In speaking of this illustrious man, Vieta, we must not omit his improvements in trigono metry, and still less his treatise on angular sections, which was a most important applica tion of Algebra to investigate the theorems, and resolve the problems of geometry. He also restored some of the books of Apollonius, in a manner highly creditable to his own ingenuity, but not perfectly in the taste of the Greek geometry ; because, though the con structions are elegant, the demonstrations are all synthetical.
About the same period, Algebra became greatly indebted to Albert Girard, a Flemish mathematician, whose principal work, Invention Nouvelle en Algebre, was printed in 1669• This ingenious author perceived a greater extent, but not yet the whole of the truth, partially discovered .by Vieta, viz. the successive formation of the coefficients of an equation from the sum of the roots ; the sum of their products taken two and two ; the same taken three and three, &c. whether the roots be positive or negative. He appears also to have been the first who understood the use of negative roots in the solution of geo metrical problems, and is the author of the figurative expression, which gives to negative quantities the name of quantities less than nothing ; a phrase that has been severely cen sured by e who forget that there are correct ideas, which correct language can hardly be made to express. The same mathematician conceived the notion of imagi nary roots, and showed that the number of the roots of an equation could not exceed the exponent of the highest power of the unknown quantity. He was also in possession of the very refined and difficult rule, which forms the sums of the powers of the roots of an equation from the coefficients of its terms. This is the greatest list of discoveries which the history of any algebraist could yet furnish.
The person next in order, as an inventor in Algebra, is Thomas Harriot, an English ma thematician, whose book, Artis Praxis, was published after his death, in 1631. This book contains the genesis of all equations, by the continued multiplication of simple equations ; that is to say, it explains the truth in its full extent, to which Vieta and Girard had been approximating. By Harriot also, the method of extracting the roots of equations was greatly improved ; the smaller letters of the alphabet, instead of the capital letters employed by Vieta, were introduced ; and by this improvement, trifling, indeed, compared with the rest, the form and exterior of algebraic expression were brought nearer to those which are now in use.
I have been the more careful to note very particularly the degrees by which the pro perties of equations were thus unfolded, because I think it forms an instance hardly paral leled in science, where a succession of able men, without going wrong, advanced, neverthe less, so slowly in the discovery of a truth which, when known, does not seem to be of a very hidden and abstruse nature. Their slow progress arose from this, that they, worked with an instrument, the use of which they did not fully comprehend, and employed a language which expressed more than they were prepared to understand ;—a language which, under the notion, first of negative and then of imaginary quantities, seemed to involve such mys teries as the accuracy of mathematical science must necessarily refuse to admit.
The distinguished author of whom I have just been speaking was born at Oxford in 1560. He was employed in the second expedition sent out by Sir Walter Ralegh to Vir ginia, and on his return published an account of that country. He afterwards devoted himself entirely to the study of the mathematics ; and it appears from some of his manu scripts, lately discovered, that he observed the spots of the sun as early as December 1610, not more than a month later than Galileo. He also made observations on Jupiter's satellites, and on the comets of 1607, and of 1618. ' The succession of discoveries, above related, brought the algebraic analysis, abstractly considered, into a state of perfection, little short of that which it has attained at the present moment. It was thus prepared for the step whioh was about to be taken by Descartes, and which forms one of the most important epochal in the history of the mathematical sciences. This was the application of the algebraic analysis, to define the nature, and investigate the properties, of curve lines, and, consequently, to represent the notion of variable quantity. It is often said, that Descartes was the first who applied algebra to geometry ; but this is in accurate ; for such applications had been 'made before, particularly by Vieta, in his trea tise on angular sections. The invention just mentioned is the undisputed property of Descartes, and opened up vast fields of discovery for those who were to come after him.