A History of Pi
Page 6
The apology that the Romans “knew no better at the time” is quite invalid. Like the Nazis, the Romans were not primitive savages, but sophisticated killers, and they certainly knew better from the people they had enslaved. The Greeks vehemently (but unsuccessfully) resisted the introduction of gladiatorism into their country by the Roman overlords; what they must have felt can perhaps be appreciated by the Czechs who, in 1968, watched the Soviet cut-throats amusing themselves by riddling the Czech Museum with machinegun fire. The Ptolemies in Alexandria also went in for pageantry to entertain the people, but they did this without bloodshed and with a sense of humor; one of the magnificent parades staged by Ptolemy II displayed the animals of the king’s private zoo, as well as a 180 foot long gilded phallus.24
The Romans’ contribution to science was mostly limited to butchering antiquity’s greatest mathematician, burning the Library of Alexandria, and slowly stifling the sciences that flourished in the colonies of their Empire. The Naturalis Historia by Gaius Plinius Secundus25 (23-73 A.D.) is an encyclopaedic compilation which is generally regarded as the most significant scientific work to have come out of Rome; and it demonstrates the Romans’ abysmal ignorance of science when compared to the scientific achievements of their contemporaries at Alexandria, even a century after the Romans had sacked it. For example, Pliny tells us that in India there is a species of men without mouths who subsist by smelling flowers.
The Roman contribution to mathematics was little more than nothing at all.26 There is, for example, Posidonius (135-51 B.C.), friend and teacher of Cicero and Pompey, who, using a method similar to that of Erastosthenes (here), calculated the circumference of the earth with high accuracy. But if one digs a little deeper, Posidonius’ original name is found to be Poseidonios; he was a Syrian who studied in Athens and settled at Rhodes, whence he was sent, in 86 B.C., to Rome as an envoy. Poseidonios was therefore as Roman as Euler was Russian. (Euler, as we shall see in Chapter 14, was a Swiss who lived some years in Russia; in Soviet textbooks, he is often referred to as “our great Russian mathematician Eyler.”) Poseidonios’ value for π must have been accurate to (the equivalent of) several decimal places, for the value he obtained for the circumference of the earth was three centuries later adopted by the great Alexandrian astronomer Ptolemy (no relation to the Alexandrian kings), and this was the value used by Christopher Columbus on his voyage to the New World.
But whatever the value of π used by Poseidonios, it was high above Roman heads. The Roman architect and military engineer Pollio Vitruvius, in De Architectura (about 15 B.C.) used the value π = 3 1/8,27 the same value the Babylonians had used at least 2,000 years earlier.
It is, of course, a simple matter to pick out the bad things in anything that is a priori to be run down; to tell the truth but not the whole truth is the basic trick of any propaganda service that has risen above outright lying. Was there, then, nothing good about ancient Rome? Of course there was; it is an ill wind that blows nobody any good. But in a brief background that is getting too far away from π already, I am not concerned with the somebodies to whom Rome may have blown some good; I am only saying that the wind that blew from Rome was an ill wind.
Yet most historians extol the achievements of Rome, and it is only fair to hear some of their reasons. For example:28
Whatever Rome’s weaknesses as ruler of empire may have been, it cannot be denied that her conquest of the Western World contributed a great deal to subsequent civilization. It accustomed the Western races to the idea of a world-state, and by pax romana (Roman peace) it demonstrated the benefits of a long absence of war, even if the price was the loss of political independence by most of the races of the world.
Simple, is it not? It appears we missed the benefits of pax germanica through Winston Churchill and similar warmongers, but all is not lost yet: We still have the chance of pax sovietica.
* * *
BEFORE Rome became a corrupt empire, it was a corrupt republic.
Across the Mediterranean, in what today is Tunisia, another city, Carthage, had risen and its dominions were expanding along the African coast and into Spain. That provoked Rome’s jealousy, and Carthage was defeated in two Punic Wars; but each time she rose again, and in the Third Punic War, after the Carthaginians had withstood a siege against hopeless odds with almost no resources, the Romans captured the city, massacred its population and destroyed the city (146 B.C.)
There had been no rational reason for the Punic Wars. In particular, the Third Punic War was fomented by a group of paranoic hawks in the Roman senate who felt threatened by Carthage’s revival. They were led by the pious superpatriot Marcus Porcius Cato (the Elder), who had distinguished himself in the Second Punic War (218-201 B.C.) and who held a number of high offices in the Roman republic, in the course of which he bloodily crushed an insurrection in Spain, raised the rents of the tax-farmers and adjusted the prices of slaves. He vigorously opposed any kind of innovation or reform, strove to stem the tide of Greek refinement, and advocated a return to the strict social life of earlier days; in an age when slaves were branded, flogged and crucified, he was known for the cruelty with which he treated his slaves.29 No matter what was being discussed in the senate, his speeches would always end with the words Ceterum censeo Carthaginem esse delendam (For the rest, I hold that Carthage must be destroyed), a sentence that has been copied in innumerable variations* by people to whom vicious bigots like Cato were presented as examples of the noble Roman spirit.
