A Strange Wilderness

by Amir D. Aczel

  Hypatia pursued mathematics and philosophy and became a lecturer in philosophy at the Neoplatonist school in Alexandria. Many of her students came from far away to hear her lectures. She was, in a way, a philosophical descendant of Plato, continuing his academy’s tradition of lecture and discussion. In addition to teaching the philosophy of Plato and mathematics, Hypatia wrote commentaries on Diophantus’s Arithmetica, Apollonius’s Conics, Euclid’s Elements, and Ptolemy’s Almagest. She also wrote a book on astronomy, entitled The Astronomical Canon.

  But Hypatia lived in turbulent political times that saw severe friction between paganism and the emerging religion of Christianity. As a woman who had achieved cultural and political influence in Alexandria, and as a freethinking intellectual who pursued a career in a world dominated by men and never married, she was vulnerable. Indeed, in March of 415 Hypatia, an alleged pagan, was publicly accused of using her political influence to prevent a reconciliation between the estranged Imperial Prefect at Alexandria and the Christian Patriarch.

  A group of monks reportedly ambushed her chariot and dragged her to the street, stripped her naked, and killed her. Some versions of the story have her flayed alive, her limbs torn and burned. To some historians her tragic death signaled the end of classical Greek mathematics—and the classical period in general—although some work in mathematics in Greece continued for several decades.

  Charles William Mitchell’s 1885 painting of Hypatia portrays the mathematician, who was stripped naked by her murderers, in front of a Christian altar, symbolizing the conflict between her paganism and the then-emerging religion of Christianity.

  The sack of Rome in the fifth century certainly marked the beginning of the economic and cultural decline of Europe. Whatever mathematical and philosophical ideas may have arrived on the main part of the continent from Greece seem to have been lost.

  PART II

  THE EAST

  Caliph Haroun al-Rashid ruled the Arab Empire during its golden age in the late seventh and early eighth centuries, and his court at Baghdad helped make the city a center of mathematical scholarship and discovery. In this illustration, the emperor Charlemagne receives an ornate water clock that the caliph had sent as a gift.

  FOUR

  THE HOUSE OF

  WISDOM

  As Europe declined in the Middle Ages, an Islamic empire was rising in the East. It included Mesopotamia—the ancient site of some of the earliest developments in mathematics—as well as the Arabian Peninsula and regions stretching eastward beyond Persia and westward to North Africa. As part of their cultural awakening, the Arabs and Persians translated many classical works of the ancient Greeks. For example, the Arab mathematician Thabit ibn Qurra translated works of Apollonius, Euclid, and Archimedes. Incidentally, ibn Qurra’s book On the Sector-Figure, translated into Latin, would allow Newton to elaborate on Apollonius’s work almost two thousand years after the Greek mathematician’s lifetime. In India mathematics also flourished during this period.

  ARYABHATA

  In 476, a date often associated with the demise of the Roman Empire, a mathematician named Aryabhata was born in India. His birthplace is believed to have been somewhere in central India. When he was twenty-three years old, Aryabhata wrote a book entitled the Aryabhatiya, which played a role in Indian mathematics somewhat akin to that of Euclid’s Elements in the West. This treatise explained much of the arithmetic and calculations used in astronomy. It described powers of ten up to ten to the tenth—a huge number to comprehend at that time—and explained how to compute the square and cube roots of integers. Amazingly, it also provided the formula for the area of a triangle, rules for the sums of the terms in arithmetic progressions, explanations of geometric progressions arising from problems about compound interest, and work related to the solution of quadratic equations. These are surprisingly extensive results and not likely to have all been derived by one man—especially one so young. Most likely, the Aryabhatiya is a compilation of the work of many mathematicians, native and foreign, or a survey of the state of the art in Indian mathematics. Greek coins found in India indicate early trade relations, so, conceivably, the derivations in the Aryabhatiya may have originated elsewhere and been elaborated on by Aryabhata, his predecessors, and his contemporaries. Aryabhata also devised an estimate of π, which he described as follows:

  Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000.1

  Aryabhata’s calculation implies an estimate of π equal to 3.1416—as good as the estimate provided by the Greek astronomer and mathematician Ptolemy. This number and the units used (ten thousands) further imply a connection between India and the Greek world of the early centuries CE—one that certainly existed through trade.

