And this brings us to the much misunderstood problem of squaring the circle, i.e., constructing a square whose area equals that of a given circle. There never was an Olympic Committee that laid down the rules of the game and the conditions under which the circle was to be squared. Yet from the way Euclid’s cathedral is built, we know exactly what the Greeks had in mind. The problem was this:
(1) Square the circle,
(2) using straightedge and compasses only,
(3) in a finite number of steps.
It is evident why Hippias’ and Dinostratus’ method of squaring the circle was not accepted as “clean” by the Greeks, even though Dinostratus proved that it did square the circle. Not because it violated the second condition; it did not. Nor (as some write) because the quadratrix was a curve other than a straight line or a circle: Euclid himself wrote a book on conic sections (lost in the Roman catastrophe), and these were generally not circles, either. But referring to the figure here, it is easily seen that the point Q can only be found by placing a French curve along the points L, which violates rule 2, or constructing infinitely many points L by straightedge and compasses. Even assuming that the Greeks would have intuitively accepted the equivalent of the modern concept of continuity (and they accepted nothing but a handful of postulates), they meticulously steered clear of anything involving the infinite, for they were still puzzled by Zeno’s paradoxes (about 500 B.C.), the best known of which is the one of Achilles and the tortoise: Achilles races the tortoise, giving it a headstart of 10 feet. While Achilles covers these 10 feet, the tortoise covers 1 foot; Achilles covers the remaining foot, but the elusive tortoise has covered another 1/10 of a foot, and so on; Achilles can never catch up with the tortoise. The stumbling block here was that the Greeks could not conceive of an infinite sum adding up to a finite number. In our own age, school children absorb the fact that this is so at the tender age when they first learn about decimal fractions:
As for the condition that only compasses and straightedge may be used in the construction, it has puzzled many, including some who ought to know better. It has been attributed to Greek esthetics and to the influence of Plato and Aristotle (from whom the Alexandrian scientists were mercifully removed by an ocean far larger than the Mediterranean); and other reasons have been suggested to explain the “Greek obsession with straightedge and compasses.”
Nonsense. The only things the Greek mathematicians were obsessed with were truth and logic. Straightedge and compasses came into the problem only incidentally and as a secondary consequence, and esthetics had nothing to do with it at all. Just as Euclid built his cathedral on five foundation stones whose simplicity made them obvious, so one can go in the opposite direction and prove a statement by demonstrating that it was entirely supported by the stones of Euclid’s building; from hereon downward it had already been proved that the building stones lie as firmly anchored as the foundations. Proving the correctness of an assertion, therefore, amounted to reducing it to the “obvious” validity of the foundation stones.
Euclid’s five foundation stones, or “obvious” axioms, were the following:
I. A straight line may be drawn from any point to any other point.
II. A finite straight line may be extended continuously in a straight line.
III. A circle may be described with any center and any radius.
IV. All right angles are equal to one another.
The fifth postulate is unpleasantly complicated, but the general idea is also conveyed by the following formulation (not used by Euclid, whose axioms do not contain the concept parallel):
V. Given a line and a point not on that line, there is not more than one line which can be drawn through the point parallel to the original line.
To prove a statement (such as that a construction attains a given object) meant the following to the Greeks: Reduce the statement to one or more of the axioms above, i.e., make the statement as obvious as these axioms are.
It is easily seen that Euclid’s axioms are visualizations of the most elementary geometrical constructions, and that they are accomplished by straightedge and compasses. (This is where the straightedge and and compasses come in as an entirely subordinate consideration.) It is evident that if a construction uses more than straightedge and compasses (e.g., a French curve), it can never be reduced to the five Euclidean axioms, that is, it could never be proved in the eyes of the ancient Greeks.23 That, and nothing else, was the reason why the Greeks insisted on straightedge and compasses.
Let us illustrate this by a specific example, the squaring of a rectangle. We are required to construct a square whose area is equal to the area of a given rectangle ABCD.
How to square a rectangle.
Building stone no. III.35.
Construction (see Figure a above): Extend AB, and make BE = BC. Bisect AE to obtain O, and with O as center, describe a circle with radius OA. Extend CB to intersect the circle at F. Then BF is the side of the required square.
Proof: Let G be the intersection of BC extended and the circle AFE.
Then by Euclid III.35 (Book III, Theorem 35, to be quoted below),
AB × BE = FB × BG.
But by Euclid III.3,
FB = BG
and by construction,
BE = BC
Hence,
AB × BC = FB2 ,
as was to be demonstrated.
The two theorems referred to above are
III.35. If two chords intersect within a circle, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.
III.3. If the diameter of a circle bisects a chord, it is perpendicular to it; or, if perpendicular to it, it bisects it.
This would usually be all that is required for the proof of the construction, for both theorems are proved by Euclid. However, let us trace the path back to the axioms. Our proof rests, among others, on building stone no. III.35, which says (figure b here) that
AB × BE = CB × BF.
