Page:Mind (New Series) Volume 15.djvu/521

SYMBOLIC REASONING. 507 If we denote negative infinity by K, negative infinitesimal T)y k, and use the symbol tan x A as an abbreviation for the statement (tanA)*, and similarly for other trigonometrical ratios, we get H (^-fc); (23) tan K (| + fc) ; (24) tan" (|); cot *" (25) cot(|) ; (26) cot(^ - fc) ; (27) (28) sec" (I); (29) cos(|) ; (30) se with numberless others on the same principle of notation. 6. The following is a geometrical illustration bearing both on the ambiguity (as commonly employed) of the word infinity, and on the (to my mind inadmissible) paradox of the non-Euclidean geometry, that a point moving always in the same straight line and in the same direction may never- theless finally find itself at the point of starting. 1 Let a M P straight line of unlimited length, such as AB produced both ways indefinitely, revolve uniformly in the unscrewing direc- tion round a fixed point C, and cut a fixed straight line, also of unlimited length, at the variable point P. Let MC, the perpendicular from C upon the fixed straight line, be our linear unit. As the moving line revolves uniformly round C, the point P moves farther and farther, and with fast increasing velocity, to the right of M, while the angle PCM {or BCM) increases continuously and uniformly. Just before this angle BCM becomes a right angle (the difference being infinitesimal) the straight line MP (which also represents tan PCM, since MC is our linear unit) passes through a 1 The paradox is, of course, perfectly admissible in regard to a line that is virtually straight but not straight absolutely (see 8). Such are some of the so-called " straight lines " of the non-Euclidean geometry.