  The Jantar Mantar observatory in New Delhi, India, constructed around 1724, contains thirteen astronomical instruments that employ technology developed centuries earlier, during the time of the mathematician Aryabhata (476–550).

  Much of Indian mathematics was derived from astronomical interests. The Indians were always great astronomers, intertwining the study of the stars with religion as early as the second millennium BCE. For example, the four-thousand-year-old Rig Veda texts reveal sophisticated astronomical calculations, including a year composed of twelve thirty-day months. At such cities as Delhi and Jaipur, one can still visit the Jantar Mantar (calculation instrument) observatories developed under the reign of Maharaja Jai Singh II in the eighteenth century, but the large stone constructions with numerical markings designed to measure the movements of stars, planets, and the sun across the sky exhibit technology that existed at the time of Aryabhata. In fact, Aryabhata worked as an astronomer, mathematician, and teacher at early institutions of learning and research in India. Before his death in 550, he published a number of mathematical and astronomical works, providing definitions for the sine and cosine functions that form the basis of trigonometry, as well as calculations for Earth’s rotation and revolution around the sun that differ from modern values by only .01 seconds and 3.33 minutes, respectively.

  BRAHMAGUPTA

  Half a century later India produced one of the greatest mathematicians who ever lived on the subcontinent. Little is known about the life of Brahmagupta, although we know that he was born in the year 598. A historian named Prthudakasvamin mentioned Brahmagupta in his ninth-century writings, referring to him as “the teacher from Bhillamala,” which we now identify as Bhinmal, near Mount Abu in Rajasthan.2 He is believed to have headed the Jantar Mantar in the city of Ujjain in central India through much of his life.

  Brahmagupta wrote a number of books on mathematics, the most important of which was the Brahmasphuta Siddhanta. A siddhanta is a system in astronomy, written in Sanskrit as a book or manual. Such treatises were prevalent in India at this time, but Brahmagupta’s was the most important because it was very complete, including many important mathematics and astronomy ideas having to do with calculation, arithmetic, and early trigonometry.

  Brahmagupta did not possess good estimates of π—he used three and the square root of ten. He did, however, devise formulas for areas, such as that of an isosceles triangle and a quadrilateral, as well as rules for the solution of quadratic equations—including negative roots! The use of negative numbers as solutions to equations was pioneered by this brilliant mathematician. He also understood trigonometry and provided solutions that required the sine function.

  Brahmagupta (598–ca. 668), the pioneering Indian mathematician, discovered many solutions to problems of mathematics and astronomy. He is also credited with being the first to identify zero as a number.

  As mentioned, Indian mathematicians were primarily interested in problems of astronomy. In particular, they wanted to know the average positions of heavenly bodies based on actual observations. Indian mathematicians working in an observatory measured the positions of the planets, the sun, and the moon over many months and years, recorded them, and then averaged
all observations for each heavenly body. Based on his computations of the positions of heavenly bodies, Brahmagupta conjectured that the universe was 4,320,000,000 years (“revolutions of the sun”) old.3 There is no explanation of how or why he obtained this number from observations of the sky, but his estimate is strikingly close to the present estimate of 4.5 billion years for the age of our solar system.

  Using the gnomon, an ancient astronomical device, Brahmagupta could also determine the latitude of his position from the shadow of the sun at noon during the equinox. However, because he was able to predict eclipses and do many astronomical computations that other mathematicians couldn’t, he became conceited about his abilities and criticized his predecessors—especially Aryabhata—in his writing.

  THE WORK OF BRAHMAGUPTA and other Indian mathematicians made its way to Arabia by 766, where it found fertile ground for growth. Arabia, the Arabic-speaking region extending roughly from central Asia to North Africa, flourished just as Europe was declining. By 775 the Indian texts had been translated into Arabic, and a few years later, Ptolemy’s Almagest was also translated. Thus, by the dawn of the ninth century, the Arabs possessed knowledge of both the Greeks and the Indians who preceded them and were ready to make their own contributions to mathematics. During this “Islamic Golden Age,” the center of the Arab empire was Baghdad, and its ruler was a caliph.

