Bisected line extended: rectangle plus square equals square
Worked example 16·4 + 6² = 10², proved both geometrically and arithmetically
The folio demonstrates a proposition (cited as 'the sixth') stating that multiplying the whole extended line a d by the added part d b and adding the square of the half b c equals the square of c d. A line a-c-b-d is marked with segments of 6, 6 and 4, and a worked example gives 16×4 = 64, 6×6 = 36, summing to 100, which equals 10×10. Five parallelograms applied to the line and a right-margin column of figures present the same equality geometrically and arithmetically, citing 'the forty-third of the first' and the common notions of Euclid's Elements.
On this page
The proposition ('the sixth'): extended rectangle plus square of the half
Leonardo states that multiplying a d by the part d b and adding the square of b c to that sum equals multiplying d c by itself. This is the geometric-algebraic identity for a line bisected and then extended.
Worked numeric example on segments a c b d
With a-c = 6, c-b = 6 and b-d = 4, he computes 4 times 16 = 64 for b d against a d, then 6 times 6 = 36 for b c, adding to make exactly 100. Multiplying d c by itself, 10 times 10, likewise makes 100.
Five parallelograms applied to the given line
The areas 6, 36, 24, 24, 24, 16, 6, 6, 4 are laid out as parallelograms applied to the line. The construction is justified 'by the forty-third of the first' and by the first and third common notions.
Right-margin arithmetic check
A column in the right margin sets 64 and 36 against 100, confirming that the two rectangle-and-square terms sum to the square of the whole. It is labelled an 'arithmetic definition.'