History of algebra

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[edit] Etymology

The word Algebra is derived from the Arabic word Al-Jabr, and this comes from the treatise written in 820 by the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī entitled, in Arabic, كتاب الجبر والمقابلة, or al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, which 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. Although no one knows with absolute certainty what the word al-jabr means, most historians agree that the word meant something like "restoration" or "completion".[1]

[edit] Stages of Algebra

[edit] Stages of Symbolic Algebra

The stages of the development of symbolic algebra are roughly[2] as follows:

  • Rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century;
  • Geometric constructive algebra, which was emphasised by the Vedic Indian and classical Greek mathematicians;
  • Syncopated algebra, as developed by Diophantus and in the Bakhshali Manuscript; and
  • Symbolic algebra, which sees its culmination in the work of Leibniz.

[edit] Quadratic Equations

Through out most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.

  • x2 + px = q
  • x2 = px + q
  • x2 + q = px

where p and q are positive. This trichotomy is due to the fact that quadratic equations of the form x2 + px + q = 0, with p and q positive, have no positive roots.[3]

[edit] History

[edit] Ancient Algebra

[edit] Babylonian Algebra

The origins of algebra can be traced to the ancient Babylonians,[4] who developed a positional number system[5] which greatly aided them in solving their rhetorical algebraic equations. The Babylonians were not interested in exact solutions but approximations, and so they would commonly use linear interpolation to approximate intermediate values. [6]

Babylonian algebra was much more advanced than the Egyptian algebra of the time; whereas the the Egyptians were mainly concerned with with linear equations the Babylonians were more concerned with quadratic and cubic equations.[6] 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.[6] They were familiar with many simple forms of factoring,[6] three-term quadratic equations with positive roots,[7] and many cubic equations[8] although it is no known if they were able to reduce the general cubic equation.[8]

[edit] Egyptian Algebra

A portion of the Rhind Papyrus
A portion of the Rhind Papyrus

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 Egyptains.[6]

The Rhind Papyrus, also known as the Ahmes Papyrus, is an ancient Egyptian papyrus written circa 1650 B.C. by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 B.C.[9] It is the most extensive ancient Egyptian mathematical document that historians know of.[10] The Rhind Papyrus contains problems where linear equations of the form x + ax = b and x + ax + bx = c are solved, where a, b, and c are known and x, which is referred to as "aha" or heap, is the unknown.[11] The solutions were arrived at by using the "method of false position," or the "rule of false" where first a specific value is substituted into the left hand side of the equation, then the require 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.[11] In some of the problems the author "checks" his solution, thereby writing one of the earliest known simple proofs.[11]

[edit] Chinese Algebra

Chinese Mathematics dates to at least 300 B.C. with the Chou Pei Suan Ching, generally considered to be one of the oldest Chinese mathematical documents. [12]

[edit] Nine Chapters on the Mathematical Art

Nine Chapters on the Mathematical Art
Nine Chapters on the Mathematical Art

Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 B.C., is one of the most influential of all Chinese math books and it is composed of some 246 problems.[12] 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.[12]

[edit] 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 Chih (or Li Yeh) (1192 - 1272 A.D.).[13] He used fan fa, or Horner's method, to solve equations of as high as degree six, althnough he did not describe his method of solving equations.[13]

[edit] 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 (ca. 1202 - ca. 1261 A.D.) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.[13]

[edit] Magic Squares

Yang Hui (Pascal's) triangle, as depicted by the ancient Chinese using rod numerals.
Yang Hui (Pascal's) triangle, as depicted by the ancient Chinese using rod numerals.

The earliest known magic squares appeared in China.[14] 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 reducing column operations on the magic square.[14] The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261 - 1275), who worked with magic squares of order as high as ten.[15]

[edit] 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 A.D.[16] 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.[16] The author uses the method of fan fa, today called Horner's method, to solve these equations.[16]

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.[17]

There are many summation series equations given without proof in the Precious mirror. A few of the summation series are:[17]

1^2 + 2^2 + 3^2 + \cdots + n^2 = {n(n + 1)(2n + 1)\over 3!}
1 + 8 + 30 + 80 + \cdots + {n^2(n + 1)(n + 2)\over 3!} = {n(n + 1)(n + 2)(n + 3)(4n + 1)\over 5!}

[edit] Greek Geometric Algebra

Hellenistic mathematician Euclid details geometrical algebra in Elements.
Hellenistic mathematician Euclid details geometrical algebra in Elements.

