  Book II of Explanation of the Difficulties in the Postulates of Euclid takes up the question of the proper definition of ratio. This is an obscure topic to the modern reader, but it was fundamental to Greek and medieval mathematics. If the quantities joined in a ratio are whole numbers, then the definition of their ratio poses no difficulty. If the quantities are geometric magnitudes, the situation is more complex because the two line segments might be incommensurable (in modern terms, their ratio corresponds to an irrational number). Euclid, following Eudoxus, asserts that A/B = C/D when, for any magnitudes x and y, the magnitudes xA and xC are both (i) greater than, (ii) equal to, or (iii) less than, the magnitudes yB and yD respectively. There is little wonder that Khayyam and others were unhappy with this definition, for while it is clearly true, it does not get at the heart of what it means for ratios to be equal.