I have read that a lost book by Fibonacci (a commentary on Book X of Euclid's Elements) gives a numerical treatment of incommensurable magnitudes.
Given that Fibonacci grew up in North Africa and travelled extensively throughout the Middle East where he learned the Hindu-Arabic number system and became familiar with the contemporary developments in mathematics, it seems reasonable to wonder if he may have encountered a numerical treatment of incommensurable magnitudes during these travels.
Q: Is there any record of Arab or Persian mathematicians developing a numerical treatment of incommensurables, or is Fibonacci’s lost commentary the first known treatment?
(N.B. A number of questions have already asked here regarding the Greek's discovery of incommensurable magintudes, but none appear to address this question.)
From the MacTutor page on Fibonacci
His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.
We know that the fourth section of Liber Abaci has Fibonacci giving numerical approximations of surds, thereby removing them from geometry. Thus, the missing text must go further than simple numerical approximations.
Further, in the 1225 work, Flos, Fibonacci details his solutions to the three questions posed to him by Johannes of Palermo during a meeting at the court of Fredrick II. In the second of these problems he finds the only real root of the equation $x^3 + 2x^2 + 10x = 20$. His argument establishes that such a root cannot be integral or rational, and that it cannot take any of the forms from Book X of Elements. As such, this represented a new type of irrational number - one not capable of construction by straight edge and compass.
And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.
Then, without justification, Fibonacci states that root is approximately
$1; 22, 07, 42, 33, 04, 40$
in sexidecimal notation.
All of these know facts suggest that the missing commentary of Book X represented a significant advance in our understanding of irrational numbers. I was hoping that any existing Middle Eastern sources might give a hint as to what this treatment may have included.