Algebra (Arab. al-jaber, the science of solution), originally, a kind of higher arithmetic in which the numbers are replaced by symbols; but by later applications the symbols are used as well for geometrical quantities in space, or in mechanics for velocities, distances, and times, so that at present algebra occupies itself with quantities in general, whatever be their nature. The oldest work on this science is that of Dio-phantus of Alexandria, a Greek writer, who possibly flourished as early as the 4th century, of which the six books that have come down to us do not contain the elements, but the theory of the evolution of powers, and the method of solving undetermined problems. Many problems of this kind were by the ancients considered determined, as they threw out all solutions in irrational quantities. The Brahmins of Hindostan also had a knowledge of algebra, as well as the Arabs; but to whom belongs the priority of the invention it is at present impossible to determine. It is only known that this science was introduced into Christian Europe by the Moors of Spain, a little before the year 1100. For the first three centuries after its introduction it was chiefly studied in Italy, and Lucas Paciolus de Burgo (Luca di Borgo) was the first European writer on the subject.

His principal work, Summa Arithmetica et Geo-metrica, was published in Venice in 1494, and republished in 1523. He mentions a Pisan merchant, Leonardo Bonaccio, who lived in the beginning of the 13th century, and learned algebra in travelling among the Arabs along the coast of Africa and in the Levant. Some historians give to him the honor of having introduced this science in Europe, while others, among them Montucla, the great historian of mathematics, mention Paolo de l'Abacco and Belmondo of Padua, who preceded Bonaccio. From the works of Luca di Borgo it appears that in 1500 the science did not go beyond equations of the second degree, the negative solutions were rejected, and the symbols consisted chiefly of abbreviations of words. Great advance was made by Jerome Cardan, who in 1545 published his Ars Magna, in which ho gave the solution of equations of the third degree, by an operation which is still known among all mathematicians as the formula of Cardan; those of the fourth degree were solved by his pupil Ludovico Ferrari, and published in the Ars Magna, in which also he makes the distinction between positive, negative, and irrational solutions.

At the same time Stifelius in Germany invented the signs +, - , and √, which did so much to simplify the formulas; he published his Arithmetica Integra in Nuremberg in 1544. In 1552 Robert Recorde published in England "The Whetstone of Witte," in which for the first time the sign of equality (=) is introduced. From that time not much progress was made till Vieta in France perfected the algebraic operations and transformations of formulas, and even advanced so far as the general solution of equations of all degrees. He first applied algebra to geometry, and he also found the remarkable expression which solved numerically the problem of the quadrature of the circle. His works were written about the year 1600, but only published long after his death, by Schooten. Among the eminent mathematicians of that time we must also mention Gerard in Flanders, who was the first to indicate the important use of the negative roots of equations in geometrical constructions, while in England Harriot introduced the signs > and <, and Oughtred first wrote the decimal fractions simply by the decimal point, as we do now, without writing the denominator always, as was customary till his time.

The 17th century was the most brilliant of all centuries in mathematical discoveries, producing the immortal Descartes, Fermat, Wallis, Galileo, Huyghens, Kepler, Newton, Leibnitz, Bernoulli, and many others not less illustrious; and that century closed with the important discovery of the logarithms and of the differential calculus. The 18th century enriched the vast domain transmitted, and men like Laplace, La Grange, D'Alembert, Maupertuis, Maclaurin, Waring, Lambert, Cutler, Stirling, De Moivre, and above all Euler, developed and perfected all the branches of the science. - The operations of algebra are founded on a mutual agreement concerning signs and symbols. The first letters of the alphabet, a, b, c, etc, are used to represent known quantities, whether of space, time, or number, and the last, z, y, x, etc, are used for the unknown quantities. They are connected by the signs + , - , x, and ÷, meaning respectively addition, subtraction, multiplication, and division. The powers of quantities are expressed by superior numbers, as a2 for a x a, a5 for axaxaxaxa; the roots by the sign √, or 3√ and 5√, etc.

The small space in which a long operation can be indicated by these signs may be illustrated by the following algebraic expression:

17a x(c - d)2 + c3 x √(a2 - b2) - a x c

+

√3ab

√(b x c) - (a - b) x (c - d) + acd

8a5

which is an ordinary expression involving so many operations that to describe them clearly would occupy a whole page.