Squaring the Gnomon and Equal Curved and Straight Parallelograms
Right-angled constructions for gnomon quadrature; equality of curvilinear and rectilinear figures
The verso is a page of plane-geometry demonstrations illustrated by five lettered diagrams. Leonardo shows how to 'square' a gnomon a b c d e f by bending a side to form a right-angled triangle h g b whose sides give the larger, middle and smaller squares, the smallest being the square equal to the gnomon; a second construction subtracts the middle square from the larger so the remaining gnomon fills the smaller square. He then argues that all parallelograms of equal frontage-spacing and equal width are equal to one another even if one has curved sides and the other straight, proving that the curvilinear parallelogram a b n m c d equals the rectilinear parallelogram a b c d by removing a common surface m from each. One trial figure is annotated 'not valid'.
On this page
Squaring the gnomon a b c d e f
Leonardo bends the side h a until it meets line b c, forming the right-angled triangle h g b. Its side h g is the side of the larger square, h b the side of the middle square, and b g the side of the smaller square — the square equal to the gnomon. Subtracting the middle square from the larger leaves the difference b c, which is the gnomon.
Extracting the middle square by right-angled triangle
To subtract the middle square from the larger and leave the gnomon in the form of the smaller square, Leonardo uses a right-angled triangle whose longest side equals the side of the larger square. Its middle side equals the middle square to be subtracted, and its smaller side is the side of the smaller square that remains — the squaring of the gnomon.
Curvilinear and rectilinear parallelograms are equal
Leonardo states that all parallelograms of equidistant frontage and the same width are equal to one another, even if one has two curved sides and the other the same sides straight. He asserts the curvilinear parallelogram a b n m c d equals the rectilinear parallelogram a b c d, proving it by removing the common surface m from each so that the remainders stay equal.