 Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of so such a definition would have been meaningless to him.Thus, a more subtle definition is needed where quantities involved are not measured directly to one another.In modern notation, Euclid's definition of equality is that, given quantities "p", "q", "r" and "s", then "p"∶"q"∷"r"∶"s" if and only if, for any positive integers "m" and "n", "np""mq" according as "nr""ms", respectively.