The Fifth Demonstration: Squaring the Line 6 = 4 + 2
A halved line and a line split 6 = 4 + 2, with rectangles and squares proving the square.
In the Fifth of his algebra ('Arcibra') demonstrations, Leonardo notes that the figure of the Sixth corresponds to the Fifth. He draws a line divided in half (6 and 6) and a line divided into 6 = 4 + 2, then a small rectangle joined to a square whose cells carry 16, 12, 8, 4 and 2. Two further analogous figures repeat the dissection with the parts 4, 2 and 1. The page continues the Euclidean squaring of a divided line by figures.
On this page
Lines halved and split 6 = 4 + 2
Leonardo draws a line divided in half into 6 and 6, and a second line divided 6 = 4 + 2. These divided lengths are the quantities whose squares the figures below decompose.
Rectangle joined to a square
A small rectangle is joined to a square, the cells labelled with values such as 16, 12, 8, 4 and 2 built from the parts 4 and 2. The figure shows the square of the divided line as the sum of its component squares and rectangles.
Analogous figures with the parts 4, 2, 1
Two further analogous figures repeat the demonstration at a smaller scale, carrying values 4, 4, 8, 4, 4, 2, 2 and 3, 2, 1, 1. They rehearse the same decomposition with reduced quantities.