Algebra (Arabic: al-jebr, from الجبر al-jabr, meaning "reunion of broken parts") is a branch of mathematics concerning the study of structure, relation, and quantity. Elementary algebra is the branch that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Many historians agree that the earliest mathematical research was done by the priest classes of ancient civilizations, most notably the Babylonians, to go along with religious rituals. The origins of algebra can thus be traced back to ancient Babylonian mathematicians roughly four thousand years ago. After further development among Hellenistic and Indian mathematicians, it was eventually the work of Islamic mathematicians that established algebra as an independent discipline in its own right.
- 1 Etymology
- 2 Stages of algebra
- 3 Early developments
- 3.1 Ancient Babylonian mathematics
- 3.2 Ancient Egyptian mathematics
- 3.3 Ancient Indian mathematics
- 3.4 Ancient Chinese mathematics
- 3.5 Greek geometric mathematics
- 3.6 Hellenistic mathematics in Egypt
- 3.7 European mathematics in Dark Ages
- 4 Classical Indian mathematics
- 5 Classical Islamic algebra
- 5.1 Al-Khwarizmi
- 5.2 Ibn Turk and Logical Necessities in Mixed Equations
- 5.3 Abu Kamil: Irrational numbers and non-linear simultaneous equations
- 5.4 Al-Karaji: Pure algebra and algebraic calculus
- 5.5 Brethren of Purity and magic squares
- 5.6 Omar Khayyám: Geometric algebra and algebraic geometry
- 5.7 Sharaf al-Dīn: Numerical analysis and dynamic functional algebra
- 5.8 Al-Hassar and symbolic notation
- 6 Late Medieval algebra
- 6.1 Indian algebra
- 6.2 Chinese algebra
- 6.3 Medieval European algebra
- 6.4 Islamic algebra
- 7 Modern algebra
- 8 See also
- 9 Footnotes and citations
- 10 References
- 11 External links
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 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 the exact meaning of the word al-jabr is still unknown, most historians agree that the word meant something like "restoration", "completion", "reuniter of broken bones" or "bonesetter." 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
Algebra did not always make use of the symbolism that is now ubiquitous in mathematics, rather, it went through three distinct stages. The stages in the development of symbolic algebra are roughly as follows:
- Rhetorical algebra, where equations are written in full sentences. For example, the rhetorical form of x + 1 = 2 is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century.
- Syncopated algebra, where some symbolism is used but which does not contain all of the characteristic of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression first appeared in Diophantus' Arithmetica, followed by Brahmagupta's Brahma Sphuta Siddhanta.
- Symbolic algebra, where full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna and al-Qalasadi, though fully symbolic algebra sees its culmination in the work of Rene Descartes.
As important as the symbolism, or lack thereof, that was used in algebra was the degree of the equations that were used. 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 p and q are positive. This trichotomy comes about because quadratic equations of the form , with p and q 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 was solved by finding the side of a square of area A.
In addition to the three stages of expressing algebraic ideas, there were four conceptual stages in the development of algebra which occurred alongside the changes in expression. These four stages were as follows:
- Geometric stage, where the concepts of algebra are largely geometric. This dates back to the Babylonians and continued with the Greeks, and was later revived by Omar Khayyám.
- Static equation-solving stage, where the objective is to find numbers satisfying certain relationships. The move away from geometric algebra dates back to Diophantus and Brahmagupta, but algebra didn't decisively move to the static equation-solving stage until Al-Khwarizmi's Al-Jabr.
- Dynamic function stage, where motion is an underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra didn't decisively move to the dynamic function stage until Gottfried Leibniz.
- Abstract stage, where mathematical structure plays a central role. Abstract algebra is largely a product of the 19th and 20th centuries.
Ancient Babylonian mathematics
The origins of algebra can be traced to the ancient Babylonians, who developed a positional number system 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. One of the most famous tablets is the Plimpton 322 tablet, created around 1900 - 1600 BCE, 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 Egyptian mathematics
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 circa 1650 BCE by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 BCE. It is the most extensive ancient Egyptian mathematical document known to historians. The Rhind Papyrus contains problems where linear equations of the form and are solved, where a, b, and c are known and x, 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.
