First — b a — m n — f e — d g c

Having here given the true squaring of the cylinder ab cg resolved into 16 cubes — n m e f the breadth and length together with the thickness e f c d — the rule of such squaring is finished, and now a start is made on the second rule, which allows any square breadth required to be reduced to any lowness, namely the square n m e f; which greater or lesser lowness we place at the front of the same square, that is in e f c d. And to make a demonstration of it, I will divide below the front e f c d in the second figure, so that I reduce it to a quarter of its thickness, that is to a b c d, and I lengthen the whole parallel accordingly, leaving to it all its first breadth given in the first figure n m e f; and then I reduce such length into a square by the second rule here (Figure).

This invention of reducing the parallel e f c d to the length of the parallel a c h i, by means of the given lowness a b c d, serves in irrational quantities just as in rational ones, et cetera.

Second — g f e — h b a — i d c

I wish to lower the square e f a b to the lowness a b c d, and to do this I will continue the two sides of the parallel e f a b in a straight line at pleasure, and I will make the parallel e c g i. Then I will lower the parallel e f c d to the aforesaid lowness a b c d, and I wish to know how far such a thinned parallel will extend in length; and to do this I will make the diagonal e b in the lesser parallel a b c d and continue this diagonal c b as far as its meeting with the line e f g; and where the diagonal line a b cuts at g the upper side of the parallel e g, there is the length into which the parallel e f c d will be converted when it is thinned to the lowness a b c d. And this is proved thus: divide the breadth of the parallel e f c d into 4 equal parallels, and of these one shall be a b c d. Then divide the parallel a c h i into 4 parts of equal length, of which one is the same parallel a b c d, and by one of the Elements our proposition is concluded, that is: if to equal things I add equal things, the joined things will be equal; and conversely, if from equal things equal things are taken, the remainders stay equal. Therefore I will remove a b c d, the lesser parallel, from the greater parallel; there will remain e f a b, and I will do the like to the parallel a c h i, removing from it the same a b c d, and so the remainders stay equal among themselves; and conversely, adding the parallel a b c d to the one and to the other, the joined things will be equal. Again, by another of the Elements, e f a b is proved to be equal to b d h i, because the triangle b f g is proved equal to the triangle g h b, and likewise the two lesser triangles a c b and b d c. Therefore, removing the greater and the lesser triangle that contain the square e f a b, that square remains; and if you remove the other greater and lesser triangle that contain the quadrilateral b d h i, that quadrilateral remains equal to the other square e f a b.