Algebra Draft for the History of Scientific Thought
M. Meo, Benson Polytechnic High School July 2006
There are various ways to define algebra, in part due to its long and complex history. One intuitive definition says algebra is simply the rules of arithmetic (which is true at the introductory level). You could enlarge that -- and improve its accuracy -- by saying that algebra is the study of the rules of arithmetic. In fact, that was how the Arabic mathematician who wrote the first book of algebra came up with the idea; he had to describe the rules of arithmetic because he wanted to use the new Hindu numerals, including zero. First, he wrote a book to explain the new numerals, and then he went on to show the usefulness of general rules for reducing equations. The new methods for the first time produced a general solution for all quadratic equations -- other mathematicians were very impressed. Factbox A quadratic equation is any equation that can be written as
ax2 +bx+c=0 ,
where a, b, and c are known constants and x is variable. End Factbox Since that time, the middle of the ninth century, algebra has become one of the largest and widely-used branches of mathematics. It hasn’t produced solutions to all equations, but it has shown how we can expand the definition of numbers to ordered pairs, quartets, and octets, and still keep reasonable addition, subtraction, multiplication, and division. Al-Khwarizmi’s innovation has become the central discipline in the modeling of the world according to mathematical laws. From a philosophic point of view, the creation of algebra may have initiated the Scientific Revolution.
The Birth of Algebra Al-Khwarizmi -- the mathematician whose combination of calculation, rules for calculation, and use of geometric models established algebra as a separate discipline -- worked as a government employee, supported by the caliph of Baghdad during his lifetime, from about 770 to 850 AD, approximately. The caliph al-Wathiq, who ruled from 842 to 847, when seriously ill called upon alKhwarizmi to predict from his horoscope how long he had yet to live. Al-Khwarizmi, who compiled a catalogue of stars and measured a degree of arc along the surface of the earth, assured the caliph that the stars predicted another fifty years of life; the ruler expired ten days later. One commentator suggests that this story shows al-Khwarizmi’s political shrewdness, in telling his boss what he wanted to hear. Marriage of Theory and Practice Like almost all of the scientists and scholars writing in Arabic, alKhwarizmi was born on the periphery of the Islamic empire, not in the Arabic peninsula. He came from a small Turkic state, Khwarizm, in the present country of Uzbekistan. It was near the mouth of the Oxus River, where it empties into the Aral Sea, wellsituated to serve as a trading center for traffic to and from China and India from the capital city Baghdad, in Iraq. Factbox [map showing Khwarizm in context of Islamic Empire] End Factbox “Just as the caliphate brought the lands of the Indian Ocean and the Mediterranean Sea into a single trading area, so too the Greek, Iranian and Indian traditions were brought together,” wrote the English historian Albert Hourani in discussing al-Khwarizmi in A History of the Arab Peoples. In writing the description of arithmetic using the ten Hindu numerals,
al-Khwarazmi wrote in the preface that he performed this work, at the request of the caliph, for practical purposes. He continued this orientation toward practice with his book on algebra. The last half of the book contains problems in the division of family legacies according to Islamic legal requirements. For example, “a woman dies, leaving her husband, a son and three daughters,” and the object is to calculate the fraction of her estate that each heir receives, given the law that the husband receives one-quarter of his wife’s legacy and sons receive twice as much as daughters. Algebraic rules existed before al-Khwarizmi, but he first provided geometric proofs for them, and then showed how they could be organized in a clear manner to solve any quadratic equation. At the birth of algebra a combination of Greek geometric step-by-step proof with rules for arithmetic calculation coincided with a concern for commercial, even monetary, application of the resulting mathematical tool. Practical needs led to generalization, and the generalization made practical problems easier to solve. Thabit ibn Qurra Nor was al-Khwarizmi the only Arabic-writing mathematician to look closely at the intersection of practical needs and theoretical mathematics (in striking contrast to the ancient Greek disdain for practical needs for, and application of, mathematics). In the ninth century Thabit ibn Qurra wrote a treatise on the Verification of the Rules of Algebra which appealed to the proofs in Euclid’s Elements -the ancient Greek textbook in geometry, dating from 300 BC; his contemporary Abu l-Wafa contributed a Book on What Is Necessary from Geometric Construction for the Artisan. Thabit personally also exemplifies the rich combination of cultures which produced the distinctively Islamic study of algebra. Three wealthy brothers, merchants of Baghdad named the Banu Musa (the sons of Moses -- the suggestion is of Jewish roots), purchased
