Let us imagine an Infinite Plane, placed upright, and facing us—as if we were standing in front of a wall, which extended to infinity, upwards, downwards, to the right-hand, and to the left. If we divide this Plane by a single horizontal straight Line, extended to infinity to the right-hand and to the left, we get half of the whole Plane above the Line, and half below it, wherever we choose to place the Line. This may be deduced from the logical principle that we have no reason for believing either to be greater than the other; or we may adopt M. Bertrand's Axiom, "Two spaces, whether finite or infinite, are equal, when one can be placed upon the other so that any point whatsoever of either coincides with a point of the other"—a condition which seems, theoretically, readily attainable, by making one portion of the Plane revolve round the boundary-line, as a hinge, till it coincides with the other portion. Each portion is, of course, an Infinity of the same order as the whole Plane. Now let us imagine two such infinite horizontal straight Lines, 'separational' from each other (i. e. never intersecting), placed at a finite distance apart, and therefore having between them a Strip, finite in width, infinite in length, and therefore infinite in area. Now it clearly is not possible, by multiplying this Infinite Strip by any finite number, however great, to make it exceed, or even equal, the whole Infinite-Plane. Here again, then, Euclid's test fails, and neither of these Magnitudes, though they are homogeneous can be said to have a 'Ratio' to the other.[Pg 44] In fact, the Infinite-Strip is an Infinity of a lower order than the Infinite-Plane. Comparing this Strip with a single square-inch, you will, I fancy, be willing to grant at once that no multiple of the latter can possibly reach—much less exceed—the former. We have, in fact, established the existence of three kinds of Area, viz. Finite Areas (e.g. a square inch), Infinities of the first order (e.g. the Infinite-Strip we have been considering), and Infinities of the second order (e.g. the upper half of the whole Infinite-Plane). Now let us go a step further. Let the two sides of this Strip be supposed to gradually approach each other—still maintaining their 'separational' character—and let us consider the effect of this process on the intervening area.