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nydus/A History of MathematicsPublic

This text examines the transition from the Middle Ages to the Modern era, highlighting how the fall of Constantinople and the invention of the printing press catalyzed a revival of classical learning. It traces the shift toward scientific inquiry through the rise of pure mathematics and astronomy, detailing the intellectual struggle against established scholastic and ecclesiastical authority.

Page 126 of 219
Table of Contents

Algebra

but Hamilton's quaternions met with immediate

appreciation in England. These algebras offer a geometrical interpretation of imaginaries. During the times of Descartes,

Newton, and Euler, we have seen the negative and the imaginary,

−1, accepted as numbers, but the latter was still regarded as an algebraic fiction. The first to give it a geometric picture, analogous to the geometric interpretation of the negative, was H. Kühn, a teacher in Danzig, in a publication of

1750–1751. He represented a−1 by a line perpendicular to the line a, and equal to a in length, and construed −1 as the mean proportional between +1 and −1. This same idea was developed further, so as to give a geometric interpretation of a+−b, by Jean-Robert Argand (1768–?) of Geneva,

in a remarkable Essai (1806).70 The writings of Kühn and Argand were little noticed, and it remained for Gauss to break

down the last opposition to the imaginary. He introduced i as an independent unit co-ordinate to 1, and a+ib as a "complex

number." The connection between complex numbers and points on a plane, though artificial, constituted a powerful aid in the further study of symbolic algebra. The mind required a visual representation to aid it. The notion of what we now call vectors was growing upon mathematicians,

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

same time.

William Rowan Hamilton (1805–1865) 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

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