The Teaching of Geometry

by David Eugene Smith

2. Angles inscribed in the same segment are equal.

By driving two nails in a board, at A and B, and taking an angle P made of rigid material (in particular, as already stated, a carpenter's square), a pencil placed at P will generate an arc of a circle if the arms slide along A and B. This is an interesting exercise for pupils.

Theorem. An angle formed by two chords intersecting within the circle is measured by half the sum of the intercepted arcs.

Theorem. An angle formed by a tangent and a chord drawn from the point of tangency is measured by half the intercepted arc.

Theorem. An angle formed by two secants, a secant and a tangent, or two tangents, drawn to a circle from an external point, is measured by half the difference of the intercepted arcs.

These three theorems are all special cases of the general proposition that the angle included between two lines that cut (or touch) a circle is measured by half the sum of the intercepted arcs. If the point passes from within the circle to the circle itself, one arc becomes zero and the angle becomes an inscribed angle. If the point passes outside the circle, the smaller arc becomes negative, having passed through zero. The point may even "go to infinity," as is said in higher mathematics, the lines then becoming parallel, and the angle becoming zero, being measured by half the sum of one arc and a negative arc of the same absolute value. This is one of the best illustrations of the Principle of Continuity to be found in geometry.

Problem. To let fall a perpendicular upon a given line from a given external point.

This is the first problem that a student meets in most American geometries. The reason for treating the problems by themselves instead of mingling them with the theorems has already been discussed.[69] The student now has a sufficient body of theorems, by which he can prove that his constructions are correct, and the advantage of treating these constructions together is greater than that of following Euclid's plan of introducing them whenever needed.

Proclus tells us that "this problem was first investigated by Œnopides,[70] who thought it useful for astronomy." Proclus speaks of such a line as a gnomon, a common name for the perpendicular on a sundial, which casts the shadow by which the time of day is known. He also speaks of two kinds of perpendiculars, the plane and solid, the former being a line perpendicular to a line, and the latter a line perpendicular to a plane.