But is not the gentle Reader saying to himself, all this time, "I cannot believe in the existence of a Strip, Infinite in length, and yet Finite in Area! No doubt, if such a Strip could exist, its width must be what you call 'Infinitesimal'; since a Finite width must give an Infinite area. But I don't believe in the existence of an Infinitesimal width! My belief is that, if you make the edges of the Infinite-Strip gradually approach till they coincide, its width will continue Finite till the last moment, and will then suddenly become Zero; and that its area will continue an Infinity of the first order till the last moment, and will then suddenly become Zero." [Pg 46] Very good. My gentle Reader has formulated his views, very clearly and definitely. Permit me now to offer to his consideration a Strip, which I will prove to be at once Infinite in length and Finite in area. Let , , be rectangular Axes of Reference; and let us trace the Curve , where = unit-line. Hence, when , ; when , ; when , ; when , ; and so on. Hence the Curve is . Also, however large becomes, can never be Zero; i. e. the Strip, intercepted between the Curve and the -Axis, and bounded at the left-hand end by , is Infinite in length. And now let us estimate its area. Its first portion, , is less than the rectangle ; its second portion is less than ; and so on. Hence its area is less than ; i. e., is less than + &c. for ever; therefore, a fortiori, it is less than 2. Now an area, which is less than '2' is surely Finite? Does the gentle Reader see any escape from admitting this? And, if he admits this, does he still maintain that the width of this Infinite-Strip (which obviously dwindles, as you go along the Strip, but never becomes Zero) never ceases to be Finite? Yet surely a Strip, Infinite in length, and nowhere less than Finite in width, must be Infinite in area? In brief, I place before my gentle Reader that savouriest of Logical dishes, a Trilemma! Either he must assert that a Strip, Infinite in length, and nowhere less than Finite in width, is only Finite in area; or he must assert that the length of this Strip is Finite, i.e. he must assert that the Curve meets the x-Axis; or else he must admit that its width ceases to be Finite[Pg 47] without becoming Zero, i.e. he must admit that its width becomes Infinitesimal! Let him take his choice, and help himself. 'May good digestion wait on appetite, And health on both!' Now let us cut off, from the infinitely-long Strip named in p. 45, whose area is a square-inch, a piece just an inch long. What will its area be? It is evident that no finite multiple of this short Strip can ever make up the infinitely-long Strip; that is, no finite multiple of its area can make up a square-inch. Hence its area must be an Infinitesimal of the first order.