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Published online by Cambridge University Press: 03 November 2016
One service which Euclid rendered to the school-teacher in the days when his supremacy was unquestioned was, that he set a logical standard.
Euclid’s logic has been attacked by many writers, and by none more severely than by a philosopher whom I have worshipped this side idolatry as fervently as any But I doubt whether even Russell, in his less controversial moods, would deny that broadly speaking Euclid pitches the standard reasonably When he is not proving the obvious, what he takes for granted is usually what we should take for granted. This is the more interesting because it is to some extent accidental. Since Euclid wrote down such axioms as “ Things which are equal to the same thing are equal to one another ” and “ Magnitudes which coincide are equal ” he seems to have thought that he had made explicit every assumption involred in his work. We know that this opinion was mistaken, but we know also that the ideal of exposing every assumption is not appropriate to school geometry—I would go so far as to say not appropriate to any subject except that which it dominates, namely, formal logic as expounded by Peano and in Principia Mathematica.
See Gazette, Vol. XVII., p. 307
page 24 note * This warning, sounded by Mr. Goodwill at the Annual Meeting in 1924, was incorporated in the Preface to later editions of the Report on the Teaching of Geometry.
page 29 note * This too was said to Dodpson, who missed the point childishly, and protested that as he had nowhere equated the angles of different tetragons, the question of whether tetragons of different sizes are similar was irrelevant. But his figure consists not of a tetragon alone, but of tetragon and circle. The question is : Why is his axiom plausible ? And the answer : Because it is true for figures sufficiently small.