f K – n K – b d – n a - c e g h

Here, in this cube root, the proof is given that the triangle g d e is exactly proportional to the larger triangle d f b: that is, the ratio the side g e of the smaller triangle has to the side d b of the larger, the other two sides of the smaller triangle have to the other two sides of the larger. This figure is admirable for having within itself four cube roots of the duplication of the cube, of which one is g e, another d K, the third K f, the fourth K b. And three of them are all semidiameters of one and the same circle drawn on the diameter d f, and therefore they are equal to one another by the definition of the circle. And most admirable is... K n is the root of one cube, and K f and K b is the cube root of a cube double to two cubes; and the same happens in the two right-angle sides d e and g e as happens in the two acute-angle sides K n and K b. And here it comes about that, where the ancients by means of the arc used to seek, with much labor, the doubtful position of the chord, here the contrary is done, because I make the position of the chord with its ends, and by means of that I find the position of the true ends of the arc, and so on. And where...

If you ask me why the semidiameter of the circle enters six times into its circumference, and why the diagonal of the square is not commensurable with its side, I will tell you why the straight line that departs from the outermost upper corner of one of two joined squares and ends at the center of the second square shows us the cube root of two cubes reduced to a single cube. Let the two joined squares lie in the space d b e c, and let the outermost upper corner be b, and let the straight line drawn from that corner b and ending at the center of the second square be the line b a. I say that this line is the cause of showing us the cube root of two cubes reduced into a single cube, because I will set that line a b onto b f, continuing the straightness of the side that the square b c has, and it will be the line f b. Then I will continue the lower side of the two joined squares, that is the line c e, in a continuous straight line as far as h, and it will be the line c h. Then I will draw from f, the end of the line b f, the line f g tangent to the other outermost upper corner d, and where that line cuts the line e h at the point g, there, between e and g, ends the line which is the true cube root of the two aforesaid cubes, which multiplied cubically generates in itself six surfaces that clothe a cube of the value of the two aforesaid cubes d b e c. Here the mechanical proof occurs, made with the curve f K g, which among the ancients is obtained with laborious effort, and by means of it the true cubication is found. But in this new invention there is no need of such a curve; yet if you should take pleasure in having it, open the compass from n to f, and carry the moving point from f to g, and so that arc will find the chord of that arc, tangent to the aforesaid corner d. But such an arc in this case serves you for nothing, because experience has shown us that it can be done without it, which could not be done before our days, but only served to give judgment of the proportion of the triangles g d e and d f b, which re-proof led to the knowledge of the truth; and for this reason, without any labor, I use it to confirm that my proof is confirmed by the ancients.

Triangle with the letters: f d c — b d — e g — f b — e d
Here are set again the same letters of the sides of the triangles, which all have to correspond proportionally one to another in the mean proportionals, as 1, 2, 4, 8, which gives the rule for every... cube root, and so on.