§3.1 THE WORD “ALGEBRA,” as everyone knows, comes from the Arabic language. This has always seemed to me somewhat unfair, for reasons I shall get to shortly. Unfair or not, it requires some explanation from the historian.

§3.2 Supposing the dates I gave for Diophantus (200–284 CE) are correct, he lived through a very unhappy period. The Roman Empire, of which Egypt was a province, was then embarked on its well-known decline and fall, of which Edward Gibbon wrote so eloquently and at such length. The sorry state of affairs in Diophantus’s time, if it was his time, is described in Chapter 7 of Gibbon’s masterpiece.

The empire rallied somewhat in the later 3rd century. Diocletian (284–305 CE) and Constantine (306–337 CE) are counted among the great Roman emperors. The first of them fiercely persecuted Christians; the second was the son of a Christian, issued the Edict of Milan (313 CE) commanding tolerance of Christianity throughout the empire, and himself accepted baptism in his last illness.

The decision first to tolerate and then to enforce the practice of Christianity did little to retard the crumbling of the empire. In some

ways it may have accelerated it. One of the great strengths of early Christianity was that it appealed to all classes. To do this, however, it had to “pay off” the sophisticated urban intellectuals with appealing but complex metaphysical theories, while maintaining its hold on the masses with a plain, clear message of salvation and divine powers, fortified with some colorful stories and concessions to older pagan beliefs. Inevitably, though, the masses got wind of the lofty metaphysical disputes and used them as fronts for social and ethnic grievances.

Diophantus’s own city of Alexandria illustrates the process well. Even after 300 years as a Greek city and a further 300 as a Roman one, Alexandria remained a glittering urban enclave that was fed and clothed from a hinterland of illiterate, Coptic-speaking Egyptian peasants. To a Christian Copt from the desert fringes, the words “Greek,” “Roman,” and “Pagan” must have been near-synonyms, and the fabulous Museion (“Temple of the Muses”), with its tradition of secular learning and its attached great library, probably seemed to be a house of Satan.

The matter was made worse in Egypt by the cult of monasticism, especially strong there, which placed several thousand vigorous but sex-starved young males at the service of anyone who wanted to whip up a religious mob. Which, of course, ambitious politicians—a category that at this point in Roman history frequently included officers of the church—often did. This is the context for the murder of Hypatia in 415 CE, which so outraged Gibbon.

Hypatia is the first female name in the history of mathematics. All her written works have been lost, so we know them only by hearsay. On this basis it is difficult to judge whether she can properly be called a significant mathematician or not. It is certain, at any rate, that she was a major intellectual. She taught at the Museion (of which her father, Theon, was the last president) and was a compiler, editor, and preserver of texts, including math texts. She was an adherent and a teacher of the philosophy called Neoplatonism, an attempt to locate in another world the order, justice, and peace so conspicuously lack-

ing in the later Roman empire.¹⁸ She was also, we are told, a beauty and a virgin.

Hypatia was active in scholarly teaching and learning during the time when Cyril of Alexandria was archbishop of the city. Cyril, later Saint Cyril, is a difficult person to judge through the mists of time and theological controversy. Gibbon gives him very unfavorable coverage, but Gibbon nursed a prejudice against Christianity and can’t be altogether trusted. Certainly Cyril launched a pogrom against Alexandria’s Jews, driving them out of the city, but the Jews seem previously to have conducted a nasty anti-Christian pogrom of their own, as even Gibbon allows.¹⁹ The Alexandrians of this time were, as we learn from the Catholic Encyclopedia, “always riotous.” At any rate, Cyril got into a church-state dispute with Orestes, the Prefect (that is, the Roman official in charge of Egypt), and it was put about that Hypatia was the main obstacle to the healing of this split. A mob was raised, or raised itself, and Hypatia was pulled from her chariot and dragged through the streets to a church, where the flesh was scraped from her bones with, depending on the authority and translator, either oyster shells or pottery shards.²⁰

Hypatia seems to have been the last person to teach at the Museion, and her appalling death in 415 is usually taken to mark the end of mathematics in the ancient European world. The Roman empire in the west limped along for another 60 years, and Alexandria continued under the authority of the eastern, Byzantine, emperors for a further 164 years (interrupted by a brief Persian²¹ occupation, 616–629), but the intellectual vitality was all gone. The next noteworthy name in the history of algebra had his home 900 miles due east of Alexandria, on the banks of the Tigris, in that same Mesopotamian plain where the whole thing had begun two and a half millennia before.