It is sometimes alleged that the Greeks had no algebra, but this is plainly false.[18] 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[19], usually lines, that had letters associated with them,[20] and with this new form of algebra they were able to find solutions to equations by using a process that they invented which is known as "the application of areas".[19] "The application of areas" is only a part of geometric algebra and it is thoroughly covered in Euclid's Elements.[19]

An example of geometric algebra would be solving the linear equation ax = bc. The ancient Greeks would solve this equation by looking at it as an equality of areas rather than as an equality between the ratios a:b and c:x. The Greeks would construct a rectangle with sides of length b and c, then extend a side of the rectangle to length a, and finally they would complete the extended rectangle so as to find the side of the rectangle that is the solution.[19]

[edit] Euclid of Alexandria

Euclid (Greek: Εὐκλείδης) was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC).[21][22] Neither the year nor place of his birth[21] 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.[21] 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.[23]

[edit] 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.[18] These propositions and their results are the geometric equivalents of our modern symbolic algebra and trigonometry.[18] 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.[18]

Many basic laws of addition and multiplication are included or proved geometrically in the Elements. For instance, proposition 1 of Book II states:

If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

But this is nothing more than the geometric version of the distributive law, a(b + c + d) = ab + ac + ad[18]; and in Books V and VII of the Elements the commutative and associative laws for multiplication are demonstrated.[18]

Many basic equations were also proved geometrically. For instance, proposition 4 in Book II proves that a2 − b2 = (a + b)(a − b)[24], and proposition 5 in Book II proves that (a + b)2 = a2 + 2ab + b2.[18]

Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation ax + x2 = b2, and proposition 11 of Book II gives a solution to ax + x2 = a2.[25]

[edit] Data

Data is a work written by Euclid for use at the university of Alexandria and it was meant to be used as a companion volume to the first six books of the Elements.[26] 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.[26] Some of these statements are geometric equivalents to solutions of quadratic equations.[26] For instance, Data contains the solutions to the equations dx2 − adx + b2c = 0 and the familiar Babylonian equation xy = a2, x ± y = b.[26]

[edit] Diophantine Algebra

Cover of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Cover of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

Diophantus was a Greek mathematician who lived circa 250 A.D., but the uncertainty of this date is so great that it may be off by more than a century.[27] He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.[27] 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.[28]

In Arithmetica, Diophantus used symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;[28] 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.[29] So, for example, what we would write as

x3 − 2x2 + 10x − 1 = 5

Diophantus would have written as

Κν ᾱ̄ ζ  ί̄ Ψ Δν  β̄ Μ  ᾱ̄ 'ίσ Μ  ε̄

where the symbols represent the following:[30][31]

Symbol Representation
 ᾱ̄ represents 1
 β̄ represents 2
 ε̄ represents 5
 ί̄ represents 10
ζ represents the unknown quantity
'ίσ (short for 'ίσΟζ) represents "equals"
Ψ (actually inverted) represents the subtraction of everything that follows it up to 'ίσ
Μ represents the zeroth power (i.e. a constant term)
Δν represents the second power, from Greek δύναμιδ, meaning strength or power
Κν represents the third power, from Greek κύβοδ, meaning a cube
ΔνΔ represents the fourth power
ΔΚν represents the fifth power
ΚνΚ represents the sixth power

Note that 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:[30]

x31x10 − x22x01 = x05

and, to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:[30]

(x31 + x10) − (x22 + x01) = x05

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.[28] 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.[28] Arithmetica also makes use of the identities:[32]

(a2 + b2)(c2 + d2) = (ac + db)2 + (bc − ad)2
= (ad + bc)2 + (ac − bd)2

[edit] Indian Algebra

The recurring themes in Indian mathematics are determinate and indeterminate linear and quadratic equations, simple mensuration, Pythagorean triples, and others.[33]

[edit] Aryabhatiya

Aryabhata (476-550 A.D.) was an India mathematician who authored Aryabhatiya.[34] In it he gave the rules,[35]

1^2 + 2^2 + \cdots + n^2 = {n(n + 1)(2n + 1) \over 6}

and,

1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2

[edit] Brahma Sphuta Siddhanta

Brahmagupta (fl. 628) was a Central Indian mathematician who authored Brahma Sphuta Siddhanta. In his work Brahmagupta solves the general quadratic equation for both positive and negative roots.[36] In indeterminate analysis Brahmagupta gives the Pythagorean triads m, {1 \over 2}({m^2\over n} - n), {1 \over 2}({m^2\over n} + n), but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with.[37] He was the first to give a general solution to the linear Diophantine equation ax + by = c, where a, b, and c are integers.[38] 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 conclude a Greek influence on Brahmagupta's work, or at least a common Babylonian source.[38]

