When the width of the Strip has been reduced to half-an-inch, you will grant, I suppose, that the area is exactly half what it was at first—since two such Strips, laid side by side, evidently make up the original Strip. And so, by reducing the width to a quarter-inch, &c., we obtain a number of different areas, all Infinite alike, and yet having finite ratios to one another: in fact, so long as the width continues to be a finite fraction (i. e. a fraction with a finite numerator and denominator) of an inch, the area continues to be an Infinity of the same order as the original Strip. (It seems obvious that Infinities of the same order have finite ratios to one another, and that any one of them can be so multiplied, by a finite number, as to exceed any other.) But this narrowing process may be continued until the two Lines absolutely coincide: and what then becomes of the area? It cannot be denied that it is then Zero. Now I have ventured (see p. 2) to lay it down, as an Axiom, that, when a Magnitude varies continuously, and has changed from a certain value to a certain other value, it must have passed through every intermediate value. And what intermediate values do we find between an Infinite Area and Zero?[Pg 45] Surely every Finite Area, that can be named, lies between them? The Reader can, if he likes, part company with me at this point: but to my Reason it seems absolutely clear—first, that the Strip does diminish continuously, and not 'per saltum'; and secondly, that its area has, at some time or other during the process, every conceivable finite value. At one time, for instance, it contains a square-mile: and, rather later in its career, it is reduced to a single square-inch. Let us contemplate it in this last-named condition. Its length? As Infinite, clearly, as it was at first. Its Area? One square-inch, undoubtedly. And what is its width? Can you, oh gentle Reader, find any reasonable answer to this question, except the following? "Its width is infinitely small—having, in fact, exactly the same relation to a linear inch which that linear inch has to an infinite Line." If this be so (and I see no way out of it), we have found two Magnitudes, both linear, neither of them infinite, and yet such that no finite multiple of the lesser can possibly exceed the greater. They are, in fact, not of the same order; one of them being Finite, and the other an Infinitesimal of the first order.