History of algebra

Elementary algebra is the branch of mathematics that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Historians know that the earliest mathematical research was done by the priest classes of ancient civilizations, such as 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.

= 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 Unicode|Muhammad ibn Mūsā al-Khwārizmī entitled, in Arabic, كتاب الجبر والمقابلة, or Unicode | 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", "completion",Harv|Boyer|1991|loc="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; the word "muqabalah" is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."] "reuniter of broken bones" or "bonesetter."

=Stages of algebra=

Algebraic expression

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: [Harv|Boyer|1991|loc="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""]
* 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 16^th 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", and then the works of several Islamic mathematicians such as Ibn al-Banna and al-Qalasadi.
* Symbolic algebra, where full symbolism is used. Symbolic algebra sees its culmination in the work of Gottfried Leibniz.

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.
* $x^2\; +\; px\; =\; q$
* $x^2\; =\; px\; +\; q$
* $x^2\; +\; q\; =\; px$where p and q are positive.This trichotomy is because quadratic equations of the form $x^2\; +\; px\; +\; q\; =\; 0$, with p and q positive, have no positive roots. [Harv|Boyer|1991|loc="Mesopotamia" p. 32 "Until modern times there was no thought of solving a quadratic equation of the form $x^2\; +\; px\; +\; q\; =\; 0$, 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)$x^2\; +\; px\; =\; q$ (2)$x^2\; =\; px\; +\; q$ (3)$x^2\; +\; q\; =\; px$"]

Conceptual stages

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: [Citation|last=Victor J. Katz|first=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|publisher=Springer Netherlands|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185-201]

*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.

=History=

Babylonian algebra

The origins of algebra can be traced to the ancient Babylonians, [cite book|first= Dirk J.|last=Struik|title=A Concise History of Mathematics|publisher=New York: Dover Publications|year=1987] 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.Harv|Boyer|1991|loc="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.""] One of the most famous tablets is the Plimpton 322 tablet, created around 1600 - 1900 B.C., which gives a table of Pythagorean triples and represents some of the most advanced mathematics prior to Greek mathematics. [cite journal | url = http://aleph0.clarku.edu/~djoyce/mathhist/plimpnote.html | title = Plimpton 322 | author = Joyce, David E. | year = 1995 | quote=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.]

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, [Harv|Boyer|1991|loc="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."] and many cubic equations although it is not known if they were able to reduce the general cubic equation.Harv|Boyer|1991|loc="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."]

Egyptian algebra

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 B.C. by Ahmes, who transcribed it from an earlier work that he dated to between 2000 and 1800 B.C. [Harv|Boyer|1991|loc="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 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."] It is the most extensive ancient Egyptian mathematical document known to historians. [Harv|Boyer|1991|loc="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."] 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.Harv|Boyer|1991|loc="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 $x\; +\; ax\; =\; b$ 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 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."] The solutions were possibly, but not likely, arrived at by using the "method of false position," or "regula falsi", where first a specific value is substituted into the left hand side of the equation, then the required arithmetic calculations are done, thirdly the result is compared to the right hand side of the equation, and finally the correct answer is found through the use of proportions. In some of the problems the author "checks" his solution, thereby writing one of the earliest known simple proofs.

Greek geometric algebra

It is sometimes alleged that the Greeks had no algebra, but this is plainly false. By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them,Harv|Boyer|1991|loc="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."] 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".Harv|Boyer|1991|loc="The Heroic Age" pp. 77-78 "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"] "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.Harv|Heath|1981a|loc="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", "x"₁, "x"₂ ... "x"_n-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."] 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.Harv|Flegg|1983|loc="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."]

or using modern notion, the solution of the following system of "n" linear equations in "n" unknowns,

x + x₁ + x₂ + ... + x_n-1 = s
x + x₁ = m₁
x + x₂ = m₂
.
.
.
x + x_n-1 = m_n-1

is,

$x=cfrac\{(m\_1\; +\; m\_2\; +\; ...\; +\; m\_\{n-1\})\; -\; s\}\{n-2\}\; =\; cfrac\{\; (sum\_\{x=1\}^n\; m\_x)\; -s\}\{n\; -2\}$

Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.

