Cubing an irregular six-faced body from twelve lines
Twelve lines of irrational proportion said to enclose a cube in 5054 ways
At the head of the page Leonardo draws an oblique six-faced solid, an 'irregular cube', with its edges lettered, and beneath it a stack of twelve horizontal lines of graded length labelled as lines of irrational proportion. The note explains how a six-faced body's 12 edges are distributed (four round the top square, four round the base, four rising between them) and argues that these twelve lines of unknown length, however combined, always enclose a cube of equal quantity and weight. He adds that the same lines may be transformed in 5054 varied ways while still enclosing the same matter.
On this page
The irregular six-faced body and its twelve edges
An oblique six-faced solid (an 'irregular cube') is drawn at the top and its edges lettered c d, a f, n e, b m. Leonardo notes that every such body has 6 square faces each bounded by 4 straight lines, and that of the 12 given lines four surround the upper square, four the base, and four extend from the base angles to the angles of the upper square.
Twelve lines of irrational proportion enclosing an equal cube
Below the solid twelve lines of graded length are lettered a b c d e f g n m o r t as 'lines of irrational proportion'. Leonardo claims that however these twelve lines of unknown length are joined, they always enclose six surfaces containing a cube of equal quantity and weight, and that the same lines may be transformed in 5054 varied ways while enclosing the same matter.