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 notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.[38]

[edit] Bhaskara

Bhaskara (1114-ca. 1185) was the leading mathematician of the twelfth century. In Algebra, he gave the general solution of the Pell equation.[38] He is the author of Lilavati and Vija-Ganita, which contain problems dealing with determinate and indeterminate linear and quadratic equations, and Pythagorean triples[33] and he fails to distinguish between exact and approximate statements.[39] 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.[39]

[edit] Islamic Algebra

A page from The Compendious Book on Calculation by Completion and Balancing.
A page from The Compendious Book on Calculation by Completion and Balancing.

The first century of the Islamic 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 eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.[40] The Muslim 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.[40] Many of these Greek works were translated by Thabit ibn-Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.[41]

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.[42]

Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra where often times even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (ex. twenty-two) with numerical numbers (ex. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra.[41]

[edit] The Compendious Book on Calculation by Completion and Balancing

The Muslim[43] 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 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind.[40] 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.[44]

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 (ax2 = bx), the second chapter deals with squares equal to number (ax2 = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax2 + bx = c), the fifth chapter deals with squares and number equal roots (ax2 + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax2).[45]

In Al-Jabr, al-Khwarizmi uses geometric proofs,[20] he does not recognize the root x = 0,[45] and he only deals with positive roots.[46] He also recognizes that the discriminant must be positive and he does not justify his procedure for completing the square.[47] The Greek influence is shown by Al-Jabr's geometric foundations[48][42] and by one problem taken from Heron.[49] 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.[20]

Al-Khwarizmi most likely did not know of Diophantus's Arithmetica,[50] which became known to the Arabs sometime before the tenth century.[51] 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.[50] So, for example, what we would write as

x2 + 10x = 39

Diophantus would have written as[52]

ΔΥ  ᾱ̄ ζ  ί̄ 'ίσ Μ λ θ

And al-Khwarizmi would have written as[52]

One square and ten roots of the same amount to thrity-nine dirhems; that is to say, what must be the square which when increased by ten of its own roots, amounts to thirty-nine?

[edit] Logical Necessities in Mixed Equations

'Abd al-Hamid 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.[51] The manuscript gives the exact 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 determinant is negative then the quadratic equation has no solution.[51] 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.[51]

[edit] Al-Karkhi

Al-Karkhi was the successor of Abu'l-Wefa and he was the first to discover the solution to equations of the form ax2n + bxn = c.[53] Al-Karkhi only considered positive roots.[53]

[edit] Omar Khayyám

Omar Khayyám (ca. 1050 - 1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.[54] 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.[54] His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám generalized the method to cover all cubic equations with positive roots.[54] He only considered positive roots and he did not go past the third degree.[54] He also saw a strong relationship between Geometry and Algebra.[54]

[edit] The Father of Algebra

The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra"[55][56] but debate now exists as to whether or not Al-Khwarizmi should take that title from Diophantus.[55] Those who support Al-Khwarizmi point to the fact that much of his work on reduction is still in use today and that he gave an exhaustive explanation of solving quadratic equations. Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.[55]

[edit] Modern Algebra

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. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues. Υ

[edit] Time Line

A timeline of key algebraic developments are as follows:

  • Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.
  • Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.
  • Circa 800 BC: Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax2 = c and ax2 + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.
  • Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.
  • Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.
  • Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.
  • Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.
  • Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.
  • Circa 150 AD: Greek mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.
  • Circa 200: Hellenistic mathematician Diophantus lived in Alexandria and is often considered to be the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
  • 499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a differential equation.
  • Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.
  • 628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell's equation, and gives rules for solving linear and quadratic equations. He discovers that quadratic equations have two roots, including both negative as well as irrational roots.
  • 820: The word algebra is derived from operations described in the treatise written by the Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.
  • Circa 850: Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.
  • Circa 850: Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations.
  • Circa 990: Persian Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and gives rules for the products of any two of these.
  • Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations.
  • 1072: Persian mathematician Omar Khayyam gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.
  • 1114: Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves quadratic equations with more than one unknown, various cubic, quartic and higher-order polynomial equations, Pell's equation, the general indeterminate quadratic equation, as well as indeterminate cubic, quartic and higher-order equations.
  • 1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.
  • Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.
  • Circa 1400: Indian mathematician Madhava of Sangamagramma finds the solution of transcendental equations by iteration, iterative methods for the solution of non-linear equations, and solutions of differential equations.
  • 1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.[57]
  • 1545: Girolamo Cardano publishes Ars magna -The great art which gives Fontana's solution to the general quartic equation.[57]
  • 1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.
  • 1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.
  • 1631: Thomas Harriot in a posthumus publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than".
  • 1682: Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.
  • 1683: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, discriminant, and Bernoulli numbers.
  • 1685: Kowa Seki solves the general cubic equation, as well as some quartic and quintic equations.
  • 1693: Leibniz solves systems of simultaneous linear equations using matrices and determinants.
  • 1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.
  • 1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.[57]
  • 1832: Galois theory is developed by Évariste Galois in his work on abstract algebra.[57]

