Where does the mirror send the light?

The expectations ask you to use the law of reflection. Decide what you must measure at the point where the ray meets the mirror, and from which line the angles are measured.

Read the problem as the record sets it. The givens are a circle representing the mirror in a plane, its center, and two points representing an object and an eye. The task is to construct a rim point satisfying equal incidence and reflection angles measured against the local normal ². A circle's radius to that point is perpendicular to its tangent, so the reference is not invariably a fixed horizontal line ². This differs from drawing a chord or checking a ray already supplied, because the meeting point itself is unknown ³. The record of the circular-mirror method sets out the rest.

The record gives no reason in Ibn al-Haytham's own words. What it shows is the problem the lemmas serve: the lemmas occur within an optical investigation but form linked geometrical problems, one assisting another, and then the results return to locating reflection points ⁴. The historian places the Optics in eleventh-century Egypt ⁵. The record adds that Egypt is the author's setting, not a workshop address verified there ⁶.

The problem carries his name, but the record limits what that means. His contribution 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 ⁷. MathWorld attributes an earlier formulation to Ptolemy ⁸. The historian's note states a modern equivalence to a fourth-degree equation and expressly says medieval Arabic mathematicians were unaware of that equivalence ⁹.

The record is open about what was read. It says it read pages 59–75 of Hogendijk's 1996 study from the author's hosted copy, not an Optics manuscript ¹⁰. The study translates al-Mu'taman's lemmas, so his adaptations must not become quotations from Ibn al-Haytham ¹⁰. The museum's Arabic Optics source covers Books 1–3, not Book V ¹¹. So what we know of Book V comes through a modern historian, who finds the discussion of certain concave configurations unsatisfactory and says multiple reflection points are possible ¹².

In the record's modern teaching example, a unit circle centred at (0,0) has the object at (2,1) and the eye at (2,−1), and the point (1,0) gives two 45° angles to the horizontal normal ¹³. The record says this checks an easy symmetric case and does not prove that guessing an arc midpoint always suffices ¹³. Move the object to a different height and the eye to a different distance: what would you have to check before trusting a guessed point, and why?

• The circular-mirror method: the givens, the auxiliary constructions and the limits of the solution.
• Ibn al-Haytham: his life and writings, and the gaps in his biography.
• The Book of Optics: the book whose fifth book holds the problem.
• Ibn al-Haytham's dark room: a different experiment by the same scholar.