 Evidently, Cartaglia had only solved cubics of the form x cubed plus px equals q, where p and q were positive. This is a type known as the cube and corsa. Remember corsa is Italian for thing, so we could view this as the cube and some unknowns. But what about other types of cubics? So while a cubic equation could have linear, qx, square, px squared, and cubic, x cubed terms, Cardano knew the square terms could always be eliminated. Viewed geometrically, we can take a cube, y cubed, and squares, p, y squared, and form it from a larger cubed, x cubed, minus cubed, q cubed, minus some sides, q squared, x. Viewed algebraically, the terms y cubed plus p, y squared can be transformed by letting y equal x minus p thirds. So for example, let's try to eliminate the square term in this cubic equation. So we'll use the transformation y equals x minus 12 thirds x minus four. And so we want to find y cubed, 12 y squared, and 36 y. And so when we put these terms together, we get 900. And when we put these terms together, our square terms drop out. And put together, this meant that Cardano only had to worry about three types of cubic equations. The cube and corsa equals number, x cubed plus px equals q. The cube equals corsa and number, x cubed equals px plus q. And the cube and number equals corsa, x cubed plus q equals px. So the cube and corsa, the equation that started it all, for this equation, Cardano gave the algorithm. Take the cube of one third the number of things, that's our x coefficient. Add the square of half the number, take the square root, then work with this twice. Add half the number to get a binomial, and subtract half the number to get an apatome. These are technical terms for when you add and when you subtract two terms. And the cube root of the binomial minus the cube root of the apatome gives the solution. So going back to our first cubic, x cubed plus 6x equals 20, we're going to take the cube of one third the number of things. So there's 6x, there's 6 things, so one third is 2, and the cube is 8. Then we're going to add the square of half the number, so the number is 20. So half is 10, and the squared is 100, and we add, and we get 108, which we'll take the square root of, and work with it twice. So we'll add half the number to get a binomial, square root 108 plus 10. We'll subtract half the number to get the apatome, square root 108 minus 10, and the cube root of the binomial minus the cube root of the apatome gives the solution. How about the cube equals cossa and number? From a modern perspective, x cubed plus px equals q, and x cubed equals px plus q are not essentially different since the latter is x cubed minus px equals q. But Cardano doesn't have this perspective, so he gives a separate rule. Take the cube of one third the number of things, subtract this from the square of half the number, take the square root, then work with this twice, add to half the number to get a binomial, and subtract from half the number to get an apatome. The cube root of the binomial plus the cube root of the apatome gives the solution. So Cardano gives the example x cubed equals 6x plus 40, so we want to find the cube of one third the number of things. So again, that's our coefficient of x, so one third of 6 cubed. We also need the square of half the number, so the number is our constant 40, half is 20, squared is 400, and we're going to subtract 400 minus 8, 392, and we'll get our square root 392, which we'll work with twice. So we'll add to half the number to get our binomial, we'll subtract from half the number to get the apatome, and the cube root of the binomial plus the cube root of the apatome gives us our solution. Now while in modern mathematics we would see no problem with handling all three types of cubic equations using the same rules for cube and corsa equals number, Cardano breaks them apart into three different cases, and cube and number equals corsa, Cardano's solution is very different from what we might expect, and it proceeds as follows. First solve the cube equal to the same number of corsa and number, what we're doing is we're solving a related equation. Then take three times the square of half the solution, subtract from the corsa and find the square root, then add or subtract from half the solution of the cube equal to the corsa and number to obtain the solutions. And Cardano solves x cubed plus 3 equals 8x. So first solve cube equal to the same number of corsa and numbers, in other words we want to solve the equation x cubed equals 8x plus 3, and we note this has solution x equal to 3. We won't go through the solution because, well, that's a surprise. And now we're going to take three times the square of half the solution. So we find half our solution and square it, then multiply by 3, and then we'll subtract this from the corsa, the number of things. That's 8 minus 27 fourths, and one of the interesting features about this is that Cardano actually gives us two solutions. We'll add or subtract from half the solution. So in this case, half the solution, three halves, we're going to add square root of five fourths to get one solution, and subtract square root of five fourths to get a different solution.