  Haroun al-Rashid (ca. 763–809) was an enlightened ruler who brought art, culture, and ideas to his people and made Baghdad an enchanted, storied city—one thinks of the timeless anthology Arabian Nights and the heroic adventurer Sinbad the Sailor. Subsequently, Caliph al-Mamun, who ruled between 809 and 833, continued in the footsteps of al-Rashid and turned Baghdad into “the new Alexandria.” Reportedly, one night the caliph had a dream in which Aristotle appeared and spoke to him. When he awoke, al-Mamun decided that he would bring to Baghdad all the works of the ancient Greeks. These texts were philosophical, astronomical, and mathematical—Euclid’s Elements being foremost among them. The caliph also founded the House of Wisdom, which he modeled after Plato’s Academy. Important thinkers of this prestigious body included Omar Khayyam, a mathematician and famous poet, and a man named Mohammad ibn Musa al-Khwarizmi, from whose name we get the modern word algorithm.

  AL-KHWARIZMI

  Al-Khwarizmi (d. ca. 850) came to Baghdad from a region in present-day Azerbaijan. Unfortunately, nothing is known about his life. Presumably, he had access to the work of Brahmagupta, and his own voluminous oeuvre seems to have been based on Indian mathematics. We don’t know exactly which works he brought to Baghdad, but all indications point to Brahmagupta’s great treatise, Brahmasphuta Siddhanta. In particular, al-Khwarizmi’s book Concerning the Hindu Art of Reckoning is believed to be based on the work of Brahmagupta. Beginning with the great Indian mathematician’s work on quadratic equations, al-Khwarizmi launched a project on the algebraic solutions of equations so extensive that today we consider him a founder of algebra. In addition to writing the first comprehensive book on algebra, he introduced Hindu numerals into Arabic mathematical literature. These Arabic-Hindu numerals would later find their way into Europe, along with his algebraic methods and astronomical calculations.

  Al-Khwarizmi expanded the algebraic work of Diophantus and popularized algebra by showing, in words rather than symbols, how equations could be set up and solved in a complete and systematic way. By simplifying the highly theoretical methodology of the Greeks, Indians, and Babylonians, he thus set a firm and practical foundation for the field of algebra for generations to come. Al-Khwarizmi’s major work on algebra, Al Gabr Wa’l Muqabala (from which the modern word algebra is derived), is a guide to solving equations, especially quadratic equations, in practical, real-world scenarios. The word al-gabr means “restoration” or “completion” and implies moving elements from one side of an equation to the other. Muqabala means “reduction” or “balancing”—canceling terms when they appear on both sides of an equation. The book’s first six chapters show how to systematically solve different kinds of equations, whereas the rest of the book uses geometry to demonstrate the validity of the methods presented in the first six chapters. The pragmatism of the first part resembles the practical Babylonian approach that we see in clay tablets from ancient Mesopotamia. Later Arabic texts even referred students to methods explained in Al Gabr Wa’l Muqabala, suggesting that the book had become a classic in the heyday of the Arabic Empire. After being translated into Latin, Al Gabr made its way to Europe.

  Muhammad ibn Musa al-Khwarizmi popularized the field of algebra with his book Al Gabr Wa’l Muqabala, published in 830. His likeness appears on this Russian commemorative stamp, issued approximately twelve hundred years after his birth.

  The House of Wisdom, founded by Caliph al-Mamun, was an active library and intellectual center in Baghdad until its destruction by Mongol invaders in 1258. This illustration of a group of scholars in an Islamic library comes from the Maqamat of al-Hariri, an illuminated manuscript of the thirteenth century.

