Geometry: doubling a sphere and dividing a triangle by proportion
The cube root of a doubled cube, and how a cut parallel to the base transforms a square
A dense geometry page from Notebook E opening on the problem of a sphere double to another and the search for the cube root of a doubled cube, illustrated by squares each enclosing a circle. The main argument concerns a triangle cut by a line parallel to its base: such a cut shrinks toward the apex exactly as much as it widens toward the base. Leonardo applies this to transform a square e c n m by narrowing its width while lengthening its side in the same proportion, and treats rational and irrational quantities alike. A marginal proposition restates the governing rule of proportional narrowing and widening in rectilinear triangles.
On this page
Doubling a sphere and the cube root of a doubled cube
The page opens by supposing one sphere double (or half) another and sets out to find the cube root of a cube double to a given one. The accompanying construction encloses within each square the circumference of the single or doubled sphere.
A cut parallel to the base of a triangle
A proposition states that a cut made parallel to the base of the triangle is smaller than that base by exactly the amount by which the base exceeds it. This reciprocity governs the transformations that follow.
Rational and irrational quantities behave alike
Using the labels a b f o and e d c n on the inverted triangle, Leonardo argues that if the whole a b becomes c d, then c n returns to a o. He insists that rational quantities behave exactly as irrational ones do under this rule.
Equal gains and losses within the angle of proportions
As the line a b descends to e d it loses three quarters of its quantity, and rising from c e to a o it gains three quarters at a f. Because such motions are equal within the angle of the proportions, gains and losses stay proportioned to the quantities where they meet.
Narrowing and lengthening the square e c n m
Taking e c as the width and c m as the length of the square, Leonardo places them on the sides and the parallel cut of triangle a o S. Made to descend and rise, the square is narrowed in the space e d and lengthened in the space a o, shrinking in proportion to its width exactly as it grows in proportion to its side.