Euclid of Alexandria

Euclid (Greek: polytonic|Εὐκλείδης) was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC).Harv|Boyer|1991|loc="Euclid of Alexandria" p. 100 "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."] [Harv|Boyer|1991|loc="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.""] 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. [Harv|Boyer|1991|loc="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."] 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. [Harv|Boyer|1991|loc="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-"]

"Elements"

The geometric work of the Greeks, typified in Euclid's "Elements", provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.

Book II of the "Elements" contains fourteen propositions, which in Euclid's time were extremely significant for doing geometric algebra. These propositions and their results are the geometric equivalents of our modern symbolic algebra and trigonometry. Today, using modern symbolic algebra, we let symbols represent known and unknown magnitudes (i.e. numbers) and then apply algebraic operations on them. While in Euclid's time magnitudes were viewed as line segments and then results were deduced using the axioms or theorems of geometry.Harv|Boyer|1991|loc="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. 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 $(a\; +\; b)^2\; =\; a^2\; +\; 2ab\; +\; b^2$,"]

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, $a(b\; +\; c\; +\; d)\; =\; ab\; +\; ac\; +\; ad$; and in Books V and VII of the "Elements" the commutative and associative laws for multiplication are demonstrated.

Many basic equations were also proved geometrically. For instance, proposition 4 in Book II proves that $a^2\; -\; b^2\; =\; (a\; +\; b)(a\; -\; b)$, [Harv|Boyer|1991|loc="Euclid of Alexandria" p. 110 "The same holds true for "Elements" II.5, which contains what we should regard as an impractical circumlocution for $a^2\; -\; b^2\; =\; (a\; +\; b)(a\; -\; b)$"] and proposition 5 in Book II proves that $(a\; +\; b)^2\; =\; a^2\; +\; 2ab\; +\; b^2$.

Furthermore, there are also geometric solutions given to many equations. For instance, proposition 6 of Book II gives the solution to the quadratic equation $ax\; +\; x^2\; =\; b^2$, and proposition 11 of Book II gives a solution to $ax\; +\; x^2\; =\; a^2$. [Harv|Boyer|1991|loc="Euclid of Alexandria" p. 111 "In an exactly analogous manner the quadratic equation $ax\; +\; x^2\; =\; b^2$ 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 $ax\; +\; x^2\; =\; a^2$"]

"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". 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 $dx^2\; -\; adx\; +\; b^2c\; =\; 0$ and the familiar Babylonian equation $xy\; =\; a^2$, x ± y = b.Harv|Boyer|1991|loc="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 $(a\; -\; x)dx\; =\; b^2c$ or $dx^2\; -\; adx\; +\; b^2c\; =\; 0$, from which $x\; =\; a/2\; +/-\; sqrt((a/2)^2\; -\; b^2c/d)$. 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 $xy\; =\; a^2$, x ± y = b., which again are the equivalents of solutions of simultaneous equations."]

Conic sections

A conic section is a curve that results from the intersection of a cone with a plane. There are three primary types of conic sections: ellipses (including circles), parabolas, and hyperbolas. The conic sections are reputed to have been discovered by Menaechmus [Harv|Boyer|1991|loc="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 B.C.E. 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 (called"oxytome") 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)."] (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 = "l"x 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.Harv|Boyer|1991|loc="The age of Plato and Aristotle" p. 94-95 "If OP=y and OD = x are coordinates of point P, we have y = R).OV, or, on substituting equals,
y²=R'D.OV=AR'.BC/AB.DO.BC/AB=AR'.BC²/AB².x
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 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 2"a". [...] It is probable that Menaechmus knew that the duplication could be achieved also by the use of a rectangular hyperbola and a parabola."]

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.

Chinese algebra

Chinese Mathematics dates to at least 300 BCE with the "Chou Pei Suan Ching", generally considered to be one of the oldest Chinese mathematical documents.Harv|Boyer|1991|loc="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."]

