hect. i.] DISSERTATION SECOND. 25 



forts of succeeding analysts have hardly been able to go 

 beyond. As to the general doctrine of equations, it ap- 

 pears that Cardan was acquainted both with the negative 

 and positive roots, the former of which he called by the 

 name of false roots. He also knew that the number of 

 positive, or, as he called them, true roofs, is equal to the 

 number of the changes of the signs of the terms ; and that 

 the coefficient of the second term is the difference between 

 the sura of the true and the false roofs. He also had per- 

 ceived the* difficulty of that case of cubick equations, which 

 cannot be reduced to his own rule. He was not able to 

 overcome the difficulty, but showed how, in all cases, an 

 approximation to the roots might be obtained. 



There is the more merit in these discoveries, that the 

 language of Algebra still remained very imperfect, and 

 consisted merely of abbreviations of words. Mathema- 

 ticians were then in the practice of putting their rules into 

 verse. Cardan has given his a poetical dress, in which, as 

 may be supposed, they are very awkward and obscure ; 

 for whatever assistance in this way is given to the memory, 

 must be entirely at the expense of the understanding. It 

 is, at the same time, a proof that the language of Algebra 

 was very imperfect. Nobody now thinks of translating an 

 algebraick formula into verse ; because, if one has acquired 

 any familiarity with the language of the science, the formu- 

 la will be more easily remembered than any thing that can 

 be substituted in its room. 



Italy was not the only country into which the algebraick 

 analysis had by this time found its way ; in Germany it 

 had also made considerable progress, and Stipbelius, in a 

 book of Algebra, published at Nuremberg in 1544, em- 

 ployed the same numeral exponents of powers, both posi- 

 tive and negative, which we now use, as far as integer 

 numbers are concerned ; but he did not carry the solution 

 of equations farther than the second degree. He introdir 



