Triangle and Square Between Parallel Lines
How the area of a triangle relates to a quadrilateral according to their base ratios
Folio 115v works out the area relations between a triangle and a quadrilateral set between the same pair of parallel lines, drawn as a series of rectangles crossed by diagonals. Leonardo states the rules: a square on the whole base makes the triangle half; when the triangle's base is half the square's, the triangle is a quarter; when double, they are equal. A closing note observes that the circles of two triangles stand in quadruple ratio and shows how to square a figure by subtracting one portion from another.
On this page
Triangle-to-quadrilateral area by base ratio, between parallels
If a quadrilateral and a triangle lie between parallel lines and the quadrilateral stands on half the triangle's base, they are equal. A square on the triangle's base makes the triangle half; if the triangle's base is half the square's the triangle is a quarter, and if double the two are equal. Generally the triangle relates to the square as half the square does to the whole base of the triangle.
Quadruple circles and squaring by subtraction
The circles of these two triangles are quadruple one to the other. Since a b equals c, c contains both a and b; drawing b out of c leaves a remainder exactly equal to a, and so a is squared.