Ancient Indian mathematics
The method known as "Modus Indorum" or the method of the Indians have become our algebra today. This algebra came along with the Hindu Number system to Arabia and then migrated to Europe. The earliest known Indian mathematical documents are dated to around the middle of the first millennium B.C.E (around the sixth century B.C.E.).
The recurring themes in Indian mathematics are, among others, determinate and indeterminate linear and quadratic equations, simple mensuration, and Pythagorean triples.
Ancient Chinese mathematics
In Nine Chapters on the Mathematical Art, 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.
Nine Chapters on the Mathematical Art
Chiu-chang suan-shu or The Nine Chapters on the Mathematical Art, written around 250 BCE, 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.
Greek geometric mathematics
While the Greeks had no algebra in the modern sense, it would be inaccurate to say there was nothing like algebra. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric equivalent of algebra, where terms were represented by sides of geometric objects, usually lines, that had letters associated with them, and with this form they were able to find solutions to equations by using a process that they invented which is 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 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.
Bloom of Thymaridas
Iamblichus in Introductio arithmatica tells us that Thymaridas (ca. 400 BCE - ca. 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 n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/ (n + 2) of the difference between the sums of these pairs and the first given sum.
or using modern notion, the solution of the following system of n linear equations in n unknowns,
x + x1 + x2 + ... + xn-1 = s
x + x1 = m1
x + x2 = m2
x + xn-1 = mn-1
Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.
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 (ca. 380 BCE – ca, 320 BCE) 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 y2 = lx holds, where l 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 BCE - 190 BCE). 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.
Hellenistic mathematics in Egypt
Euclidean geometric mathematics
Euclid (Greek: Εὐκλείδης) was a Hellenistic Egyptian mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). 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.
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:
- 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 (left) distributive law, ; and in Books V and VII of the Elements the commutative and associative laws for multiplication are demonstrated.
Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation , and proposition 11 of Book II gives a solution to .
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. 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 and the familiar Babylonian equation , x ± y = b.
Diophantus was a Hellenized Babylonian mathematician who lived in Alexandria, Egypt, circa 250 AD, 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, but resembles Babylonian mathematics to a much greater extent, though it is also quite different from it in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.
In Arithmetica, Diophantus used 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
Diophantus would have written as
- ΚΥ α̅ς ι̅ ⫛ ΔΥ β̅ Μ α̅ ἴσ Μ ε̅
|ς||represents the unknown quantity (i.e. the variable)|
|ἴσ||(short for ἴσος) represents "equals"|
|⫛||represents the subtraction of everything that follows it up to ἴσ|
|Μ||represents the zeroth power of the variable (i.e. a constant term)|
|ΔΥ||represents the second power of the variable, from Greek δύναμις, meaning strength or power|
|ΚΥ||represents the third power of the variable, from Greek κύβος, meaning a cube|
|ΔΥΔ||represents the fourth power of the variable|
|ΔΚΥ||represents the fifth power of the variable|
|ΚΥΚ||represents the sixth power of the variable|
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:
and, 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:
It seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to earlier Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”
According to mathematics historian Odile Kouteynikoff:
According to the fact that Al-Khwarizmi founded Algebra during the 9th century, it is not surprising that, when being translated into Arabic in the late 9th century by Lebanese Ibn Luqa whose native language was Greek, Diophante’s Arithmetics seemed to be considered as a treatise about Algebra since algebraic vocabulary and way of thinking were most widely shared. Only few people understood that it was actually an arithmetic treatise: Al-Khazin (900–971) did, and therefore he is one of those who laid the foundations for the integer Diophantine analysis.
European mathematics in Dark Ages
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.
