"Between any two points on a line in our continuum, however close they may be, we have... interposed an indefinite number of rational fractions defining points; yet, despite this fact, we have by no means eliminated gaps between the various points along our line. Pythagoras was the first to draw attention to this deficiency after studying certain geometrical constructions. He remarked, for instance, that if we considered a square whose sides were of unit length, the diagonal of the square (as a result of his famous geometrical theorem of the square of the hypotenuse) would be equal to √2. Now √2 is an irrational number and differs from all ordinary fractional or rational numbers. Hence, since all points of a line would correspond to rational or ordinary fractional numbers, it was obvious that the opposite corner of the square would define a point which did not belong to the diagonal. In other words, the sides of the square meeting at the opposite corner to that whence the diagonal had been drawn, would not intersect the diagonal; and we should be faced with the conclusion that two continuous lines could cross one another in a plane and yet have no point in common. The only way to remedy this situation was to assume that the point corresponding to √2 and in a general way points corresponding to all irrational numbers (such as π, e and radicals) were after all present on a continuous mathematical line. ...the mathematical continuum, and with it mathematical continuity, are as near an approach to the sensory continuum and to sensory continuity as it is possible for the mathematician to obtain. The sensory continuum itself is barred from mathematical treatment owing to its inherent inconsistencies."