"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.

"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.). 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.Harv|Boyer|1991|loc="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."]

"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.

Magic squares

[
Yang Hui (Pascal's) triangle, as depicted by the ancient Chinese using rod numerals.] The earliest known magic squares appeared in China.Harv|Boyer|1991|loc="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."] In "Nine Chapters" the author solves a system of simultaneous linear equations by placing the coefficients and constant terms of the linear equations into a magic square (i.e. a matrix) and performing column reducing operations on the magic square. The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261 - 1275), who worked with magic squares of order as high as ten. [Harv|Boyer|1991|loc="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."]

"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. 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.Harv|Boyer|1991|loc="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."]

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:Harv|Boyer|1991|loc="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."]

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

Diophantine algebra

Diophantus was a Hellenistic mathematician who lived 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.Harv|Boyer|1991|loc="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.}"] "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.Harv|Boyer|1991|loc="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."]

In "Arithmetica", Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as "syncopated" algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.Harv|Boyer|1991|loc="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."] So, for example, what we would write as :$x^3\; -\; 2x^2\; +\; 10x\; -1\; =\; 5$Diophantus would have written as:Κ^υunicode| ᾱ̄ ζ unicode| ί̄ unicode| ⫛ Δ^υ unicode| β̄ Μ unicode| ᾱ̄ polytonic|ἴσ Μ unicode| ε̄where the symbols represent the following:Harv|Derbyshire|2006|loc="The Father of Algebra" pp. 35-36] [Harv|Cooke|1997|loc="Mathematics in the Roman Empire" pp. 167-168]

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:

:$\{x^3\}1\{x\}10\; -\; \{x^2\}2\{x^0\}1\; =\; \{x^0\}5$

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

:$(\{x^3\}1+\{x\}10)\; -\; (\{x^2\}2+\{x^0\}1)\; =\; \{x^0\}5$

"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: [Harv|Boyer|1991|loc="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."]

:

Indian algebra

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.). [Harv|Boyer|1991|loc="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."]

The recurring themes in Indian mathematics are, among others, determinate and indeterminate linear and quadratic equations, simple mensuration, and Pythagorean triples.Harv|Boyer|1991|loc="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."]

"Aryabhatiya"

Aryabhata (476–550 A.D.) was an Indian mathematician who authored "Aryabhatiya". In it he gave the rules, [Harv|Boyer|1991|loc="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."] :$1^2\; +\; 2^2\; +\; cdots\; +\; n^2\; =\; \{n(n\; +\; 1)(2n\; +\; 1)\; over\; 6\}$and:$1^3\; +\; 2^3\; +\; cdots\; +\; n^3\; =\; (1\; +\; 2\; +\; cdots\; +\; n)^2$

"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. [Harv|Boyer|1991|loc="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."] In indeterminate analysis Brahmagupta gives the Pythagorean triads $m$, $\{1\; over\; 2\}(\{m^2over\; n\}\; -\; n)$, $\{1\; over\; 2\}(\{m^2over\; n\}\; +\; n)$, but this is a modified form of an old Babylonian rule that Brahmagupta may have been familiar with. [Harv|Boyer|1991|loc="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."] 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 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.Harv|Boyer|1991|loc="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 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."]

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. The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.

Bhāskara II

Bhāskara II (1114-ca. 1185) was the leading 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 and he fails to distinguish between exact and approximate statements.Harv|Boyer|1991|loc="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."] 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

:$(\; -x\; -\; 1\; )\; +\; (\; 2x\; -\; 8\; )\; =\; x\; -\; 9$

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.

Islamic algebra

The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.Harv|Boyer|1991|loc="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."] 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.Harv|Boyer|1991|loc="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."]

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.Harv|Boyer|1991|loc="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."]

Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where often even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs never adopted or developed 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.

