Proportions and proportionality, 16th century. This page shows the 'Tree of Proportions and Proportionality' as published in 'On the Divine Proportion' (1509) by Italian mathematician Luca Pacioli (c.1447-1517). This tree is a complex classification and naming of types of proportions and ratios, including continuous and discontinuous,and rational and irrational numbers. The terms include: ratios of equality and inequality; of greater and lesser inequality; multiplexes or multiples; single, double and triple ratios; and particulars, alterates, and partiens. Pacioli's 'On the Divine Proportion' was about mathematical proportions such as the 'golden ratio' referred to in the title.