At the turn of the third millennium, the field of cosmology—broadly defined as the study of the mechanisms and structure governing the universe across different historical frameworks—faces challenges that echo those encountered at the turn of the second millennium in two foundational issues with both epistemological and methodological dimensions. Today, we grapple with substantiated ideas regarding the uncertainty surrounding the constancy of fundamental cosmological parameters, such as the speed of light, the cosmological constant, and the gravitational constant, which are presently considered invariant. While, from a Poincaréan perspective, we have not yet been compelled to accept their variability, it remains conceivable that, at some point in the future—whether sooner or later—we may need to abandon the notion of a finely tuned universe. A thousand years ago, during the culmination of astronomical inquiry in the medieval Islamic world, empirical investigations led some astronomers to lay the foundations for a new astronomy. This emerging framework challenged the long-standing assumption—rooted in Greek and Ptolemaic traditions—that certain fundamental parameters were immutable. Instead, it proposed that these parameters might undergo linear, periodic, or secular variations. At the time, the primary “cosmological” constants under scrutiny included precession, the solar orbital elements, and the obliquity of the ecliptic. The astronomical community remained deeply engaged with this issue until the revolutionary shift formed by the transition from mere kinematic constructions to dynamic explanations in the post-Newtonian era confirmed their variability and periodicity. The second issue concerns the geometrical shape and physical structure of the universe. Our current observable universe extends across a sphere with a radius of about 46 billion light-years, very likely falling short of the true boundaries of the cosmos. Neither the local geometry of the universe—dependent on its curvature (Flat: Euclidean, Closed: Spherical, or Open: Hyperbolic)—nor its global shape has yet been ascertained. While the flat model remains the prevailing consensus, debates over topology and curvature persist. A mathematically viable possibility is that a flat, finite universe with multiply-connected topology could take the form of a torus—a doughnut-shaped structure. A millennium ago, Abū al-Rayḥān al-Bīrūnī (d. 1048), a practical astronomer from Khurāsān (a pre-Islamic Persian territory later incorporated into the Islamic domain) proposed that the long-standing principle of a spherical cosmos, upheld for some 1,400 years since Eudoxus and firmly trenched in the Aristotelian tradition, might be nothing more than the consequence of a mere optical illusion stemming from the effect of the sky’s color and the gradual weakening of the human eyesight, occurring equidistantly from the earthbound observer. He also put forward the intriguing hypothesis of elliptical planetary orbits, which led him to envision a cylindrical universe, and noted the simple fact that other solid shapes could satisfy the arguments and conditions set forth by both the Stagirite and Ptolemy just as well as a sphere. The core of Bīrūnī’s groundbreaking ideas has reached us indirectly through the medium of a short essay titled Fī kuriyyat al-samāʾ (On the sphericity of the heavens) authored by his mentor, Abū Naṣr Manṣūr b. ʿAlī b. ʿIrāq (b. probably in 960 in Guilan, northern Iran, near the Caspian Sea), as a response to his ideas. This paper aims to introduce and examine the various technical peculiarities and philosophical aspects of this overlooked episode in the context of medieval astronomical thought.