"Al-jabr wa'l muqabalah"

The Muslim [Harv|Boyer|1991|loc="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.""] Persian mathematician Unicode|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". 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. [Harv|Boyer|1991|loc="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."] 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-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).Harv|Boyer|1991|loc="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."]

In "Al-Jabr", al-Khwarizmi uses geometric proofs, he does not recognize the root x = 0, and he only deals with positive roots. [Harv|Boyer|1991|loc="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."] He also recognizes that the discriminant must be positive and he does not justify his procedure for completing the square. [Harv|Boyer|1991|loc="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,"] The Greek influence is shown by "Al-Jabr"'s geometric foundations [Harv|Boyer|1991|loc="The Arabic Hegemony" p. 231 "The "Algebra" of al-Khwarizmi betrays unmistakable Hellenic elements,"] and by one problem taken from Heron. [Harv|Boyer|1991|loc="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."] 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",Harv|Boyer|1991|loc="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."] 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 :$x^2\; +\; 10x\; =\; 39$Diophantus would have written asHarv|Derbyshire|2006|loc="The Father of Algebra" p. 49] :Δ^Υ unicode| ᾱ̄ ζ unicode| ί̄ 'ίσ Μ λ θAnd 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?

"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".Harv|Boyer|1991|loc="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."] 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 and al-Karkhi

Arabic mathematicians treated irrational numbers as algebraic objects. [MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999 "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated 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. [Jacques Sesiano, "Islamic mathematics", p. 148, in citation|title=Mathematics Across Cultures: The History of Non-western Mathematics|first1=Helaine|last1=Selin|first2=Ubiratan|last2=D'Ambrosio|year=2000|publisher= [Springer Science+Business Media|Springer|isbn=1402002602]

Al-Karkhi (953-1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940-998) and he discovered the first numerical solution to equations of the form ax²ⁿ + bxⁿ = c.Harv|Boyer|1991|loc="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),"] 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. [MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji]

Omar Khayyám, Sharaf al-Dīn, and al-Kashi

Omar Khayyám (ca. 1050 - 1123) wrote a book on Algebra that went beyond "Al-Jabr" to include equations of the third degree.Harv|Boyer|1991|loc="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.""] Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible. His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Ibn al-Haytham (Alhazen), but Omar Khayyám generalized the method to cover all cubic equations with positive roots. He only considered positive roots and he did not go past the third degree. He also saw a strong relationship between Geometry and Algebra.

In the 12th century, Sharaf al-Dīn al-Tūsī (1135-1213) wrote the "Al-Mu'adalat" ("Treatise on Equations"), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed the concepts of the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. [MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi] He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula [Citation | last1=Rashed | first1=Roshdi | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=Springer | isbn=0792325656 | pages=342-3] to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes.Citation|first=J. L.|last=Berggren|year=1990|title=Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat|journal=Journal of the American Oriental Society|volume=110|issue=2|pages=304-9|quote=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.]

Sharaf al-Din also developed the concept of a function. In his analysis ofthe equation $x^3\; +\; d\; =\; bx^2$ for example, he begins by changing the equation's form to $x^2\; (b\; -\; x)\; =\; d$. 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 $d$. To determine this, he finds a maximum value for the function. He proves that the maximum value occurs when $x\; =\; frac\{2b\}\{3\}$, which gives the functional value $frac\{4b^3\}\{27\}$. Sharaf al-Din then states that if this value is less than $d$, there are no positive solutions; if it is equal to $d$, then there is one solution at $x\; =\; frac\{2b\}\{3\}$; and if it is greater than $d$, then there are two solutions, one between $0$ and $frac\{2b\}\{3\}$ and one between $frac\{2b\}\{3\}$ and $b$. [Citation|last=Victor J. Katz|first=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|publisher=Springer Netherlands|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185-201 [192] ]

In the early 15th century, Jamshīd al-Kāshī developed an early form of Newton's method to numerically solve the equation $x^P\; -\; N\; =\; 0$ to find roots of $N$. [Tjalling J. Ypma (1995), "Historical development of the Newton-Raphson method", "SIAM Review" 37 (4): 531–51, doi|10.1137/1037125]

Al-Hassār, Ibn al-Banna, and al-Qalasadi

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.cite web|title= Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa|author=Prof. Ahmed Djebbar|publisher=FSTC Limited|url=http://muslimheritage.com/topics/default.cfm?ArticleID=952|date=June 2008|accessdate=2008-07-19]

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.MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi] 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."

