History of Algebra
The history of algebra began in ancient Egypt and Babylon, where people learned to solve
linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations
such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians
solved arbitrary quadratic equations by essentially the same procedures taught today. They
also could solve indeterminate equations.
The Alexandrian mathematicians Hero and Diophantus continued the traditions of Egypt and
Babylon, but Diophantus' book Arithmetica is on a much higher level and gives many
surprising solutions to difficult indeterminate equations. This ancient knowledge of
solutions of equations in turn found a home early in the Islamic world, where it was known
as the “science of restoration and balancing.” (The Arabic word for restoration,
al-jabru, is the root of the word algebra, and algebra as a science is an Arabic
contribution.) In the 9th century, al-Khwarizmi wrote one of the first Arabic algebras, a
systematic exposé of the basic theory of equations, with both examples and proofs. By the
end of the 9th century the Egyptian Abu Kamil (850-930) stated and proved the basic laws
and identities of algebra and solved such complicated problems as finding x, y, and z such
that x + y + z = 10, x2 + y2 = z2, and xz = y2.
Ancient civilizations wrote out algebraic expressions, using only occasional
abbreviations, but by medieval times Islamic mathematicians were able to talk about
arbitrarily high powers of the unknown x, and work out the basic algebra of polynomials
(without yet using modern symbolism). This included the ability to multiply, divide, and
find square roots of polynomials as well as a knowledge of the binomial theorem. The
Persian Omar Khayyam showed how to express roots of cubic equations by line segments
obtained by intersecting conic sections, but he could not find a formula for the roots. A
Latin translation of al-Khwarizmi's Algebra appeared in the 12th century, and in the early
13th century appeared the writings of the great Italian mathematician Leonardo Fibonacci
(1170-1230), among whose achievements was a close approximation to the solution of the
cubic equation x3 + 2x2 + cx = d. Because Fibonacci had travelled in Islamic lands, he
probably used an Arabic method of successive approximations.
Early in the 16th century, the Italian mathematicians Scipione del Ferro (1465-1526),
Niccolò Tartaglia (1500-57), and Gerolamo Cardano (1501-76) solved the general cubic
equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico
Ferrari (1522-65), soon found an exact solution to equations of the fourth degree, and as
a result, mathematicians for the next several centuries tried to find a formula for the
roots of equations of degree five, or higher. Early in the 19th century, however, the
Norwegian Niels Abel and the French Évariste Galois proved that no such formula exists.
An important development in algebra in the 16th century was the introduction of symbols
for the unknown and for algebraic powers and operations. Thus Book III of the French
philosopher and mathematician René Descartes' La géometrie (1637) looks much like a
modern algebra text. Descartes is most significant in mathematics, however, for his
discovery of analytic geometry, which reduces the solution of geometric problems to the
solution of algebraic ones. His geometry text also contained the essentials of a course on
the theory of equations, including his so-called rule of signs for counting the number of
what Descartes called the “true” (positive) and “false” (negative)
roots of an equation. Work continued through the 18th century on the theory of equations,
but not until 1799 was the proof published, by the German mathematician Carl Friedrich
Gauss, that every polynomial equation has at least one root.
By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving
polynomial equations to the structure of systems of elements that arose as abstractions of
systems, such as the complex numbers, that mathematicians had met in their study of
polynomial equations. Two examples of such systems are groups (see Group) and quaternions,
which share some of the properties of number systems but also depart from them in
important ways. Groups began as systems of permutations and combinations of roots of
polynomials, but they became one of the chief unifying concepts of 19th-century
mathematics. Important contributions to their study were made by the French mathematicians
Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian
mathematicians Niels Abel and Sophus Lie (1842-99). Quaternions were the discovery of the
British mathematician William Rowan Hamilton, who extended the arithmetic of complex
numbers (see Complex Number) to quaternions (quadruples of real numbers).
Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann
(1809-77) began investigating vectors (see Vector). Despite its abstract character,
American physicist J. W. Gibbs recognized in vector algebra a system of great utility for
physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread
influence of this abstract approach led to the publication in 1854 of George Boole's The
Laws of Thought, an algebraic treatment of basic logic. Since that time, modern
algebra—also called abstract algebra—has continued to develop. Important new
results have been discovered, and the subject has found applications in all branches of
mathematics and in many of the sciences, as well.
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