Rational and medial products on a divided line
Squares and rectangles labelled r a and 'binomio', with Euclid citations and a numeric check
The page develops the algebra of rational and irrational magnitudes. A line divided at a, b, c is called rational where its parts stand in ratio; a square and an applied rectangle are labelled r a and 'binomio' (binomial). The left note states that multiplying a b by b c 'makes a surface' which is 'medial, that is irrational', checked by the arithmetic 12 x 12 = 144 broken into 64 + 16 + 32 + 32. The right margin cites the propositions used, including 'the ninth of this and the fourth of the second' and 'the 16th of this'.
On this page
The compound line is rational
A line divided at a, b, c is labelled r a r and its compound declared rational. It fixes the rational case before the medial (irrational) product is treated on the left.
Rectangle applied to the square: a binomial
A rectangle labelled i n and r a is applied to the square r a and named a binomial. The construction realises the binomial magnitude geometrically as an applied area.
Product a b by b c is medial (irrational)
The left note argues that multiplying a b by b c makes a surface that is medial, that is irrational. It is checked by the figures 12 and 144 decomposed as 64, 16, 32, 32.
Propositions cited in the margin
The right margin records the authorities for the steps: 'the ninth of this and the fourth of the second', 'the ninth of this and its converse', and 'the 16th of this'.