ABSURDUM, a term made use of by mathematicians when they demonstrate any truth, by showing that its contrary is impossible, or involves an absurdity. Thus, Euclid demonstrates the truth of the fourth proposition of the first book of his Elements, by showing that its contrary implies this obvious absurdity—"that two straight lines may inclose a space."
This mode of demonstration is called reductio ad absurdum, and is every whit as conclusive as the direct method; because the contrary of every falsehood must be truth, and of every truth, falsehood.
The young geometrician, however, does not, we believe, feel himself so perfectly satisfied with a demonstration of this kind, as with those which, proceeding from a few self-evident truths, conducts him directly, by necessary consequences, to the truth of the proposition to be proved. The reason is, that he has not yet learned to distinguish accurately between the words false and impossible, different and contrary. Many different assertions may be made relating to the same thing, and yet be all true or all false; but it is impossible to make two assertions directly contrary to each other, of which the one shall not be true and the other false. Thus, "snow is white," "snow is cold," are different assertions relating to the same thing, and both true; as, "snow is black," "snow is red," are both false: but let it be remembered, that of the first and second, and of the third and fourth of these assertions, neither is directly contrary to the other; nor is any one of them, abstractly considered, impossible, or such as a blind man, who had never felt nor heard of snow, might not believe upon ordinary testimony. But were all the men in Europe to tell a native of the interior parts of Africa that snow is a thing at once white and not white, cold and not cold, the woolly-headed savage would know as well as the most sagacious philosopher, that of these contrary assertions the one must be true and the other must be false. Just so it is with respect to Euclid's fourth proposition. Had he proved its truth by showing that its contrary involves this proposition, that "the diagonal of a square is commensurate with its side," the skilful geometrician would indeed have admitted the demonstration, because
he knows well that the diagonal of a square is not commensurate with its side; but the tyro in geometry would have been no wiser than before. He knew from the beginning, that the proposition and its contrary cannot both be true; but which of them is true, and which false, such a demonstration could not have taught him, because he is ignorant of the incommensurability of the diagonal and side of a square. No man, however, is ignorant, that two straight lines cannot inclose a space; and since Euclid shows that the contrary of his proposition implies this absurdity, no man of common sense can entertain a doubt but that the proposition itself must be true.