Next, as to Nos. 4, 5. (I take these next, because they will help us to investigate the properties of No. 3.) It is evident that, so long as No. 5 continues to exist, the two together constitute—and that, when No. 5 has ceased to exist, No. 4 by itself constitutes—an Infinite-Sector, which is the exact counterpart of No. 1—the two having what Euclid calls 'opposite vertical angles.' Hence, while the lower part of the revolving-Line ranges between and [Pg 52], the area of No. 4 continues to be an Infinity of the second order, and entirely declines to be 'cribb'd, cabin'd, and confined' within such narrow limits as our Infinite-Strip! Hence the revolving-Line continues, all this time, to intersect -produced. (N.B. Here we have a proof of the truth of Euclid's Axiom, when amended, as I have done at p. 28, by inserting the words 'the defect being a finite angle.') Let us now, reserving for future consideration the phenomena of the period while the upper part of the revolving-Line is crossing from to , suppose it to have passed , so that the angle, which it makes with , has become an Infinitesimal of the second order. Will its lower part continue to intersect -produced? It may easily be shown, by a reductio ad absurdum, that it will not: for, if it did, Nos. 1, 2, 3 would then make up an Infinite Sector, whose vertical angle would be an Infinitesimal of the second order, and whose area would therefore be finite. But the area of No. 2 is always an Infinity of the first order. Which is absurd. Hence, after the upper part of the revolving-Line has passed , its lower part does not intersect -produced. We have now to answer a far more puzzling question, namely, what happens while the upper part of the revolving-Line is crossing from to ? Does its lower part intersect [Pg 53]-produced, all the while? Or does it fall clear of it, all the while? Or does it at first intersect it, and afterwards cease to do so? First, suppose it to intersect -produced. In this case No. 3 is clearly a closed Triangle, whose vertical-angle is an Infinitesimal of the first order. This looks as if its area must also be an Infinitesimal of the first order: but this, we know, cannot be, since it contains within it the finite area possessed by No. 3 while the upper part of the revolving-Line was passing from to . Hence its area must be, at least, finite. The only way I can see out of this difficulty is to assume that the sides of this Triangle have become infinite; i.e. that the revolving-Line intersects -produced at an infinite distance. Next, suppose it to fall clear of -produced. In this case No. 3 is a semi-Infinite-Strip, but we cannot be certain that its area is, like such Strips when of a uniform width, infinite; for its width dwindles so much towards the right-hand that it may possibly be, in the early part of the period, finite. I see no way of settling this question; but, luckily, it is not relevant to the question of intersection.