[edit] See Also

[edit] Footnotes and Citations

  1. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 229. ISBN 0471543977. “It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. 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.” 
  2. ^ Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 180. ISBN 0471543977. “It has been said that three stages of in the historical development of algebra can be recognized: (1) the rhetorical or early stage, in which everything is written out fully in words; (2) a syncopated or intermediate state, in which some abbreviations are adopted; and (3) a symbolic or final stage. Such an arbitrary division of the development of algebra into three stages is, of course, a facile oversimplification; but it can serve effectively as a first approximation to what has happened” 
  3. ^ Boyer, Carl B. (1991). "Mesopotamia", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 32. ISBN 0471543977. “Until modern times there was no thought of solving a quadratic equation of the form x2 + px + q = 0, where p and q are positive, for the equation has no positive root. Consequently, quadratic equations in ancient and Medieval times - and even in the early modern period - were classified under three types: (1)x2 + px = q (2)x2 = px + q (3)x2 + q = px” 
  4. ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. 
  5. ^ Boyer, Carl B. (1991). "Mesopotamia", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 27. ISBN 0471543977. “By about the time of the conquest by Alexander the Great, however, a special sign, consisting of two small wedges placed obliquely, was invented to serve as a placeholder where a numeral was missing. ... The Babylonian zero symbol apparently did not end all ambiguity, for the sign seems to have been used for intermediate empty positions only.” 
  6. ^ a b c d e Boyer, Carl B. (1991). "Mesopotamia", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 30. ISBN 0471543977. “Babylonian mathematicians did not hesitate to interpolate by proportional parts to approximate intermediate values. Linear interpolation seems to have been a commonplace procedure in ancient Mesopotamia, and the positional notation lent itself conveniently to the rile of three. ... a table essential in Babylonian algebra; this subject reached a considerably higher level in Mesopotamia than in Egypt. Many problem texts from the Old Babylonian period show that the solution of the complete three-term quadratic equation afforded the Babylonians no serious difficulty, for flexible algebraic operations had been developed. They could transpose terms in an equations by adding equals to equals, and they could multiply both sides by like quantities to remove fractions or to eliminate factors. By adding 4ab to (a - b)2 they could obtain (a + b)2 for they were familiar with many simple forms of factoring. ...Egyptian algebra had been much concerned with linear equations, but the Babylonians evidently found these too elementary for much attention. ... In another problem in an Old Babylonian text we find two simultaneous linear equations in two unknown quantities, called respectively the "first silver ring" and the "second silver ring."” 
  7. ^ Boyer, Carl B. (1991). "Mesopotamia", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 31. ISBN 0471543977. “The solution of a three-term quadratic equation seems to have exceeded by far the algebraic capabilities of the Egyptians, but Neugebauer in 1930 disclosed that such equations had been handled effectively by the Babylonians in some of the oldest problem texts.” 
  8. ^ a b Boyer, Carl B. (1991). "Mesopotamia", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 33. ISBN 0471543977. “There is no record in Egypt of the solution of a cubic equations, but among the Babylonians there are many instances of this. ... Whether or not the Babylonians were able to reduce the general four-term cubic, ax3 + bx2 + cx = d, to their normal form is not known.” 
  9. ^ Boyer, Carl B. (1991). "Egypt", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 11. ISBN 0471543977. “It had been bought in 1959 in a Nile resort town by a Scottish antiquary, Henry Rhind; hence, it often is known as the Rhind Papyrus or, less frequently, as the Ahmes Papyrus in honor of the scribe by whose hand it had been copied in about 1650 B.c. The scribe tells us that the material is derived from a prototype from the Middle Kingdom of about 2000 to 1800 B.C.” 
  10. ^ Boyer, Carl B. (1991). "Egypt", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 19. ISBN 0471543977. “Much of our information about Egyptian mathematics has been derived from the Rhind or Ahmes Papyrus, the most extensive mathematical document from ancient Egypt; but there are other sources as well.” 
  11. ^ a b c Boyer, Carl B. (1991). "Egypt", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 15-16. ISBN 0471543977. “The Egyptian problems so far described are best classified as arithmetic, but there are others that fall into a class to which the term algebraic is appropriately applied. These do not concern specific concrete objects such as bread and beer, nor do they call for operations on known numbers. Instead they require the equivalent of solutions of linear equations of the form x + ax = b or x + ax + bx = c, where a and b and c are known and x is unknown. The unknown is referred to as "aha," or heap. ... The solution given by Ahmes is not that of modern textbooks, but is characteristic of a procedure now known as the "method of false position," or the "rule of false." A specific value, most likely a false one, is assumed for heap, and the operations indicated on the left hand side of the equality sign are performed on this assumed number. The result of these operations then is compared with the result desired, and by the use of proportions the correct answer is found. ... Ahmes then "checked" his result by showing that if to 16 + 1/2 + 1/8 one adds a seventh of this (which is 2 + 1/4 + 1/8), one does indeed obtain 19. Here we see another significant step in the development of mathematics, for the check is a simple instance of a proof.” 
