The Teaching of Geometry

by David Eugene Smith

1. In attacking either a theorem or a problem, take the most general figure possible. Thus, if a proposition relates to a quadrilateral, take one with unequal sides and unequal angles rather than a square or even a rectangle. The simpler figures often deceive a pupil into feeling that he has a proof, when in reality he has one only for a special case.

2. Set forth very exactly the thing that is given, using letters relating to the figure that has been drawn. Then set forth with the same exactness the thing that is to be proved. The neglect to do this is the cause of a large per cent of the failures. The knowing of exactly what we have to do and exactly what we have with which to do it is half the battle.

3. If the proposition seems hazy, the difficulty is probably with the wording. In this case try substituting the definition for the name of the thing defined. Thus instead of thinking too long about proving that the median to the base of an isosceles triangle is perpendicular to the base, draw the figure and think that there is given

AC = BC,
AD = BD,

and that there is to be proved that

∠CDA = ∠BDC.

Here we have replaced "median," "isosceles," and "perpendicular" by statements that express the same idea in simpler language.

Bibliography. Petersen, Methods and Theories for the Solution of Geometric Problems of Construction, Copenhagen, 1879, a curious piece of English and an extreme view of the subject, but well worth consulting; Alexandroff, Problèmes de géométrie élémentaire, Paris, 1899, with a German translation in 1903; Loomis, Original Investigation; or, How to attack an Exercise in Geometry, Boston, 1901; Sauvage, Les Lieux géométriques en géométrie élémentaire, Paris, 1893; Hadamard, Leçons de géométrie, p. 261, Paris, 1898; Duhamel, Des Méthodes dans les sciences de raisonnement, 3^e éd., Paris, 1885; Henrici and Treutlein, Lehrbuch der Elementar-Geometrie, Leipzig, 3. Aufl., 1897; Henrici, Congruent Figures, London, 1879.