The Teaching of Geometry

by David Eugene Smith

Proclus (410-485 A.D.) tells us that Eudemus, who lived just before Euclid (or probably about 325 B.C.), affirmed that the theorem was due to the Pythagoreans, although this does not necessarily mean to the actual pupils of Pythagoras. The proof as he gives it consists in showing that a = a´, b = b´, and a´ + c + b´ = two right angles. Since the proposition about the exterior angle of a triangle is attributed to Philippus of Mende (ca. 380 B.C.), the figure given by Eudemus is probably the one used by the Pythagoreans.

There is also some reason for believing that Thales (ca. 600 B.C.) knew the theorem, for Diogenes Laertius (ca. 200 A.D.) quotes Pamphilius (first century A.D.) as saying that "he, having learned geometry from the Egyptians, was the first to inscribe a right triangle in a circle, and sacrificed an ox." The proof of this proposition requires the knowledge that the sum of the angles, at least in a right triangle, is two right angles. The proposition is frequently referred to by Aristotle.

There have been numerous attempts to prove the proposition without the use of parallel lines. Of these a German one, first given by Thibaut in the early part of the eighteenth century, is among the most interesting. This, in simplified form, is as follows:

Suppose an indefinite line XY to lie on AB. Let it swing about A, counterclockwise, through ∠A, so as to lie on AC, as X'Y'. Then let it swing about C, through ∠C, so as to lie on CB, as X''Y''. Then let it swing about B, through ∠B, so as to lie on BA, as X'''Y'''. It now lies on AB, but it is turned over, X''' being where Y was, and Y''' where X was. In turning through ⦞A, B, and C it has therefore turned through two right angles.

One trouble with the proof is that the rotation has not been about the same point, so that it has never been looked upon as other than an interesting illustration.

Proclus tried to prove the theorem by saying that, if we have two perpendiculars to the same line, and suppose them to revolve about their feet so as to make a triangle, then the amount taken from the right angles is added to the vertical angle of the triangle, and therefore the sum of the angles continues to be two right angles. But, of course, to prove his statement requires a perpendicular to be drawn from the vertex to the base, and the theorem of parallels to be applied.

Pupils will find it interesting to cut off the corners of a paper triangle and fit the angles together so as to make a straight angle.

This theorem furnishes an opportunity for many interesting exercises, and in particular for determining the third angle when two angles of a triangle are given, or the second acute angle of a right triangle when one acute angle is given.