must conform, namely, that "a thing is what it is!" Therefore the proposition "two straight lines can enclose a space" being in this way convicted of absurdity, its opposite is established as a necessary truth. Such is an illustration of the manner in which the law of contradiction has to be applied. It has usually been regarded merely as an example of necessary truth. These remarks may serve to explain not only how it is an instance, but (what is of far more importance) how it is the criterion of necessary truth.
The law of contradiction is the immediate test of all necessary truths, even of the conclusion of the longest demonstration in Euclid: nevertheless, demonstration cannot be dispensed with; for this law is their immediate test only when every previous step in the demonstration has been immediately tested by the same criterion. Of course the first principle or starting-point (or points, if there are more than one) not only may, but must, be known without demonstration. But "a felt necessity of believing them" is not their immediate test, and therefore they do not stand out of all relation to each other, in so far, that is, as their reasoned exhibition is concerned, and, of course, it is only in this respect that they stand related. Who would maintain that there was any "felt necessity of believing" the 47th proposition of the first book of Euclid? The law of contradiction is its test, but it