Classical Indian mathematics
Aryabhata and Aryabhatiya
Brahmagupta and 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 , , , 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 ax + by = c, where a, b, and c are integers. Unlike Diophantus who only gave one solution to an indeterminate equation, Brahmagupta gave all integer solutions; but that Brahmagupta used some similar examples as Diophantus has led some historians to consider the possibility of influence on Brahmagupta's work, or at least a common Babylonian source.
Like the work 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. 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.
Classical Islamic algebra
The first century of the Islamic Arab Empire saw very little mathematical achievements, since the Arabs, with their newly conquered empire, had not yet gained the intellectual drive, while research in other parts of the world had faded. In the eighth 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.
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 Hellenistic-Egyptian influence. Many scholars believe that it is the result of a combination of all three sources.
The Arabs initially used a fully rhetorical algebra, where often even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs did not initially develop a syncopated or symbolic algebra, until the work of Ibn al-Banna in the 13th century and Abū al-Hasan ibn Alī al-Qalasādī in the 15th century.
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.
Al-jabr wa'l muqabalah
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.
R. Rashed and Angela Armstrong write:
"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns 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. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, 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."
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).
In Al-Jabr, al-Khwarizmi uses geometric proofs, he does not recognize the root x = 0, 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 tenth 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
- ΔΥα̅ ςι̅ 'ίσ Μ λ̅θ̅
And al-Khwarizmi would have written as
- One square and ten roots of the same amount to thirty-nine dirhems; that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?
Father of algebra
The Islamic Persian mathematician Al-Khwarizmi is widely considered the father of algebra, though some have also given that title to the Hellenistic Babylonian mathematician Diophantus. Many agree that Al-Khwarizmi deserves this title most.
Those who support Diophantus point to the algebra found in Al-Jabr being more elementary than Arithmetica, and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to earlier Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”
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." Al-Khwarizmi's work established algebra as a mathematical discipline that is independent of geometry and arithmetic. In addition, R. Rashed and Angela Armstrong write:
"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns 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. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, 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."
Ibn Turk and 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 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 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: Irrational numbers and non-linear simultaneous equations
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-Karaji: Pure algebra and algebraic calculus
Al-Karkhi (953-1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940-998) and he was the first to discovered the numerical solution to equations of the form ax2n + bxn = c. 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.
Brethren of Purity and magic squares
Magic squares (an early form of matrix) were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian culture, and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Brethren of Purity's Rasa'il Ikhwan al-Safa (Encyclopedia of the Brethren of Purity); simpler magic squares were known to several earlier Arab mathematicians.
Islamic mathematicians also solved more complex examples of magic squares. They developed two basic methods to solve odd-order magic squares: the "diamond" technique, and a more sophisticated magic torus method understood in terms of a virtual torus.
Omar Khayyám: Geometric algebra and algebraic geometry
Omar Khayyám (ca. 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 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.
Geometric solution of cubic equation
As shown in this graph, to solve the third-degree equation where Omar Khayyám constructed the parabola the circle with diameter having its center on the positive x-axis and intersecting the origin, and a vertical line through the point above the x-axis where the circle and parabola intersect. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.
Sharaf al-Dīn: Numerical analysis and dynamic functional 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 have pointed out that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance.
Sharaf al-Din also developed the concept of a function. In his analysis of the equation for example, he begins by changing the equation's form to . 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 . To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when , which gives the functional value . Sharaf al-Din then states that if this value is less than , there are no positive solutions; if it is equal to , then there is one solution at ; and if it is greater than , then there are two solutions, one between and and one between and .
Al-Hassar and symbolic notation
Al-Hassār, a mathematician from the Maghreb (North Africa) 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.
The symbol now commonly denote an unknown variable. Even though any letter can be used, is the most common choice. This usage can be traced back to the Arabic word šay' شيء = “thing,” used in Arabic algebra texts such as the Al-Jabr, and was taken into Old Spanish with the pronunciation “šei,” which was written xei, and was soon habitually abbreviated to . (The Spanish pronunciation of “x” has changed since). Some sources say that this is an abbreviation of Latin causa, which was a translation of Arabic شيء. This started the habit of using letters to represent quantities in algebra. In mathematics, an “italicized x” () is often used to avoid potential confusion with the multiplication symbol.
