The Eighth Demonstration: Squaring 8 + 4 + 4 into 256
A note defining algebra, with a square dissected into sub-squares and rectangles totalling 256.
Leonardo heads the sheet with the name 'Alberto da Imola' and a short definition of algebra ('Alcibra'), which he says shows how a number and a 'thing' (the unknown) are set equal to the thing. The Eighth demonstration then divides a line into 8 + 4 + 4 and squares it: the accompanying square is cut into rectangles and squares labelled 64, 32, 32, 16, 16 and so on, which sum in the marginal tally to 256. The page is a worked instance of the Euclidean method of squaring a divided line, here 16 = 8 + 4 + 4 giving 256.
On this page
A definition of algebra
Beneath the name Alberto da Imola, Leonardo notes that algebra 'shows how a number and a thing are made equal to the thing' – the medieval framing of an equation in which a number plus the unknown ('cosa') is set equal to the unknown. It frames the geometric demonstrations that follow as algebra done with figures.
Square dissected into rectangles and squares
A square is divided into rectangles and squares carrying the values 64, 32, 32, 16 and 16, built from the line divided 8, 4, 4. The dissection shows how the square of the whole line is composed of the squares and rectangles of its parts.
The parts summed to 256
A side column adds the cells of the square – 64, 32, 32, 32, 32, 16, 16, 16, 16 – to reach the total 256. This is the square of 16 (= 8 + 4 + 4), confirming the dissection arithmetically.