A History of Pi Page 8
Scientific research, wrote Tertullian (160-230), had become superfluous since the Gospel of Jesus Christ had been received. Tomaso d’Aquino (St. Thomas, 1225-1274) wrote that there was no conflict between science and religion; but by “science” he understood Aristotle’s tiresome speculations. The mediaeval Church never made St. Thomas’ mistake of confusing the two: It promoted Aristotle to some kind of pre-Christian saint, but (except for a very modest amount of third-rate work within its own cloisters) persecuted science almost wherever it appeared.
Scientific works and entire libraries were set to the torch kindled by the insane religious fanatics. We have already mentioned the Bishop of Yucatan, who burned the entire native literature of the Maya in the 1560’s, and Bishop Theophilus, who destroyed much of the remnants of the Library of Alexandria (391). The Christian Roman emperor Valens ordered the burning of non-Christian books in 373. In 1109, the crusaders captured Tripoli, and after the usual orgy of butchery typifying the crusades (though this one did not yet include the murderous Teutonic Knights), they burned over 100,000 books of Muslim learning. In 1204, the fourth crusade captured Constantinople and sacked it with horrors unparalleled even in the bloody age of the crusades; the classical works that had survived until then were put to the torch by the crusaders in what is generally considered the biggest single loss to classical literature. In the early 15th century, Cardinal Ximenes (Jimenez), who succeeded Torquemada as Grand Inquisitor and was directly responsible for the cruel deaths of 2,500 persons, had a haul of 24,000 books burned at Granada.*
In 1486, Torquemada sentenced the Spanish mathematician Valmes to be burned at the stake because Valmes had claimed to have found the solution of the quartic equation. It was the will of God, maintained the Grand Inquisitor of the Holy Office of the Inquisition Against Heretical Depravity, that such a solution was inaccessible to human understanding.39
Not only scientific theory was condemned as the work of the devil. The devil also seems to have known a lot more about navigation than the bloodthirsty Men of God. Many (perhaps most) ships sailing the Mediterranean in the Middle Ages had Jewish navigators, for the Christian captains and crews were not supposed to meddle with the devilish science of mathematics.40 In the 10th century, Raud the Strong, a Viking chieftain, escaped the fanatical Christianizing king of Norway Olaf Trygvasson by sailing into the wind (i.e., maintaining a zig-zag course whose average advances against the wind); the pious king, who was better acquainted with witchcraft than with the triangle of forces, thereupon accused Raud of being in alliance with the devil, and when he finally caught him, he had him killed by stuffing a viper down his throat.41
GIORDANO BRUNO (1548-1600)
The Middle Ages are usually considered to have ended with the fall of Constantinople (1453) or the discovery of the New World (1492); but the insane persecution of science continued well beyond that time, and it is difficult to give a date when it ceased. For more than a century longer, the Church tolerated no deviation from the literal word of the Bible, or from the teachings of its idol Aristotle. In 1600, Giordano Bruno was burned alive in Rome for claiming that the earth moves round the sun. In 1633, the 70-year old Galileo Galilei went through the torture chambers of the Inquisition until he was willing to sign that
I, Galileo Galilei,… aged seventy years, being brought personally to judgement, and kneeling before you Most Eminent and Most Reverend Lords Cardinals, General Inquisitors of the universal Christian republic against depravity … swear that … I will in future believe every article which the Holy Catholic and Apostolic Church of Rome holds, teaches, and preaches … I held and believed that the sun is the center of the universe and is immovable, and that the earth is not the center and is movable; willing, therefore, to remove from the minds of your Eminences, and of every Catholic Christian, this vehement suspicion [of heresy] rightfully entertained against me,… I abjure, curse and detest the said errors and heresies,… and I swear that I will never more in future say or assert anything verbally, or in writing, which may give rise to a similar suspicion of me … But if it shall happen that I violate any of my said promises, oaths and protestations (which God avert!), I subject myself to all the pain and punishments which have been decreed … against delinquents of this description.42
Thereupon he was sentenced to life imprisonment in a Roman dungeon. The sentence was later commuted and he died, a blind and broken man, in 1642. But not even in death did the pious inquisitors leave him in peace. They destroyed many of his manuscripts, disputed his right to burial in consecrated ground, and denied him a monument in the ludicrous hope that this brilliant thinker and his work would be forgotten. 250 years later, Sir Oliver Lodge commented:
Poor schemers! Before the year was out, an infant was born in Lincolnshire, whose destiny it was to round and complete and carry forward the work of their victim, so that, until man shall cease from the planet, neither the work nor its author shall have need of a monument.
* * *
SUCH was the ugly face of the Middle Ages. It is not surprising that mathematics made little progress; toward the Renaissance, European mathematics reached a level that, roughly, the Babylonians had attained some 2,000 years earlier43 and much of the progress made was due to the knowledge that filtered in from the Arabs, the Moors and other Muslim peoples, who themselves were in contact with the Hindus, and they, in turn, with the Far East.
The history of π in the Middle Ages bears this out. No significant progress in the method of determining π was made until Viète discovered an infinite product of square roots in 1593, and what little progress there was in the calculation of its numerical value, by various modifications of the Archimedean method, was due to the decimal notation which began to infiltrate from the East through the Muslims in the 12th century.
