A History of Pi Page 7
as was to be demonstrated. In particular, for θ = π/2,
* * *
SUCH were some of the contributions of antiquity’s greatest genius to the history of π and squaring the circle. Though later investigators found closer numerical approximations, Archimedes’ polygonal method remained unsurpassed until 19 centuries later an infinite product and an infinite continued fraction were found in England just before the discovery of the differential calculus led to a totally new approach to the problem.
In physics, no one came near Archimedes’ stature for 18 more centuries, until Galileo Galilei dared to challenge Aristotle’s humbug.
ARCHIMEDEAN SCIENCE32
ARISTOTELIAN PRATTLE33
Shadows of that humbug are with us even now. There is hardly a history of mathematics that does not apologize for Archimedes for getting his hands dirty with experimental work. As late as 1968, we are told that “he placed little value in his mechanical contrivances,” and (1965) that “he regarded his contrivances and inventions as sordidly commercial.” Similar comments will be found in almost all books on the subject, bearing out Philip Guedella’s remark “History repeats itself; historians repeat each other.” The repetitious myth was started by Plutarch, who in the 1st century A.D. wrote that
Regarding the business of mechanics and every utilitarian art as ignoble and vulgar, he gave his zealous devotion only to those subjects whose elegance and subtlety are untrammeled by the necessities of life.
Now Plutarch could not possibly have known what Archimedes regarded as ignoble and vulgar; his guess was as good as yours or mine. But just as for 19 centuries historians have been echoing Plutarch, so Plutarch echoed the attitude of Plato and Aristotle, the fathers of intellectual snobbery. They taught that experimentation was fit only for slaves, and that the laws of nature could be deduced merely by man’s lofty intellect; and Aristotle used his lofty intellect to deduce that heavier bodies fall to the ground more rapidly; that men have more teeth than women; that the earth is the center of the universe; that heavenly bodies never change; and much more of such wisdom, for he was a very prolific writer. As a matter of fact, Aristotle was defeated on his own grounds, by sheer intellectual deduction unaided by experimental observation. Long before Galileo Galilei dropped the wooden and leaden balls from the leaning tower of Pisa, he asked the following question: “If a 10 lb stone falls ten times as fast as a 1 lb stone, what happens if I tie the two stones together? Will the combination fall faster than the 10 lb stone because it weighs 11 lbs, or will it fall more slowly because the 1 lb stone will retard the 10 lb stone?”
Plutarch’s comment on Archimedes’ attitude to engineering is a concoction with no foundation other than Aristotelian snobbery. Archimedes’ book The Method was discovered only in the present century, and it sheds some interesting light on the point. A reductio ad absurdum is to this day a popular method of proof; it does, however, have a drawback, and that is that one must know the result beforehand in order to know the alternative that is to be disproved. How this result was originally derived does not transpire from the proof. Since Archimedes very often used reductiones ad absurdum, historians have often marveled by what method he originally discovered his wonderful results.
The law of the lever.
W1r1 = W2r2, where W1, W2 are weights (or other forces).
The lever principle applied to geometry. A plane PS, perpendicular to GF at any point P will cut the sphere, cone and cylinder in circles with radii PR, PQ and PS, respectively. Archimedes proved that the first two circles (their weights proportional to their areas) placed on the lever GEF with fulcrum E would exactly balance the third circle at P. From this he derived the volume of a spherical segment, as well as the volume of a whole sphere (4 π r3/3).
It turns out that in many cases he did this by a method of analogy or modeling (as it is called today), based on the principles of none other than the ignoble, vulgar and commercially sordid contrivances that he is alleged to have invented with such reluctance. In The Method, he writes to Erastosthenes, the Librarian at Alexandria whom we have already met (here):34
Archimedes to Erastosthenes, greetings.
I sent you on a former occasion some of the theorems discovered by me, merely writing out the enunciations and inviting you to discover the proofs, which at the time I did not give. [ … ] The proofs then of these theorems I have written in this book and now send to you. Seeing moreover in you, as I say, an earnest student, [ … ] I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proofs of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.
