My own inclination is to believe that, during this second period of revolution, the lower part of the revolving-Line at first intersects -produced, at an infinite distance, and then ceases to intersect it. After the revolving-Line has once ceased to intersect -produced, No. 4 is of course equal to No. 1. That is to say, the surface, lying between the whole revolving-Line and , is, from that moment until the revolving-Line coincides with , of a constant area, since it is the sum-total of Nos. 1, 2, 3, which is equal to the sum-total of Nos. 2, 3, 4, i.e. to the whole Infinite-Strip. And this will continue true, while No. 1 and No. 4 dwindle down, through all finite and infinitesimal values, till they finally reach Zero. [Pg 54] The results we have arrived at will perhaps be more easily understood by examining the following Table of values. The symbols used in it are as follows:— Symbols. Meanings. a right angle. a large finite magnitude a small " " a large Infinitesimal of 1st order. " " 2nd " a large Infinity of 1st order. area of the semi-Infinite-Strip No. 2. [an Infinity of 1st order.] area of one-fourth of the Infinite-Plane. [an Infinity of 2nd order.] Position of upper part of revolving- Line. Vertical angle of Space No. 1. Areas of Spaces. No. 1. No. 2. No. 3. No. 4. No. 5. Zero " " " " " ? ? ? " " does not " " exist " " " " " " &c. &c. " " &c. " Zero. Zero. " " Zero. " [Pg 55] The sum of the whole matter appears to be this. If a Pair of Lines make, with a certain transversal, two interior angles on the same side of it together less than two right angles, then, so long as the defect is finite, there is no doubt that the Lines intersect: also, if the theory be true, that the area of an Infinite-Sector, whose vertical-angle is finite, and whose area is therefore undoubtedly an Infinity of the second order, passes, as its vertical angle dwindles to Zero, through infinite values of the first order, and then through finite values, its vertical angle meanwhile passing through infinitesimal values of the first and second order—if all this be true, it follows that, when the 'defect from two right angles' becomes an Infinitesimal of the first order, the Lines may possibly intersect, but can only do so at an infinite distance; and that, when the defect has become an Infinitesimal of the second order, the Lines have ceased to intersect. The theory, here discussed, may be bewildering; but it is at least consistent with itself: and it seems to me to be quite as credible as the theory that 'Infinitesimals' are mythical, and that a Finite Magnitude, dwindling down to Zero, continues Finite to its last gasp. [Pg 56] Another process—a negative one—has occurred to me, for disproving the absolute truth of Euclid's Axiom. It is a very simple process, and has the great recommendation of not requiring any belief in the existence of Infinitesimals.