Arab mathematics came to Europe through the trade in the Mediterranean, mainly via Italy; ironically, the other stream of mathematics was the Church itself.44 Not only because the mediaeval priests had a near monopoly of learning, but also because they needed mathematics and astronomy as custodians of the calendar. Like the Soviet High Priests who publish Pravda for others but read summaries of the New York Times themselves, so the mediaeval Church condemned mathematics as devilish for others, but dabbled quite a lot in it itself. Gerbert d’Aurillac, who ruled as Pope Sylvester II from 999 to 1003, was quite a mathematician; so was Cardinal Nicolaus Cusanus (1401-1464); and much of the work done on π was done behind thick cloister walls. And just like the Soviets did not hesitate to spy on the atomic secrets of bourgeois pseudo-science, so the mediaeval Church did not hesitate to spy on the mathematics of the Muslim infidels. Adelard of Bath (ca.1075-1160) disguised himself as a Muslim and studied at Cordoba;45 he translated Euclid’s Elements from the Arabic translation into Latin, and Ptolemy’s Almagest from Greek into Latin. When Alfonso VI of Castile captured Toledo from the Moors in 1085, he did not burn their libraries, containing a wealth of Muslim manuscripts. Under the encouragement of the Archbishop of Toledo, a veritable intelligence evaluation center was set up. A large number of translators, the best known of whom was Gerard of Cremona (1114-1187), translated from Arabic, Greek and Hebrew into Latin, at last acquainting Europe not only with classical Greek mathematics, but also with contemporary Arab algebra, trigonometry and astronomy. Before the Toledo leak opened, mediaeval Europe did not have a mathematician who was not a Moor, Greek or a Jew.46
GALILEO GALILEI (1564-1642)
Discovered the law of the pendulum; radically improved the telescope; discovered the satellites of Jupiter, sunspots, the rotation of the Sun, and the libration of the Moon; proved the uniform acceleration of all bodies falling to earth; doubted the infinite velocity of light and suggested how to measure its velocity; established the concept of relative velocity. Above all, he discovered the laws of motion, though he was not yet able to formulate them quantitatively.
One of the significant mediaeval European mathematicians was Leonardo of Pisa (ca. 1180-1250), better known by his nick
name Fibonacci (“son of Bonaccio”). Significantly, he was an Italian merchant, so that he worked within one of the Arabic infiltration routes. In 1202, he published a textbook using algebra and the (present) Hindu-Arabic numerals. He is best known for the Fibonacci sequence
where each number (after the initial 1’s) is the sum of the two preceding ones. This sequence turns up in the most surprising places (see figure here), and its applications range from the growth of pineapple cells to the heredity effects of brother-sister incest. There is a Fibonacci Quarterly in the United States in our own day, for the fertility of this sequence appears to be endless.
But Fibonacci also worked on π, though his progress on this point was not as impressive as his other achievements. Like Archimedes, Leonardo used a 96-sided polygon, but he had the advantage of calculating the corresponding square roots by the new decimal arithmetic. Theoretically, his work (published in Practica geometriae, 1220) is not as good as Archimedes’, whose approximations to the square roots are always slightly on the low side for the circumscribed polygon, and slightly on the high side for the inscribed polygon; since the square roots appeared in the denominator, this assured Archimedes of getting the correct bounds. Fibonacci used no such care, extracting the square roots as nearly as he could, but he was lucky: His bounds turned out to be more accurate than Archimedes’, and the mean value between them,
is correct to three decimal places.
Otherwise there was little progress. Gerbert (Pope Sylvester) used the Archimedean value π = 22/7, but in the next 400 years we also find the Babylonian 3 1/8 and the Egyptian π = (16/9)2 in various cloister correspondence.47 The documents of the time mostly show the low level of mediaeval mathematics. Franco von Lüttich, for example, wrote a long treatise on the squaring of the circle (ca. 1040), which shows that he did not even know how to square a rectangle47 (here), and Albert von Sachsen (died 1390) wrote a long treatise De quadratura circuli consisting for the most part of philosophical polemics. The crux of the problem is brushed away by saying that “following the statement of many philosophers,” the ratio of circumference to diameter is exactly 22/7; of this, he says, there is proof, but a very difficult one.47 Dominicus Parisiensis, author of Practica geometriae (1346), distinguishes himself above his contemporaries by at least knowing that π = 22/7 was an approximation. So did the Viennese Georg Peurbach (1423-1461), who knew Greek and some of the history of π; he understood Archimedes’ derivation of π > 22/7, and knew both the Ptolemeian π = 377 : 120, and the Indian value π = √10.
Perhaps the only interesting contribution of this sorry time is that of Cardinal Nicolaus Cusanus (1401-1464), a German who worked in Rome from 1448 until his death. Although his work on π was not very successful and his approaches were more ecclesiastic than mathematical,48 he did find a good approximation for the length of a circular arc. His derivation is somewhat tiresome,49 but in modern terminology, it amounts to
To find the quality of this approximation, we multiply (1) by (2 + cosθ) and expand both sides in series (even though Brook Taylor is still two and a half centuries away), obtaining
so that up to the third-order term the two expressions are identical, and the fifth order term differs only by 3/5. The approximation is therefore excellent for
for θ = 36°, the error is about 2 minutes of arc. Well done, Cardinal.
