II. THE MATHEMATICAL REVOLUTION

The first great name in the science of this period is Leonardo Fibonacci of Pisa.

Sumerian mathematics, born of forgotten parentage, had descended through Babylonia to Greece; Egyptian geometry, still visible in the pyramids, had passed, perhaps through Crete and Rhodes, to Ionia and Greece; Greek mathematics had gone to India in the wake of Alexander, and had played a part in the Hindu development that culminated in Brahmagupta (588?-660); about 775, translations were made of Hindu mathematicians, and soon afterward of Greek mathematicians, into Arabic; about 830 the Hindu numerals entered Eastern Islam; about 1000 Gerbert brought them to France; in the eleventh and twelfth centuries Greek, Arabic, and Hebrew mathematics streamed into Western Europe through Spain and Sicily, and came with Italian merchants to Venice and Genoa, Amalfi and Pisa. Transmission is to civilization what reproduction is to life.

Another line of transmission appeared in the sixth century B.C. in the form of the Chinese abacus (Greek abax, a board), an instrument for counting by transferring little bamboo rods from one group to another; its descendant, the suanpan, is still used by the Chinese. In the fifth century B.C., says Herodotus, the Egyptians reckoned with pebbles, “bringing the hand from right to left”; the Greeks proceeded contrariwise. The Romans used several forms of the abacus; in one form the counters slid in grooves; they were made of stone, metal, or colored glass, and were called calculi, little stones.29 Boethius, about 525, mentioned the abacus as enabling one to count by tens; but this invitation to a decimal system was ignored. The merchants of Italy used the abacus, but wrote the results in clumsy Roman numerals.

Leonardo Fibonacci was born at Pisa in 1180. His father was manager of a Pisan trade agency in Algeria; Leonardo in adolescence joined him there, and was taught by a Moslem master. He traveled in Egypt, Syria, Greece, and Sicily, studied the methods of the merchants, and learned to reckon, he tells us, “by a marvelous method through the nine figures of the Indians”;30 here at the outset of their European career the new numerals were properly called Hindu, and what is now a bore and chore of our childhood was then a wonder and delight. Perhaps Leonardo learned Greek as well as Arabic; in any case we find him well acquainted with the mathematics of Archimedes, Euclid, Hero, and Diophantus. In 1202 he published his Liber abaci; it was the first thorough European exposition of the Hindu numerals, the zero, and the decimal system by a Christian author, and it marked the rebirth of mathematics in Latin Christendom. The same work introduced Arabic algebra to Western Europe, and made a minor revolution in that science by occasionally using letters, instead of numbers, to generalize and abbreviate equations.31 In his Practica geometriae (1220) Leonardo, for the first time in Christendom so far as we know, applied algebra to the treatment of geometrical theorems. In two smaller works of the year 1225 he made original contributions to the solution of equations of the first and second degree. In that year Frederick II presided at Pisa over a mathematical tournament in which different problems were set by John of Palermo and solved by Fibonacci.

Despite his epoch-making work, the new method of calculation was long resisted by the merchants of Europe; many of them preferred to finger the abacus and write the results with Roman numerals; as late as 1299 the abacists of Florence had a law passed against the use of the “new-fangled figures.”32 Only a few mathematicians realized that the new symbols, the zero, and the decimal alignment of units, tens, hundreds… opened the way to such developments of mathematics as were almost impossible with the old letter numerals of Greeks, Romans, and Jews. Not till the sixteenth century did the Hindu numerals finally replace the Roman; in England and America the duodecimal system of reckoning survives in many fields; 10 has not finally won its thousand-year-long war against 12.

Mathematics in the Middle Ages had three purposes: the service of mechanics, the keeping of business accounts, and the charting of the skies. Mathematics, physics, and astronomy were closely allied, and those who wrote on one of them usually contributed to the others as well. So John of Holy wood (in Yorkshire), known to the Latin world as Joannes de Sacrobosco, studied at Oxford, taught at Paris, wrote a Tractatus de sphaera—Treatise on the (Earthly) Sphere—and an exposition of the new mathematics,Algorismus vulgaris—Mathematics for the Millions (c. 1230). Algorismus, a corruption of the name al-Khwarizmi, was the Latin term for an arithmetical system using the Hindu numerals. John credited the “Arabs” with the invention of this system, and was partly responsible for the misnomer “Arabic numerals.”33 Robert of Chester, about 1149, in adapting the astronomical tables of al-Battani and al-Zarqali, brought Arabic trigonometry to England, and introduced the word sinus (bay, sine) into the new science.

Interest in astronomy was maintained by the needs of navigation and the passion for astrology. The immense authority of the oft-translated Almagest petrified the astronomy of Christian Europe into the Ptolemaic theory of eccentrics and epicycles, with the earth at the hub of the world; alert minds like Albertus Magnus, Thomas Aquinas, and Roger Bacon felt the force of the criticisms that the Moorish astronomer al-Bitruji had aimed at this system in the twelfth century; but no satisfactory alternative to Ptolemy’s celestial mechanics was found before Copernicus. Christian astronomers in the thirteenth century pictured the planets as revolving about the earth; the fixed stars, snared in a crystal firmament, and steered by divine intelligences, revolved as a regimented host around the earth; the center and summit of the universe was that same man whom the theologians described as a miserable worm tainted with sin and mostly doomed to hell. The suggestion offered by Heracleides Ponticus, four centuries before Christ, that the apparent daily motion of the heavens was due to the axial rotation of the earth, was discussed by Semitic astronomers in the thirteenth century, but was quite forgotten in Christendom. Another notion of Heracleides, that Mercury and Venus revolve about the sun, had been handed down by Macrobius and Martianus Capella; John Scotus Erigena had seized upon it in the eighth century, and had extended it to Mars and Jupiter; the heliocentric system was on the verge of victory;34 but these brilliant hypotheses were among the casualties of the Dark Ages, and the earth held the center of the stage till 1521. All astronomers, however, agreed that the earth is a sphere.35

The astronomical instruments and tables of the West were imported from Islam, or were modeled on Islamic originals. In 1091 Walcher of Lorraine, later Prior of Malvern Abbey, observed lunar eclipses in Italy with an astrolabe; this is the earliest known case of observational astronomy in the Christian West; but even two centuries later (c. 1296) William of St. Cloud had to remind astronomers, by precept and example, that the science grew best on observation rather than on reading or philosophy. The best contribution to Christian astronomy in this period was the Alfonsine Tables of celestial movements, prepared for Alfonso the Wise by two Spanish Jews.

The accumulation of astronomic data revealed the imperfections of the calendar established by Julius Caesar (46 B.C.) from the work of Sosigenes, which made the year too long by eleven minutes and fourteen seconds; and the increasing intercourse of astronomers, merchants, and historians across frontiers exposed the inconvenience of conflicting calendars. Al-Biruni had made a useful study of the rival systems of dividing time and dating events (c. 1000); Aaron ben Meshullam and Abraham bar Hiyya furthered the study in 1106 and 1122; and Robert Grosseteste and Roger Bacon followed with constructive proposals in the thirteenth century. The Computus (c. 1232) of Grosseteste—a set of tables for calculating astronomic events and movable dates (e.g., Easter)—was the first step toward the Gregorian calendar (1582) that guides and confuses us today.

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