Late Medieval algebra
Bhāskara II: Lilavati and Vija-Ganita
Bhāskara II (1114-ca. 1185) was the leading Indian mathematician of the twelfth century. In Algebra, he gave the general solution of the Pell 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, though 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
- . _ .
- ya 1 ru 1
- ya 2 ru 8
- Sum ya 1 ru 9
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.
Citrabhanu and simultaneous equations
Citrabhanu (c. 1530) was a 16th-century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:
For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.
Li Zhi and 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 - 1272 A.D.). 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.
Ch'in Chiu-shao and 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.
Yang Hui and magic squares
Chu Shih-chieh and 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 series equations given without proof in the Precious mirror. A few of the summation series are:
Medieval European algebra
In the thirteenth century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra in the 13th century.
Al-Kashi and numerical analysis
In the early 15th century, Jamshīd al-Kāshī developed an early form of Newton's method to numerically solve the equation to find roots of . 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.
Ibn al-Banna and Al-Qalasadi: Algebraic symbolism
Abū al-Hasan ibn Alī al-Qalasādī (1412–1482) 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."
As the Islamic world was declining after the fifteenth century, the European world was ascending. And it is here that Algebra was further developed.
A 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.
Modern European algebra
The symbol commonly denotes an unknown variable. Even though any letter can be used, is the most common choice. This usage can be traced back to the Arabic word šay' شيء = “thing,” used in Arabic algebra texts such as the Al-Jabr, and was taken into Old Spanish with the pronunciation “šei,” which was written xei, and was soon habitually abbreviated to . (The Spanish pronunciation of “x” has changed since). Some sources say that this is an abbreviation of Latin causa, which was a translation of Arabic شيء. This started the habit of using letters to represent quantities in algebra. In mathematics, an “italicized x” () is often used to avoid potential confusion with the multiplication symbol.
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.
Footnotes and citations
- "algebra". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=algebra&allowed_in_frame=0.
- (Boyer 1991, "Origins" p. 7) "It has been suggested that both Indian and Egyptian geometry may derive from a common source - a protogeometry that is related to primitive rites in somewhat the same way in which science developed from mythology and philosophy from theology. We must bear in mind that the theory of the origin of geometry in a secularization of ritualistic practice is by no means established. The development of geometry may just as well have been stimulated by the pratical needs of construction and surveying or by an aesthetic feeling for design and order."
- (Boyer 1991, "The Arabic Hegemony" p. 229) "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, which is evident in the treatise; the word muqabalah is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."
- (Boyer 1991, "Revival and Decline of Greek Mathematics" p.180) "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""
- (Boyer 1991, "Mesopotamia" p. 32) "Until modern times there was no thought of solving a quadratic equation of the form , 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) (2) (3)"
- Victor J. Katz, Bill Barton (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201, doi:10.1007/s10649-006-9023-7
- Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.
- (Boyer 1991, "Mesopotamia" p. 30) "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.""
- Joyce, David E. (1995). Plimpton 322. http://aleph0.clarku.edu/~djoyce/mathhist/plimpnote.html. "The clay tablet with the catalog number 322 in the G. A. Plimpton Collection at Columbia University may be the most well known mathematical tablet, certainly the most photographed one, but it deserves even greater renown. It was scribed in the Old Babylonian period between -1900 and -1600 and shows the most advanced mathematics before the development of Greek mathematics."
- (Boyer 1991, "Mesopotamia" p. 31) "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."
- (Boyer 1991, "Mesopotamia" p. 33) "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."
- (Boyer 1991, "Egypt" p. 11) "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 BCE. The scribe tells us that the material is derived from a prototype from the Middle Kingdom of about 2000 to 1800 BCE."
