History of algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property). This article describes the history of the theory of equations, called here "algebra", from the origins to the emergence of algebra as a separate area  of mathematics. Etymology. The word "algebra" is derived from the Arabic word الجبر "al-jabr", and this comes from the treatise written in the year 830 by the medieval Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, whose Arabic title, "Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala", can be translated as "The Compendious Book on Calculation by Completion and Balancing". The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "[i]t is not certain just what the terms "al-jabr" and "muqabalah" mean, but the usual interpretation is similar to that implied  in the previous translation. The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation. Arabic influence in Spain long after the time of al-Khwarizmi is found in "Don Quixote", where the word 'algebrista' is used for a bone-setter, that is, a 'restorer'." The term is used by al-Khwarizmi to describe the operations that he introduced, "reduction  and balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. Stages of algebra. Algebraic expression. Algebra did not always make use of the symbolism that is now ubiquitous in mathematics; instead, it went through three distinct stages. The stages in the development of symbolic algebra are approximately as follows: Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most  of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories. where formula_5 and formula_6 are positive. This trichotomy comes about because quadratic equations of the form formula_7with formula_5 and formula_6 positive, have no positive roots. In between the rhetorical and syncopated stages of symbolic algebra, a geometric constructive algebra was developed by classical Greek and Vedic Indian mathematicians in which algebraic equations were solved through geometry. For instance, an equation of the form formula_10 was solved by finding the side of a square of area formula_11 Conceptual stages. In addition to  the three stages of expressing algebraic ideas, some authors recognized four conceptual stages in the development of algebra that occurred alongside the changes in expression. These four stages were as follows: Babylon. The origins of algebra can be traced to the ancient Babylonians, who developed a positional number system that greatly aided them in solving their rhetorical algebraic equations. The Babylonians were not interested in exact solutions, but rather approximations, and so they would commonly use linear interpolation to approximate intermediate values. One of the most famous tablets is the Plimpton 322 tablet, created around 1900–1600 BC, which gives a  table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek mathematics. Babylonian algebra was much more advanced than the Egyptian algebra of the time; whereas the Egyptians were mainly concerned with linear equations the Babylonians were more concerned with quadratic and cubic equations. The Babylonians had developed flexible algebraic operations with which they were able to add equals to equals and multiply both sides of an equation by like quantities so as to eliminate fractions and factors. They were familiar with many simple forms of factoring, three-term quadratic equations with positive roots, and many  cubic equations, although it is not known if they were able to reduce the general cubic equation. Ancient Egypt. Ancient Egyptian algebra dealt mainly with linear equations while the Babylonians found these equations too elementary, and developed mathematics to a higher level than the Egyptians. The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written c. 1650 BC by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 BC. It is the most extensive ancient Egyptian mathematical document known to historians. The Rhind Papyrus contains problems where linear  equations of the form formula_12 and formula_13 are solved, where formula_14 and formula_15 are known and formula_16 which is referred to as "aha" or heap, is the unknown. The solutions were possibly, but not likely, arrived at by using the "method of false position", or "regula falsi", where first a specific value is substituted into the left hand side of the equation, then the required arithmetic calculations are done, thirdly the result is compared to the right hand side of the equation, and finally the correct answer is found through the use of proportions. In some of the problems the  author "checks" his solution, thereby writing one of the earliest known simple proofs. Greek mathematics. It is sometimes alleged that the Greeks had no algebra, but this is inaccurate. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them, and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas". "The application of areas" is only a part  of geometric algebra and it is thoroughly covered in Euclid's "Elements". An example of geometric algebra would be solving the linear equation formula_17 The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios formula_18 and formula_19 The Greeks would construct a rectangle with sides of length formula_20 and formula_21 then extend a side of the rectangle to length formula_22 and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution. Bloom of Thymaridas. Iamblichus in  Introductio arithmatica says that Thymaridas (c. 400 BCE – c. 350 BCE) worked with simultaneous linear equations. In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that: If the sum of formula_23 quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to formula_24 of the difference between the sums of these pairs and the first given sum. or using modern notation, the solution of the following system of formula_23 linear equations in formula_23 unknowns, is , formula_32 Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form. Euclid of Alexandria. Euclid (Greek: ) was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BCE). Neither the year nor place of his birth have been established, nor the circumstances of his death. Euclid is regarded as the "father of geometry". His "Elements" is the most successful textbook in the history of mathematics. Although he is one of the most famous mathematicians in history there are no new  discoveries attributed to him; rather he is remembered for his great explanatory skills. The "Elements" is not, as is sometimes thought, a collection of all Greek mathematical knowledge to its date; rather, it is an elementary introduction to it. Section::::"Elements". The geometric work of the Greeks, typified in Euclid's "Elements", provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations. Book II of the "Elements" contains fourteen propositions, which in Euclid's time were extremely significant for doing geometric algebra. These propositions and their results are the geometric  equivalents of our modern symbolic algebra and trigonometry. Today, using modern symbolic algebra, we let symbols represent known and unknown magnitudes (i.e. numbers) and then apply algebraic operations on them, while in Euclid's time magnitudes were viewed as line segments and then results were deduced using the axioms or theorems of geometry. Many basic laws of addition and multiplication are included or proved geometrically in the "Elements". For instance, proposition 1 of Book II states: But this is nothing more than the geometric version of the (left) distributive law, formula_33; and in Books V and VII of the "Elements  the commutative and associative laws for multiplication are demonstrated. Many basic equations were also proved geometrically. For instance, proposition 5 in Book II proves that formula_34 and proposition 4 in Book II proves that formula_35 Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation formula_36 and proposition 11 of Book II gives a solution to formula_37 Section::::Data". "Data" is a work written by Euclid for use at the schools of Alexandria and it was meant to be used as a companion volume to the first  six books of the "Elements". The book contains some fifteen definitions and ninety-five statements, of which there are about two dozen statements that serve as algebraic rules or formulas. Some of these statements are geometric equivalents to solutions of quadratic equations. For instance, "Data" contains the solutions to the equations formula_38 and the familiar Babylonian equation formula_39 Conic sections. A conic section is a curve that results from the intersection of a cone with a plane. There are three primary types of conic sections: ellipses (including circles), parabolas, and hyperbolas. The conic sections are reputed to have been discovered  by Menaechmus (c. 380 BC – c. 320 BC) and since dealing with conic sections is equivalent to dealing with their respective equations, they played geometric roles equivalent to cubic equations and other higher order equations. Menaechmus knew that in a parabola, the equation formula_40 holds, where formula_41 is a constant called the latus rectum, although he was not aware of the fact that any equation in two unknowns determines a curve. He apparently derived these properties of conic sections and others as well. Using this information it was now possible to find a solution to the problem of the duplication  of the cube by solving for the points at which two parabolas intersect, a solution equivalent to solving a cubic equation. We are informed by Eutocius that the method he used to solve the cubic equation was due to Dionysodorus (250 BC – 190 BC). Dionysodorus solved the cubic by means of the intersection of a rectangular hyperbola and a parabola. This was related to a problem in Archimedes' "On the Sphere and Cylinder". Conic sections would be studied and used for thousands of years by Greek, and later Islamic and European, mathematicians. In particular Apollonius of Perga's famous "Conics  deals with conic sections, among other topics. China. Chinese Mathematics dates to at least 300 BC with the Zhoubi Suanjing", generally considered to be one of the oldest Chinese mathematical documents. Section::::"Nine Chapters on the Mathematical Art". "Chiu-chang suan-shu" or "The Nine Chapters on the Mathematical Art", written around 250 BC, is one of the most influential of all Chinese math books and it is composed of some 246 problems. