BY GIROLAMO CARDANO

...[con]siderations, and an innumerable series of chapters could be added, but we will nevertheless make an end to our careful consideration at the cube, treating the rest, even if generally, yet as if in passing original: "per transennam," meaning looking through a lattice or cursorily.. For since the position unknown original: "positio," the first power of the unknown, which we call x. represents a line, the square a surface, and the cube a solid body, it would indeed be foolish for us to progress further where nature does not permit. Therefore, he will seem to have taught perfectly enough who has handed down everything as far as the cube; the rest which we add, as if compelled or incited, we do not hand down further. In all the preceding matters, and especially in the third and fourth books, it will be worthwhile to remember them, lest I become tedious by teaching them again, or more obscure by omitting them.

For we remember that we have already taught which are odd or even denominations. For the square, the square of the square x⁴, the cube of the square x⁶, and so on, with one always omitted in between, we call even denominations. We call the position x¹, the cube x³, and the first and second names odd denominations. But truly, just as 9 is produced from 3, so it is from minus 3, because minus multiplied by minus produces plus. But in odd denominations the same nature is preserved: plus is not made except from a true number; nor can a cube, whose value estimation original: "æstimatio," the value of the unknown root. is minus (or what we call a debt), be produced from any position of a true number; this, it must be remembered, has already been explained more clearly.

If therefore an even denomination is equal to a number, the value of the thing is twofold: minus and plus, the one being equal to the other. For example, if a square is equal to 9, the thing is 3, or minus 3; and if it is equal to 16, the thing is 4, or minus 4; and if a square-square is equal to 81, the value of the thing is 3, or minus 3. To combine even denominations is not very necessary, because the square-square the fourth power belongs to derivative chapters; but if you carefully observe what I write, you will satisfy your wish with this rule. For when a square and a square-square are equal to a number, the ratio will be the same as in the simple case—namely, a double equation, one plus and the other minus, and equal to each other. For example, 1 square-square plus 3 squares equals 28, the position is worth 2 or minus 2. But truly, if a square-square and a number are equal to squares, we will indeed demonstrate in the eighth chapter that there are two values of the thing in true numbers, and it will have just as many by minus, each corresponding equal to the other. For example, if I say 1 square-square plus 12 equals 7 squares, the value of the position is either 2, or minus 2, or the root original: "Rz," the symbol for radix. of 3, or minus the root of 3, and thus there are four equations meaning four possible roots/solutions. But if it should lack a true value, it will also lack that which is by minus; for example, 1 square-square plus 12 equals 6 squares; because it cannot have a true equation, it will also lack a fictitious one Cardano uses "fictitious" (ficta) to refer to negative roots/estimations.—for so we call that which is of debt or less. But truly, if a square-square is equal to a number and squares, there is always one true value of the thing, and another equal to it which is fictitious or by minus. For example, if 1 square-square is equal to 2 squares plus 8, the value of the thing is 2, or minus 2. The same reasoning applies to the others...