The First Proposition of 10th book of Euclid's Elements and Islamic Scholar's Alternatives

The first proposition of the tenth book of Euclid's Elements is the basis of Archimedes’ “method of exhaustion”, which is used by Euclid himself in book XII. Euclid proves this proposition for variable proportion. Ibn al-Haytham (4th AH) claims that in his ḥall-i shukūk he has proved this proposition for a fixed proportion as the general solution, whereas Euclid had done it just for particular cases. Ṭūsī (597-672 AH) in his recension of Euclid’s Elements, in which seems has been used fī ḥall-i shukūk a lot, repeats the same idea as Ibn al-Haytham’s. Ibn al-Ṣalāḥ al-Ḥamadanī (d. 548 AH) has a treatise in which he criticizes Ibn al-Haytham’s idea and brings three main objections. Muḥammad Bāqir Yazdī (alive in 1047 AH) in his commentary on tenth book of Euclid’s Elements criticizes Ṭūsī’s idea and explains that the general condition which Ṭūsī applies disrupts the proof of second proposition of the tenth book.