§3.3 The northern and western territories of the Roman empire were lost to Germanic barbarians in the 5th century. The southern

and eastern ones, except for Greece, Anatolia, and some southern fragments of Italy and the Balkans, fell to the armies of Islam in the 7th. Alexandria itself fell on December 23, 640 CE, breaking the heart of the Byzantine emperor Heraclius, who had spent his entire working life recovering territories that had been lost to the early 7th-century Persian resurgence.²²

The actual conqueror of Alexandria was a person named Amr ibn al-’As.²³ He was a direct report to Omar, the second Caliph (that is, leader of the Muslims after Mohammed’s death). Alexandria put up a fight—the siege lasted 14 months—but most of Egypt was a pushover for the Muslims. The Egyptians had persisted in a Christian heresy called Monophysitism, and after wresting the province back from its brief subjection to Persia, the Byzantine emperor Heraclius persecuted them savagely for this.²⁴ The result was that the native Egyptians came to detest the Byzantines and were glad to exchange a harsh master for a more tolerant one.

The third Caliph, Othman, belonged to the Omayyad branch of Mohammed’s clan. After some complicated civil wars between his faction and that of the Prophet’s son-in-law Ali (one indirect result of which was the Sunni–Shiite split that still divides Islam today), the Omayyads established a dynasty that ruled the Islamic world from Damascus for 90 years, 661–750 CE. Then a revolt led to a change of dynasty, the Omayyads keeping only Spain, where they hung on for another 300 years.

The new dynasty traced its descent to Mohammed’s uncle al-Abbas, and so its rulers are known to history as the Abbasids. They founded a new capital, Baghdad, in 762 CE, plundering the old Babylonian and Persian ruins for building materials. The English word “algebra” is taken from the title of a book written in this Baghdad of the Abbasid dynasty around 820 CE by a man who rejoiced in the name Abu Ja’far Muhammad ibn Musa al-Khwarizmi. I shall refer to him from here on just as al-Khwarizmi, as everyone else does.²⁵

§3.4 Baghdad under the fifth, sixth, and seventh Abbasid Caliphs (that is, from 786 to 833 CE) was a great cultural center, dimly familiar to modern Westerners as the world of viziers, slaves, caravans, and far-traveling merchants pictured in the Arabian Nights stories. Arabs themselves look back on this as a golden age, though in fact the Caliphate no longer had the military strength to keep together all the conquests won in the first flush of Islamic vigor and was losing territories to rebels in North Africa and the Caucasus.

Persia was part of the Abbasid domain, under the spiritual and temporal authority of the Caliph. However, Persia had been the home of high civilizations since the Median empire of 1,400 years earlier, while the Arabs of 800 CE were only half a dozen generations away from their roots as desert-dwelling nobodies. The Abbasids therefore nursed something of a cultural inferiority complex toward the Persians, rather as the Romans did toward the Greeks.

Beyond the Persians were the Indians, whom the first great Muslim expansion left untouched. Northern India had been united under the Gupta dynasty in the 4th and 5th centuries CE but thereafter was generally divided into petty states until Turkish conquerors arrived in the late 10th century. These medieval Hindu civilizations were fascinated by numbers, especially very big numbers, for which they had special names. (The Sanskrit term tallakchana, should you ever encounter it, means a hundred thousand trillion trillion trillion trillion.) It is to the Indians—probably to the mathematician Brahmagupta, 598–670 CE—that the immortal honor of having discovered the number zero belongs, and our ordinary numerals, which we call Arabic, are actually of Indian origin.

Beyond the Indians were of course the Chinese, with whom India had been in cultural contact since at least the travels of the Buddhist monk Xuan-zang in the middle 7th century and with whom the Persians conducted a busy trade along the Silk Route. The Chinese had long had a mathematical culture of their own—I shall say something about it in §9.1.

Those inhabitants of Baghdad who had the leisure and inclination could therefore acquaint themselves with everything that was known anywhere in the civilized world at that time. The culture of the Greeks and Romans was familiar to them through Alexandria, now one of their cities, and through their trading contacts with the Byzantine empire. The cultures of Persia, India, and China were easily accessible.

All that was needed to make Abbasid Baghdad an ideal center for the preservation and enrichment of knowledge was an academy, a place where written documents could be consulted and lectures and scholarly conferences held. Such an academy soon appeared. It was called Dar al-Hikma, the “House of Wisdom.” This academy’s greatest flourishing was in the reign of the seventh Abbasid Caliph, al-Mamun. In the words of Sir Henry Rawlinson, Baghdad under al-Mamun “in literature, art, and science … divided the supremacy of the world with Cordova; in commerce and wealth it far surpassed that city.” This was the time when al-Khwarizmi lived and worked.

§3.5 We know very little about al-Khwarizmi’s life. His dates are known only approximately. There are some fragmentary dry notices in the works of Islamic historians and bibliographers, for details of which I refer the reader to the DSB. We do know that he wrote several books: one on astronomy, one on geography, one on the Jewish calendar, one on the Indian system of numerals, one a historical chronicle.

The work on Indian numerals survives only in a later Latin translation, whose opening words are “Dixit Algorithmi…” (“According to al-Khwarizmi …”). This book lays out the rules for computing with the modern 10-digit place value system of arithmetic, which the Indians had invented, and it was tremendously influential. Because of those opening words, medieval European scholars who had mastered this “new arithmetic” (as opposed to the old Roman numeral system, which was hopeless for computation) called themselves “algorithmists.” Much later the word “algorithm” was used to mean

any process of computing in a finite number of well-defined steps. This is the sense in which modern mathematicians and computer scientists use it.

The book that really concerns us is the one titled al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala (“A Handbook of Calculation by Completion and Reduction”). This is a textbook of algebra and arithmetic, the first significant work in the field since Diophantus’s Arithmetica of 600 years earlier. The book is in three parts, with these topics: solution of quadratic equations, measurement of areas and volumes, and the math required to deal with the very complicated Islamic laws of inheritance.

Only the first of these three parts is strictly algebraic, and it is something of a disappointment. For one thing, al-Khwarizmi has no literal symbolism—no way to lay out equations in letters and numbers, no sign for the unknown quantity and its powers. The equation we would write as

and which Diophantus would have written as

appears in al-Khwarizmi’s book as

(Dirhem was a unit of money. Al-Khwarizmi uses it to refer to what we nowadays call the constant term, the term in x⁰.)

For another thing, Diophantus’s historic turn away from the geometrical method toward manipulation of symbols is nowhere visible in al-Khwarizmi’s work. This is not very surprising, since he had no symbols to manipulate, but it is still a sliding back from Diophantus’s great breakthrough of 600 years earlier. Says van der Waerden: “[W]e may exclude the possibility that al-Khwarizmi’s work was much influenced by classical Greek mathematics.”

Al-Khwarizmi’s main algebraic achievement, in fact, was to bring forward the idea of equations as objects of interest, classifying all equations of the first and second degrees in one unknown and giving rules for manipulating them. His classification was into six fundamental types, which we would write in modern symbolism as

Some of these look trivially the same type to us but that is because we have negative numbers to help us. Al-Khwarizmi had no such aids. He could speak of subtraction, of course, and of one quantity exceeding another, or falling short of another, but his natural arithmetic tendency was to see everything in terms of positive quantities.

As for techniques of manipulation, that is where “completion” (al-jabr) and “reduction” (al-muqabala) come in. Once you have an equation like

(or, as al-Khwarizmi says, “a square which is equal to forty things less four squares”), how do you manipulate it into one of those six standard forms? By al-jabr, that’s how—“completing” the equation by adding 4x² to each side, leaving us with a type-1 equation:

That is adding equal terms on both sides. The opposite thing, where you need to subtract equal terms from both sides, is al-muqabala, for example, turning the equation

into a type 5

by subtracting 29 from both sides.

§3.6 None of this is really new. In fact, al-jabr and al-muqabala can both be found in Diophantus—together with, of course, a rich literal symbolism to aid the manipulations. “Al-Khwarizmi’s scientific achievements were at best mediocre, but they were uncommonly influential,” says Toomer in the DSB.

I fear, in fact, that at this point the reader may be slipping into the conviction that these ancient and medieval algebraists were not very bright. We started in 1800 BCE with the Babylonians solving quadratic equations written as word problems, and now here we are 2,600 years later with al-Khwarizmi … solving quadratic equations written as word problems.

It is, I agree, all a bit depressing. Yet it is also inspiring, in a way. The extreme slowness of progress in putting together a symbolic algebra testifies to the very high level at which this subject dwells. The wonder, to borrow a trope from Dr. Johnson, is not that it took us so long to learn how to do this stuff; the wonder is that we can do it at all.

And in fact things began to pick up a little in these middle Middle Ages.²⁶ Al-Khwarizmi was followed by other mathematicians of note operating in Muslim lands, both eastern and western. Thabit ibn Qurra, of the generation after al-Khwarizmi and also based in Baghdad, did notable work in mathematical astronomy and the pure theory of numbers. A century and a half later, Mohammed al-Jayyani of Cordova in Muslim Spain wrote the first treatise on spherical trigonometry. None of them made significant progress in algebra, though. In particular, none attempted to replicate Diophantus’s great leap into literal symbolism. All spelled out their problems in words, words, words.

I am going to give detailed coverage to only one other mathematician from medieval Islam, partly because he is worth covering, but also as a bridge to the Europe of the early Renaissance, where things really begin to pick up.

§3.7 Omar Khayyam is best known in the West as the author of the Rubaiyat, a collection of four-line poems offering a highly personal view of life—a sort of death-haunted hedonism with an alcoholic thread, somewhat prefiguring A. E. Housman. Edward Fitzgerald turned 75 of these into English quatrains, each rhymed a-a-b-a, in a translation published in 1859, and Fitzgerald’s The Rubaiyat of Omar Khayyam was a great favorite all over the English-speaking world up to World War I. (An elaborate jeweled copy of the original went down with the Titanic.)

The DSB gives 1048–1131 as the most probable dates for Khayyam, and those are the dates I shall use faute de mieux. This puts Khayyam at least 250 years after al-Khwarizmi. It is worth bearing in mind these great gulfs of time when surveying intellectual activity in the Middle Ages.

The region in which Khayyam lived and worked was at the eastern end of the first great zone of Islamic conquest. That eastern-most region included Mesopotamia, the northern part of present-day Iran, and the southern part of Central Asia (present-day Turkmenistan, Uzbekistan, Tajikistan, and Afghanistan). In Khayyam’s time this was a region of both ethnic and religious conflict. The principal ethnies involved were the Persians, the Arabs, and the Turks. The religious conflict was all within Islam: first between Sunnis and Shias, then between two factions of Shias, the main body and the split-off Ismailites.²⁷

The Turks, originally nomads from farther Central Asia, had been hired as mercenaries by the declining Abbasids. Of course, the Turks soon realized what the true balance of power in the relationship was, and the later Abbasid caliphs, excepting a short-lived revival in the late 9th century, were puppets of their Turkish guards. Their only consolation was that the Turks had at least converted to orthodox (that is, Sunni) Islam. The farther eastern territories, beyond Mesopotamia, were anyway lost by the Abbasids to a Persian (and Shiite) revival in the 10th century. These short-lived Persian dynasties hired Turkish troops just as the Abbasids had. In due course a

Turkish general overthrew his Persian masters, establishing the Ghaznavid dynasty—the first Turkish empire. Wisely deferring to the now-subjugated Persians in matters of statecraft and high culture, the Ghaznavids ran a decent court, adorned with famous Persian scholars and poets. They also conducted several invasions of south Asia, carrying Islam to the region occupied by present-day Pakistan and India.

In 1037, a few years before Omar Khayyam was born, a man named Seljuk, one of the Ghaznavids’ own Turkish mercenaries, rebelled and defeated the Ghaznavid armies. This new Turkish power expanded very fast. In 1055, when Omar was seven years old, Seljuk’s grandson took Baghdad and gave himself the title of Sultan, meaning “ruler.” The implication here was that the Caliph’s power was now to be merely spiritual, like the Pope’s.

The Seljuks ruled all the eastern territories of Islam through the remainder of the 11th century and much of the 12th. Their dominions extended all the way westward to the Holy Land and the borders of Egypt (at this time under Shiite rulers, of the Ismailite subpersuasion—the Seljuks were orthodox Sunnis). It was the Seljuk defeat of the Byzantine Emperor at the battle of Manzikert in 1071 that won them Anatolia, laying the first foundations of modern Turkey. It was the loss of Anatolia that caused the Byzantines to call on Western Europe for aid, thus precipitating the Crusades. And it was Seljuk Turks that the Crusaders faced on their trek across Anatolia to the Holy Land and at Antioch and Jerusalem.

