 The solutions to the cubic raised some intriguing questions. For x cubed plus 6x equals 20, the solution found by Cardano is, but in fact, we can just look at this equation and find that a solution can be found by inspection, x equal to 2. And what this means is that this horrifying mess here has to simplify to 2, and it should be possible to simplify the left-hand side. But how? Similarly, the equation x cubed equals 6x plus 40, we can figure out a solution by inspection and see that x equals 4 works, but Cardano found, and again, somehow there must be a way of simplifying this expression so it becomes an integer. And this suggests, at the very least, that the problem of identifying when something is an integer is not as easy as it might seem to be. Something even more disturbing happens with one of Cardano's other examples. For x cubed equals 8x plus 3, Cardano gives the solution x equal to 3, which obviously works. But if we apply Cardano's algorithm, this is cube equal cosa and number. So we take the cube of one-third the number of things, so that's the cube of one-third of our x coefficient, and we're going to subtract from the square of half the number, so that's 9 fourth minus 512 27ths, and now we'll take the square root and work with it twice, adding and subtracting and then taking the cube root. I don't feel very good about this answer, but wait, the square root of a negative number doesn't exist. And here's a truly peculiar situation. We obviously have a solution, because we know what a solution is, and our method gives us this mass. So somehow we have to reconcile this bizarre expression has to be equal to 3. Now, I'd like to say that this is the first time the square root of a negative number shows up in the history of mathematics. But Cardano doesn't solve the cubic this way, and so he doesn't get these expressions. Again, in this problem, he just says that the solution is x equal to 3. Now, it's possible Cardano didn't realize he'd end up with the square root of a negative number, but he probably knew something strange was going on, because a little while later he gives the following problem, divide 10 into two parts, so their product is 40. And Cardano's solution is essentially the same as that of the Mesopotamian canal problems. He uses the identity we'd write as u plus v squared minus u minus v squared equals u v. Now, we have the sum u plus v equal to 10, and the product u v equal to 40, so we can proceed. So we need the square of half the sum. That's 25. And if we subtract our product, we get negative 15. And this will be the square of half the difference. And so we know half the sum is 5. Half the difference is square root of negative 15. Well, we'll ignore that for a second. Adding and subtracting these expressions, if we add them, we get 5 plus square root negative 15. And if we subtract them, we get 5 minus square root of negative 15, which are the solutions to the problem. And, as Cardano says, putting aside the mental tortures involved, we can verify that the solution is correct, even if we can't imagine what square root of negative 15 is. And we've already seen that Cardano did a number of things we associate with the modern practice of mathematics. Well, here's one more. He got an answer that didn't make sense. And rather than simply declaring that the solution was impossible, Cardano recognized that something interesting was going on, even if he himself wasn't able to pursue it.