Transforming Boards into Tetragons and Cylinders
A program of solid transformations, plus reducing a square to a triangle
This leaf collects a short program of transformations of flat solids: turning a rectangular board (tavola) into an equilateral one, a square board into a rectangle of a given length, and boards into cylinders, each keyed to an earlier proposition (the fifth or sixth of this book). A further note explains how to reduce a square to a triangle of a given length by doubling it, forming a long rectangle and halving it along the diagonal into two equal triangles. Two small solids and a rectangle divided by its diagonal are sketched at the lower right and lower left; two of the cataloguer's headings mark figures whose own text was left blank.
On this page
A program of board transformations
A list of problems: to make a rectangular board equilateral (by the fifth of this book), to make a square board into a rectangle of given length (by the sixth), to make a square board into a cylinder according to a side's length or width (by the fifth), and to make a board into a cylinder of a given height.
Reducing a square to a triangle of given length
To reduce the square to the length of a given triangle, double the square, make of it a rectangle as long as that triangle, then divide it along the diagonal into two equal triangles; either one equals the required triangle, and its base may be reshaped with any angles provided the length is not diminished.
Sketches of solids and a divided rectangle
At the lower right two three-dimensional solids are drawn, a slab-like board and a wedge-shaped body, illustrating the board-to-cylinder transformations. A small rectangle crossed by its diagonal, and further divided, is sketched at the lower left.