Proportion of Equal Squares Equals Their Heights
Geometric proofs on squares, rhombi and triangles of equal base or height, amid structural sketches
Continuing the reasoning of the recto, Leonardo repeats in several forms the theorem that among equilateral squares the proportion is the same as that of their heights, and extends it to quadrilaterals and rhombi of equal sides and to triangles of equal height but different base. Figures of squares set within rhombi and of triangles illustrate each statement. Around the geometric text the sheet carries structural and mechanical sketches—stepped and beam-like constructions above and a leaning wall or tower at lower left—which are not transcribed in the bundle.
On this page
Equilateral squares proportional to their heights
Leonardo states and restates that among equilateral squares there is the same proportion as that of their heights, and that all squares with sides equal among themselves obey this relation. The proposition is repeated several times with slight variations as he refines the wording. It is the governing theorem of the whole sheet.
Triangles of equal height and different base
Beside figures of triangles sharing one height but differing in base, he notes that among bases of equal height the magnitude is as that of the base. The rule ties area to base when height is held constant. It is the triangular counterpart to the theorem on squares.
Quadrilaterals and rhombi of equal sides
For a square set among rhombi of equal base he argues that quadrilaterals whose sides are all equal share the same proportions as their heights, and that among bases of equal width the proportion of magnitude follows the height. The equal-sided rhombus is thus compared with the square. The point extends the theorem from squares to all equilateral quadrilaterals.
Structural and leaning-wall sketches
Around the geometry the leaf carries small structural drawings: stepped and beam-like constructions in the upper area and a tall leaning wall or tower at the lower left. These sketches are faint and not transcribed in the bundle. They sit alongside the proportional proofs without accompanying legible captions here.