Reformed Logic / A System Based on Berkeley's Philosophy with an Entirely New Method of Dialectic

by D. B. McLachlan

XXXIV—GEOMETRICAL DEMONSTRATION

This exercise is regarded by logicians as one of the purest forms of argument. It is nothing more than an aid to a certain kind of perception.

Take, for instance, the fifth proposition of the first book of Euclid—'The angles at the base of an isosceles triangle are equal, and if the equal sides be produced the angles on the other side shall also be equal.' The proposition is accompanied by a diagram of an isosceles triangle with the equal sides already produced, so that the conditional phrasing of the proposition does not mean that the production of the sides, and what results therefrom, are future or possible events which neither Euclid nor anybody else has yet experienced, and the probability of which is an argumentative conclusion.

What the proposition means is this: an isosceles triangle of which the equal sides have been produced, has equal angles on the same side of the base both within and without the triangle. It is an affirmation of what is, not of what we must believe to be for reasons to be given.

The truth of the proposition is seen at once from simple inspection of the diagram. It is an association of properties related in a certain manner. It has many relations which the geometer does not mention in this proposition, but those which he mentions are seen to be correctly described as soon as we direct attention to them. If we have any doubt on the subject we remove it by measuring the angles.

Euclid however does not appeal to the powers of inspection we can exercise in this case, and he ignores our facilities for measurement. He appeals to simpler and easier kinds of perception expressed in his axioms, which he began by assuming we were capable of exercising without demonstration. They constitute what he considers the minimum power of relational perception, which if a man have not he cannot be taught geometry. Euclid also in this proposition refers to the result of a prior demonstration, the relation in which he supposes we have seized. By means of these antecedents he prompts our perceptive faculty up to the point of seeing the relations expressed in this proposition. If we saw them without the prompting, the latter is superfluous; if the relations do not stand the test of measurement, the prompting goes for nothing.

All Euclid's demonstrations are of this sort. They are pointings-out of what can be seen by inspection and sufficient attention. He is not bringing a case under a precedent—he is describing relations in things, that may serve as precedents in concrete or applied geometry. The service he performs is that of a connoisseur who points out the beauties of a picture or landscape to a careless or uninterested spectator. Relations are sometimes difficult to see—much more difficult than colours or masses—and there is a legitimate sphere of usefulness for people who point out what others are apt to overlook. There is no prediction in this. We are not asked to conceive anything that is not before us. Geometrical demonstration thus assists perception, but does not imply reasoning. Euclid does not argue—he prompts.

Those who maintain that Euclid is syllogistic do so on the ground that the axioms are generalisations, and that as often as one is cited there occurs the subsumption of an object under a class-notion. That would not be argument; but let us suppose it means bringing a case under a precedent. Then if the axioms be precedents and the demonstration an application of them to new cases, the theorem is a fallacy—a useless argument written to prove a foregone certainty, for the conclusion can be and generally is perfectly known without reference to the demonstration.

It appears to me more true to regard the axioms as the simplest relations, which everybody may be supposed capable of perceiving, and that geometrical demonstration consists in showing that other relations not so apparent are really varieties or combinations of the simpler relations. By using in concert with the axioms the relations already demonstrated, we are enabled to grasp relations that would not have been at all obvious on first beginning the geometrical praxis. Euclid's geometry is thus a series of graduated lessons in a special sort of observation, not a system of deductive arguments.