(B) HISTORY OF ALGEBRA
(ix) Franciscus Vieta (François Viète) (1540-1603). Albert Girard.
By such gradual steps did algebra advance in improvement from its first introduction by Leonardo, each succeeding writer making some change for the better; but with the exception of Tartalea, Cardan, and Ferrari, hardly any one rose to the rank of an inventor. At length came Vieta, to whom his branch of mathematical learning, as well as others, is highly indebted. His improvements in algebra were very considerable; and some of his inventions, although not then fully developed, have yet been the germs although not then fully developed, have yet been the germs of later discoveries. He was the first that employed general characters to represent known as well as unknown quantities. Simple as this step may appear, it has yet led to important consequences. He must also be regarded as the first that applied algebra to the improvement of geometry. The older algebraists had indeed resolved geometrical problems, but each solution was particular; whereas Vieta, by introducing general symbols, produced general formulae, which were applicable to all problems of the same kind, without the trouble of going over the same process of analysis for each.
This happy application of algebra to geometry produced great improvements: it led Vieta to the doctrine of angular sections, one of the most important of his discoveries, which is now expanded into the arithmetic of sines or analytical trigonometry. He also improved the theory of algebraic equation, and he was he first that gave a general method of r esolving them by approximation. As he loved between the years 1540 and 1603, his writing belong to the latter half of the 16th century. He printed them at his own expense, and liberally bestowed them on men of science.
The Flemish mathematician Albert Girard was one of the improvers of algebra. He extended the theory of equations somewhat further than Vieta, but he did not completely unfold their composition; he was the first that showed the use of the negative sign in the resolution of geometrical problems, and the first to speak of imaginary quantities. He also inferred by induction that every equation has precisely as many roots as there are units in the number that expresses its degree. His algebra appeared in 1629.
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