Mathematics

A

5

Roman Mathematics

The Alexandrian period of Greek civilization ended in 31 bc with Rome’s conquest of Egypt, the last of Alexander’s kingdoms. Roman orator Cicero boasted that the Romans were not dreamers like the Greeks but applied their study of mathematics to the useful. Nothing mathematically significant was accomplished by the Romans. The Roman numeration system was based on Roman numerals, which were cumbersome for calculation. Despite this drawback, the use of Roman numerals continued in some European schools until about 1600 and in bookkeeping for another century.

B

Indian and Islamic Mathematics

After the decline of Greece and Rome, mathematics flourished for hundreds of years in India and the Islamic world. Mathematics in India was largely a tool for astronomy, yet Indian mathematicians discovered a number of important concepts. Their mathematical masterpieces and those of the Greeks were translated into Arabic in centers of Islamic learning, where mathematical discoveries continued during the period known in the West as the Middle Ages. Our present numeration system, for example, is known as the Hindu-Arabic system.

B

1

Indian Mathematics

The system of numbers that we use today, with each number having an absolute value and a place value (units, tens, hundreds, and so forth) originated in India. Mathematicians in India also were the first to recognize zero as both an integer and a placeholder. When the Indian numeration system was developed is not known, but digits similar to the Arabic numerals used today have been found in a Hindu temple built about 250 bc.

In the 5th century Hindu mathematician and astronomer Aryabhata studied many of the same problems as Diophantus but went beyond the Greek mathematician in his use of fractions as opposed to whole numbers to solve indeterminate equations (equations that have no unique solutions). Aryabhata also figured the value of p accurately to eight places, thus coming closer to its value than any other mathematician of ancient times. In astronomy, he proposed that Earth orbited the sun and correctly explained eclipses of the Sun and Moon.

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The earliest known use of negative numbers in mathematics was by Hindu mathematician Brahmagupta about ad 630. He presented rules for them in terms of fortunes (positive numbers) and debts (negative numbers). Brahmagupta’s understanding of numbers exceeded that of other mathematicians of the time, and he made full use of the place system in his method of multiplication. Brahmagupta headed the leading astronomical observatory in India and wrote two works on mathematics and astronomy. The works dealt with topics such as eclipses, risings and settings, and conjunctions of the planets with each other and with fixed stars.

Writing in the 9th century, Jain mathematician Mahavira stated rules for operations with zero, although he thought that division by zero left a number unchanged. The best-known Indian mathematician of the early period was Bhaskara, who lived in the 12th century. Bhaskara supplied the correct answer for division by zero as well as rules for operating with irrational numbers. Bhaskara wrote six books on mathematics, including Lilavati (The Beautiful), which summarized mathematical knowledge in India up to his time, and Karanakutuhala, translated as “Calculation of Astronomical Wonders.”

B

2

Islamic Mathematics

Indian mathematics reached Baghdād, a major early center of Islam, about ad 800. Supported by the ruling caliphs and wealthy individuals, translators in Baghdād produced Arabic versions of Greek and Indian mathematical works. The need for translations was stimulated by mathematical research in the Islamic world. Islamic mathematics also served religion in that it proved useful in dividing inheritances according to Islamic law; in predicting the time of the new moon, when the next month began; and in determining the direction to Mecca for the orientation of mosques and of daily prayers, which were delivered facing Mecca.

In the 9th century Arab mathematician al-Khwārizmī wrote a systematic introduction to algebra, Kitab al-jabr w’al Muqabalah (Book of Restoring and Balancing). The English word algebra comes from al-jabr in the treatise’s title. Al-Khwārizmī’s algebra was founded on Brahmagupta’s work, which he duly credited, and showed the influence of Babylonian and Greek mathematics as well. A 12th-century Latin translation of al-Khwārizmī’s treatise was crucial for the later development of algebra in Europe. Al-Khwārizmī’s name is the source of the word algorithm.

By the year 900 the acquisition of past mathematics was complete, and Muslim scholars began to build on what they had acquired. Alhazen, an outstanding Arab scientist of the late 900s and early 1000s, produced algebraic solutions of quadratic and cubic equations. Al-Karaji in the 10th and early 11th century completed the algebra of polynomials (mathematical expressions that are the sum of a number of terms) of al-Khwārizmī. He included polynomials with an infinite number of terms.

Later scholars, including 12th-century Persian mathematician Omar Khayyam, solved certain cubic equations geometrically by using conic sections. Arab astronomers contributed the tangent and cotangent to trigonometry. Geometers such as Ibrahim ibn Sinan in the 10th century continued Archimedes’s investigations of areas and volumes, and Kamal al-Din and others applied the theory of conic sections to solve problems in optics. Astronomer Nasir al-Din al-Tusi created the mathematical disciplines of plane and spherical trigonometry in the 13th century and was the first to treat trigonometry separately from astronomy. Finally, a number of Muslim mathematicians made important discoveries in the theory of numbers, while others explained a variety of numerical methods for solving equations.

Many of the ancient Greek works on mathematics were preserved during the Middle Ages through Arabic translations and commentaries. Europe acquired much of this learning during the 12th century, when Greek and Arabic works were translated into Latin, then the written language of educated Europeans. These Arabic works, together with the Greek classics, were responsible for the growth of mathematics in the West during the late Middle Ages.

C

Medieval and Renaissance Mathematics

Few advances in mathematics took place in Europe during the early Middle Ages, before about 1100. Most learning was concentrated in monasteries and focused on questions of theology. A most important application of mathematics during the Middle Ages was in astrology; astrologers were called mathematici. Inasmuch as the practice of medicine was based largely on astrological determination of the proper treatment, physicians had to become mathematicians.

The introduction of Greek and Arabic works starting about 1100 played a major role in the rebirth of secular (worldly) learning in Europe. The translation of these works into Latin led to an upsurge in mathematical study in the West. English philosopher Adelard translated al-Khwārizmī’s astronomical tables and an Arabic version of Euclid’s Elements into Latin in the 12th century. Italian mathematicians such as Leonardo Fibonnaci and Luca Pacioli depended heavily on Arabic sources in improving business mathematics used for accounting and trade. Fibonnaci’s Liber abaci (1202, Book of the Abacus) introduced Arabic numbers, the Hindu-Arabic place-value decimal system, and Arabic algebra to Europe.

The late Middle Ages saw some fruitful mathematical considerations of infinite series by French prelate Nicole d’Oresme and others. But not until the 16th century did the first truly important mathematical discovery in Europe occur. This discovery was an algebraic formula for the solution of both cubic and quartic equations—equations with terms raised to the third or fourth powers. Italian mathematician Gerolamo Cardano published the formula in 1545 in his Ars Magna (Great Art). The discovery drew the attention of mathematicians to complex numbers and stimulated a series of solutions to equations of degrees higher than four.

During the 16th century mathematicians began to use symbols to make algebraic thinking and writing more concise. These symbols included +, -, ×, =, > (greater than), and < (less than). The most significant innovation, by French mathematician François Viète, was the systematic use of letters for variables in equations. Viète’s remarkable work on solving equations to the third and fourth degree influenced many mathematicians of the following century, including Fermat in France and Newton in England.

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