  12. ^ a b c Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 195-197. ISBN 0471543977. “estimates concerning the Chou Pei Suan Ching, generally considered to be the oldest of the mathematical classics, differ by almost a thousand years. ... A date of about 300 B.C. would appear reasonable, thus placing it in close competition with another teatise, the Chiu-chang suan-shu, composed about 250 B.C., that is, shortly before the Han dynasty (202 B.C.). ... Almost as old at the Chou Pei, and perhaps the most influenctial of all Chinese mathematical books, was the Chui-chang suan-shu, or Nine Chapters on the Mathemtical Art. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. ... Chapter eight of the Nine chapters is significant for its solution of problems of simultaneous linear equations, using both positive and negative numbers. The last problem int the chapter involves four equations in five unknowns, and the topic of indeterminate equations was to remain a favorite among Oriental peoples.” 
  13. ^ a b c Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 204. ISBN 0471543977. “Li Chih (or Li Yeh, 1192-1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His Ts'e-yuan hai-ching (Sea-Mirror of the Circle Measurements) includes 170 problems dealing with...some of the problems leading to equations of fourth degree. Although he did not describe his method of solution of equations, including some of sixth degree, it appears that it was not very different form that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202-ca.1261) and Yang Hui (fl. ca. 1261-1275_. The former was an unprincipled governor and minister who acquired immense wealth within a hundred days of assuming office. His Shu-shu chiu-chang (Mathematical Treatise in Nine Sections) marks the high point of Chinese indeterminate analysis, with the invention of routines for solving simultaneous congruences.” 
  14. ^ a b Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 197. ISBN 0471543977. “The Chinese were especially fond of patters; hence, it is not surprising that the first record (of ancient but unknown origin) of a magic square appeared there. ... The concern for such patterns left the author of the Nine Chapters to solve the system of simultaneous linear equations ... by performing column operations on the matrix ... to reduce it to ... The second form represented the equations 36z = 99, 5y + z = 24, and 3x + 2y + z = 39 from which the values of z, y, and x are successively found with ease.” 
  15. ^ Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 204-205. ISBN 0471543977. “The same "Horner" device was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten.” 
  16. ^ a b c Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 203. ISBN 0471543977. “The last and greatest of the Sung mathematicians was Chu Chih-chieh (fl. 1280-1303), yet we known little about him-, ...Of greater historical and mathematical interest is the Ssy-yüan yü-chien(Precious Mirror of the Four Elements) of 1303. In the eighteenth century this, too, disappeared in China, only to be rediscovered in the next century. The four elements, called heaven, earth, man, and matter, are the representations of four unknown quantities in the same equation. The book marks the peak in the development of Chinese algebra, for it deals with simultaneous equations and with equations of degrees as high as fourteen. In it the author describes a transformation method that he calls fan fa, the elements of which to have arisen long before in China, but which generally bears the name of Horner, who lived half a millennium later.” 
  17. ^ a b Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 205. ISBN 0471543977. “A few of the many summations of series found in the Precious Mirror are the following:... However, no proofs are given, nor does the topic seem to have been continued again in China until about the nineteenth century. ... The Precious Mirror opens with a diagram of the arithmetic triangle, inappropriately known in the West as "pascal's triangle." (See illustration.) ... Chu disclaims credit for the triangle, referring to it as a "diagram of the old method for finding eighth and lower powers." A similar arrangement of coefficients through the sixth power had appeared in the work of Yang Hui, but without the round zero symbol.” 
  18. ^ a b c d e f g Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 109. ISBN 0471543977. “Book II of the Elements is a short one, containing only fourteen propositions, not one of which plays any role in modern textbooks; yet in Euclid's day this book was of great significance. This sharp discrepancy between ancient and modern views is easily explained - today we have symbolic algebra and trigonometry that have replaced the geometric equivalents from Greece. For instance, Proposition 1 of Book II states that "If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments." This theorem, which asserts (Fig. 7.5) that AD(AP + PR + RB) = AD·AP + AD·PR + AD·RB, is nothing more than a geometric statement of one of the fundamental laws of arithmetic known today as the distributive law: a(b + c + d) = ab + ac + ad. In later books of the Elements (V and VII) we find demonstrations of the commutative and associative laws for multiplication. Whereas in our time magnitudes are represented by letters that are understood to be numbers (either known or unknown) on which we operate with algorithmic rules of algebra, in Euclid's day magnitudes were pictured as line segments satisfying the axions and theorems of geometry. It is sometimes asserted that the Greeks had no algebra, but this is patently false. They had Book II of the Elements, which is geometric algebra and served much the same purpose as does our symbolic algebra. There can be little doubt that modern algebra greatly facilitates the manipulation of relationships among magnitudes. But it is undoubtedly also true that a Greek geometer versed in the fourteen theorems of Euclid's "algebra" was far more adept in applying these theorems to practical mensuration than is an experienced geometer of today. Ancient geometric "algebra" was not an ideal tool, but it was far from ineffective. Euclid's statement (Proposition 4), "If a straight line be cut at random, the square on the while is equal to the squares on the segments and twice the rectangle contained by the segments, is a verbose way of saying that (a + b)2 = a2 + 2ab + b2,” 
  19. ^ a b c d Boyer, Carl B. (1991). "The Heroic Age", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 77-78. ISBN 0471543977. “Whether deduction came into mathematics in the sixth century B.C. or the fourth and whether incommensurability was discovered before or after 400 B.C., there can be no doubt that Greek mathematics had undergone drastic changes by the time of Plato. ... A "geometric algebra" had to take the place of the older "arithmetic algebra," and in this new algebra there could be no adding of lines to areas or of areas to volumes. From now on there had to be strict homogeneity of terms in equations, and the Mesopotamian normal form, xy = A, x +or- y = b, were to be interpreted geometrically. ... In this way the Greeks built up the solution of quadratic equations by their process known as "the application of areas," a portion of geometric algebra that is fully covered by Euclid's Elements. ... The linear equation ax = bc, for example, was looked upon as an equality of the areas ax and bc, rather than as a proportion - an equality between the two ratios a:b and c:x. Consequently, in constructing the fourth proportion x in this case, it was usual to construct a rectangle OCDB with the sides b = OB and c = OC (Fig 5.9) and then along OC to lay off OA = a. One completes the rectangle OCDB and draws the diagonal OE cutting CD in P. It is now clear that CP is the desired line x, for rectangle OARS is equal in area to rectangle OCDB.” 
  20. ^ a b c Boyer, Carl B. (1991). "Europe in the Middle Ages", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 258. ISBN 0471543977. “In the arithmetical theorems in Euclid's Elements VII-IX, numbers had been represented by line segments to which letters had been had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry.” 
  21. ^ a b c Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 100. ISBN 0471543977. “but by 306 B.C. control of the Egyptian portion of the empire was firmly in the hands of Ptolemy I, and this enlightened ruler was able to turn his attention to constructive efforts. Among his early acts was the establishment at Alexandria of a school or institute, known as the Museum, second to none in its day. As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written - the Elements (Stoichia) of Euclid. Considering the fame of the author and of his best seller, remarkably little is known of Euclid's life. So obscure was his life that no birthplace is associated with his name.” 
  22. ^ Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 101. ISBN 0471543977. “The tale related above in connection with a request of Alexander the Great for an easy introduction to feometry is repeated in the case of Ptolemy, who Euclid is reported to have assured that "there is no royal road to geometry."” 
  23. ^ Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 104. ISBN 0471543977. “Some of the faculty probably excelled in research, others were better fitted to be administrators, and still some others were noted for teaching ability. It would appear, from the reports we have, that Euclid very definitely fitted into the last category. There is no new discovery attributed to him, but he was noted for expository skills.” 
  24. ^ Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 110. ISBN 0471543977. “The same holds true for Elements II.5, which contains what we should regard as an impractical circumlocution for a2 − b2 = (a + b)(a − b)” 
  25. ^ Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 111. ISBN 0471543977. “In an exactly analogous manner the quadratic equation ax + x2 = b2 is solved through the use of II.6: If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole (with the added straight line) and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line. ... with II.11 being an important special case of II.6. Here Euclid solves the equation ax + x2 = a2” 
