Elementary geometry is commonly taught in the schools for two main purposes: to instil a knowledge of the geometrical figures (triangle, rectangle, circle) met in common experience, and also to develop their properties by logical argument proceeding step by step from the most primitive conceptions. The supreme exponent of the subject is Euclid, whose authority remained almost unchallenged until very recent times.

It is probable that Euclid's own system of geometry is not now used in many schools, but children studying geometry become familiar with a number of the standard theorems and with the proofs of several of them. The general idea of a geometrical proof, if not of the details, will thus be familiar to anyone likely to read this book. We give in illustration a typical example, which we shall, in fact, find it necessary to criticise later. The proof will first be stated in standard form, and the nature of the geometrical arguments leading towards it will then be discussed.

To prove that the exterior angle of a triangle is greater than the interior opposite angles.

GIVEN: A triangle ABC whose side BC is produced beyond C to P (Fig. 6).

REQUIRED: To prove that the angle PCA is greater than the angle BAC.

CONSTRUCTION: Let O be the middle point of AC, and produce BO beyond O to D so that OD = BO.