Where does light reflect on a curved mirror?

Look at the expectation: it asks for the laws of reflection and for ray diagrams. Ask against what line the angles are measured when the mirror is curved, and why a flat mirror does not raise the question.

Read how the problem is set, and what its first steps look like. In the record the givens are a circle representing the mirror in a plane, its center, and two points representing object and eye. The task is to construct a rim point satisfying equal incidence and reflection angles measured against the local normal ². This differs from drawing a chord or checking a ray already supplied, because the meeting point itself is unknown ³. In the first two lemmas, for example, Ibn al-Haytham treats a segment cut off between a circle and its diameter, separately handling its location inside or outside ⁴. Some work uses circle–hyperbola intersections, and drawing both curves does not make the construction a straightedge-and-compass operation ⁵.

The museum's records preserve no sentence in which Ibn al-Haytham says why he took up this problem, and no preface that states a purpose. What they show is where it sits. The lemmas occur within an optical investigation but form linked geometrical problems: one lemma assists another, then the results return to locating reflection points ⁶. The Encyclopaedia of Islam article says that in the fifth book of the Optics his mathematical genius reached its height when he resolved the problem that now bears his name ⁷. The same article says he held that rays of light travel from the object to the eye, and not the reverse as Euclid, Ptolemy and al-Kindi maintained ⁸.

A famous name can suggest that one man began everything. The record cautions against that. His contribution here is geometrical lemmas and applications in Book V of the Book of Optics, not invention of mirrors or discovery of reflection's law from nothing ⁹. The name does not erase the earlier question attributed to Ptolemy, which the record reports through MathWorld, not from Ptolemy's own text ^10 11. Nor should he be credited with a general algebraic quartic formula: the historian's note states a modern equivalence to a fourth-degree equation and expressly says medieval Arabic mathematicians were unaware of that equivalence ¹².

The museum's reading is partial and says so. It read pages 59–75 of Hogendijk's 1996 study from the author's hosted copy and inspected images 59–65, which was not reading the entire paper, an Optics manuscript or the complete Sabra 1982 article ¹³. The Arabic Optics source the museum holds covers Books 1–3, not Book V ¹⁴. The study translates al-Mu'taman's lemmas, and his adaptations must not become quotations from Ibn al-Haytham ¹³. Hogendijk compares al-Mu'taman's sequence with that of the Optics and regards the correspondence as evidence of use, although al-Mu'taman does not name his sources; the record calls the attribution of the simplifications historical inference, not a signed declaration ¹⁵.

For the claim that this problem is Ibn al-Haytham's, the record relies on two kinds of source: MathWorld, which attributes an earlier formulation to Ptolemy, and Hogendijk's study, which reads and compares the lemmas. The record says the first is reported through the reference actually read, not from Ptolemy's text. Which source should carry more weight on who first posed the problem, and which on who first solved it, and why? What would you need to see to change your answer?

• Ibn al-Haytham's circular-mirror reflection method: the lemmas, the notation and the later simplifications by al-Mu'taman.
• Ibn al-Haytham: the Encyclopaedia of Islam's account of his Optics and of the problem that bears his name.
• Kitab al-Manazir: the Book of Optics, in which the problem appears in the fifth book.
• Optics: the science the problem belongs to.