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 and the geometric addition of vectors in space was discovered independently by Hamilton, Grassmann, and others, about the same time.

William Rowan Hamilton (1805–1866) was born of Scotch parents in Dublin. His early education; carried on at home, was mainly in languages. At the age of thirteen he is said to have been familiar with as many languages as he had lived years. About this time he came across a copy of Newton's Universal Arithmetic. After reading that, he took up successively analytical geometry, the calculus, Newton's Principia, Laplace's Mécanique Céleste. At the age of eighteen he published a paper correcting a mistake in Laplace's work. In 1824 he entered Trinity College, Dublin, and in 1827, while he was still an undergraduate, he was appointed to the chair of astronomy. His early papers were on optics. In 1832 he predicted conical refraction, a discovery by aid of mathematics which ranks with the discovery of Neptune by Le Verrier and Adams. Then followed papers on the Principle of Varying Action (1827) and a general method of dynamics (1834–1835). He wrote also on the solution of equations of the fifth degree, the hodograph, fluctuating functions, the numerical solution of differential equations.

The capital discovery of Hamilton is his quaternions, in which his study of algebra culminated. In 1835 he published in the Transactions of the Royal Irish Academy his Theory of Algebraic Couples. He regarded algebra "as being no mere art, nor language, nor primarily a science of quantity, but rather as the science of order of progression." Time appeared to him as the picture of such a progression. Hence his definition of algebra as "the science of pure time." It was the subject of years' meditation for him to determine what he should regard as the product of each pair of a system of perpendicular directed lines. At last, on the 16th of October,