The Greek mathematician 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 hypothenuse) would be equal [Pg 29] to . Now, is an irrational number and differs from all ordinary fractional or rational numbers. Hence, since all the 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 difficulty was to assume that the point corresponding to and in a general way points corresponding to all irrational numbers (such as , and radicals) were after all present on a continuous mathematical line. Accordingly, mathematical continuity along a line was defined by the inclusion of all numbers whether rational or irrational, and a similar procedure was followed for a mathematical continuum of any number of dimensions. In this way mathematicians obtained what is known as the Grand Continuum, or Mathematical Continuum.[4] Now, it is obvious that although the mathematical continuum is still called a continuum, it differs considerably from the popular conception of a continuum, where every element merges into its neighbour. However this may be, 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 mathematicians to obtain. The sensory continuum itself is barred from mathematical treatment owing to its inherent inconsistencies. And here an important point must be noted. In a sensory continuum considered as a chain of elements, an understanding of nextness or contiguity, hence an understanding of order, was imposed upon us by judgments of identity in our sensory perceptions. But the same no longer holds in the case of a mathematical aggregate of points, owing to the absence of that merging condition which guided us in the sensory manifolds. Theoretically, we may with equal justification order these points in whatever way we choose, and by varying the order in which we pass from point to point, we should find that the dimensionality of the aggregate varied in consequence.[5] [Pg 30] Dimensionality is thus a property of order, and order must be imposed before dimensionality can be established.