Geometric Proof: Figures Equal to Their Squares
Lettered figures a-h shown equal to the squares below them, combining into square K equal to surface m
A geometric demonstration of the equivalence of areas: Leonardo labels a row of figures a, b, c, d, e, f, g, h and argues that a b equals its square below (e f) and c d equals its square (g h). He concludes that the parts a d of each figure equal their parts e h below, which joined together make the square K equal to the surface m. The proof sits among several drawn figures at the head of the page, including a vase-like profile, a trapezoid and curved arch-like shapes, and two boxed diagrams carrying further numerical values that are not transcribed here. The blue oval is the Biblioteca Nacional stamp.
On this page
Each figure equal to its square below
Leonardo states that a b is equal to its square below, that is e f, and likewise c d equals its square g h. Therefore the parts a d of both figures equal their lower parts e h, which joined together make the square K equal to the surface m. The reasoning transforms curved or composite figures into equal squares.
Lettered construction a-h
The figures at mid-page are keyed with the letters a, b, c, d, e, f, g, h, which the demonstration pairs into upper members and their squares below.
Square K and surface m
At the left a boxed square is labeled K and a curved figure m is drawn beside it; the proof concludes that the two are equal in area.