This rests, among others, on the theorem that similar triangles (CAB and BFE) have proportional sides. To show that these triangles are similar, we have to show, among other things, that the angles CAB and BFE are equal. This theorem, in turn, rests on the theorem that the angle at the circumference equals half the angle at the center, subtended by the same chord. This, in turn, rests on the theorem that the angles at the base of an isosceles triangle are equal. This again rests on the theorem that if one triangle has two sides and the included angle equal to two sides and the included angle of another, then the two triangles are congruent. And the congruency theorems finally rest almost immediately on the Axioms.
We have traced but a single path from stone no. III.35 down to the foundation stones of Euclid’s cathedral. This path has many other branches (branching off where it says “among others” above), and if we had the patience, these could be traced down to the foundation stones, too. So could the paths from stone no. III.3 and from the operations used in the construction (whose feasibility need not be taken for granted).
Thus, the proof that the construction for squaring a rectangle as shown in the figure here is correct, is essentially a reduction to Euclid’s axioms.
No such reduction is possible with Hippias’ construction for squaring the circle. For the reasons given above, it cannot find a resting place in Euclid’s cathedral; the paths of proof (such as the derivation given here) will not lead to Euclid’s foundation stones, and Hippias’ stone will tumble to the ground.
This does not mean that Hippias’ construction is incorrect. Hippias’ stone can find rest in a cathedral built on different foundations. What Lindemann proved in 1882 was not that the squaring of the circle (or its rectification, or a geometrical construction of the number π) was impossible; what he proved (in effect) was that it could not be reduced to the five Euclidean axioms.
* * *
LET us now take a brief look at the later fate of these five foundation stones, the five Euclidean axioms or p
ostulates. There were a few flaws in them which were later mended, but we have no time for hairsplitting here. We will take a look at a more significant aspect.
The attitude of the ancient Greeks to Euclidean geometry was essentially this: “The truth of these five axioms is obvious; therefore everything that follows from them is valid also.”
The attitude of modern mathematics is somewhat different: “If we assume that these axioms are valid, then everything that follows from them is valid also.”
What does “straight” mean?
At first sight the difference between the two seems to be a chicken-hearted technicality. But in reality it goes much deeper. In the 19th century it was discovered that if the fifth postulate (here) was pulled out from under Euclid’s cathedral, not all of the building would collapse; a part of the structure (called absolute geometry) would remain supported by the other four axioms. It was also found that if the fifth axiom was replaced by its exact opposite, namely, that it is possible to draw more than one straight line through a point parallel to a given straight line, then on this strange fifth foundation stone (together with the preceding four) one could build all kinds of weird and wonderful cathedrals. Riemann, Lobachevsky, Bolyai and others built just such crazy cathedrals; they are known as non-Euclidean geometry.
The non-Euclidean axiom may sound ridiculous. But an axiom is unprovable; if we could prove it, it would not be an axiom, for it could be based on a more primitive (unprovable) axiom. We just assume its validity or we don’t; all we ask of an axiom is that it does not lead to contradictory consequences. And non-Euclidean geometry is just as free of contradictions as Euclidean is. One is no more “true” than the other. The fact that we cannot draw those parallel lines in the usual way proves nothing.
Nevertheless, some readers may feel that all this is pure mathematical abstraction with no relation to reality. Not quite. Reality is what is confirmed by our experience. It is our experience that when we join two stakes A and B by a rope, there is a certain configuration of the rope for which the length of rope between A and B is minimum. We find that configuration of the rope by stretching it, and we call it a straight line. Take a third stake C and stretch a rope from A to B via C. Our experience tells us that this rope is longer than the rope stretched between A and B directly. This experience is in agreement with the Euclidean theorem that the sum of two sides of a triangle cannot be less than the third side. Euclidean geometry is convenient for describing this kind of experience; which is not the same thing as saying it is “universally true.”
Three who chose to reject Euclid’s fifth postulate: Nikolai Ivanovich LOBACHEVSKI (1792-1856), Georg Friedrich Bernhard RIEMANN (1826-1866), and Albert EINSTEIN (1879-1955).
For there are other experiences for whose description Euclidean geometry is extremely inconvenient. Suppose point A is on this page of the book and point B is on some star in a distant galaxy; then what does “straight” mean? In that case we have no experience with ropes, but we do have experience with light rays. And this experience shows that light rays traveling through gravitational fields do not behave like ropes stretched between stakes. Their behavior is described by Einstein’s General Theory of Relativity, which works with non-Euclidean geometry. This is more convenient in describing the laws that govern our experience. If we were to express these laws in Euclidean space, they would assume very complicated forms, or alternatively, we would have to revamp all of our electromagnetic theory from scratch (without guarantee of success), and this is not considered worth while (by the few physicists who have given this alternative any thought).
And so the chicken-hearted technicality of saying “if” is neither chicken-hearted nor a technicality.
5
THE ROMAN PEST
Ave Caesar, morituri te salutant!
(Hail Caesar, we who are about to die salute you)
Führer befiehl, wir folgen!
(Führer, command us, we shall follow)
WHILE the quest for knowledge was storming ahead at the University of Alexandria, the ominous clouds of the coming Roman Empire were already gathering. Whilst Alexandria had become the world capital of thinkers, Rome was rapidly becoming the capital of thugs.
