Euclid, Book V: the definitions of part and multiple
Leonardo sets down the opening definitions of proportion theory
This page opens Leonardo's study of the Fifth Book of Euclid, the theory of proportion. He records the first definition, in which a 'part' is the lesser quantity that numbers (measures) the greater, and distinguishes a 'multiplicative' part, which exactly recomposes its whole, from a 'common' or 'aggregative' part, which does not. A second definition follows, treating the 'multiple' as the greater quantity that the lesser measures.
On this page
First definition: a 'part' that measures the whole
A part is defined as a quantity of a greater quantity, the lesser of the two, provided that the lesser numbers (measures) the greater. This is the foundational notion of ratio between magnitudes.
Multiplicative part versus common aggregative part
A 'multiplicative' part, multiplied by some number, exactly recomposes its whole. A 'common' or 'aggregative' part, however many times it is taken, makes more or less than the whole, and needs the help of another quantity to remake it.
Second definition: the multiple
The multiple is the greater quantity, defined as greater than the lesser when the lesser measures it. Leonardo notes that the lesser extreme was defined above and the greater is defined here.