- (Boyer 1991, "Egypt" p. 19) "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."
- (Boyer 1991, "Egypt" pp. 15-16) "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 or , 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 one proposed characteristic of a procedure now known as the "method of false position," or the "rule of false." A specific false value has been proposed by 1920's scholars and the operations indicated on the left hand side of the equality sign are performed on this assumed number. Recent scholarship shows that scribes had not guessed in these situations. Exact rational number answers written in Egyptian fraction series had confused the 1920's scholars. The attested result shows that Ahmes "checked" result by showing that 16 + 1/2 + 1/8 exactly added to a seventh of this (which is 2 + 1/4 + 1/8), does obtain 19. Here we see another significant step in the development of mathematics, for the check is a simple instance of a proof."
- (Boyer 1991, "The Mathematics of the Hindus" p. 197) "The oldest surviving documents on Hindu mathematics are copies of works written in the middle of the first millennium B.C.E., approximately the time during which Thales and Pythagoras lived. [...] from the sixth century B.C.E."
- (Boyer 1991, "China and India" p. 222) "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."
- (Boyer 1991, "China and India" p. 197) "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."
- Swaney, Mark. . Cite error: Invalid
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- (Boyer 1991, "China and India" pp. 195-197) "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 treatise, 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 influential of all Chinese mathematical books, was the Chui-chang suan-shu, or Nine Chapters on the Mathematical 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."
- (Boyer 1991, "Euclid of Alexandria" p.109) "Book II of the Elements is a short one, containing only fourteen propositions, not one of which plays any role in modern textbooks the OTIN and The TORJAK are the main books; 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."
- (Boyer 1991, "The Heroic Age" pp. 77-78) "Whether deduction came into mathematics in the sixth century BCE or the fourth and whether incommensurability was discovered before or after 400 BCE, 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"
- (Boyer 1991, "Europe in the Middle Ages" p. 258) "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."
- (Heath 1981a, "The ('Bloom') of Thymaridas" pp. 94-96) Thymaridas of Paros, an ancient Pythagorean already mentioned (p. 69), was the author of a rule for solving a certain set of n simultaneous simple equations connecting n unknown quantities. The rule was evidently well known, for it was called by the special name [...] the 'flower' or 'bloom' of Thymaridas. [...] The rule is very obscurely worded , but it states in effect that, if we have the following n equations connecting n unknown quantities x, x1, x2 ... xn-1, namely [...] Iamblichus, our informant on this subject, goes on to show that other types of equations can be reduced to this, so that they rule does not 'leave us in the lurch' in those cases either."
- (Flegg 1983, "Unknown Numbers" p. 205) "Thymaridas (fourth century) is said to have had this rule for solving a particular set of n linear equations in n unknowns:
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/ (n + 2) of the difference between the sums of these pairs and the first given sum."
- (Boyer 1991, "The Euclidean Synthesis" p. 103) "Eutocius and Proclus both attribute the discovery of the conic sections to Menaechmus, who lived in Athens in the late fourth century BCE. Proclus, quoting Eratosthenes, refers to "the conic section triads of Menaechmus." Since this quotation comes just after a discussion of "the section of a right-angled cone" and "the section of an acute-angled cone," it is inferred that the conic sections were produced by cutting a cone with a plane perpendicular to one of its elements. Then if the vertex angle of the cone is acute, the resulting section (calledoxytome) is an ellipse. If the angle is right, the section (orthotome) is a parabola, and if the angle is obtuse, the section (amblytome) is a hyperbola (see Fig. 5.7)."
