The importance of Khayyam's contribution is that he examined both Euclid 's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions . Khayyam proved that the two definitions are equivalent.
Ang kahalagahan ng Khayyam's kontribusyon na siya ay napagmasdan parehong Euclid 's kahulugan ng pagkapantay-pantay ng ratios (na kung saan ay na unang iminungkahi ng Eudoxus) at ang mga kahulugan ng pagkapantay-pantay ng ratios tulad ng mga iminungkahing sa pamamagitan ng mas maaga Islamic mathematicians tulad ng al-Mahani na kung saan ay nakabatay sa patuloy fractions. Khayyam na ang ang dalawang kahulugan ay katumbas ng mga ito.