A vivid example of this method is Archimedes’ application of the lever principle to derive the volume of a spherical segment as well as that of a whole sphere, as indicated in the figure here. So highly did Archimedes value this discovery that he asked for a sphere inscribed in a cylinder to be engraved in his tombstone, and this was done, for though the tombstone has been lost, we have a description of the tomb by Cicero, who visited it in the 1st century A.D. during his office as quaestor in Sicily.
It is not without interest to recall how The Method was recovered. It was found in 1906 in Constantinople on a palimpsest, that is, a parchment whose original text has been washed off (to salvage the parchment) and replaced by different text. If the original has been washed off imperfectly, it can be restored by special photography. In this case, the original text was a 10th century copy of some known Archimedean works, but also including the only surviving text of The Method. The mediaeval zealots did not always, like the Bishop of Yucatan or the Crusaders at Constantinople, burn scientific books as work of the devil. Sometimes they would only wash off the text for the sake of the parchment, so that they might besmirch it with their superstitious garbage.
7
DUSK
Theodotus: What is burning there is the memory of mankind.
Caesar: A shameful memory. Let it burn.
George Bernard Shaw,
Caesar and Cleopatra
THE University of Alexandria remained the intellectual center of the ancient world even after its library had been burned during the occupation by the Roman hordes. The details of that catastrophe are roughly as follows. When Caesar occupied Alexandria in 48 B.C., the Quisling Queen Cleopatra (a far cry from Hollywood’s puerile fantasies) offered him not only her bed, but also the Library. What made a thug like Julius Caesar accept the offer (of the library) is not altogether clear, but he helped himself to a large part of the rolls, which were readied for shipment to Rome. Then the Alexandrians, in an early instance of campus unrest, rose against Caesar and Cleopatra. They were, of course, crushed by Roman might, and the shipment of books for Rome, or the remaining library, or both, were burned in the fighting.
When Cleopatra bedded with the next Roman war lord Marcus Antonius (Mark Anthony), he compensated her with Roman generosity: He stole the 200,000 roll library of Pergamon and graciously presented it to Cleopatra.
The remaining story of the Library of Alexandria is typical for what was in store for science at the hands of political rulers and religious fanatics in the years to come. It was again damaged when the Roman emperor Aurelianus quelled an Egyptian revolt in 272 A.D., and again in 295 when the emperor Diocletian suppressed another revolt. In 391, a Christian mob led by the fanatical Bishop Theophilus destroyed the Temple of Serapis, where some of the books were kept. Another zealous bishop, Cyril, led a Christian mob against the astronomer and mathematician Hypatia; as a devotee of p
agan learning, she was hacked to death in 415, and her death marks the end of Alexandria as a center of mathematical learning. The end came in 646, when the Arabs captured the city. It is related that their general Amr ibn-al-As wrote to his Khalif to ask what to do with the books of the library. The reply was that if the books agreed with the Koran, they were superfluous; if they disagreed, they were pernicious. So they were burned.
Many famous mathematicians worked in the Greek world under Roman rule. Ptolemy, the great astronomer, worked in Alexandria in 139-161 A.D. He used the value
which he may have taken over from Apollonius of Perga, an outstanding mathematician some 30 years younger than Archimedes. Eutokios, in the 5th century A.D., comments that Apollonius refined Archimedes’ method, but the book he refers to has been lost.
There were others. Heron (1st century A.D.), Diophantus (3rd century), Pappus (late 3rd century), all of Alexandria, were famous mathematicians, but none of them appears to have contributed significantly to the history of π. This was the “Silver Age” of Greek mathematics; still high above anything Rome ever squinted at, but well below the golden age of Euclid, Archimedes and Apollonius.
But already Greek mathematics, like the other sciences, were slowly dying under the cold breath of Rome; it was getting dusk, and the stage was set for a disaster even greater than that of the Roman Empire.
In this age of dusk, when Rome began to change from ferocity to decadence, and when science was beginning to drown in the oncoming flood of mysticism, superstition and dogma, we find the curious figure of Nehemiah, a Rabbi and mathematician, who is perhaps not very significant in the history of mathematics, but who is symbolic for this age, for he made a gallant attempt to reconcile science with religion.