Such is the background of the Punic Wars, which lead us back to the story of π. During the Second Punic War, the Romans sent an expeditionary force under Claudius Marcellus to Sicily in 214 B.C. For one thing, the king of Syracuse had renewed his alliance with Carthage; for another, the Romans specialized in winning easy victories over small foes.
But this time it was not so easy. Roman brute force, assaulting the city of Syracuse by land and sea, ran into scientific engineering; the engineering that is not bigger, but smarter. The Syracusans had been taught the secret of the lever and of the multiple pulley, and they put it to use in their artillery and marine defenses. The Roman land forces reeled back under the storm of catapult balls, catapult darts, sling bullets and crossbow bolts. The attack by sea fared no better. Syracusan grapnels were lowered from cranes above the cliffs until they caught the bows of Roman ships, which were then hoisted by multiple pulleys until the ships hung vertically and the proud warriors of mighty Rome tumbled into the sea; Roman devices to scale the walls of the city from the ships were battered to pieces by boulders suspended from cranes that swung out over the city walls as the Roman fleet approached. What was then left of the crippled Roman fleet withdrew, and Marcellus hatched a new plan. Under cover of darkness, the Romans sneaked by land to the walls of Syracuse, thinking that the defenders’ catapults could not be used at close quarters. But here they ran into more devilish machines. Plutarch reports that “the wall shot out arrows at all points,” and that “countless evils were poured upon them from an unseen source” even after they had fled and tried to regroup. Once more the haughty Roman warriors withdrew to lick their wounds, and Marcellus ranted against this foe “who uses our ships like cups to ladle water from the sea … and outdoes all the hundred-handed monsters of fable in hurling so many missiles against us all at once.” In the end the invincible Roman legions became so filled with fear that they would run as soon as they saw a piece of rope or wood projecting over the wall. Marcellus had to settle for a siege that was to last the better part of three years.
But, as Bernard Shaw said, God is on the side of the big batallions;30 and the city finally fell to the Roman cut-throats (212 B.C.), who sacked, plundered and looted it by all the rules of Roman civilization. Inside the city was the 75-year old thinker who had grasped the secret of the lever, the pulley and the principle of mechanical advantage. Plutarch tells us that “it chanced that he was alone, examining a diagram closely; and having fixed both his mind and his eyes on the object of his inquiry, he perceived
neither the inroad of the Romans nor the taking of the city. Suddenly a soldier came up to him and bade him follow to Marcellus, but he would not go until he had finished the problem and worked it out to the proof.”
“Do not touch my circles!” said the thinker to the thug. Thereupon the thug became enraged, drew his sword and slew the thinker.
The name of the thug is forgotten.
The name of the thinker was Archimedes.
6
ARCHIMEDES OF SYRACUSE
Greek scholars are privileged men; few of them know Greek, and most of them know nothing else.
George Bernard SHAW
(1856-1950)
WHEN Newton said “If I have seen further than others, it is because I stood on the shoulders of giants,” one of the giants he must have had in mind was Archimedes of Syracuse, the most brilliant mathematician, physicist and engineer of antiquity.
Little is known about his life. He was born about 287 B.C. in Syracuse, the son of the astronomer Pheidias, and apparently spent most of his life in Syracuse. He studied at the University of Alexandria either under Euclid’s immediate successors, or perhaps under Euclid himself. He was a kinsman and friend of Hieron II, king of Syracuse, for whom he designed the machines of war used against the Roman aggressors, and whose crown was involved in the discovery of the law of upthrust that bears his name. Hieron suspected (correctly) that his crown was not pure gold and asked Archimedes to investigate without damaging the crown. Archimedes is said to have pondered the problem while taking a bath, and to have found the answer as he observed the water level rising on submerging his body into the bath. Shouting Heureka! (I have found it), says this legend, he ran naked through the streets of Syracuse to tell Hieron of his discovery.
His book On Floating Bodies goes far beyond Archimedes’ Law, and includes complicated problems of buyoancy and stability. Likewise, On the Equilibrium of Planes goes beyond the principle of the lever and solves complicated problems such as finding the center of gravity of a parabolic segment. In these, as in his other works, Archimedes used the Euclidean approach: From a set of simple postulates, he deduced his propositions with unimpeachable logic. As the first writer who consistently allied mathematics and physics, Archimedes became the father of physics as a science. (Aristotle’s Physics was published a century earlier, but it is only a long string of unfounded speculations, totally void of any quantitative relations.)
ARCHIMEDES OF SYRACUSE (ca. 287–212 B.C.)
Archimedes was also the first scientific engineer, the man searching for general principles and applying them to specific engineering problems. His application of the lever principle to the war machines defending Syracuse are well known; yet he also applied the same principle to find the volume of the segment of a sphere by an unusually beautiful balancing method about which we shall have more to say later in this chapter. He used the same method to determine the volumes of other solids of revolution (ellipsoid, paraboloid, hyperboloid) and to find the center of gravity of a semicircle and a hemisphere. It is not known how many of Archimedes’ works have been lost (one of the most important, The Method, came to light only in 1906), but his extant books, including On Spirals, On the Measurement of the Circle, Quadrature of the Parabola, On Conoids and Spheroids, On the Sphere and Cylinder, Book of Lemmas and others, are unmatched by anything else produced in antiquity.
