Abstract:
Cardano′s calculations in Artis Magnae of a positive root of two quartic equations in problem 34.2 and 34.3 respectively are reconstructed, and the rule of five quantities in continued proportions is revealed as an algorithm which transforms the problem of the solution of some special quartic equation to that of the ratio of five quantities in continued proportions. It is pointed out that by means of this algorithm Cardano could solve special quartic equations containing both the first and the third powers on condition that the product of the constant in the equation and the square of the coefficient of the third power is equal to the square of the coefficient of the first power. Besides, Cardano′s obscure statements are cleared up: firstly, the two numbers in the questions are not included in the aforementioned five quantities; secondly, the purpose of his choice of a definite number when searching the ratio is considered as to diminish the number of the unknowns and to simply the calculation; thirdly, the choice of the number is arbitrary and the value of the root of the quartic equation has nothing to do with the chosen number.