  THE HOUSE OF WISDOM was home to a number of important mathematicians, astronomers, and scholars. Arabia is conveniently located between the world of ancient Greece and that of India and East Asia. As a result, the Arab empire borrowed from both West and East, building upon the straight-line geometry developed by the Greeks while employing trigonometric sine tables created by the Indians. For example, the Arab astronomer al-Battani (ca. 850–929), known in the West as Albategnius, used the Hindu sine function in his book On the Motion of the Stars, which enabled the medieval European astronomers who followed him to use trigonometry in calculating motions of heavenly bodies. He is credited with deriving the relation that the tangent of an angle is equal to its sine divided by its cosine. Computations were accurate to eight decimal places, and angles were accurate to a quarter of a degree.4

  By the tenth century, Arab mathematicians derived further trigonometric identities, such as the law for the doubling of an angle, sin(2x) = 2sin(x)cos(x). But the heyday of Arab mathematics came in the eleventh century. An Arab mathematician named al-Karkhi, who is known to have lived around 1029, extended the work in Diophantus’s Arithmetica to equations of higher orders and ones in which coefficients and solutions were not restricted to being rational numbers. In addition to al-Karkhi, that period saw the emergence of Ibn Sina (980–1037), known in the West as Avicenna. One of the most prominent scholars of this period, he made contributions not only to mathematics but also to medicine and philosophy.

  Avicenna’s contemporary al-Biruni (973–1048) traveled widely. His famous book, Indica, described Indian science and showed how a nonagon—a polygon with nine sides—could be inscribed in a circle, using the trigonometric formula for the cosine of 30 degrees to show that the problem of inscribing the nonagon in a circle was equivalent to solving the equation x3 = 1 + 3x. (He solved the problem to an amazing accuracy of six decimal places.) Indica also included a discussion of the Indian heliocentric theories of Aryabhata and Brahmagupta, noting that the idea of Earth revolving around the sun and rotating about its own axis were consistent with his astronomical calculations and, thus, could not be refuted.

  Abu Rayhan al-Biruni (973–1048) was a Persian mathematician, astronomer, and linguist whose many important books include the astrological treatise Kitab al-Tafhim, which includes this illustration of the phases of the moon.

  An Egyptian mathematician named Ibn Yunus (ca. 950–1008) extended trigonometry even further by introducing a powerful formula, 2cos(x)cos(y) = cos(x + y) + cos(x – y), which translates a product into a sum. Because this formula turns the product (on the left side of the equation) into a sum (on the right), it lowers the order of computation (because adding is easier than multiplying, and subtraction is easier than division). Ibn Yunus’s formula thus offered a tool for simplifying calculations. This prelogarithmic method for aiding computation was given the name prost-haphaeresis, which in Greek means “addition and subtraction.” Later, and until the advent of the modern calculator, loga
rithms were used to carry out multiplications as sums and divisions as subtractions through a slide rule and similar devices.

  OMAR KHAYYAM

  Omar Khayyam (1048–1123) was a Persian mathematician and poet, and it was he who took al-Khwarizmi’s pioneering ideas on algebraic equations and solutions to a new level. In his seminal work, Algebra, this poet-mathematician showed how to solve quadratic equations both algebraically and geometrically. He studied cubic equations, revealing the steps to their geometric solution, but he assumed (falsely) that a purely algebraic solution to cubic equations was not possible. Geometrically, he solved cubic equations using a method known to the ancient Greeks—the intersections of conics—but he generalized this method to any given arbitrary cubic equation with positive roots. Because we live in three-dimensional space, Khayyam was stumped in his attempts to solve equations of a higher order than three, not being able to envision the space in which such equations lived. He was equally hampered in his work by the fact that he and his contemporaries did not understand that negative solutions to equations have meaning. Thus he worked with only positive coefficients. The recognition that negative numbers are meaningful would come later, as mathematics matured, as would the so-called imaginary numbers.

  In general, the Arabs followed the Indian approach, which emphasized algebra and trigonometry over geometry, the realm of the ancient Greeks. Khayyam’s great genius was that he was able to use and understand both approaches. His thinking along the parallel lines of geometry and algebra foretold Descartes’s unification of these two mathematical fields in the seventeenth century. Presciently, Omar Khayyam wrote, “No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.”5 The connection between geometry and algebra, however, would not be pursued to a great extent by the Arabs. Regarding algebraic solutions of higher-order equations, Khayyam conceded, “they are impossible for us and even for those who are experts in this science. Perhaps one of those who will come after us will find them.”6