On Proportion: Continuous and Discontinuous Proportionality
Definitions of arithmetic and geometric proportion among three or four terms
This text page sets out Leonardo's definitions of proportion and proportionality. He states that a proportion always holds between two terms of the same kind, that continuous proportionality requires at least three terms (containing two ratios), and that discontinuous proportionality requires at least four, of which two may be of one kind and two of another. He then distinguishes continuous from discontinuous arithmetic proportional quantities by their equal differences, giving 6 to 4 and 10 to 8 as examples, and notes these may span different kinds as in geometric proportion. Faint underlying box-like sketches are visible lower on the sheet.
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How many terms proportion and proportionality require
A proportion must always be found between two terms of the same kind. Continuous proportionality must be constituted among at least three terms, among which two ratios always fall; discontinuous proportionality requires at least four terms, of which two may be of one kind and two of another.
Continuous and discontinuous arithmetic proportion
In continuous arithmetic proportion each term exceeds the previous by an equal amount, the third exceeding the second as the second the first. In discontinuous arithmetic proportion the second exceeds the first by the same amount that the fourth exceeds the third and the sixth the fifth, as in 6 to 4 and 10 to 8.