i n — e – f – g – 9 – h – i – 1 7 1 6 x – K – 1 5 1 0 – L – m – 7 18 13 4 b a – t p o S – 8 1 9 1 4 5 c

If the cylinder of square base is lowered by half, the base grows by half of its first value; and if the cylinder is lowered by a quarter of its first height, that base grows by a quarter of its first value; and if such lowering is an eighth, that base grows by an eighth; and if it were seven eighths, the base gains 7/8 of its first value.
Conclusion.
Having here given a square body as 10, 15, 4 and 13 with the thickness 4, 13, 5, 14, equal to half the height of this cylinder n i c 5, which is 9 cubes — this quantity being 4 and a half cubes — and having then given the squaring of the whole cylinder with its thickness, which is the square 16, 11, 4 and 18 with the thickness 4, 5, 18, 19: I wish to divide that thickness into as many various irrational heights as can be given, and I want to know how much that body widens when lowered wholly to any given thickness. And its rule is here behind this page, which of itself gives oblong squares equal to any square dilation that can be given, which oblong squares are reduced to perfect squares by one of the Elements, that is this figure. And know that the parallels marked here behind serve only to give the true thickness of the squaring, and also give the length of the oblong square, which oblong square is squared as is said above.
As much as the height of the cylinder diminishes, so much it gains in width of base. It is proved. Let the cylinder be n c, which I reduce to the lowness b c, and I have taken from it eight ninths of its height, that is 8 ninths of nine sides of cubes, that is e f g h i K L m. There remains the side S, to which I add the two sides o p, which are sides of 3 cubes, which 3 cubes I multiply by themselves and I have 9 cubes, and thus I have remade, in the height of the cube S o p, the 9 cubes of the whole cylinder e f g h i K L m S, etcetera. But by the rule of the great square made from the side n c, side of the cylinder, I find the line c, diameter of that great square, which gives me at b 4 the height of the diminished cylinder, and at 4 75 8 gives me the quantity of the increased base of the diminished cylinder, to which — by one proposition of Euclid — I shall increase the side b a according to the side b c, and I shall take the middle of the a 7 line, and make the semicircle a 9 7, and where the periphery cuts the side n b, will be found the part x b, cut off by that semicircle, to be one of the 4 sides of the base of the lowered cylinder, in which will stand a square that will contain 9 cubes similar to the cubes included in the cylinder n c. And always, by rule, the diameter of the great square will cut the line b 7, equal to the height of the cylinder; it will cut into proportional parts whatever lowness you wish to reduce the cylinder to. For example, you see b c to be a ninth part of the line n c, to which the height of the cylinder n c has to diminish, and you see in the remainder of the same line b 7 the part 4 7, added with e 8, make the quadrilateral 4 c 7 8, which, when it is squared, will be equal to the base of the aforesaid cylinder lowered in height and increased in base.

Here you need only the infinite parts of the square at the foot, to mark on them, within the axes of the pyramid, the last lownesses and widths, by always squaring the remainder of the 9 squares of the cylinder, which must always stand, either wholly or in part, as base of the said cylinder ... [gap] ... side ... 9 squares ... when the surface ...
* a b 8 high 1.
* b c 2.