ON ARITHMETIC, BOOK X.

...the solutions are considered as one type; however, because they are joined together, we will say there are only three chapters. But truly, there are three [chapters] of the cube, things, and number, yet since one of them has two equations, they result in four. Just as many come from the cube, squares, and number, so now there are twelve. Then there are seven [chapters] of the cube of squares, original: "cubi quadratorum," likely referring to $x^6$. positions, and number, but in four of them there are twin equations, wherefore all will become eleven chapters. Therefore, there are twenty-three primary and general chapters. Neglecting the first of these, any of them joins two derivatives to itself, one by reason of the square and the other by reason of the cube; there will therefore be forty-four general derivatives. After these, there are two others of an unknown quantity, one when it is multiplied, the other when it is taken by itself. There is, furthermore, one general [chapter] of means. meansIn this context, middle terms or proportional values in a sequence. Thus, the number of all notable primary chapters is twenty-six, the derivatives are forty-four, and the collection of all is seventy. After these, we have added many other individual ones, but there is greater pleasure than necessity in them; therefore, we shall not number them among these.

a

The necessity of these is gathered as follows: when lines are known by surfaces, or surfaces by lines, the chapters of squares, positions, and number are appropriate. But if from a square original: "Tetragonico," from the Greek for "four-angled" or square. or solid side, Cardano is referring to square roots and cube roots as "sides." it is a simple chapter. However, when two unknowns of the three are assumed, and these pertain to surfaces and lines, the chapters of the unknown quantity and of the "thing" original: "rei." In 16th-century algebra, the "thing" (res) is the unknown x. must be explored. And these are simple if lines are compared to lines; but products are used when compared with surfaces. If bodies bodiesthree-dimensional volumes or solids. are to be compared to lines, we use [chapters] of the cube of things and number; if to bodies, the surfaces of the cube of squares and number. But if the ratio of surface, bodies, and lines must be sought, the chapters of the cube of squares, positions, and number are more useful. Furthermore, in all of these, a comparison will always be made to a number. This method is the most important, although it is often necessary to use all of them in each case; nevertheless, it will be worthwhile to describe each of these individually and to join the derivatives to their primitives. They are as follows:

1

Number equal to things, or number equal to squares, or number equal to cubes, or number equal to biquadrates, original: "$qd^{ra}qd^{ra}$" or squared-squares, meaning $x^4$. or number equal to the name or first relatum, first relatumThe fifth power of the unknown, or $x^5$. and so on by comparing the number to any denomination.

2

Number and squares equal to things, or number and cube equal to things, or number and cube equal to squares, or number and biquadrate equal to things, or number and the square of a biquadrate original: "qdrati qdra," meaning $x^8$. equal to the square, or number and biquadrate equal to cubes, or number and the first name original: "nomen primum," referring again to the fifth power. equal to things or squares or cubes, and so on without end.