Constructing a square equal to a triangle
Quadrature of a triangle: a parallelogram f g d e equal in area to a given triangle
This page works toward turning a triangle into an equal-area rectilinear figure. A right triangle a, b, c and a separate open angle n accompany the instruction to 'make a square equal to a triangle.' Two triangles of equal base set between parallels (vertices a, b, c sharing point d) lead, by adding a pair of parallel diagonals, to a parallelogram f, g, d, e that equals the given triangle. Marginal citations 'by the twenty-third' and 'by the forty-first' name the propositions Leonardo invokes, closing with the reasoning that a figure twice the half is equal to the whole.
On this page
Make a square equal to a triangle
A right triangle with vertices a, b, c is set beside an open angle n. The note directs the reader to construct a square equal to the triangle, arguing that 'if it has one of them, it has two of them.'
Two triangles of equal base between parallels
Two adjacent triangles of equal base stand between two parallels, their apices labeled a, b, c and the shared base-point d. Triangles on equal bases between the same parallels are equal in area.
Parallelogram f g d e equal to the given triangle
Adding a second diagonal parallel to the first closes a parallelogram with vertices f, g, d, e, with points a and n at the far end of the parallels. By the forty-first proposition this parallelogram equals the given triangle: being twice the half, it is equal to the whole.