- (Boyer 1991, "The age of Plato and Aristotle" p. 94-95) "If OP=y and OD = x are coordinates of point P, we have y<sup2 = R).OV, or, on substituting equals,
Inasmuch as segments AR', BC, and AB are the same for all points P on the curve EQDPG, we can write the equation of the curve, a "section of a right-angled cone," as y2=lx, where l is a constant, later to be known as the latus rectum of the curve. [...] Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a string resemblance to the use of coordinates, as illustrated above, it has sometimes been maintains that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. [...] He had hit upon the conics in a successful search for curves with the properties appropriate to the duplication of the cube. In terms of modern notation the solution is easily achieved. By shifting the curring plane (Gig. 6.2), we can find a parabola with any latus rectum. If, then, we wish to duplicate a cube of edge a, we locate on a right-angled cone two parabolas, one with latus rectum a and another with latus rectum 2a. [...] It is probable that Menaechmus knew that the duplication could be achieved also by the use of a rectangular hyperbola and a parabola."
- (Boyer 1991, "Euclid of Alexandria" p. 100) "but by 306 BCE 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."
- (Boyer 1991, "Euclid of Alexandria" p. 101) "The tale related above in connection with a request of Alexander the Great for an easy introduction to geometry is repeated in the case of Ptolemy, who Euclid is reported to have assured that "there is no royal road to geometry.""
- (Boyer 1991, "Euclid of Alexandria" p. 104) "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."
- (Boyer 1991, "Euclid of Alexandria" p. 104) "The Elements was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all elementary mathematics-"
- (Boyer 1991, "Euclid of Alexandria" p. 110) "The same holds true for Elements II.5, which contains what we should regard as an impractical circumlocution for "
- (Boyer 1991, "Euclid of Alexandria" p. 111) "In an exactly analogous manner the quadratic equation 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 "
- (Boyer 1991, "Euclid of Alexandria" p. 103) "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 or , from which . 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 , x ± y = b., which again are the equivalents of solutions of simultaneous equations."
- (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) 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.}"
- (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 180-182) "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."
- (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "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."
- (Derbyshire 2006, "The Father of Algebra" pp. 35-36)
- (Cooke 1997, "Mathematics in the Roman Empire" pp. 167-168)
- (Boyer 1991, "Europe in the Middle Ages" p. 257) "The book makes frequent use of the identities [...] which had appeared in Diophantus and had been widely used by the Arabs."
- Harald Kittel, Übersetzung: ein internationales Handbuch zur Übersetzungsforschung, Volume 2 p. 1123, 1124
- History of Mathematics from Medieval Islam to Renaissance Europe: Guillaume Gosselin, an algebraist in Renaissance France
- (Boyer 1991, "Euclid of Alexandria pp. 192-193) "The death of Boethius may be taken to mark the end of ancient mathematics in the Western Roman Empire, as the death of Hypatia had marked the close of Alexandria as a mathematical center; but work continued for a few years longer at Athens. [...] When in 527 Justinian became emperor in the East, he evidently felt that the pagan learning of the Academy and other philosophical schools at Athens was a threat to orthodox Christianity; hence, in 529 the philosophical schools were closed and the scholars dispersed. Rome at the time was scarcely a very hospitable home for scholars, and Simplicius and some of the other philosophers looked to the East for haven. This they found in Persia, where under King Chosroes they established what might be called the "Athenian Academy in Exile."(Sarton 1952; p. 400)."
- (Boyer 1991, "The Mathematics of the Hindus" p. 207) "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."
- (Boyer 1991, "China and India" p. 219) "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."
- (Boyer 1991, "China and India" p. 220) "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."
- (Boyer 1991, "China and India" p. 221) "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 similar examples as Diophantus, we see again the possibility of 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."
- (Boyer 1991, "The Arabic Hegemony" p. 227) "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."
- (Boyer 1991, "The Arabic Hegemony" p. 234) "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."
- (Boyer 1991, "The Arabic Hegemony" p. 230) "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."
- (Boyer 1991, "The Arabic Hegemony" pp. 228-229): the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful."
- (Boyer 1991, "The Arabic Hegemony" p. 228) "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."
- Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11–2, ISBN 0792325656, OCLC 29181926
- (Boyer 1991, "The Arabic Hegemony" p. 229) "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 more 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."
- (Boyer 1991, "The Arabic Hegemony" pp. 229-230) "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 quadratic equations having positive roots."
