The Eighth Rule: Multiples in Continued Proportion
d taken twice — as consequent, then as antecedent
Headed 'Eighth' and twice labelled 'multiples' (multiplici), the page sets out a chain of proportional line-segments carrying the values h 18, K 18, f 9, g, and a 6, b 6, with e noted as the triple of c, alongside further terms d 3, L 18 and m 21. A closing note explains that the term d occurs twice, 'that is, as antecedent and consequent' — the mark of a continued proportion, where one magnitude ends one ratio and begins the next. The figures are drawn as stacked horizontal segments in pencil reinforced with pen. It forms part of the notebook's Euclid-based studies on ratio and proportion.
On this page
d as both antecedent and consequent
The note records that d appears twice: once as the consequent (end term) of one ratio and once as the antecedent (lead term) of the next. This double role is what links successive ratios into a single continued proportion. The heading 'multiples' (multiplici) marks the whole set as a study of multiple-part ratios.
The proportional values
The segments carry the numbers 18, 9 and 6, with e marked as the triple (tripla) of c, and further terms d 3, L 18 and m 21. Read together they express the ratios being compared under the 'eighth' rule.