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns. Section::::"Sea -Mirror of the Circle Measurements". "Ts'e-yuan hai-ching", or "Sea-Mirror of the Circle Measurements", is a collection of some 170 problems written by Li Zhi (or Li Ye) (1192 – 1279 CE). He used "fan fa", or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations. Section::::"Mathematical Treatise in Nine Sections". "Shu-shu chiu-chang", or "Mathematical Treatise in Nine Sections", was written by the wealthy governor and minister Ch'in Chiu-shao (c. 1202 – c. 1261) and with the invention of a method of solving simultaneous  congruences, now called Chinese remainder theorem, it marks the high point in Chinese indeterminate analysis. Magic squares. The earliest known magic squares appeared in China. In "Nine Chapters" the author solves a system of simultaneous linear equations by placing the coefficients and constant terms of the linear equations into a magic square (i.e. a matrix) and performing column reducing operations on the magic square. The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. c. 1261 – 1275), who worked with magic squares of order as high as ten. Section::::"Precious Mirror of the Four Elements . Ssy-yüan yü-chien"《四元玉鑒》, or "Precious Mirror of the Four Elements", was written by Chu Shih-chieh in 1303 and it marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. The "Ssy-yüan yü-chien" deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of "fan fa", today called Horner's method, to solve these equations. The "Precious Mirror" opens with a diagram of the arithmetic triangle (Pascal's triangle) using a round  zero symbol, but Chu Shih-chieh denies credit for it. A similar triangle appears in Yang Hui's work, but without the zero symbol. There are many summation equations given without proof in the "Precious mirror". A few of the summations are: Diophantus. Diophantus was a Hellenistic mathematician who lived c. 250 CE, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written "Arithmetica", a treatise that was originally thirteen books but of which only the first six have survived. "Arithmetica" has very little in  common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations. It is usually rather difficult to tell whether a given Diophantine equation is solvable. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. Also, no general method may be abstracted from all Diophantus' solutions. In "Arithmetica", Diophantus is the first to use symbols for unknown numbers as well  as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as "syncopated" algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So, for example, what we would write as which can be rewritten as would be written in Diophantus's syncopated notation as where the symbols represent the following: Unlike in modern notation, the coefficients come after the variables and that addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated  equation into a modern symbolic equation would be the following: where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as: "Arithmetica" is a collection of some 150 solved problems with specific numbers and there is no postulational development nor is a general method explicitly explained, although generality of method may have been intended and there is no attempt to find all of the solutions to the equations. "Arithmetica" does contain solved problems involving several unknown quantities, which are solved, if possible, by expressing the unknown quantities in terms of only one  of them. "Arithmetica" also makes use of the identities: India. The Indian mathematicians were active in studying about number systems. The earliest known Indian mathematical documents are dated to around the middle of the first millennium BC (around the 6th century BC). The recurring themes in Indian mathematics are, among others, determinate and indeterminate linear and quadratic equations, simple mensuration, and Pythagorean triples. Section::::"Aryabhata". Aryabhata (476–550) was an Indian mathematician who authored "Aryabhatiya". In it he gave the rules, and Section::::"Brahma Sphuta Siddhanta". Brahmagupta (fl. 628) was an Indian mathematician who authored "Brahma Sphuta Siddhanta". In his work Brahmagupta solves  the general quadratic equation for both positive and negative roots. In indeterminate analysis Brahmagupta gives the Pythagorean triads formula_52 formula_53 but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with. He was the first to give a general solution to the linear Diophantine equation formula_54 where formula_14 and formula_15 are integers. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave "all" integer solutions; but that Brahmagupta used some of the same examples as Diophantus has led some historians to consider the possibility of a Greek influence on Brahmagupta 's work, or at least a common Babylonian source. Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our modern notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source. Bhāskara  II. Bhāskara II (1114 – c. 1185) was the leading mathematician of the 12th century. In Algebra, he gave the general solution of Pell's equation. He is the author of "Lilavati" and "Vija-Ganita", which contain problems dealing with determinate and indeterminate linear and quadratic equations, and Pythagorean triples and he fails to distinguish between exact and approximate statements. Many of the problems in "Lilavati" and "Vija-Ganita" are derived from other Hindu sources, and so Bhaskara is at his best in dealing with indeterminate analysis. Bhaskara uses the initial symbols of the names for colors as the symbols of  unknown variables. So, for example, what we would write today as Bhaskara would have written as where "ya" indicates the first syllable of the word for "black", and "ru" is taken from the word "species". The dots over the numbers indicate subtraction. Islamic world. The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the 8th century, Islam had a cultural awakening, and research in  mathematics and the sciences increased. The Muslim Abbasid caliph al-Mamun (809–833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's "Almagest" and Euclid's "Elements". Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace. Many of these Greek works were translated by Thabit ibn Qurra (826–901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius. Arabic mathematicians  established algebra as an independent discipline, and gave it the name "algebra" ("al-jabr"). They were the first to teach algebra in an elementary form and for its own sake. There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources. Throughout their time in power, the Arabs used a fully rhetorical algebra, where often even the numbers were spelled out in words. The Arabs would eventually  replace spelled out numbers (e.g. twenty-two) with Arabic numerals (e.g. 22), but the Arabs did not adopt or develop a syncopated or symbolic algebra until the work of Ibn al-Banna, who developed a symbolic algebra in the 13th century, followed by Abū al-Hasan ibn Alī al-Qalasādī in the 15th century. Section::::"Al-jabr wa'l muqabalah". The Muslim Persian mathematician Muhammad ibn Mūsā al-Khwārizmī was a faculty member of the "House of Wisdom" ("Bait al-Hikma") in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 CE, wrote more than half a dozen  mathematical and astronomical works, some of which were based on the Indian "Sindhind". One of al-Khwarizmi's most famous books is entitled "Al-jabr wa'l muqabalah" or "The Compendious Book on Calculation by Completion and Balancing", and it gives an exhaustive account of solving polynomials up to the second degree. The book also introduced the fundamental concept of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as "al -jabr". The name "algebra" comes from the ""al-jabr"" in the title of his book. R. Rashed and Angela Armstrong write: "Al-Jabr" is divided into six chapters, each of which deals with a different type of formula. The first chapter of "Al-Jabr" deals with equations whose squares equal its roots formula_58 the second chapter deals with squares equal to number formula_59 the third chapter deals with roots equal to a number formula_60 the fourth chapter deals with squares and roots equal a number formula_61 the fifth chapter deals with squares and number equal roots formula_62 and the sixth  and final chapter deals with roots and number equal to squares formula_63 In "Al-Jabr", al-Khwarizmi uses geometric proofs, he does not recognize the root formula_64 and he only deals with positive roots. He also recognizes that the discriminant must be positive and described the method of completing the square, though he does not justify the procedure. The Greek influence is shown by "Al-Jabr"'s geometric foundations and by one problem taken from Heron. He makes use of lettered diagrams but all of the coefficients in all of his equations are specific numbers since he had no way  of expressing with parameters what he could express geometrically; although generality of method is intended. Al-Khwarizmi most likely did not know of Diophantus's "Arithmetica", which became known to the Arabs sometime before the 10th century. And even though al-Khwarizmi most likely knew of Brahmagupta's work, "Al-Jabr" is fully rhetorical with the numbers even being spelled out in words. So, for example, what we would write as Diophantus would have written as And al-Khwarizmi would have written as Section::::"Logical Necessities in Mixed Equations". 'Abd al-Hamīd ibn Turk authored a manuscript entitled "Logical Necessities in  Mixed Equations", which is very similar to al-Khwarzimi's "Al-Jabr" and was published at around the same time as, or even possibly earlier than, "Al-Jabr". The manuscript gives exactly the same geometric demonstration as is found in "Al-Jabr", and in one case the same example as found in "Al-Jabr", and even goes beyond "Al-Jabr" by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the  time of al-Khwarizmi and 'Abd al-Hamid. Abu Kamil and al-Karkhi. Arabic mathematicians treated irrational numbers as algebraic objects. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. He was also the first to solve three non-linear simultaneous equations with three unknown variables. Al-Karkhi (953–1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940–998) and he discovered the  first numerical solution to equations of the form formula_68 Al-Karkhi only considered positive roots. Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in "Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi" (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this , Al-Karaji investigated binomial coefficients and Pascal's triangle. Omar Khayyám, Sharaf al-Dīn, and al-Kashi. Omar Khayyám (c. 1050 – 1123) wrote a book on Algebra that went beyond "Al-Jabr" to include equations of the third degree. Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible. His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Ibn al-Haytham (Alhazen), but Omar Khayyám generalized the method to cover all cubic  equations with positive roots. He only considered positive roots and he did not go past the third degree. He also saw a strong relationship between geometry and algebra. In the 12th century, Sharaf al-Dīn al-Tūsī (1135–1213) wrote the "Al-Mu'adalat" ("Treatise on Equations"), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed the concepts of the maxima and minima of  curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes. Sharaf al-Din also developed the concept of a function. In his analysis of the equation formula_69 for example, he begins by changing  the equation's form to formula_70. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value formula_71. To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when formula_72, which gives the functional value formula_73. Sharaf al-Din then states that if this value is less than formula_71, there are no positive solutions; if it is equal to formula_71, then there is one solution at formula_72; and if it is greater than formula_71, then there  are two solutions, one between formula_78 and formula_79 and one between formula_79 and formula_20. In the early 15th century, Jamshīd al-Kāshī developed an early form of Newton's method to numerically solve the equation formula_82 to find roots of formula_83. Al-Kāshī also developed decimal fractions and claimed to have discovered it himself. However, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. Al-Hassār, Ibn al-Banna, and al-Qalasadi. Al-Hassār, a mathematician from  Morocco specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. This same fractional notation appeared soon after in the work of Fibonacci in the 13th century. Abū al-Hasan ibn Alī al-Qalasādī (1412–1486) was the last major medieval Arab algebraist, who made the first attempt at creating an algebraic notation since Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since Diophantus and Brahmagupta in ancient times. The syncopated notations of his predecessors , however, lacked symbols for mathematical operations. Al-Qalasadi "took the first steps toward the introduction of algebraic symbolism by using letters in place of numbers" and by "using short Arabic words, or just their initial letters, as mathematical symbols." Europe and the Mediterranean region. Just as the death of Hypatia signals the close of the Library of Alexandria as a mathematical center, so does the death of Boethius signal the end of mathematics in the Western Roman Empire. Although there was some work being done at Athens, it came to a close when in 529 the Byzantine emperor Justinian closed  the pagan philosophical schools. The year 529 is now taken to be the beginning of the medieval period. Scholars fled the West towards the more hospitable East, particularly towards Persia, where they found haven under King Chosroes and established what might be termed an "Athenian Academy in Exile". Under a treaty with Justinian, Chosroes would eventually return the scholars to the Eastern Empire. During the Dark Ages, European mathematics was at its nadir with mathematical research consisting mainly of commentaries on ancient treatises; and most of this research was centered in the Byzantine Empire. The end of the medieval period  is set as the fall of Constantinople to the Turks in 1453. Late Middle Ages. The 12th century saw a flood of translations from Arabic into Latin and by the 13th century, European mathematics was beginning to rival the mathematics of other lands. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. As the Islamic world was declining after the 15th century, the European world was ascending. And it is here that algebra was further developed. Symbolic algebra. Modern notation for arithmetic operations was introduced between  the end of the 15th century and the beginning of the 16th century by Johannes Widmann and Michael Stifel. At the end of 16th century, François Viète introduced symbols, now called variables, for representing indeterminate or unknown numbers. This created a new algebra consisting of computing with symbolic expressions as if they were numbers. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten  years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. The symbol "x". By tradition, the first unknown variable in an algebraic problem is nowadays represented by the symbol formula_84 and if there is a second or a third unknown, then these are labeled formula_85 and formula_86 respectively. Algebraic formula_87 is conventionally printed in italic type to distinguish it from the sign of multiplication. Mathematical historians generally agree that the use of formula_87 in algebra was introduced by René Descartes and was  first published in his treatise "La Géométrie" (1637). In that work, he used letters from the beginning of the alphabet formula_89 for known quantities, and letters from the end of the alphabet formula_90 for unknowns. It has been suggested that he later settled on formula_87 (in place of formula_92) for the first unknown because of its relatively greater abundance in the French and Latin typographical fonts of the time. Three alternative theories of the origin of algebraic formula_87 were suggested in the 19th century: (1) a symbol used by German algebraists and thought to be derived from a cursive letter  formula_94 mistaken for formula_87; (2) the numeral "1" with oblique strikethrough; and (3) an Arabic/Spanish source (see below). But the Swiss-American historian of mathematics Florian Cajori examined these and found all three lacking in concrete evidence; Cajori credited Descartes as the originator, and described his formula_96 and formula_92 as "free from tradition[,] and their choice purely arbitrary." Nevertheless, the Hispano-Arabic hypothesis continues to have a presence in popular culture today. It is the claim that algebraic formula_87 is the abbreviation of a supposed loanword from Arabic in Old Spanish. The theory originated in 1884 with the German  orientalist Paul de Lagarde, shortly after he published his edition of a 1505 Spanish/Arabic bilingual glossary in which Spanish "cosa" ("thing") was paired with its Arabic equivalent, شىء ("shay"), transcribed as "xei". (The "sh" sound in Old Spanish was routinely spelled formula_99) Evidently Lagarde was aware that Arab mathematicians, in the "rhetorical" stage of algebra's development, often used that word to represent the unknown quantity. He surmised that "nothing could be more natural" ("Nichts war also natürlicher...") than for the initial of the Arabic word—romanized as the Old Spanish formula_87—to be adopted for use in algebra . A later reader reinterpreted Lagarde's conjecture as having "proven" the point. Lagarde was unaware that early Spanish mathematicians used, not a "transcription" of the Arabic word, but rather its "translation" in their own language, "cosa". There is no instance of "xei" or similar forms in several compiled historical vocabularies of Spanish. Gottfried Leibniz. Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Gottfried Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate , tangent, chord, and the perpendicular. In the 18th century, "function" lost these geometrical associations. Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination. Leibniz also discovered Boolean algebra and symbolic logic, also relevant to algebra. Abstract algebra. The ability to do algebra is a skill cultivated in mathematics education. As explained by Andrew Warwick, Cambridge University students in the early 19th century practiced "mixed mathematics", doing exercises based  on physical variables such as space, time, and weight. Over time the association of variables with physical quantities faded away as mathematical technique grew. Eventually mathematics was concerned completely with abstract polynomials, complex numbers, hypercomplex numbers and other concepts. Application to physical situations was then called applied mathematics or mathematical physics, and the field of mathematics expanded to include abstract algebra. For instance, the issue of constructible numbers showed some mathematical limitations, and the field of Galois theory was developed. The father of algebra. The title of "the father of algebra" is frequently credited to the Persian mathematician Al-Khwarizmi , supported by historians of mathematics, such as Carl Benjamin Boyer, Solomon Gandz and Bartel Leendert van der Waerden. However, the point is debatable and the title is sometimes credited to the Hellenistic mathematician Diophantus. Those who support Diophantus point to the algebra found in "Al-Jabr" being more elementary than the algebra found in "Arithmetica", and "Arithmetica" being syncopated while "Al-Jabr" is fully rhetorical. However, the mathematics historian Kurt Vogel argues against Diophantus holding this title, as his mathematics was not much more algebraic than that of the ancient Babylonians. Those who support Al-Khwarizmi point to the fact  that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers. Al-Khwarizmi also introduced the fundamental concept of "reduction" and "balancing" (which he originally used the term "al-jabr" to refer to), referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. Other supporters of Al-Khwarizmi point to his algebra  no longer being concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." They also point to his treatment of an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems." Victor J. Katz regards "Al-Jabr" as the first true algebra text that is still extant