ON ARITHMETIC, BOOK 10

Subsequently, the true solution is, by [the rule of] the chapter, 1 cube plus 3 things plus 18 equal to 6 squares. original: "1 cub^i p: 3 rebus p: 18 æqualiũ 6 quadratis," modern form: $x^3 + 3x + 18 = 6x^2$ But when the number of squares is too small to be equal to the cube, things, and number, then in the chapter of the cube, squares, and things equal to a number, there is one true solution and no feigned one. Cardano uses the term "feigned" (ficta) to refer to what we now call negative roots. However, in the chapter of squares equal to the cube, things, and number, there is one feigned solution and no true one; for example, if we say 1 cube plus 1 square plus 2 things are equal to 16, the true value of the thing is 2. And this is the feigned solution for the cube, two things, and 16 equal to 1 square. It is clear, therefore, that the chapters of the cube, squares, and things equal to a number and the cube, things, and number equal to squares correspond to each other.

10

Similarly, the chapter of the cube equal to squares, things, and number corresponds to the chapter of the cube, squares, and number equal to things. Therefore, where the things are very few, there is one feigned solution, which corresponds to the true solution of the other chapter of the cube equal to the same number of squares, things, and number. For example: if a cube is equal to 2 squares, 1 position, and the number 6, the thing is worth 3, and no more or less. In modern terms: $x^3 = 2x^2 + x + 6$. If $x=3$, then $27 = 18 + 3 + 6$. This is because if the cube, 2 squares, and the number 6 were equal to one position, there could be no true solution, but there would be a feigned one of minus 3, which was the true one in the other chapter. But if there are so many things that the chapter of the cube, squares, and number equal to things can have a true solution, then the true solution will be double, and there will be one feigned one, corresponding to the two feigned and one true solution of the other chapter. For example: if the cube, 3 squares, and the number 6 are equal to 20 things, there will be two true solutions, namely 3 and the root of 11 minus 3, and one feigned one, namely the root of 11 plus 3, minus. Cardano expresses the negative root $-(3 + \sqrt{11})$ as "$\sqrt{11}$ plus 3, minus." Therefore, the true solution for the cube equal to 3 squares, 20 things, and the number 6 is the root of 11 plus 3, and there will be two feigned ones: 3 minus and the root of 11 minus 3, minus.

11

By the same reasoning, the chapters of the cube and squares equal to things and number, and the cube and number equal to squares and things, correspond to each other. Thus, where the chapter of the cube and number equal to things and squares does not have a true solution, it will have only one feigned one, equal to the true one of the other chapter. For example: 1 cube plus 72 is equal to 6 squares plus 3 things. The feigned value of the thing is minus 3, and this is the true solution for one cube and 6 squares equal to 3 things and 72. And just as the chapter of 1 cube plus 72 equal to 6 squares plus 3 things lacks a true solution, so the chapter of 1 cube plus 6 squares equal to 3 things and 72 lacks a feigned one. But where the chapter of the cube and number equal to squares and things has a true solution, it will have two, and one feigned one, corresponding to the two feigned and one true solution of the other chapter. For example: let a cube plus 4 be equal to 3 squares plus 5 things. Then the true values are 4, or the root of 1 1/4 minus 1/2. $\sqrt{1.25} - 0.5 \approx 0.618$ The feigned one is the root of 1 1/4 plus 1/2, minus, and this is the true value for the chapter of the cube...

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