HISTORY OF ALGEBRA The earliest known treatise containing problems which would at present be called algebraic is the Ahmes Papyrus (also called, from the name of its former owner, the Rhind Papyrus) now in the British Museum, and written c. 1700-1600 B.C. The first of these problems reads as follows : "Ahe (or hau, transliterated by Egyptologists as `h`w, meaning heap, mass or quantity), its whole, its seventh, it makes I 9,"—that is, x-I-? x=19. The method of solving was by estimating the result and then correcting the error by repeated trials. The Greek mathematicians were primarily geometers, and they made various kinds of geometric constructions by which they found a line segment corresponding to the root of an equation. Euclid (c. 30o B.e. ), for example, solved problems equivalent to x±y=a, and to The only Greek to write extensively on algebra was Diophantus (c. 275). He was also the first writer to introduce any algebraic symbols of special significance, these including a character to represent the unknown quantity, a symbol for subtraction, the use of initial letters for equality, square and cube, and a scheme of combinations for other powers. In the Orient there is evidence of an early interest in problems which would now be solved by algebra, and the Chinese were able to solve the quadratic equation before the Christian era. The Hindu works of Brahmagupta (c. 628), Mahavira (c. 85o) and Bhaskara (c. contain a large number of problems solved by algebraic methods, and show considerable ability in analysis. In the Muslim world, and par ticularly at Baghdad in the time of the caliphs, two lines of mathematical thought converged, the first from Greek sources and the second from India. The result was the preparation of such textbooks as those of Mohammed ibn Musa al-Khowarizmi (c. 825), Abu Kamil (c. goo) and al-Karkhi (c. Imo). Of these, the algebra of al-Khowarizmi had the greatest influence upon European mathe matics, being translated by Robert of Chester (c. 114o) and other mediaeval scholars. The Oriental writers mentioned were able to solve the quadratic equation, but whether their method originated in the East or was suggested by the Greeks is uncertain.
The beginning of printing in Europe found algebraists pos sessed of no convenient symbolism but, as already stated, able to solve the quadratic. Italy was the centre of learning and her scholars devoted much attention to solving the cubic equation. The latter solution was finally effected, with substantial complete ness, by Tartaglia (q.v., 1535) being published by Cardan (q.v.) in his Ars Magna • The biquadratic was solved by Ferrari (154o), a pupil of Cardan's, and was also published by the latter in the same work. The needed improvements in symbolism were generally made outside of Italy by such writers as Vieta (q.v., c. 1590), Harriot (q.v., c. 1600) and Descartes (q.v., 1637), with the notable extension of the exponential notation by John Wallis (q.v., 1655). The proof of the insolubility of the general equation of the fifth degree by algebraic methods is due to the investiga tions of Ruffin (1803, 1805), Abel (q.v., 1824) and Galois (q.v., 1831; posthumous publication, 1846). Elementary algebra may be said to have been substantially completed by the close of the I 7th century, by which time higher algebra, especially through the