European algebra

As the Islamic world was declining from the twelfth century onwards, the European world was ascending. And it is here that Algebra was further developed.

Dark ages

Just as the death of Hypatia signals the close of the University 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".Harv|Boyer|1991|loc="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)."] 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.

The twelfth century saw a flood of translations from Arabic into Latin and by the thirteenth century, European mathematics was beginning to rival the mathematics of other lands. In the thirteenth century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra.

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.

The symbol $mathit\{x\}$ commonly denotes an unknown variable. Even though any letter can be used, $mathit\{x\}$ 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 $mathit\{x\}$. (The Spanish pronunciation of “x” has changed since). Some sources say that this $mathit\{x\}$ 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” ($x!$) is often used to avoid potential confusion with the multiplication symbol.

Gottfried Leibniz

Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Gottfried Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular. [Struik (1969), 367] In the 18th century, "function" lost these geometrical associations.

Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which can be manipulated to find the solution of the system, if any. This method was later called Gaussian elimination. Leibniz also discovered Boolean algebra and symbolic logic, also relevant to algebra.

Abstract algebra

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

The father of algebra

The Hellenistic mathematician Diophantus was traditionally known as "the father of algebra"Harv|Boyer|1991|loc="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."] Harv|Derbyshire|2006|loc="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."] but debate now exists as to whether or not Al-Khwarizmi deserves this title instead. 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.

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, [Harv|Boyer|1991|loc="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."] 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. [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".] 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."

= See also =
*Algebra
*Timeline of algebra
*History of Mathematics

= Footnotes and citations =

= References =
* 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.
*Harvard reference
first=Carl B.
last=Boyer
authorlink=Carl Benjamin Boyer
title=A History of Mathematics
edition=Second Edition
publisher=John Wiley & Sons, Inc.
year=1991
isbn=0471543977
*Harvard reference
first=Roger
last=Cooke
authorlink=Roger Cooke
title=The History of Mathematics: A Brief Course
publisher=Wiley-Interscience
year=1997
isbn=0471180823
*Harvard reference
first=John
last=Derbyshire
authorlink=John Derbyshire
title=Unknown Quantity: A Real And Imaginary History of Algebra
publisher=Joseph Henry Press
year=2006
isbn=030909657X
*Harvard reference
first=John
last=Stillwell
authorlink=John Stillwell
title=Mathematics and its History
edition=Second Edition
publisher=Springer Science + Business Media Inc.
year=2004
isbn=0387953361
*Harvard reference
first=David M.
last=Burton
authorlink=David M. Burton
title=The History of Mathematics: An Introduction
edition=Third Edition
publisher=The McGraw-Hill Companies, Inc.
year=1997
isbn=0070094659
*Harvard reference
first=Thomas Little
last=Heath
authorlink= T. L. Heath
title=A History of Greek Mathematics, Volume I
publisher=Dover publications
year=1981a
isbn=0486240738
*Harvard reference
first=Thomas Little
last=Heath
authorlink= T. L. Heath
title=A History of Greek Mathematics, Volume II
publisher=Dover publications
year=1981b
isbn=0486240746
*Harvard reference
first=Graham
last=Flegg
authorlink= Graham Flegg
title=Numbers: Their History and Meaning
publisher=Dover publications
year=1983
isbn=0486421651

= External links =
* [http://mathdl.maa.org/convergence/1/?pa=content&sa=browseNode&resultsPerPage=20&categoryId=8&subjectId=1&searchKeywords=&imageField.x=116&imageField.y=23 History of Algebra Articles] at [http://mathdl.maa.org/convergence/1/ Convergence]