable to Hobbes. Any infringement on the absolute power of the civil sovereign would lead to disagreements, divisions, and, inevitably, civil war.
Clavius died in 1612, long before Hobbes published his tracts and almost certainly without ever having heard the Englishman’s name. But if he had had a chance to read any of Hobbes’s works—De cive, De corpore, Leviathan, De homine—there is no question what his reaction would have been. To a devout Jesuit such as Clavius, Hobbes was a godless materialist and a heretic, an enemy of the Cat
holic Church whose books should be banned. If Hobbes had ever been so unfortunate as to fall into the hands of Clavius and his brethren, he would have been lucky to escape the stake. Meanwhile, Hobbes’s verdict on the Jesuits was no less harsh: their goal, he argued, was to scare men and “fright them from obeying the laws of their country.” This, as far as Hobbes was concerned, was true of all clergymen, but he reserved special scorn for the Catholic Church. The Jesuit dream of a universal and all-powerful Church, ruled by the Pope, was to Hobbes the darkest of nightmares.
Only on the role of geometry were these two natural enemies in perfect agreement. Euclidean geometry, Clavius believed, was a model of correct logical reasoning, which would ensure the triumph of the Roman Church and the establishment of a universal Christian kingdom on earth, with the Pope at its apex. Hobbes’s Leviathan state was, in many ways, the precise opposite of the Jesuits’ Christian kingdom: it was ruled by a civic magistrate who embodied the will of the people, not by the Pope, who derived his authority from God; its laws were derived from the Leviathan’s will, not from divine or scriptural injunctions, and the Leviathan would never tolerate any clerical infringement on his absolute powers. But in their deep structure, the Jesuit papal kingdom and the Hobbesian commonwealth are strikingly similar. Both are hierarchical, absolutist states where the will of the ruler, whether Pope or Leviathan, is the law. Both deny the legitimacy, or even the possibility, of dissent, and each assigns to each person a fixed and unalterable place in the order of the state. Finally, both rely on the same intellectual scaffolding to guarantee their fixed hierarchy and eternal stability: Euclidean geometry.
Today, Euclidean geometry is just one narrowly defined area of mathematics, albeit one with an extraordinarily long and impressive pedigree. Not only is it just one among many mathematical fields, it is also, since the nineteenth century, just one among an infinite number of geometries. It is taught to high school students today partly because of tradition and partly because it is thought to impart the powerful method of rigorous deductive reasoning. Beyond that, it is of little interest to practicing mathematicians. But things were very different in the early modern world, when Euclidean geometry was viewed by many as one of the towering achievements of humanity, the unassailable bastion of reason itself. To Clavius, Hobbes, and their contemporaries, it seemed natural that geometry would have implications far broader than for objects such as triangles and circles. As the science of reason, it should apply to any field in which chaos threatened to eclipse order: religion, politics, and society, all of which were in a state of profound disarray in this period. All that was needed was to use its methods in the afflicted fields, and peace and order would replace chaos and strife.
Euclidean geometry thus came to be associated with a particular form of social and political organization, which both Hobbes and the Jesuits strived for: rigid, unchanging, hierarchical, and encompassing all aspects of life. To us, who can look back on the rise and fall of bloody totalitarian regimes in recent centuries, it is a chilling, repellent vision. But at the dawn of the modern age, with the old medieval world in shambles and nothing to replace it, perspectives were different. To Clavius, to Hobbes, and to many others, it seemed that the answer to uncertainty and chaos was absolute certainty and eternal order. And the key to both, they believed, was geometry.
THE PROBLEM THAT WOULD NOT BE SOLVED
Beautiful and powerful as it was, Euclidean geometry was not free from flaws, as Hobbes discovered to his dismay when he began studying the subject in depth in the years after his encounter with the Pythagorean theorem. The difficulty was that certain classical problems of mathematics, known since ancient times, still defied solution: the squaring of the circle, the trisection of an angle, the doubling of the cube. Despite the efforts of the greatest mathematicians over a span of nearly two millennia, these classical conundrums still defeated every effort to solve them.
This was very bad news for Hobbes’s science of politics. If geometry is fully known, as he declared it must be, then it should have no unsolved, not to mention insoluble, problems. The fact that it does suggests that it possesses dark corners where the light of reason does not shine. And if geometry, which deals with simple points and lines, is not fully understood, how can one expect the theory of the commonwealth, which deals with the thoughts and passions of men, to be perfectly known? If geometry has its blind spots, then the science of politics may well have some, too, and they are likely to be far greater and more significant than those of geometry. For Hobbes, as long as the classic geometrical problems remained unsolved, his entire philosophical edifice remained insecure, and the Leviathan state a political house built on sand.
In order to secure the foundations of his political theory, Hobbes set out to solve the three classical outstanding problems of geometry. Initially he seems to have believed that this should not be too difficult. Surely, he thought, just as he had corrected the errors of all past philosophers, he could also correct the errors of all past geometers. And he can perhaps be excused for his unwarranted optimism, because part of the reason the problems had attracted the attention of the greatest mathematicians over centuries was that they were easily stated, and appeared deceptively simple. “The quadrature of the circle” means constructing a square equal in area to a given circle; “the trisection of the angle” means dividing any given angle into three equal parts; and “the doubling of the cube” means constructing a cube of double the volume of a given cube. How hard could it be to solve such questions? As it turns out, very hard. In fact, impossible.
Figure 7.1. The Quadrature of the Circle 1. The Case of the Inscribed Polygon.
To understand why this is so, consider the problem that most interested Hobbes and to which he devoted an entire chapter in De corpore: the quadrature of the circle. Already in the second century BCE, the polymath Archimedes of Syracuse proved that the area enclosed in a circle is equal to that of a right triangle whose two perpendicular legs are equal to the radius and to the circumference of a circle—that is, PQ and QR in Figure 7.1. Archimedes proved this result by looking at polygons inscribing and circumscribing a circle. The greater the number of sides of a polygon, the closer its enclosed area is to that of the circle. Now, Archimedes reasoned, let us consider the circumscribed octagon AHDGCFBE. Its area is equal to that of a right triangle whose two perpendicular sides are equal to the apothem and the sum of the octagon’s sides (the “apothem” being a vertical line from the center of the polygon to its side). This is obvious if we think of the area of the octagon to be the sum of eight triangles with bases on the sides of the octagon and the apex at the center. The area of each triangle is half its base times the apothem, and consequently the area of the entire octagon is half all the bases together times the apothem—that is, the area of the right triangle in question.
Let us look, Archimedes continued, at the area of the circle, which we call C, and the area of the right triangle whose perpendicular sides are its radius and circumference, which we call T, and the area of an inscribed polygon with n sides, which we call In. Let us assume for the moment that the circle is greater than the triangle—that is, C > T. Archimedes had already shown that as we increase the number of sides of the inscribed polygon, it approaches the area of the circle as closely as we please. Consequently, there is a number of sides n for which the area of the polygon falls between the area of the circle and the triangle that is greater than the triangle but (being inscribed) still smaller than the circle. In modern notation: In > T.
This, however, is impossible, because the area of In 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 In < 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 circum
scribing 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.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Page 25