  26. ^ a b c d Boyer, Carl B. (1991). "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 103. ISBN 0471543977. “Euclid's Data, a work that has come down to us through both Greek and the Arabic. It seems to have been composed for use at the university of Alexandria, serving as a companion volume to the first six books of the Elements in much the same way that a manual of tables supplements a textbook. ... It opens with fifteen definitions concerning magnitudes and loci. The body of the text comprises ninety-five statements concerning the implications of conditions and magnitudes that may be given in a problem. ... There are about two dozen similar statements serving as algebraic rules or formulas. ... Some of the statements are geometric equivalents of the solution of quadratic equations. For example... Eliminating y we have (a − x)dx = b2c or dx2 − adx + b2c = 0, from which x = a / 2 + / − sqrt((a / 2)2 − b2c / d). The geometric solution given by Euclid is equivalent to this, except that the negative sign before the radical us used. Statements 84 and 85 in the Data are geometric replacements of the familiar Babylonian algebraic solutions of the systems xy = a2, x ± y = b., which again are the equivalents of solutions of simultaneous equations.” 
  27. ^ a b Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 178. ISBN 0471543977. “Uncertainty about the life of Diophantus is so great that we do not know definitely in which century he lived. Generally he is assumed to have flourished about A.D. 250, but dates a century or more earlier or later are sometimes suggested... If this conundrum is historically accurate, Diophantus lived to be eighty-four-years old. ... The chief Diophantine work known to us is the Arithmetica, a treatise originally in thirteen books, only the first six of which have survived.” 
  28. ^ a b c d Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 180-182. ISBN 0471543977. “In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with approximate solutions of determinate equations as far as the third degree, the Arithmetica of Diophantus (such as we have it) is almost entirely devoted to the exact solution of equations, both determinate and indeterminate. ... Throughout the six surviving books of Arithmetica there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter ζ (perhaps for the last letter of arithmos). ... It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them.” 
  29. ^ Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 178. ISBN 0471543977. “The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation.” 
  30. ^ a b c Derbyshire, John (2006). "The Father of Algebra", Unknown Quantity: A Real And Imaginary History of Algebra. Joseph Henry Press, 35-36. ISBN 030909657X. 
  31. ^ Cooke, Roger (1997). "Mathematics in the Roman Empire", The History of Mathematics: A Brief Course. Wiley-Interscience, 167-168. ISBN 0471180823. 
  32. ^ Boyer, Carl B. (1991). "Europe in the Middle Ages", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 257. ISBN 0471543977. “The book makes frequent use of the identities ... which had appeared in Diophantus and had been widely used by the Arabs.” 
  33. ^ a b Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 222. ISBN 0471543977. “The Livavanti, like the Vija-Ganita, contains numerous problems dealing with favorite Hindu topics; linear and quadratic equations, both determinate and indeterminate, simple mensuration, arithmetic and geometric progretions, surds, Pythagorean triads, and others.” 
  34. ^ Cooke, Roger (1997). "The Mathematics of the Hindus", The History of Mathematics: A Brief Course. Wiley-Interscience, 204. ISBN 0471180823. “Aryabhata himself (one of at least two mathematicians bearing that name) lived in the late fifth and the early sixth centuries at Kusumapura (now Pataliutra, a village near the city of Patna) and wrote a book called Aryabhatiya.” 
  35. ^ Cooke, Roger (1997). "The Mathematics of the Hindus", The History of Mathematics: A Brief Course. Wiley-Interscience, 207. ISBN 0471180823. “He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes.” 
  36. ^ Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 219. ISBN 0471543977. “Brahmagupta (fl. 628), who lived in Central India somewhat more than a century after Aryabhata ... in the trigonometry of his best-known work, the Brahmasphuta Siddhanta, ... here we find general solutions of quadratic equations, including two roots even in cases in which one of them is negative.” 
  37. ^ Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 220. ISBN 0471543977. “Hindu algebra is especially noteworthy in its development of indeterminate analysis, to which Brahmagupta made several contributions. For one thing, in his work we find a rule for the formation of Pythagorean triads expressed in the form m, 1/2(m2/n - n), 1/2(m2/n + n); but this is only a modified form of the old Babylonian rule, with which he may have become familiar.” 
  38. ^ a b c d Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 221. ISBN 0471543977. “he was the first one to give a general solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. ... It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India - or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words. ... Bhaskara (1114-ca. 1185), the leading mathematician of the twelfth century. It was he who filled some of the gaps in Brahmagupta's work, as by giving a general solution of the Pell equation and by considering the problem of division by zero.” 
  39. ^ a b Boyer, Carl B. (1991). "China and India", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 222-223. ISBN 0471543977. “In treating of the circle and the sphere the Lilavati fails also to distinguish between exact and approximate statements. ... Many of Bhaskara's problems in the Livavati and the Vija-Ganita evidently were derived from earlier Hindu sources; hence, it is no surprise to note that the author is at his best in dealing with indeterminate analysis.” 
