ON ARITHMETIC, BOOK X

A 4

...[when they are] equal in all their denominations among themselves, and are joined with a number, how this may be done through depression depression original: "depressionem," the algebraic process of reducing the degree of an equation, such as dividing $x^3$ by $x$ to get $x^2$., we have fully taught in the fourth book.

4

But for odd denominations powers of the unknown like $x^1, x^3, x^5$., only one true estimation estimation original: "æstimatio," meaning the value of the root or solution for $x$. exists, and none is feigned feigned original: "ficta," Cardano’s term for negative roots, which were considered "false" or "fictitious" at the time because you cannot have a negative physical length., when they are compared to a number alone. For instance, if two things original: "res," the unknown $x$. are equal to 16, the estimation of the thing is 8. If two cubes are equal to 16, the estimation of the thing is 2. However, the number to which the denominations are compared in this chapter is always assumed to be true positive, not feigned; for what would be so foolish as to weaken the foundation itself? Nevertheless, the opposite reasoning must be followed in opposite cases. The logic is therefore the same where many denominations are compared to a number; even if there were a thousand of them, there will be one true estimation of the thing, and none feigned. For example, 1 cube plus 6 positions original: "positionibus," another term for the unknown $x$. is equal to 20; the estimation of the thing is none other than 2, neither true nor feigned.

5

When, however, two denominations are compared with a number, either both are odd, and the comparison is made to the extreme or to the middle (for the case made to the number has already been discussed in the preceding rule), or one is odd and the other even (for we spoke generally about both being even in the third rule). If, therefore, the extreme denomination—namely the cube—is compared with the number and the middle denomination—that is, the positions—see whether by multiplying two-thirds of the number of Things by the root of one-third of that same number, the resulting number is equal to, greater than, or less than the proposed number. If it results in the proposed number precisely, the estimation of the thing is twofold: one is true, namely the root itself which was multiplied.

Example: A cube plus 16 is equal to 12 positions $x^3 + 16 = 12x$. By multiplying 8 (which is $\frac{2}{3}$ of 12, the number of things) by 2 (the root of 4, which is $\frac{1}{3}$ of the number of things), the result is 16, the proposed number of the equation. Therefore, the estimation is 2 (the root of 4). And there is another feigned estimation, which corresponds to the true one of a cube equal to those same things and the same number. As in the example: if a cube is equal to 12 things plus the number 16 $x^3 = 12x + 16$, the true estimation is 4. Therefore, if a cube plus 16 is equal to 12 positions, the estimation of the thing is minus 4 original: "m: 4". For 12 things are minus 48, and the cube of minus 4 is minus 64; when 16 is added to this, it becomes minus 48.

But if the product of $\frac{2}{3}$ of the number of things multiplied by the root of one-third of that same number exceeds the proposed number of the equation, then the chapter will have three equations: two true ones and a third feigned one.

Example: 1 cube plus 9 is equal to 12 things $x^3 + 9 = 12x$. One true equation root is 3; another is the root of $5 \frac{1}{4}$ minus $1 \frac{1}{2}$ $\sqrt{5.25} - 1.5$. The third feigned one is always the sum of these two, and it corresponds to the true estimation of a cube equal to those same things and that same number; it is the root of $5 \frac{1}{4}$ plus $1 \frac{1}{2}$, and thus the remaining feigned one, which we spoke of in the other example, is the sum of the two true ones. But because the true ones are equal to each other in that case, the feigned one is always double the true one. It is clear, therefore, that the false or feigned equations of the chapter "cube and number equal to things" correspond to the true equations of the chapter "cube equal...