Now let us suppose that the two sides of this Infinite Sector gradually approach each other until they coincide. It cannot be denied that the area then becomes Zero. It has dwindled then, from an Infinity of the second order (when its angular magnitude was finite), down to absolute Zero. And there seems no room to doubt that it has done this continuously, and not 'per saltum.' What values, then, has it gone through on the way? Can we reasonably doubt that it has gone through, first, all Infinities of the first order (during which process its angular magnitude would be an Infinitesimal of the first order), secondly, all Finite values (its angular magnitude being an Infinitesimal of the second order), thirdly, all Infinitesimals of the first order (its angular magnitude being an Infinitesimal of the third order), and so on—the angular magnitude being always two degrees ahead of the area, in this long and fatiguing competition in the Dwindling-Race. § 4. [Pg 49] Pairs of Lines. We are now in a position to examine the phenomena of intersection, or non-intersection, with regard to a Pair of Lines, by imagining one of them to revolve about a fixed Point. Let be one of the 2 Lines: and let be, at first at right angles to it: and let it then revolve, about , so as to take the successive positions , , , its final position being at right angles to its first position, and therefore parallel to . Let us further suppose that the angle, contained between and the upper part of the revolving Line, dwindles, from a right angle, through all possible finite lesser values, while its[Pg 50] upper edge revolves from , to ; that, the moment the upper edge goes below , this angle becomes an Infinitesimal of the first degree, and so dwindles, through all such values, while its upper edge revolves from to ; that, the moment the upper edge goes below , this angle becomes an Infinitesimal of the second degree; and so on.