The Teaching of Geometry

by David Eugene Smith

3. To describe a circle with any center and radius.

4. That all right angles are equal to one another.

While this postulate asserts the essential truth that a right angle is a determinate magnitude so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration. If the statement is to be proved, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the invariability of figures, which would have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal; and hence his postulate must be taken as equivalent to the principle of invariability of figures, or its equivalent, the homogeneity of space.[48]

It is better educational policy, however, to assert this fact more definitely, and to state the additional assumption that figures may be moved about in space without deformation. The fourth of Euclid's postulates is often given as an axiom, following the idea of the Greek philosopher Geminus (who flourished in the first century B.C.), but this is because Euclid's distinction between

axiom and postulate is not always understood. Proclus (410-485 A.D.) endeavored to prove the postulate, and a later and more scientific effort was made by the Italian geometrician Saccheri (1667-1733). It is very commonly taken as a postulate that all straight angles are equal, this being more evident to the senses, and the equality of right angles is deduced as a corollary. This method of procedure has the sanction of many of our best modern scholars.

5. That, if a straight line falling on two straight lines make the interior angle on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

This famous postulate, long since abandoned in teaching the beginner in geometry, is a remarkable evidence of the clear vision of Euclid. For two thousand years mathematicians sought to prove it, only to demonstrate the wisdom of its author in placing it among the assumptions.[49] Every proof adduced contains some assumption that practically conceals the postulate itself. Thus the great English mathematician John Wallis (1616-1703) gave a proof based upon the assumption that "given a figure, another figure is possible which is similar to the given one, and of any size whatever." Legendre (1752-1833) did substantially the same at one time, and offered several other proofs, each depending upon some equally unprovable assumption. The definite proof that the postulate cannot be demonstrated is due to the Italian Beltrami (1868).

Of the alternative forms of the postulate, that of Proclus is generally considered the best suited to beginners. As stated by Playfair (1795), this is, "Through a given point only one parallel can be drawn to a given straight line"; and as stated by Proclus, "If a straight line intersect one of two parallels, it will intersect the other also." Playfair's form is now the common "postulate of parallels," and is the one that seems destined to endure.