On Proportion: Its Names, Members, and Three Species
Proportion defined between quantities of one kind; arithmetic, geometric and harmonic proportion
Folio 47r opens Leonardo's treatise on proportion, defining it as a relation considered between two quantities of the same kind that are greater, lesser, or equal to one another, and naming its three species: superposition, supposition, and equiparance. He lists proportion's alternative names (relation, habitude, respect, mediety) and divides it into 'commonly said' (comparing things of different kinds, like a sweet voice to honey) and 'properly said,' which is further split into arithmetic, geometric, and harmonic. Setting harmonic aside, he notes that geometric proportion compares one continuous quantity to another and can be continuous or discontinuous. The page is dense mirror-script with faint marginal glosses.
On this page
The three species of proportion: superposition, supposition, equiparance
Of two unequal quantities, referring the greater to the lesser yields superposition, and the lesser to the greater yields supposition; referring an equal thing to an equal yields equiparance. Leonardo asserts that in no other way can proportion occur.
Commonly said versus properly said proportion
'Commonly said' proportion compares things not of the same kind under one term, as when one says the voice is sweeter than honey or sharper than a knife. Leonardo dismisses this and pursues the 'properly said' proportion, which holds among things of the same kind and divides into arithmetic, geometric, and harmonic.
Geometric proportion between continuous quantities
Geometric proportion compares one continuous quantity to another and is either continuous or discontinuous. Continuous proportion (as in 1-2-2-2-4) holds when the first term is to the second as the second is to the third, needing at least three terms; discontinuous proportionality needs at least four.