Geometry¶
Custodians Who Renovated the House¶
The Islamic tradition inherited Greek geometry entire, Euclid, Archimedes, Apollonius, and did three things with it: preserved it (parts survive only in Arabic, including Apollonius Conics V–VII and books of Diophantus), taught it (Euclid was the universal curriculum), and extended it along several research lines.
Research Line 1: Measurement and Proto-Calculus¶
The Archimedean program of areas and volumes by exhaustion continued with real novelty:
- The Banu Musa recast Archimedes computationally, treating \(\pi\) as a magnitude.
- Thabit ibn Qurra computed the parabolic segment's area by summing over unequal subdivisions, and volumes of paraboloids.
- Ibrahim ibn Sinan gave the most elegant pre-modern parabola quadrature via affine invariance.
- Ibn al-Haytham summed fourth powers to find the volume of a paraboloid rotated about its base, using exactly the \(\sum_{k=1}^{n} k^4\) machinery that seventeenth-century Europe (Cavalieri, Fermat, Pascal, Wallis) needed for \(\int_0^a x^n dx\), the arithmetical on-ramp to the integral calculus.
Whether Europe's integrators knew the Arabic precedents is mostly undocumented; the point is that the mathematics of "slice, sum, refine" was carried further between 850 and 1050 than anywhere else before the 1600s.
Research Line 2: The Parallel Postulate¶
An unbroken chain of Islamic geometers attacked Euclid's fifth postulate: al-Jawhari, Thabit, Ibn al-Haytham (the tri-rectangular quadrilateral later named for Lambert), Khayyam (the equal-sides quadrilateral later named for Saccheri, with the explicit trichotomy of acute/right/obtuse summit angles), and Nasir al-Din al-Tusi and his school. All "proofs" fail, necessarily, by smuggling in an equivalent assumption; but the accumulated analysis of what the postulate is equivalent to, and the quadrilateral configurations themselves, passed into Europe when a pseudo-Tusi recension was printed in Rome in 1594, was discussed by John Wallis (1663 lecture), and was cited on the first pages of Saccheri's Euclides Vindicatus (1733), the book from which the road runs through Lambert and Legendre to Gauss, Bolyai, and Lobachevsky. Non-Euclidean geometry has a documented Islamic chapter in its prehistory.
Research Line 3: Constructions Beyond the Ruler¶
The tradition treated the classical problems (trisection, heptagon, cube-type problems) not as taboos but as a hierarchy of tools: what needs conics, what needs neusis, what can be approximated. al-Sijzi's heptagon by conics, al-Quhi's perfect compass for drawing conics in one motion, al-Saghani's trisections, and Abu al-Wafa's constructions with a rusty (fixed-opening) compass belong here; the last anticipates the 20th-century Mohr–Mascheroni-adjacent results on restricted construction tools and connects directly to artisanal practice, the girih strapwork and muqarnas vaulting of Islamic architecture, whose quasi-periodic girih-tile patterns (Lu & Steinhardt, Science, 2007) have been compared to Penrose tilings.
Research Line 4: Foundations of Ratio and Magnitude¶
Commentary on Euclid Book V (proportion) and Book X (irrationals), by al-Mahani, Khayyam, and others, pushed toward treating ratios as numbers, i.e., toward the real-number continuum. Khayyam's discussion of compounding ratios and his complaint that Euclid's definition hides the quantitative nature of ratio are way-stations on the road that ends with Dedekind's construction of the reals (1872), and Dedekind-style "cut" reasoning has been noted by historians in Eudoxus and in the Arabic commentators alike.
Research Line 5: Geometry of the Sphere¶
Spherical geometry, driven by astronomy and the qibla, was transformed from Menelaus-theorem gymnastics into systematic trigonometric solution of spherical triangles: see the Trigonometry chapter.
Sources: Berggren, Episodes, ch. 3; Rashed, Les mathématiques infinitésimales; Rosenfeld, A History of Non-Euclidean Geometry; Katz; Lu & Steinhardt, "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture," Science 315 (2007).