28 DISSERTATION SECOND. [part i. 



cients 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 geometrical pro- 

 blems, 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 censured 

 by those who forget that there are correct ideas, which 

 correct language can hardly be made to express. The 

 same mathematician conceived the notion of imaginary 

 roots, and showed that the number of the roots of an equa- 

 tion 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 mathematician, whose book, 

 Artis Analyticae 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 im- 

 proved; 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 exteriour of algebraick 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 properties of equations were 

 thus unfolded, because I think it forms an instance hardly 



