Quadrature proofs and the light of the moon
Triangles equal to squares, with a note on the sun, the eye, the moon and waves of water
The sheet is mostly geometry: several quadrature diagrams in which triangles are shown to be exactly half of, or equal to, given squares, culminating in a longer proof that the triangle a c S equals the square a b c d. At the top right a cylinder shedding rays onto a wavy surface is annotated 'sun, eye' and 'Moon, or rather earth, that is, waves of water', a study of reflected light. A closing aphorism about lending and returning what was lent reads as an allegory of adding and removing equal areas. The mirror-script notes were recorded in the transcription.
On this page
A triangle proved half of its square
Beneath the quadrature figure Leonardo states that triangle a e m is proved to be exactly half of the square. The construction reduces the curved or composite figure to an equal rectilinear area.
Squaring a surface by height times length
To square the surface a e c b f d, Leonardo multiplies the height a b by the length a c, obtaining the quadrature a a c b d n m exactly. The rule converts the area into an equivalent rectangle.
Sun, eye, and the moon's light on water
A diagram of a luminous body casting lines onto a rippled surface is labelled 'sun, eye'. The caption below adds 'Moon, or rather earth, that is, waves of water', treating the moon's brightness as sunlight reflected from the waves.
Maxim on lending and restitution
A prose aphorism holds that one should lend and then take back what was lent: he who returns what was lent to him keeps nothing of another's, and to him to whom what was taken is restored, the original quantity is not lacking. The saying doubles as an allegory of the equal cutting and replacing of areas in the quadrature figures.