By Alhazen (lbn al-Haytham), 965-1039 ; A. Mark Smith (editor, translation, commentary)
Read Online or Download Alhacen on the principles of reflection. A Critical Edition, with English Translation and Commentary, of Books 4 and 5 of Alhacen’s De aspectibus. Volume One - Introduction and Latin Text ; Volume two - English Translation PDF
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Extra resources for Alhacen on the principles of reflection. A Critical Edition, with English Translation and Commentary, of Books 4 and 5 of Alhacen’s De aspectibus. Volume One - Introduction and Latin Text ; Volume two - English Translation
But we have already established that angle XDQ (which = angle XDP + angle PDQ) = one-half angle BGA. , angle XDP + angle PDQ) = angle BGA. It follows, then, that angle XDB + angle PDQ = one half angle BGA, so angle XDB = angle XDP. The next step is to demonstrate that, if PD is extended toward point A, it will intersect GA at that very point. That it does rests on showing that the resulting triangle PAG will be similar to triangle PDS, which follows from the fact that angle PDS = angle BGA by construction, while angle APG is common to both triangles.
420-422—see esp. 169, p. 421). Proposition 20, lemma 2, pp. 419-420, shows how to drop a line from some point on the circumference of a circle through the diameter such that the segment of this line from where it intersects the diameter to where it intersects the opposite arc on the circle is equal to some randomly chosen line. Thus, as illustrated in figure 5, p. 525, the objective is to drop line AD from randomly chosen point A through diameter GB of the circle so that ED equals randomly chosen line HZ.
415-419, however, there follows a set of six lemmas that are instrumental in one way or another to the subsequent determination of reflection-points in convex and concave mirrors. , lemmas 3-6—are directly implicated in those determinations; the other two play ancillary yet critical roles in laying the requisite groundwork for them. In the first lemma, which is dealt with in proposition 19, Alhacen shows how to generate a line extending from some randomly chosen point on a circle to the extension of its diameter such that the segment of this line between where it intersects the circle and where it intersects the diameter is equal to some randomly chosen line.