On progression. Say: 8 times 8 makes 64 for the whole of its square; taking away half of it, there remain 32 squares, of which 28 are whole and 8 are halves. Adding another 8 halves, they will make 8 whole ones, and thus you will have made 36 squares. And this rule applies to the numbers of progressions, saying: I want to know how many units there are from 1 to 8. Multiply 8 by itself, and take away half of the resulting sum; and then add to it half of the multiplier, which was 8; and thus you will have made 36, as you first did in the other rule.

Way to find all the roots of numbers geometrically, and by experiment we will give the root of 9. Place your 9 upon the line b e, and add one more by rule, which will then be the whole line d e. Then take the middle of that line, which will be at c, and set there the point of the compass, and make the semicircle d f e. Then set the perpendicular that falls upon the point b; and where that perpendicular cuts the circle at point a, there, between a and b, the root of 9 is determined. It is proved by the right triangle a b c, which has its three sides composed of 3, 4 and 5; and that 3 remains the root of 9, because, as is shown in geometry, the multiplication of each of the two sides of the right triangle which contain the right angle are equal to the multiplication made of the third side. Which third side is set at 5, which, multiplied by itself, makes 25; the second side b c, which is 4, multiplied by itself, makes 16. Subtracted from 25, there remain 9, of which the root is the 3 of the side a b.