§3.8 Omar Khayyam’s life was therefore spent under the rule of the Seljuk Turks. His great patron was the third Seljuk sultan, Malik Shah, who ruled from 1073 to 1092 from his capital city of Esfahan in present-day Iran, 440 miles east of Baghdad.²⁸ Malik Shah is less famous than his vizier Nizam al-Mulk, one of the greatest names in the history of statecraft, a genius of diplomacy. Al-Mulk, like Khayyam, was a Persian. The two of them are sometimes spoken of together

with Hasan Sabbah, founder of the Assassins sect,²⁹ as “the three Persians” of their time—that is, the three important men of the Persian ethny in the Seljuk empire.

Malik Shah’s court seems to have been easygoing in matters of religion, as such things went in medieval Islam. This probably suited Khayyam very well. His poems show a skeptical and agnostic attitude to life, and his contemporaries often spoke of him as a freethinker. Anxious mainly to get on with his work—he was director of the great observatory as Esfahan—and studies, Khayyam did his best to stay out of trouble, writing orthodox religious tracts to order and probably performing the pilgrimage to Mecca that is every Muslim’s duty. From the poems and such biographical facts as are available, Khayyam strikes the modern reader as rather simpatico.

His main interest for algebraists is a book he wrote in his 20s before going to Esfahan. The book’s title is Risala fi’l-barahin ’ala masa’il al-jabr wa’l-muqabala, or “On the Demonstration of Problems in Completion and Reduction.”

Like al-Khwarizmi and all the other medieval Muslim mathematicians, Khayyam ignores, or was altogether ignorant of, Diophantus’s great breakthrough into literal symbolism. He spells out everything in words. Also, like the older Greeks, he has a strongly geometric approach, turning naturally to geometric methods for the solution of numerical problems.

Khayyam’s main importance for the development of algebra is that he opened the first serious assault on the cubic equation. Lacking any proper symbolism and apparently unwilling to take negative numbers seriously, Khayyam was laboring under severe handicaps. The equation that we would write as x³ + ax = b, for example, was expressed by Khayyam as: “A cube and a number of sides are equal to a number.” Nonetheless he posed and solved several problems involving cubic equations, though his solutions were always geometrical.

This is not quite the first appearance of the cubic equation in history. Diophantus had tackled some, as we have seen. Even before that, Archimedes had bumped up against cubics when deliberating

on such problems as the division of a sphere into two parts whose volumes have some given ratio.³⁰ (The connection with Archimedes’ interest in floating bodies will occur to you if you think about this for a minute.) Khayyam seems to have been the first to recognize cubic equations as a distinct class of problems, though, and offered a classification of them into 14 types, of which he knew how to solve four by geometrical means.

As an example of the kind of problem Khayyam reduced to a cubic equation, consider the following:

The answer is that the ratio of the triangle’s shortest side to the next shortest—a ratio that completely determines the triangle’s shape—must satisfy the cubic equation

The only real-number solution of this equation is 0.647798871 …, an irrational quantity very close to the rational number . So a right-angled triangle with short sides 103 and 159 very nearly fills the bill, as the reader can easily verify.³¹ Khayyam took an indirect approach, ending up with a slightly different cubic, which he solved numerically via the intersection of two classic geometric curves.

§3.9 To offer an extremely brief summary of events to date:

• The ancient Babylonians developed some techniques for solving a limited range of linear and quadratic equations with one unknown.

• The later Ancient Greeks tackled similar equations geometrically.

• Diophantus, in the 3rd century, broadened the scope of inquiry to many other kinds of equations, including equations of higher degree, equations with many variables, and systems of simultaneous equations. He also developed the first literal symbolism for algebraic problems.

• Medieval Islamic scholars gave us the word “algebra.” They began to focus on equations as worthwhile objects of inquiry in themselves and classified linear, quadratic, and cubic equations according to how difficult it was to solve them with the techniques available.

In discussing Omar Khayyam, I mentioned the terrible battle of Manzikert, the great retreat of Eastern Christendom that followed it, and the strange, disorderly, and still controversial reaction to that retreat: the Crusades. By the time of these events—Khayyam’s time—Western European culture was beginning to struggle to its feet after the Dark Ages. The lights came on earliest and brightest in Italy, and it is there that we meet our next few algebraists.