Squares and Triangles Made Equal Between Parallels
Adding or removing equal parts keeps equals; a square on half the base equals the triangle
Through a series of short Euclidean-style arguments, Leonardo shows how adding or removing equal parts preserves the equality of areas. Given a and b equal, adding d e to each makes a d e equal to d b e; and where b alone had equalled d, the whole figures a b c and a d c become equal once the common part is added. The page concludes with a general rule: a square and a triangle drawn between two parallel lines are always equal when the square's side is made on half the triangle's base — as the square a b equals the triangle b c, since removing b from each leaves the equal parts a c. Small labelled diagrams accompany each step.
On this page
Equals added to equals remain equal
Given a b equal, adding d e to each quantity makes a d e equal to d b e. The addition of a common part to equal figures preserves their equality.
Whole figures equal once the common part is joined
Where before b alone equalled d, now the whole figure a b c is equal to a d c. Building the equal parts onto a shared base extends a partial equality to the complete figures.
A square equal to a triangle between two parallels
A square and a triangle drawn between two parallel lines are always equal when the square's side is made on half the triangle's base. The square a b equals the triangle b c, because removing b from each leaves the equal parts a c.