98 JEROME CARDAN 



with Ferrari, an algebraist as famous as Tartaglia or 

 himself. The opening chapters of the book show that 

 Cardan was well acquainted with the chief properties of 

 the roots of equations of all sorts. He lays it down that 

 all square numbers have two different kinds of root, one 

 positive and one negative, 1 vera and ficta : thus the 

 root of 9 is either 3. or 3. He shows that when a case 

 has all its roots, or when none are impossible, the 

 number of its positive roots is the same as the number 

 of changes in the signs of the terms when they are all 

 brought to one side. In the case of X s + ^bx = 2c, 

 he demonstrates his first resolution of a cubic equation, 

 and gives his own version of his dealings with Tartaglia. 

 His chief obligation to the Brescian was the information 

 how to solve the three cases which follow, i. e. x 3 + bx 

 = c. x? = bx + c. and x+*c=bx, and this he freely 

 acknowledges, and furthermore admits the great service 

 of the system of geometrical demonstration which 

 Tartaglia had first suggested to him, and which he 

 always employed hereafter. He claims originality for 

 all processes in the book not ascribed to others, assert- 

 ing that all the demonstrations of existing rules were 

 his own except three which had been left by Mahommed 

 ben Musa, and two invented by Ludovico Ferrari. 



With this vantage ground beneath his feet Cardan 

 raised the study of Algebra to a point it had never 

 reached before, and climbed himself to a height of fame 

 to which Medicine had not yet brought him. His name 

 as a mathematician was known throughout Europe, and 

 the success of his book was remarkable. In the De 



1 Montucla, who as a historian of Mathematics has a strong bias 

 against Cardan, gives him credit for the discovery of the ficta 

 radices, but on the other hand he attributes to Vieta Cardan's dis- 

 covery of the method of changing a complete cubic equation into 

 one wanting the second term. Ed. 1729, p. 595. 



