"The obstacle... is that geometrical magnitudes when compared together are in many cases found to be incommensurable;—that is to say, two such magnitudes may be quite incapable of a common measurement—they may be of a nature not to admit of being both measured by one and the same unit of measurement, however minute the measuring unit be taken, and consequently, both cannot be represented by numbers. I have already adverted to an instance of this kind... in the side and diagonal of a square, and to another in the diameter and circumference of a circle. ...That they are so, could not have been found out by such practical or experimental tests as those here adverted to for illustration: they are proved to be so by geometrical reasoning."