GIROLAMO CARDANO

[of the] chapters, they subsequently have only one solution in the rest, just as the chapter of the cube equal to things and a number original: "cubi æqualis rebus & numero," modern form: $x^3 = ax + n$ in the lower part, and the chapter of the cube and squares equal to a number $x^3 + ax^2 = n$, and the chapter of the cube and number equal to squares or things $x^3 + n = ax^2$ or $x^3 + n = ax$. For in one part they have three equations, and in the other subsequently only one. Similarly, the chapter of the square-square and number equal to the square $x^4 + n = ax^2$ has four equations in one part, and subsequently none in the other. Some indeed have two throughout, such as the chapter of the square and things equal to a number $x^2 + ax = n$, or the chapter of the square equal to things and a number $x^2 = ax + n$. Those which have only one are such as the chapter of the cube and things equal to a number $x^3 + ax = n$, and the chapter of the square and number equal to things $x^2 + n = ax$, which has two equations in one part, and subsequently none in the other.

Note well.

And you should know that the equations of the chapters of the cube and squares equal to a number, and likewise the cube and number equal to squares, are related such that the difference between the true and feigned solutions feigned solutions original: "æquationum uerarum & fictarum," meaning positive and negative roots. Cardano calls negative roots "feigned." is always the number of the squares. For example, if the cube and 72 are equal to 11 squares $x^3 + 72 = 11x^2$, the feigned solution is the root of 40 minus 4 $\sqrt{40} - 4$, and the true solutions are the root of 40 plus 4 $\sqrt{40} + 4$ and 3. The difference between the root of 40 minus 4 and [the sum of] 7 plus the root of 40 is 11, the number of the squares Cardano is observing that the sum of the roots equals the coefficient of the $x^2$ term, accounting for the sign of the "feigned" root: $(\sqrt{40} + 4) + 3 - (-(\sqrt{40} - 4)) = 11$ is not his logic here; rather he is looking at the magnitude of the difference.. And it is the same if the cube and 11 squares were equal to the number 72.

9

In these chapters, however, which consist of a double denomination, one odd and one even, plus a number Referring to equations with terms like $x^3$ (odd power) and $x^2$ (even power), if the cube and things are equal to squares and a number $x^3 + ax = bx^2 + n$, the solutions can be three, and all of them true, and none feigned. This is because, as has been said, when a "minus" is led to a solid i.e., when a negative number is cubed, it becomes minus, and thus a minus would be equal to a plus, which cannot be.

Where indeed the cube, square, and thing are equal to a number $x^3 + ax^2 + bx = n$, then there will also be three solutions: one plus and two minus. This occurs if, under the same denominations, the squares can be equal to the things, the number, and the cube; and the true solutions here are the feigned ones in that example. Thus, if 1 cube plus 6 squares and 3 things is equal to 18 $x^3 + 6x^2 + 3x = 18$, then the true valuation of the thing is found from its own chapter. Then it has the feigned valuations of the chapter 1 cube plus 3 things plus 18 equal to 6 squares $x^3 + 3x + 18 = 6x^2$. One of these is 3, another is the root of $8\frac{1}{4}$ plus $1\frac{1}{2}$. Therefore, minus 3, or minus (root of $8\frac{1}{4}$ plus $1\frac{1}{2}$), is the feigned valuation of 1 cube plus 6 squares plus 3 positions equal to 18. And with this, there is also a third true solution.

From this are obtained the three solutions of the chapter of the cube, things, and number equal to squares, where the solution is possible. This is known from its own chapters. Two of these, therefore, are true and equal, as was said, to the solutions of the chapter of the same number of squares, things, and the cube equal to the same number. As in the said example, the third [solution] corresponds to the true one of the other chapter and is feigned. Therefore, the solution of the chapter 1 cube plus 6 squares plus 3 [things]...