Rome was not the first state of organized gangsterdom, nor was it the last; but it was the only one that managed to bamboozle posterity into an almost universal admiration. Few rational men admire the Huns, the Nazis or the Soviets; but for centuries, schoolboys have been expected to read Julius Caesar’s militaristic drivel (“We inflicted heavy losses upon the enemy, our own casualties being very light”) and Cato’s revolting incitements to war. They have been led to believe that the Romans had attained an advanced level in the sciences, the arts, law, architecture, engineering and everything else.
It is my opinion that the alleged Roman achievements are largely a myth; and I feel it is time for this myth to be debunked a little. What the Romans excelled in was bullying, bludgeoning, butchering and blood baths. Like the Soviet Empire, the Roman Empire enslaved peoples whose cultural level was far above their own. They not only ruthlessly vandalized their countries, but they also looted them, stealing their art treasures, abducting their scientists and copying their technical know-how, which the Romans’ barren society was rarely able to improve on. No wonder, then, that Rome was filled with great works of art. But the light of culture which Rome is supposed to have emanated was a borrowed light: borrowed from the Greeks and the other peoples that the Roman militarists had enslaved.
There is, of course, Roman Law. They scored some points here, a layman must assume. Yet the ethical substance of our law comes from Jewish Law, the Old Testament; as for the ramifications, the law in English speaking countries is based on the Common Law of the Anglo-Normans. Trial by jury, for example, was an Englishman’s safeguard against tyranny, an institution for which he was, and perhaps still is, envied by the people of continental Europe, whose legal codes are based on Roman Law. Even today, this provides for trial by jury only in important criminal cases. Roman Law never had such vigorous safeguards against tyranny as, for example, the Athenian constitution had in the device of ostracism (in a meeting in which not less than 6,000 votes were cast, the man with the highest number of votes was exiled from Athens for 10 years). So I would suspect that what the Romans mainly supplied to our modern lawyers in abundant quantities are the phrases with which they impress their clients and themselves: Praesumptio innocentiae sounds so much more distinguished than “innocent until proven guilty,” and a maxim like Ubi non accusator, ibi non judex shows profound learning and real style. Freely translated, it means “where there is no patrol car, there is no speed limit.”
Then there is Roman engineering: the Roman roads, aequeducts, the Colosseum. Warfare, alas, has always been beneficial to engineering. Yet there are unmistakable trends in the engineering of the gangster states. In a healthy society, engineering design gets smarter and smarter; in gangster states, it gets bigger and bigger. In World War II, the democracies produced radar and split the atom; German basic research was far behind in these fields and devoted its efforts to projects like lenses so big they could burn Britain, and bells so big that their sound would be lethal. (The lenses never got off the drawing board, and the bells, by the end of the war, would kill mice in a bath tub.) Roman engineering, too, was void of all subtlety. Roman roads ran absolutely straight; when they came to a mountain, they ran over the top of the mountain as pigheadedly as one of Stalin’s frontal assaults. Greek soldiers used to adapt their camps to the terrain; but the Roman army, at the end of a days’ march, would invariably set up exactly the same camp, no matter whether in the Alps or in Egypt. If the terrain did not correspond to the one and only model decreed by the military bureaucracy, so much the worse for the terrain; it was dug up until it fitted into the Roman Empire. The Roman aequeducts were bigger than those that had been used centuries earlier in the ancient world; but they were administered with extremely poor knowledge of hydraulics. Long after Heron of Alexand
ria (1st century A.D.) had designed water clocks, water turbines and two-cylinder water pumps, and had written works on these subjects, the Romans were still describing the performance of their aequeducts in terms of the quinaria, a measure of the cross-section of the flow, as if the volume of the flow did not also depend on its velocity. The same unit was used in charging users of large pipes tapping the aequeduct; the Roman engineers failed to realize that doubling the cross-section would more than double the flow of water. Heron could never have blundered like this.
The architecture of the thugs also differs from that of normal societies. It can often be recognized by the megalomaniac style of their public buildings and facilities. The Moscow subway is a faithful copy of the London Underground, except that its stations and corridors are filled with statues of homo sovieticus, a fictitious species that stands (or sits on a tractor), chin up, chest out, belly in, heroically gazing into the distance with a look of grim determination. The Romans had similar tastes. Their public latrines were lavishly decorated with mosaics and marbles. When a particularly elaborately decorated structure at Puteoli was dug up by archaeologists in the last century, they thought at first that they had discovered a temple; but it turned out to be a public latrine.24
The architecture of the Colosseum and other places of Roman entertainment are difficult to judge without recalling what purpose they served. It was here that gladiators fought to the death; that prisoners of war, convicts and Christians were devoured by as many as 5,000 wild beasts at a time; and that victims were crucified or burned alive for the entertainment of Roman civilization. When the Romans screamed for ever more blood, artificial lakes were dug and naval battles of as many as 19,000 gladiators were staged until the water turned red with blood. The only Roman emperors who did not throw Christians to the lions were the Christian emperors: They threw pagans to the lions with the same gusto and for the same crime — having a different religion.
A History of Pi Page 5