- (Boyer 1991, "The Arabic Hegemony" p. 230) "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 multiply 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,"
- (Boyer 1991, "The Arabic Hegemony" p. 231) "The Algebra of al-Khwarizmi betrays unmistakable Hellenic elements,"
- (Boyer 1991, "The Arabic Hegemony" p. 233) "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."
- (Boyer 1991, "The Arabic Hegemony" p. 228) "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."
- (Boyer 1991, "The Arabic Hegemony" p. 234) "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."
- (Derbyshire 2006, "The Father of Algebra" p. 49)
- (Boyer 1991, "The Arabic Hegemony" p. 228) "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 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."
- (Derbyshire 2006, "The Father of Algebra" p. 31) "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."
- (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."
- Gandz and Saloman (1936), The sources of al-Khwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
- Roshdi Rashed (November 2009), Al Khwarizmi: The Beginnings of Algebra, Saqi Books, ISBN 0863564305
- Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11–2, ISBN 0792325656, OCLC 29181926
- O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html . "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects"."
- Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-Western Mathematics, Springer, ISBN 1402002602
- Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 9780691114859.
- (Boyer 1991, "The Arabic Hegemony" p. 239) "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),"
- O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Karaji.html .
- History of Mathematics from Medieval Islam to Renaissance Europe, Canadian Mathematical Society
- (Boyer 1991, "The Arabic Hegemony" pp. 241-242): 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."
- O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf.html .
- Rashed, Roshdi; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 342–3, ISBN 0792325656
- Berggren, J. L. (1990), "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2): 304–9, "Rashed has argued that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance for investigating conditions under which cubic equations were solvable; however, other scholars have suggested quite difference explanations of Sharaf al-Din's thinking, which connect it with mathematics found in Euclid or Archimedes."
- Victor J. Katz, Bill Barton (October 2007), "Stages in the History of Algebra with Implications for Teaching", Educational Studies in Mathematics (Springer Netherlands) 66 (2): 185–201 , doi:10.1007/s10649-006-9023-7
- Prof. Ahmed Djebbar (June 2008). "Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa". FSTC Limited. http://muslimheritage.com/topics/default.cfm?ArticleID=952. Retrieved 2008-07-19.
- (Boyer 1991, "China and India" pp. 222-223) "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."
- (Boyer 1991, "China and India" p. 204) "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."
- (Boyer 1991, "China and India" pp. 204-205) "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."
- (Boyer 1991, "China and India" p. 203) "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."
- (Boyer 1991, "China and India" p. 205) "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."
- Tjalling J. Ypma (1995), "Historical development of the Newton-Raphson method", SIAM Review 37 (4): 531–51, doi:10.1137/1037125
- O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html .
- Struik (1969), 367
- Bashmakova, I., and Smirnova, G. (2000) The Beginnings and Evolution of Algebra, Dolciani Mathematical Expositions 23. Translated by Abe Shenitzer. The Mathematical Association of America.
- Boyer, Carl B. (1991), A History of Mathematics (Second Edition ed.), John Wiley & Sons, Inc., ISBN 0471543977
- Cooke, Roger (1997), The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0471180823
- Derbyshire, John (2006), Unknown Quantity: A Real And Imaginary History of Algebra, Joseph Henry Press, ISBN 030909657X
- Stillwell, John (2004), Mathematics and its History (Second Edition ed.), Springer Science + Business Media Inc., ISBN 0387953361
- Burton, David M. (1997), The History of Mathematics: An Introduction (Third Edition ed.), The McGraw-Hill Companies, Inc., ISBN 0070094659
- Heath, Thomas Little (1981a), A History of Greek Mathematics, Volume I, Dover publications, ISBN 0486240738
- Heath, Thomas Little (1981b), A History of Greek Mathematics, Volume II, Dover publications, ISBN 0486240746
- Flegg, Graham (1983), Numbers: Their History and Meaning, Dover publications, ISBN 0486421651