I must ask the Reader's pardon for this long digression: but the fallacy, which (as I believe) lies at the root of the Direction-Theory, is a very subtle one, and has cost me a great many hours of hard thinking to un-earth it. Yet another process has been invented—quite fascinating in its brevity and its elegance—which, though involving the same fallacy as the Direction-Theory, proves Euc. I. 32 without even mentioning the dangerous word 'Direction.' We are told to take any Triangle ; to produce to ; to make part of , viz. , revolve, about , into the position ; then to make part of this Line, viz. , revolve, about , into the position ; and lastly to make part of this Line, viz. , revolve, about , till it lies along , of which it originally formed a part. We are then assured that it must have revolved through 4 right angles: from which it easily follows that the interior angles of the Triangle are together equal to 2 right angles. [Pg 71] The disproof of this fallacy is almost as brief and elegant as the fallacy itself. We first quote the general principle that we cannot reasonably be told to make a Line fulfil two conditions, either of which is enough by itself to fix its position: e.g. given 3 Points, , , , we cannot reasonably be told to draw a Line, from , which shall pass through and : we can make it pass through , but it must then take its chance of passing through ; and vice versâ. Now let us suppose that, while one part of , viz. , revolves into the position , another little bit of it, viz. , revolves, through an equal angle, into the position ; and that, while revolves into the position of lying along , revolves—and here comes the fallacy. You must not say "revolves, through an equal angle, into the position of lying along ," for this would be to make fulfil two conditions at once. If you say that the one condition involves the other, you are virtually asserting that the Lines , are equally inclined to —and this in consequence of having been so drawn that these same Lines are equally inclined to . That is, you are asserting § 2. (2). (See p. 63.) One other proof—a very beautiful one, though largely dealing with Infinities and Infinitesimals—may here be mentioned, that of M. Bertrand. It rests on the principle that an infinite Sector (with a vertical angle which has a finite ratio to a right angle) is an Infinity of the same order as an infinite Plane, whereas an infinite Strip (i. e. the area contained between 2 'separational' Lines) is an Infinity of a lower order. Hence he concludes that no such Sector, however small its vertical angle, will lie wholly between 2 such 'separational' Lines, however far apart: hence, if a Line intersect one of 2 'separational' Lines, it must, if produced, intersect the other. He thus proves the Theorem numbered (8) in § 4: but his results are, of course, only partially, and not absolutely, true. [Pg 72]