THE WORKS OF GIROLAMO CARDANO

...[will be] equal to things and a number, where the things and number are the same. But truly, when from such a multiplication of the root of a third part of the number of things, into two-thirds of the same number, there results less than the proposed number, then there will be no true equation original: "æquatio uera," a positive solution but only a feigned one feigned original: "ficta," meaning a negative solution, which Cardano considered "fictitious" even as he used them.. This feigned solution is equal to the true solution of the chapter on the cube equal to just as many things and the same number. For example, 1 cube plus 21 is equal to 2 things $x^3 + 21 = 2x$; although it lacks a true equation, the feigned one is nonetheless minus 3, and this is the true estimation of the cube equal to two things and the number twenty-one $x^3 = 2x + 21$.

6 From these things, it is not difficult to hunt out how many equations the chapter on the cube equal to things and a number $x^3 = ax + n$ may have. If, therefore, from two-thirds of the number of things multiplied by the root of a third part of that same number, the proposed number is produced, the chapter has two equations: a true one equal to the feigned one of the preceding rule, and a feigned one equal to the true one. Thus, the true one is double the feigned one, because in that case the feigned was double the true. For example, if 1 cube is equal to 12 things and the number 16 $x^3 = 12x + 16$, the true equation is 4, and the feigned is minus 2; because if 1 cube plus 16 is equal to 12 positions $x^3 + 16 = 12x$, the true estimation is 2, and the feigned is minus 4.

But if from the said multiplication there should come forth more than the number of the equation, the true estimation will be one, corresponding to the false one of the preceding rule, and the false one will be double, each responding to the true one of the preceding rule. For example, if a cube is equated to 12 positions plus 9 $x^3 = 12x + 9$, both false estimations are the root of $5 \frac{1}{4}$ minus $1 \frac{1}{2}$ $\sqrt{5.25} - 1.5 \approx 0.791$ and 3, and the true one is the root of $5 \frac{1}{4}$ plus $1 \frac{1}{2}$ $\sqrt{5.25} + 1.5 \approx 3.791$. Thus you see how the false respond to the true, and the true to the false in turn; moreover, the true is composed of both false ones, for from the root of $5 \frac{1}{4}$ minus $1 \frac{1}{2}$ and 3, comes the root of $5 \frac{1}{4}$ plus $1 \frac{1}{2}$ Note that Cardano is observing the relationship between roots: $x_1 + x_2 + x_3 = 0$ for $x^3 + px + q = 0$.. But if from such a product there results less than the number of the equation, the estimation is only one and true, just as in the preceding rule there is only one and feigned—such as if a cube is equal to two things and the number 21, the equation is 3, just as in the cube plus 21 equal to two things, the feigned estimation is minus 3.

7 However, in the chapters in which a number and both even and odd denominations are equated to each other—either the even is the extreme the highest power, like $x^2$ in $x^2 + x = n$, such as when the square and the position and the number are equated to each other; or the extreme denomination is odd, such as when the cube and square are equated to a number—if therefore the square is equated to positions and a number $x^2 = ax + n$, it will have two equations: one true, equal to the feigned one of the chapter of the square and the same things being equal to the same number, and another feigned one, equal to the true one of that other chapter.

Example: If a square and 4 positions are equal to 21 $x^2 + 4x = 21$, the true estimation is 3, and the feigned is minus 7. And if the square is equal to 4 positions and 21 $x^2 = 4x + 21$, the true estimation is 7, and the feigned is minus 3. Therefore, once the true ones are found, the feigned ones are mutually found, just as in the preceding rule, but in a different way: for here the extremes are compared to extremes, while there the middle terms are compared to the extremes. For there, the chapter of the cube and number equal to things is compared to the chapter of the cube equal to things and a number; here, the chapter of the square and things equal to a number is compared to the chapter of the square equal to things and a number.