The cube root of 3 is 3/9; 3 times 3 makes 9, that is 9/9 which is one whole, and 3 times 9 makes 27 ninths, that is 3 wholes. Therefore the cube root of 3 is 3/9, because saying 3 ninths times 3 ninths makes 9 ninths, that is one whole, and 3 times 9 ninths makes 27 ninths which are 3 wholes. Therefore it is true that 3/9 is the cube root of 3, that is of three.

The root of 4 is 4/16, and you will say 4/16 times 4/16 makes 16/16, that is 1, and 4 times 16/16 makes 64/16, that is 4.

The root of 6 is 6/36, and you will say thus: 6/36 times 6/36 makes 36/36, one; and 36/36 times 6/36 makes 216/36, that is 6; therefore this is the way to find cube roots by means of broken numbers, and they will all be rational; before me there is no record of their having been found.

What is said is nothing other than to diminish a measure and to ask for its half, or third, or fourth, or as you please.

If we multiply the broken numbers as is prescribed for the whole numbers, we shall have the cube roots, both surd and rational, of all numbers; and first the root of 1: we shall say, 1 times 1 makes 1, that is one square, and to make it a cube we say once more: 1 times 1 makes 1, that is one cube.

Then 2/4 times 2/4 makes 4/4, and 2 times 4/4 makes 8/4, that is two wholes, which by yourself you can recognise as 8 dice, each being the capacity of a quarter of a braccio cubed.

Again: 3/9 times 3/9 makes 9/9, and 3 times 9/9 makes 27/9.

1/4 times 1/4 makes 1/16; 1/4 times 1/4 makes one of those quarters. And here I use only the numbers that are above the line, and the rest only for the naming of the parts that I use. And when I say 1/4, I mean a cube that contains the fourth part of another larger cube, and so I shall continue.