It has been remarked before (here) that the biblical verse I Kings vii, 23 was a curious pebble on the road to the confrontation between science and religion. Nehemiah noticed this pebble, picked it up and contemplated it.
Nehemiah was a Hebrew Rabbi, scholar and teacher who lived in Palestine and wrote about 150 A.D., after the last Judean revolt against the Romans (132-135) led by Bar-Kokba and resulting in the Diaspora of the Jews. Nehemiah was the author of the Mishnat ha-Middot, the earliest Hebrew geometry known to us. Not all of it has been preserved; but it deals with the elements of plane and solid geometry, and with the measurements and construction of the Tabernacle. Our concern here is only with matters touching on π, in particular, the Old Testament verse (I Kings vii, 23, also 2 Chronicles iv,2),
Also, he made a molten sea ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.
which implies π = 3.
Nehemiah writes in his textbook:
The circle has three aspects: the circumference, the thread and the roof. Which is the circumference? That is the rope surrounding the circle; for it is written: And a rope of thirty cubits did encompass it round about. And the thread? That is the straight line from brim to brim; for it is written: From brim to brim. And the roof itself is the area.
[ … ]
And if you want to know the circumference all around, multiply the thread into three and one seventh …
That is the Archimedean value, π = 3 1/7. But Nehemiah was also a Rabbi; so how does he get round I Kings vii, 23? Like this:
Now it is written: And he made the molten sea of ten cubits from brim to brim, round in compass, and yet its circumference is thirty cubits, for it is written: And a line of thirty cubits did compass it round about. What is the meaning of the verse And a line of thirty cubits and so forth? Nehemiah says: Since the people of the world say that the circumference of a circle contains three times and one seventh of the thread, take off that one seventh for the thickness of the walls of the sea on the two brims, then there remain thirty cubits did compass it round about.
Hats off to the crafty old fox! The “people of the world” say π = 3 1/7, but the scriptures say π = 3; so you measure the inner circumference of the walls, whereas the diameter is measured from outer rim to outer rim; and the thickness of the walls, you blockheads, makes up exactly for that secular one seventh! Certainly the Rabbi had more wits than the dogmatic commentators of the Bible in Germany in the 18th century; crudely ignoring the description “round in compass,” they claimed that the molten sea must have been hexagonal.35
Nevertheless, the dear Rabbi swindled quite brazenly, for the width of the molten sea walls is given three verses further on (I Kings vii, 26):
And it was an hand breadth thick, and the brim thereof was wrought like the brim of a cup, with flowers of lilies; it contained two thousand baths.36
This was the age of dusk, when it was still possible to attempt compromising between science and religion. No such compromise was tolerated in the mediaeval night. Whoever tampered with what the Bible said, risked torture chamber and the stake.
* * *
BEFORE we go on into that night, let us pause to admire the ancients for their tenacity in tackling mathematical problems without the benefit of algebraic symbolism (which was introduced much later by the Arabs). Nehemiah, for example, states the area of a circle as follows:
If one wants to measure the area of a circle, let him multiply the thread (diameter) into itself and throw away from it the one seventh and the half of one seventh; the rest is the area, its roof.
That is, the area is
which equals (3 1/7) × (d/2)2, so that if the Archimedean value π = 3 1/7 is accepted, the formula is correct.
There was also no single symbol, such as π, for the circle ratio. In mediaeval Latin, it was described by the following mouthful: quantitas, in quam cum multiplicetur diameter, proveniet circumferentia (the quantity which, when the diameter is multiplied by it, yields the circumference), and this clause was inserted in the long sentences that stated the equivalent of our formulas; for example, the area of a circle was given as follows:
Multiplicatio medietatis diametri in se ejus, quod proveniet, in quantitatem, in quam cum multiplicatus diameter provenit circumferentia, aequalis superficies circuli. (Multiplication of half the diameter with itself and of that which results by the quantity, which when the diameter is multiplied by it, yields the circumference, equals the area of the circle.)
