 Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5.In modern notation it says that given quantities "p", "q", "r" and "s", "p"∶"q">"r"∶"s" if there are positive integers "m" and "n" so that "np">"mq" and "nr"≤"ms".As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors.