As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms p, q and r to be in proportion when p:q∷q:r. This is extended to four terms p, q, r and s as p:q∷q:r∷r:s, and so on. Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and if p, q, r and s are in proportion then p:s is the triplicate ratio of p:q.
As with definition 3, definition 8 is regarded by some as being a later insertion by Eu?clid's editors. It defines three terms p, q and r to be in proportion when p:q∷q:r. This is extended to four terms p, q, r and s as p:q∷q:r∷r:s, and so on. Se.quences that have the property that the ratios of consecutiv.e terms are equal are called geometric progressions. Definitions 9 and 10 apply th.is, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and! if p, q, r and s are in proportion then p:s is the triplicate ratio of p:q.