Disjunct and conjunct proportionality
Euclidean proportion theory proved on 8, 4, 2 and 12, 6, 3
This recto sets out two theorems of the theory of proportion with worked numerical proofs and bracket diagrams. The first, 'disjunct proportionality', states that quantities proportional when taken together stay proportional when separated, illustrated with a b : b c as d e : e f. Leonardo checks it on the numbers 8, 4, 2 and 12, 6, 3, showing that 10 : 6 equals 15 : 9 because each divisor goes into its dividend once and two-thirds. The second theorem, 'conjunct proportionality', gives the converse, with a further diagram of the same terms.
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Disjunct proportionality stated
Quantities that are proportional conjointly remain proportional disjointly. Let the proportion of a b to b c be as d e to e f; then a c is to c b as d f is to f e.
Numerical proof on 8, 4, 2 and 12, 6, 3
Taking a, b, c as 8, 4, 2 and d, e, f as 12, 6, 3: 8 and 2 make 10, 4 and 2 make 6, 12 and 3 make 15, 3 and 6 make 9. Then 10 : 6 equals 15 : 9, since 6 goes into 10 once and two-thirds and 9 goes into 15 once and two-thirds, so the proportions are equal.
Conjunct proportionality, the converse
Quantities that are proportional disjointly, once conjoined, become proportional. Let a c be to c b as d f to f e; then a b is to b c as d e to f e. The result holds because if a b and b c are two doubles, d e and f e are two doubles as well.