PROP. III.————
which could not be known—a supposition which is contradictory and absurd.
Distinction between ignorance and nescience of the opposites of necessary truth.3. Ignorance, properly so called—that is, the ignorance which is a defect—must not be confounded with a nescience of the opposites of the necessary truths of reason; in other words, with a nescience of that which it would contradict the nature of all intelligence to know. Such nescience is no defect or imperfection—it is, on the contrary, the very strength or perfection of reason; and therefore such nescience is not to be regarded as ignorance. This simple but very important distinction must be explained and illustrated, for it is one which is very apt to be lost sight of; or confounded; indeed, it has been altogether overlooked until now.
There can be no ignorance of the opposite of the geometrical axioms.4. When boys at school are taught Euclid, they learn that "the enclosure of space by two straight lines" is what cannot be known,—that "if equals be added to equals the wholes are unequal" is what cannot be known,—that "a part is greater than the whole" is what cannot be known, and so forth; but they do not learn that they are equally incapable of being ignorant of such matters. It is not necessary to apprise them of this in order to carry them forward in the study of mathematics. Nothing in geometry depends on the circumstance that we cannot be ignorant of what is deponed to
