The Pyramid as One-Third of the Cube; the Order of the Sciences
A cube dissected by its diagonals into pyramids, with Leonardo's ranking of the mathematical sciences and the senses
A large cube crossed by internal diagonals, with a small shaded pyramid beside it, illustrates the proof that a pyramid on the cube's face is one-third of the cube: four such pyramids take 4/6 of the solid, leaving 1/3. A long note then sets out Leonardo's hierarchy of knowledge, naming arithmetic and geometry as the only two sciences of first-degree certainty, from which perspective, astronomy and music descend. The page grounds the geometry of the drawing in the accompanying transcription and closes with the ordering of the senses. Further mirror-script text fills the sheet in blocks matching the transcribed passages.
On this page
The pyramid is one-third of the cube
The base a c d has its apex at b, and this pyramid is one-third of the cube. Four such pyramids are lifted from the four faces, each being 1/6 of the whole cube, so together they hold 4/6; the remaining 2/6, that is 1/3, stays in the first pyramid.
The two mathematical sciences of certainty
The mathematical sciences are those which, through the senses, are in the first degree of certainty, and they are only two: arithmetic, extending to discontinuous quantity, and geometry, to the continuous. From these is born perspective, dedicated to all the offices of the eye. If one of the three is lacking, nothing is achieved.
Astronomy, music and the ordering of the senses
From arithmetic, geometry and perspective is born astronomy, which by the visual ray, with number and measure, concludes the distances and measures of the celestial as of the terrestrial bodies. There follows music, born of continuous and discrete quantity, dedicated to the ear, a sense less worthy than the eye, and then smell, taste and touch.