The Great Art

for in both ways 104 is collected; we shall therefore say deservedly, on account of two things, that 2 is the evaluation evaluation: aestimatio, the root or value that solves the equation of the thing, and when you have operated correctly in the evaluation or equation, either experience succeeds.

DEMONSTRATION.

2

In order that the truth of the thing thing: res, the unknown variable, commonly 'x' might be more clearly grasped, and with it the reason—for to know is to understand through demonstration, as they say—let there be, for example, three cubes equal to 24, and let A C be placed as the side of one cube, and C D of the second, and D B of the third. Therefore, because the cubes are equal to each other, the lines A C, C D, and D B will also be equal. Since, therefore, 24 is divided according to the number by which A C is in A B (which is 3), the quantity of the cubes will be made from the 19th [proposition] of the fifth [book] or the 17th of the seventh [book] of the Elements original: "elementorum"; referring to Euclid's Elements, the foundation of geometry, and the 31st [proposition] of the 11th [book] of the same: the cube of A C is equal to 8. Therefore, the side A C will be 2, the evaluation of the thing, from which a general rule is gathered.

RULE.

3

Reduce the proposed two denominations denominations: different powers of the unknown, such as $x^2$ or $x^3$ to a number, if a number is not present, by decreasing them equally. When one becomes a denomination and the other a number, divide the number by the number of the denomination; the result is the evaluation of that denomination. If that denomination is the "position" position: positio, another term for the unknown 'x', you have the evaluation of the position. If it is another denomination, take the side or root of that number according to the quality of the denomination: if it is a square, take the square root; if a cube, the cube root; if a square-square original: "q̃d' q̃dᵗⁱ", quadrato-quadratum, or $x^4$, the root of the root; and so on. That side or root is the true evaluation of the position.

Example: 20 cubes are equal to 180 first relates first relates: relata prima, the fifth power of the unknown, $x^5$. Because there is no simple number here, you shall set the lowest denomination—the cubes—as a simple number, namely 20. And the greater or higher denomination—the relates—you shall reduce by the cubes, and they become 180 squares. Therefore, divide the number 20 by the number of squares, 180; the result is 1/9, the evaluation of the square. But we seek the evaluation of the position, not the square; therefore, take the square root of 1/9, and it is 1/3, for the true evaluation.

Another Example: 7 squares are equal to 21 cube-squares original: "cub' q̃dᵗⁱ", cubo-quadratum, or $x^6$. Reduce equally to a number, and they become 7 equal to 21 square-squares. Divide 7 by 21; the result is 1/3, and the square-square root original: "℞' ℞' 1/3", the fourth root of 1/3, which is the side of the square-square, is the evaluation of the thing.

Another: 2 cubes are equal to 20 square-squares. Having reduced the cubes to a number, the square-square will arrive at the position; therefore, 20 positions are equal to 2. Divide 2 by 20; the result is 1/10. And because you divided with the number of positions, the evaluation of the position will be 1/10.

Another: 20 are equal to 5 squares. Divide 20 by 5; the result is 4, the evaluation of the square; therefore, the evaluation of the thing is 2.

4

And so that I may satisfy all future chapters, you shall divide all the remaining terms and the number by the number of the greatest denomination; by the greatest