  c   =   0 ▁to ▁cub ic ▁equations ▁without ▁a ▁quadratic ▁term , ▁but , ▁again , ▁he ▁has ▁to ▁consider ▁several ▁cases . ▁In ▁all , ▁Card ano ▁was ▁driven ▁to ▁the ▁study ▁of ▁th ir teen ▁different ▁types ▁of ▁cub ic ▁equations ▁( ch ap ters ▁XI – XX III ). ▁ ▁In ▁Ar s ▁Mag na ▁the ▁concept ▁of ▁multiple ▁root ▁appears ▁for ▁the ▁first ▁time ▁( chapter ▁I ). ▁The ▁first ▁example ▁that ▁Card ano ▁provides ▁of ▁a ▁polynomial ▁equation ▁with ▁multiple ▁roots ▁is ▁x 3   =   1 2 x   +   1 6 , ▁of ▁which ▁− 2 ▁is ▁a ▁double ▁root . ▁ ▁Ar s ▁Mag na ▁also ▁contains ▁the ▁first ▁occurrence ▁of ▁complex ▁numbers ▁( chapter   XXX V II ). ▁The ▁problem ▁mentioned ▁by ▁Card ano ▁which ▁leads ▁to ▁square ▁roots ▁of ▁negative ▁numbers ▁is : ▁find ▁two ▁numbers ▁whose ▁sum ▁is ▁equal ▁to ▁ 1 0 ▁and ▁whose ▁product ▁is ▁equal ▁to ▁ 4 0 . ▁The ▁answer ▁is ▁ 5   +   √ − 1 5 ▁and ▁ 5   −   √ − 1 5 . ▁Card ano ▁called ▁this ▁" s oph istic ," ▁because ▁he ▁saw ▁no ▁physical ▁meaning ▁to ▁it , ▁but ▁bold ly ▁wrote ▁" ne ver theless ▁we ▁will ▁operate " ▁and ▁formally ▁calculated ▁that ▁their ▁product ▁does ▁indeed ▁equal ▁ 4 0 . ▁Card ano ▁then ▁says ▁that ▁this ▁answer ▁is ▁“ as ▁subt le ▁as ▁it ▁is ▁useless ”.