manuscript copies of the ancient Greek works of science on their travels to the Byzantine empire. At the border between the Islamic and the Byzantine empires they encountered Thabit, working as a money changer. Impressed with his skills in Greek, Syriac, and Arabic, they brought him back to Baghdad and sponsored his activity as translator -- he translated Euclid, Archimedes, Ptolemy, and Appolonius of Perga -- and as mathematician. Thabit generalized the Pythagorean theorem, creating a relation that applied to triangles even if they were not right triangles. He was honored for his medical skill by the caliph with an appointment at court. But Thabit was a member of a sect of star worshippers who used an assumed name for their religion to escape forced conversion to Islam, which condemned all polytheistic religions. This theological dodge, which referred to a name in the Koran, let him rise to high official status without changing his religion. One expert points to the suffix “al-Magi” in al-Khwarizmi’s name as evidence that he was from a family of priests of Zoroastrianism -- fire worshippers -which pagan belief was common in Khwarizm. Islam in the eighth, ninth, and tenth centuries was a more tolerant environment than the contemporary Christian world. Diffusion into Europe In subsequent centuries the extended central authority of the early Islamic empire, as well as the intellectual confidence that promoted active investigation of foreign ideas, collapsed. Mathematical thinkers with the Islamic lands continued to develop new tools: they showed how to take roots of higher order than square roots; they developed fractions in decimal form; they identified and explored problems in the geometry and astronomy of the ancient Greek authorities Euclid and Ptolemy. The framework of algebra, its scope and method, however, remained as it had been defined about 830 AD. It is interesting to note that the Christian peoples of Europe, once they
became aware of the work of al-Khwarizmi (in the course of the twelfth century, approximately), adopted the Hindu-Arabic numerals within a couple of centuries, a shorter period of time than the Hindu-Arabic numerals took to be adopted throughout medieval Islam. In contrast, the creative development of algebraic formulas took much longer, until the work of René Descartes (15951650) during the Scientific Revolution of the seventeenth century. Two cultural causes made the adoption of decimal numerals more rapid in Christian Europe than the assimilation and use of algebra. First, it is easier to change the way you do arithmetic than it is to make a detailed study of the rules of arithmetic: that is, decimals were not so much of a challenge for students of mathematics in Renaissance Italy as was the quadratic formula. Factbox The Quadratic Formula states if
ax2 +bx+c=0 ,
where a, b, and c are known constants and x is variable, then
End Factbox
−b ± b2 − 4ac x= 2a
Second, merchants who used the new decimal notation in their accounts were highly motivated to develop a system everyone could understand; it was during the fourteenth century that Italian bankers began the use of double-entry bookkeeping to display the assets and liabilities of every commercial transaction. On the other hand, mathematicians, who in Europe were scholars at universities, tried hard to keep their ideas secret. Solution of the Cubic When Niccolo Tartaglia (1499 -1559) [born Niccolo Fontana] managed to derive, by algebraic methods similar to those of Arabic writers
on algebra, the complete solution to the general cubic, he employed his discovery in order to defeat Antonio Maria Fior in an academic contest, held in Venice in public, to see who could solve the most difficult cubic equations. Shortly afterwards the academic entrepreneur Girolamo Cardano, under pledge of secrecy, persuaded Tartaglia to explain how he had done it. Factbox The general cubic equation is any equation that can be written as
ax3 +bx2 +cx+d=0 ,
where a, b, c, and d are known constants and x is variable. End Factbox Cardano then published the secret in 1545 in his book Ars Magna (that is, in Latin, the “great art,” namely algebra). Even though Cardano had given him full credit, Tartaglia protested that he had been robbed. Cardano, who was as rough as his Renaissance Italian peers -- his favorite son murdered his own wife, for which crime Cardano blamed the victim, his daughter-in-law -- gave Tartaglia no satisfaction. Indeed, Cardano’s friends threatened Tartaglia’s life. The violence of his society had already taken a toll on Tartaglia: as a teenaged boy he had been in Brescia when the besieged city was taken by the French army, and all within the city (the family had taken refuge in the cathedral) were subjected to a massacre. Tartaglia’s father was killed, and he was so severely slashed with a saber that his skull and jaw were both cleaved open. The only medical care he received was that provided by his grieving mother: imitating what she had seen done by injured dogs, she licked his wounds with her tongue. Although Tartaglia lived, he always spoke with a stammer: his palate never knit together. In his autobiography he recalls that his mother scraped together enough money to send him to school, but only for
fifteen days. He stole a copybook and taught himself to read and write. From a modest social background and sunk in poverty, he nevertheless mastered the mathematics of the time and produced the first Italian translation of the ancient Greek geometer Euclid, who lived in Egypt in 300 BC, as well as a Latin translation of the Greek mathematician Archimedes, who lived in Sicily around 200 BC. Tartaglia was an unusually gifted mathematician, who responded to the taunts of that fashion-conscious age by adopting the nickname that had been hurled at him -- Tartaglia, “the stammerer”. Reformation and Scientific Revolution Thanks to its central location Italy profited substantially from the growth of trade throughout the Mediterranean basin in the centuries which followed the failure of Christian armies to subjugate Syria and Palestine to military rule during the Crusades, which ended in 1291 with the evacuation of Acre in what is now Israel. The pious contributions from all parts of Christian Europe, from Iceland to Poland, flowing into Rome for the support of the Church further aided Italian prosperity. The outpouring of wonderful art and architecture, the marvelous works of poetry, sculpture, and music came with the decisive victory of the elites in Italian communes in their centuries-long struggle with the “lower” -- less wealthy, less powerful -- classes. Although contemporaries announced the discovery of the natural world and of the dignity of man, in reality the paid minstrels of the rich sang for their supper in beautiful harmony. “There was intense cultivation of every aspect of classical learning,” wrote Norman F. Cantor in The Civilization of the Middle Ages, speaking of Italy during the fifteenth and sixteenth centuries, “except the two that would have had the greatest social impact -- mathematical literacy and republicanism. The former would have ignited the scientific revolution, the latter a democratic upheaval.” The Reformation, the revolt of the founders of Protestant Christianity