  40. ^ a b c Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 227. ISBN 0471543977. “The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. ... It was during the caliphate of al-Mamun (809-833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's Almagest and a complete version of Euclid's Elements. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the Sindhad derived from India.” 
  41. ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 234. ISBN 0471543977. “but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. ... Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.” 
  42. ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 230. ISBN 0471543977. “Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories.” 
  43. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 228-229. ISBN 0471543977. “the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful."” 
  44. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 228. ISBN 0471543977. “The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization - respects in which neither Diophantus nor the Hindus excelled.” 
  45. ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 229. ISBN 0471543977. “in six short chapters, of the six types of equations made up from the three kinds of quantities: roots, squares, and numbers (that is x, x2, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as x2 = 5x, x2/3 = 4x, and 5x2 = 10x, giving the answers x = 5, x = 12, and x = 2 respectively. (The root x = 0 was not recognized.) Chapter II covers the case of squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term is equal to, more than, or less than one. Chapters IV, V, and VI are mor interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares.” 
  46. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 229-230. ISBN 0471543977. “The solutions are "cookbook" rules for "completing the square" applied to specific instances. ... In each case only the positive answer is give. ... Again only one root is given for the other is negative. ...The six cases of equations given above exhaust all possibilities for linear and guadratic equations having positive roots.” 
  47. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 230. ISBN 0471543977. “Al-Khwarizmi here calls attention to the fact that what we designate as the discriminant must be positive: "You ought to understand also that when you take the half of the roots in this form of equation and then mutliply the half by itself; if that which proceeds or results from the multiplication is less than the units above mentioned as accompanying the square, you have an equation." ... Once more the steps in completing the square are meticulously indicated, without justification,” 
  48. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 231. ISBN 0471543977. “The Algebra of al-Khwarismi betrays unmistakable Hellenic elements,” 
  49. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 233. ISBN 0471543977. “A few of al-Khwarizmi's problems give rather clear evidence of Arabic dependence on the Babylonian-Heronian stream of mathematics. One of them presumably was taken directly from Heron, for the figure and dimensions are the same.” 
  50. ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 228. ISBN 0471543977. “the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers.” 
  51. ^ a b c d Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 234. ISBN 0471543977. “The Algebra of al-Khwarizmi usually is regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in Mixed Equations," was part of a book on Al-jabr wa'l muqabalah which was evidently very much the same as that by al-Khwarizmi and was published at about the same time - possibly even earlier. The surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's Algebra and in one case the same illustrative example x2 + 21 = 10x. In one respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ... Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century.” 
  52. ^ a b Derbyshire, John (2006). "The Father of Algebra", Unknown Quantity: A Real And Imaginary History of Algebra. Joseph Henry Press, 49. ISBN 030909657X. 
  53. ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 239. ISBN 0471543977. “Abu'l Wefa was a capable algebraist aws well as a trionometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax2n + bxn = c (only equations with positive roots were considered),” 
  54. ^ a b c d e Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 241-242. ISBN 0471543977. “Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."” 
  55. ^ a b c Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 228. ISBN 0471543977. “Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to al-Khwarizmi. It is true that in two respects the work of al-Khwarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in in the Diophantine problems and, second, the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers.” 
  56. ^ Derbyshire, John (2006). "The Father of Algebra", Unknown Quantity: A Real And Imaginary History of Algebra. Joseph Henry Press, 31. ISBN 030909657X. “Diophantus, the father of algebra, in whose honor I have named this chapter, lived in Alexandria, in Roman Egypt, in either the 1st, the 2nd, or the 3rd century CE.” 
  57. ^ a b c d Stewart, Ian (2004). Galois Theory, Third Edition, Chapman & Hall/CRC Mathematics. 
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