Archimedes screw or helical pump. It is still used 23 centuries later by the Egyptian felahin, whose rulers think it more important to destroy Israel than to provide their people with modern irrigation.
Archimedes’ method of calculating π.
Not only because of the marvelous results contained in these books. But also because Archimedes was a pioneer of method. He took the step from the concept of “equal to” to the concept of “arbitrarily close to” or “as closely as desired” (which Euclid had enunciated, but not actively used) and thus reached the threshold of the differential calculus, just as his method of squaring the parabola reached the threshold of the integral calculus (some consider he crossed it).
He was also the first to give a method of calculating π to any desired degree of accuracy. It is based on the fact that the perimeter of a regular polygon of n sides inscribed in a circle is smaller than the circumference of the circle, whereas the perimeter of a similar polygon circumscribed about the circle is greater than its circumference (see figure above). By making n sufficiently large, the two perimeters will approach the circumference arbitrarily closely, one from above, the other from below. Archimedes started with a hexagon, and progressively doubling the number of sides, he arrived at a polygon of 96 sides, which yielded
or in decimal notation,
That Archimedes did this without trigonometry, and without decimal (or other positional) notation is an illustration of his tenacity (see Heath’s translation of On the Measurement of the Circle, Proposition 3 and following). However, we shall use both of these to go through the calculation.
If θ = π/n is half the angle subtended by one side of a regular polygon at the center of the circle, then the length of the inscribed side is
A regular polygon of 40 sides. No internal circle has been drawn.31
and that of a circumscribed side is
For the circumference C of the circle we therefore have
or dividing by 2r,
If the original number of sides n is doubled k times, this yields
and by making k sufficiently large, the lower and upper bounds will approach π arbitrarily closely.
Archimedes did not, of course, use trigonometric functions; however, for n = 6, he had sinθ = 1/2, tanθ = √(1/3) by Pythagoras’ Theorem, and the remaining functions in (6) can be obtained by the successive use of the half-angle formulas (which correspond to finding proportions in right-angled triangles). For k = 4, the two polygons will have 96 sides, and this will lead to the limits (1), if the square roots involved in the half-angle formulas are approximated by slightly smaller rational numbers for the lower limit, and by slightly larger rational numbers for the upper limit.
This again is easier described in modern terminology than actually done without trigonometry or a decimal system for the accompanying arithmetic. The half-angle formulas most closely resembling Archimedes’ procedure are
which enabled him (in effect) to find cot(θ/2) and cosec(θ/2) from cotθ and cosec θ. For a hexagon, Archimedes approximated √3 by the slightly smaller value 265/153; a 12-sided polygon already involved him in the ratio √(349450): 153, which he simplified to 591 1⁄8 :153; and the final 96-sided polygon involved a square root of a number that in the decimal system had ten digits! How he managed to extract his square roots with such accuracy, always taking care to keep slightly on the small or large side as demanded by the bounds, is one of the puzzles that this extraordinary man has bequeathed to us.
But it appears that Archimedes went even further. Heron of Alexandria, in his Metrica (about 60 A.D., but not disovered until 1896), refers to an Archimedes work that has since been lost, where Archimedes gives the bounds
or
where the upper limit u, in the copy of Heron’s work found in Constantinople in 1896, is given as
however, this is evidently an error that must have crept in during the transcription of the copy, for this is far coarser than the upper bound 3 1/7 found by Archimedes earlier. Heron adds “Since these numbers are inconvenient for measurements, they are reduced to the ratio of the smaller numbers, namely 22 : 7.”
Rectification of the circle by the Archimedean Spiral
The bounds attained by Archimedes in the Measurement of the Circle were
or in decimal notation,
Archimedes also showed that a curve discovered by Conon of Alexandria could, like Hippias’ quadratrix, be used to rectify (and hence square) the circle. The curve is today called the Archimedean Spiral; it is defined as the plane locus of a point moving uniformly along a ray while the ray rotates uniformly about its end point. It will thus be traced by a fly c
rawling radially outward on a turning phonograph record.
Let P be any point on the spiral (see figure above), and let the tangent at P intersect the line OP at R. Archimedes showed in his book On Spirals that the segment OR (i.e. the polar subtangent at the point P) equals the length of the circular arc PS described with radius OP, where S is the intersection with the initial ray OU. It follows that OU equals ¼ of the circumference of a circle with radius OT, so that
and the area of the triangle OTU is
whence
area of circle with radius OT = 4 times area of ΔOTU,
and the circle is squared once more, though not to the liking of Greek geometry; the objections are the same as in the case of Hippias’ quadratrix.
Archimedes used a double reductio ad absurdum to prove that the segment OR and the arc PS were equal in length. We shall prove it more quickly by differential geometry. Let OP = ρ and the angle SOP = θ. Then from the definition of the Archimedean Spiral,
where k is a constant equal to the ratio of angular and linear velocities. As shown in textbooks of differential geometry, the length of a subtangent in polar coordinates is
and since ρ′(θ) = k, we have from (9), (10), and the figure