Reflection in a convex mirror: locating the image
A geometric construction for the angle of incidence, with two supporting propositions
Two semicircle diagrams support a study of reflection in a convex mirror. A first note states geometric propositions, that unequal parts taken in the same proportion leave a proportional remainder, and that all angles of contingence of a circle with a straight line are equal, and remarks that the upper and lower figures are the same, the lower being less intricate to demonstrate. The main construction locates the image of an object in a convex mirror: from the eye a and the object b, lines a d and b d run to the centre d, while tangents a e and b f touch the circle at m and n, and their intersection c marks the angle of incidence. The reasoning is geometric optics throughout.
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Locating an image in a convex mirror
From the eye and the object (points a and b) the lines a d and b d are drawn to the mirror's centre d; two tangents are then drawn, a e touching the circle at m and b f touching at n, and where they cross at c lies the angle of incidence that fixes the image's place. If that angle sits too high above the circle it is lowered by the method shown alongside.
Angles of contingence and proportional remainders
If unequal parts are taken from two unequal quantities so that the inequality keeps the same proportion as the whole, the remainder stays in that proportion. All the angles of contingence of one circle with a straight line are equal, and at equal distance from the angle they make triangles of equal base.