against Roman Catholicism, which was led by Luther and Calvin, stripped the elites of their complacency. Galileo Galilei, discoverer of the satellites of Jupiter, was sentenced to life-long house arrest for the crime of heresy; Giordano Bruno, convinced of the existence of infinitely many solar systems, was burned alive for it. A European-wide witch hunt, conducted with equal fervor by Protestant and Catholic alike, snuffed out the lives of hundreds of thousands of tortured and mutilated innocents. The very violence of the dispute over the truth of the Christian religion, the brutality of the wars which flared in Holland, France, and Germany in the last half of the sixteenth and first half of the seventeenth century, promoted a willingness to construct an experimentally-tested, mathematically-proved world picture. That is, it generated the Scientific Revolution which the manifest artistic beauty of the Italian Renaissance did not. The Birth of Analytic Geometry In these circumstances René Descartes, who considered himself just as loyal a Roman Catholic as Galileo, prudently left officially Catholic France to live in exile in Calvinist, but tolerant, Holland. Much more than Galileo, Descartes aspired to reformulate philosophy. In place of an experimental inquiry into the secrets of nature, he took as his model of all of reality the certainty available in algebraic deduction. In a 1630 manuscript which he never published, he proposed 1) to transform any sort of problem to a problem in mathematics; 2) to transform any mathematical problem to one in algebra; and 3) to reduce any algebraic problem to a single equation. Only in an environment where all manner of beliefs were in question could such a research program have had any appeal. Al-Khwarizmi had used the step-by-step method of geometric proof, and had shown with geometric diagrams how “completing the
square” led to a solution of quadratic equations; but Descartes envisioned mathematical relationships of all sorts as displayed along two co-ordinate axes -- one to represent the possible values of the unknown, and the other to show how the conditions placed on the unknown would look for each value of the unknown. For all mathematicians prior to Descartes, x 2 could only be an area : for him it was a distance along an axis showing the value of the the algebraic function of x. For all before Descartes’ time, x 3 had to be the volume of a cube, with x a distance along one edge: for him it was exactly on the same footing as x 2 -- a distance along the y-axis. The fourth and higher powers of the unknown, previously impossible to represent, for the first time joined straight lines, quadratics, and cubics as curves in the co-ordinate plane. The treatment of all polynomials became immediately unified. Using his new system Descartes solved with relative ease problems that had remained unsolved by any ancient Greek geometer. On the strength of that unquestionable success he went on to a host of dubious assertions in physics, astronomy, physiology, and psychology, not to mention philosophy and theology. Algebra was at the center of the scientific revolution which replaced Aristotelian philosophy with Cartesianism. Personally Descartes was not very congenial -- he quarreled with most of his contemporaries, and on his independent income he chose to live alone in a small town. Despite his published “life maxim” to “follow the laws and customs of my country,” he fathered an illegitimate daughter with the woman who worked as his housekeeper; it is true, however, that for a nobleman of modest wealth in those days to have married his housekeeper because she had borne him a child would not have been customary. In any case, Descartes never married; the scholars who have found signs of nervous breakdowns on at least two occasions in his life note his unusual good cheer and productivity during the five years of his daughter’s life (indeed, it was then that Descartes published his analytic geometry). Popular legend imagines the aloof man of
reason constructing a robot in the shape of a girl, naming it after his daughter, and carrying it with him in his travels. The Revolution Exploited The same dedication to the cause of recasting all of philosophy on the basis of “clear and distinct ideas” that led Descartes to his great insights into the flexibility of algebra also led him to slight the opportunities for exploiting those insights. The mathematicians who read him, however, adopted his advances immediately. It is hard to imagine Isaac Newton (1642-1727) or G. W. Leibniz (16461716), working fifty years after Descartes, developing the differential and integral calculus without making use of Descartes’ analytical geometry. The most intuitive picture we have of the derivative of a function is that of a tangent line in the Cartesian coordinate plane; that of an integral is the area under the curve -again, only where the curve of the function is represented in the Cartesian co-ordinate system. The philosophic and theological considerations on which Descartes focused his attention in the years following his invention of analytic geometry hampered his younger contemporary Blaise Pascal (1623-1662) even more.