94 JEROME CARDAN 



of Milan, and amongst them was one involving the 

 equation x^ + 6;tr 2 + 36 = 60^, one which he probably 

 found in some Arabian treatise. Cardan tried all his 

 ingenuity over this combination without success, but his 

 brilliant pupil, Ludovico Ferrari, worked to better pur- 

 pose, and succeeded at last in solving it by adding to 

 each side of the equation, arranged in a certain fashion, 

 some quadratic and simple quantities of which the 

 square root could be extracted. 1 Cardan seems to have 

 been baffled by the fact that the equation aforesaid 

 could not be solved by the recently-discovered rules, 

 because it produced a bi-quadratic. This difficulty 

 Ferrari overcame, and, pursuing the subject, he dis- 

 covered a general rule for the solution of all bi-quadratics 

 by means of a cubic equation. Cardan's subsequent 

 demonstration of this process is one of the masterpieces 

 of the Book of the Great Art. It is an example of the 

 use of assuming a new indeterminate quantity to intro- 

 duce into an equation, thus anticipating by a considerable 

 space of time Descartes, who subsequently made use of a 

 like assumption in a like case. 



How far this discovery of Ferrari's covered the rules 

 given by Tartaglia to Cardan, and how far it relieved 

 Cardan of the obligation of secresy, is a problem fitted 

 for the consideration of the mathematician and the 

 casuist severally. 2 An apologist of Cardan might affirm 

 that he cannot be held to have acted in bad faith in 

 publishing the result of Ferrari's discovery. If this 



1 Montucla, Histoire de Math. i. 596, gives a full account of 

 Ferrari's process. 



2 In the De Vita Propria, Cardan dismisses the matter briefly : 

 " Ex hoc ad artem magnam, quam collegi, dum Jo. Colla certaret 

 nobiscum, et Tartalea, a quo primum acceperam capitulum, qui 

 maluit semulum habere, et superiorem, quam amicum et beneficio 

 devinctum, cum alterius fuisset inventum." ch. xlv. p. 175. 



