SYMBOLIC REASONING. 509 point P to the right of M, and a real infinite distance MP. If, on the contrary, it revolves from its parallel position through an infinitesimal angle in the unscrewing direction, we get the point Q to the left of M, and a real infinite negative distance MQ. 8. It would conduce to logical accuracy in dealing with these questions if we introduced the term virtually into our reasoning and defined it as follows. Two straight lines are said to be virtually parallel l when they meet at some real infinite distance, H or - H. Suppose, for example, we have three straight lines, A, B, C, and that A meets B at an in- finite distance H lf and meets C at another but still infinite distance H 2 . Here we may accurately assert that A, B, C are virtually parallel, for, by our very definition of the words infinite and infinitesimal, it follows that the error of deviation from the parallel position, though theoretically real, must for ever remain too small for the most perfect instrument to detect, and for the most powerful notation accurately or approximately to express in finite terms. In like manner, if we suppose A and B to be two ratios, finite, infinite or infinitesimal, the statement that "A and B are virtually* equal " means = 1 h, in which h denotes any infinites- B imal ratio, as defined in 4. In the infinitesimal calculus (including the differential and integral) an equation (A = B) often asserts virtual and not real equality ; but logical accuracy may be secured by the tacit convention that when- ever we have the statement (A = B), it is to be understood as asserting that "A is either absolutely or virtually equal to B ". Let A, B, C be three points on the surface of a sphere of radius K, forming the spherical triangle ABC. Whether B be finite, infinite, or infinitesimal, as defined in 3, 4 A B Bf C 1 A if the ratios 15 -, -', - - be all three infinitesimal, the sum JrC Jtv -tvi of the three angles A, B, C is virtually but not really equal to two right angles. If the three points A, B, C be on a real plane surface, then the sum of the three angles A, B, C is really equal to two right angles. Again, a finite section AB of a curve may be called virtually straight when at every 1 Similarly two straight lines are really parallel when their (non-exist- ent) point of intersection is at some pseudo-infinite distance, such as 1 2, oo o' or 5 or 3 2 For example, the statement ^1 + - + ^ + !+ . . . +^=2 asserts virtual and not absolute equality. 34