It is doubtful whether some two centuries later, the great Dutch physicist Christiaan Huygens (1629-1693) was familiar with Cusanus’ work, but in his treatise De circuli magnitude inventa, which we shall meet again in Chapter 11, he derived a theorem suggesting a construction for the rectification of a circular arc as shown in the figure below, and which makes arc AB ≈ AB′. The construction was first suggested by Snellius, and it is equivalent to the Cardinal’s approximation: We have
Approximate rectification of an arc suggested by one of Huygens’ theorems, and equivalent to Cardinal Cusanus’ approximation:
AB′ ≈ AB.
and from the triangle OCB,
From (3) we find tanβ, substitute in (2), and obtain after some trigonometric manipulation
If this is set equal to arc AB = rθ, it will be seen that Huygens’ construction is equivalent to Cusanus’ approximation (1).
But Huygens already lived in happier times.
9
AWAKENING
E pur si muove!
(And nonetheless it moves!)
Giordano Bruno’s last cry from the burning stake, 16th February, 1600.50
THE classification of history into ancient, mediaeval and modern is but a symptom of the white man’s arrogance; for while Europe was suffering the imbecility of the Dark Ages, the rest of the world went on living. Yet in the struggles of oriental science we find the same type of patterns and tendencies as in Europe, though not necessarily at corresponding times. Just as the torch of Alexandria was extinguished by militaristic Rome, so the intellectual life of Babylon was wiped out by the militaristic Assyrians, and the golden age of Muslim science was stifled by the militaristic Turks; in India and China the story is similar. And again; we find the confrontations between religion and science. In the vein of Tertullian, the Muslim philosopher al-Ghazzali (1058-1111) wrote that scientific studies shake men’s faith in God and undermine religion, and that they lead to loss of belief in the origin of the world and the Creator;51 and several Muslim fanatics, like their Christian counterparts, proved their piety by burning libraries. In China, too, the emperor Tsin Shi Hwang-di (3rd century B.C.) was warned by his advisors of the idling scholars whose influence was founded on books; and he ordered all literature in China burned, excepting only books on medicine, husbandry and divination, and those in the hands of the seventy official scholars. And again, we find the quaint mixture of mathematics and mumbo-jumbo that can be found in the mysticism of the Pythagoreans. For example, the able Chinese mathematician Sun-Tsu (probably 1st century A.D.) wrote a textbook containing, for the most part, sound mathematics; but it also contains this problem:
A pregnant woman, who is 29 years of age, is expected to give birth to a child in the 9th month of the year. Which shall be her child, a son or a daughter?
The solution is truly Pythagorean:
Take 49; add the month of her child-bearing; subtract her age. From what remains, subtract the heaven 1, subtract the earth 2, subtract the man 3, subtract the four seasons 4, subtract the five elements 5, subtract the six laws 6, subtract the seven stars 7, subtract the eight winds 8, subtract the nine provinces 9. If the remainder be odd, the child shall be a son; and if even, a daughter.
Note that the remainder is negative, so that a Chinese student of the 1st century would be stunned, because he could scarcely know the concept of a negative number, let alone its parity. To say very impressively nothing at all is the secret of all such oracles; it is also the secret of computerized astrology and the trumpets of Madison Avenue.
* * *
AS we have seen, European mathematics was far behind that of the Muslim world. Yet as the Middle Ages ended, Europe was catching up, and by the 17th century it was so far ahead that the rest of the world never caught up with it again, and to Europe we now return.
The end of the Middle Ages was heralded by three events, all of which took part in the latter half of the 15th century.
First, in 1453 the Turks captured Constantinople. The significance of this event is not that they grabbed and sacked a city ruled by those who had grabbed and sacked it before them, but that they had breached its walls with guns. The era of gun powder mean the end of the feudal fiefs who often defied their kings behind the walls of their castles. The local tyranny of the feudal aristocracy was replaced by the centralized tyranny of the divine kings and emperors. And the absolute power of the Church was broken; where the Church had patronized the nominal ruler, the central and powerful ruler was now patronizing the Church.
Christopher Columbus (1451–1506)
Vasco da Gama (1469–1525)
Ferdinand Magellan (1480–1521)
Second, the turn of the fifteenth century saw the great discoveries of the rest of the world. Christopher Columbus discovered America; Vasco da Gama rounded the Cape of Good Hope (1499) on his voyage to India; and Ferdinand Magellan’s flotilla circumnavigated the globe (1519-1522). The far-reaching consequences of these events included the need for better clocks, better astronomy, better trigonometry, and the resulting stimulation of the exact sciences. It is probably not accidental that the mathematicians appearing on the scene come increasingly from England, France and Holland, countries that suddenly found themselves in the center of the maritime world, and where (unlike Spain and Portugal) the influence of the Church was rapidly waning.