This monster sentence states (correctly) that
Perhaps the Greeks made such enormous progress in mathematics because their geometry steered clear of numerical calculations, and thus did not get bogged down in expressing algebraic relations intelligibly. Euclid’s statements that
In a circle, equal arcs subtend equal chords
or
If corresponding angles of two triangles are equal, then corresponding sides are proportional
have not been improved on in 2,200 years.
8
NIGHT
What ye have done unto the last of my brethren, ye have done unto me
JESUS OF NAZARETH
THE fall of Rome in 476 A.D. marks the time when the literate barbarians of Rome were replaced by the illiterate barbarians of Germany, and this event is generally considered as the beginning of the Middle Ages.
Disaster was followed by catastrophe: The Roman Empire was followed by the Roman Church. In the Eastern or Byzantine Empire, the Roman Empire continued hand in hand with the Eastern Orthodox church for a thousand more years.
What made the bloody tyrants of the time accept Christianity is debatable. Perhaps they welcomed a religion that would teach their subjects to turn the other cheek, and that promised salvation for humility and eternal damnation for rebellion. Perhaps they were impressed by the unparalleled organization and iron discipline of the early Christians. What is certain is that once the mediaeval kings and emperors had adopted Christianity as the official faith, their executioners saw to it that nothing else was allowed.
And “nothing else,” of course, included science, the staunchest opponent of the Supernatural.
In the early days of civilization,
science and religion were in the same hands, in the hands of the priests, who had a monopoly on learning. In Egypt, Babylon, the Yucatan, and everywhere else in the Belt, they were the timekeepers and surveyors, the astronomers and geometers. They soon learned that knowledge was power: power over the gullible and over the uneducated. Five thousand years ago, the Chaldaean priests knew how to predict an eclipse, and they would impress the uneducated with much mumbo-jumbo about the impending catastrophe. The Egyptian priests had nilometers in their temples, instruments that communicated with the Nile by secret, underground channels; they learned to predict the rising and falling of the Nile, and they would impress the uneducated with much much mumbo-jumbo about the imminent floods or recessions of the Sacred River.
Mathematics and mumbo-jumbo remained allied for a long time. Pythagoras, for example, was a mathematician to a much lesser degree than he was a mumbo-jumboist. He taught that one stood for reason, two for opinion, three for potency, four for justice, five for marriage, seven held the secret of health, eight the secret of marriage, the even numbers were female, the odd numbers male, and so on. “Bless us, divine number,” the Pythagoreans would incant to the number four, “who generatest Gods and men, O holy tetraktys, that containest the root and source of the eternally flowing creation.”37 Immerged in this mumbo-jumbo, Pythagoras discovered that 32 + 42 = 52, and that there were other triplets of numbers x, y, z that satisfy the relation x2 + y2 = z2. Translated into geometry, this led him to the theorem which is quite unjustly called after him, for it had been known to every rope-stretcher (surveyor) from the Nile to the Yang-Tse Kiang for a thousand years before Pythagoras’ witchcraft.38
But witchcraft is easier than mathematics; specialization set in, and mumbo-jumbo became more and more separated from mathematics. Those interested in knowledge took the path to science; others took the road to the occult, the mystic and the supernatural. Thousands of cults and religions developed. The vast majorities of these did not recruit converts by force; not even the Roman enslavers forced their religion on the peoples they had conquered. But in three major cases religion became militant, and its fanatics disseminated it by fire and sword, plunging large parts of humanity into prolonged nightmares of horror. They were mediaeval Christianity, mediaeval Islam, and modern communism. For communism is a religion, too, with its church (party), its priests, its scriptures, its rituals, its dogmas, and its liturgy; the communist claims about dialectical materialism being a science are just so much more mumbo-jumbo. Communism uses science for its own ends; Muslim science, especially mathematics, flourished (on the whole) under the rule of the Mohammedan fanatics; but mediaeval Christianity, in many ways the most terrible of the three catastrophies, persecuted science with torture chamber and the stake, and almost succeeded in extinguishing it completely.