484 Mr. Moou on the Symbols sin co and cos oo, 



been clearly resolved ; and if we take up with the second, we 

 have the dissatisfaction of resting in a conclusion manifestly 

 opposed to one's common sense notions, which it must be re- 

 membered may indeed fail to conduct us to the truth, but can- 

 not mislead us. 



A full investigation of the subject is therefore necessary ; 

 and if there be some to whom this discussion may appear of 

 obscure interest, I would observe that we shall be led by it 

 not merely to settle the point immediately at issue, but to ex- 

 pose the fallacy of principles and results now universally re- 

 ceived and occasionally of the utmost importance, and to give 

 to a great branch of mathematical inquiry a degree of truth 

 and certainty which it has never hitherto possessed. 



The conclusion that sin cd and cos cx> represent indefinite 

 quantities, is so obvious from the simple consideration of the 

 symbols themselves, and is apparently so incontrovertible, that 

 it is difficult to conceive how any person could have been 

 brought to believe the contrary ; yet of all the writers who 

 liave held the opposite opinion, Prof. De Morgan is the only 

 one who has seen the propriety of attempting to obviate the 

 force of the reasoning upon which the rival theory is grounded, 

 though till that is done no other can be said to be established. 

 I shall therefore give Mr. De Morgan's aigument upon this 

 point in his own words : — 



" The continental mathematicians with one voice pronounce 

 these symbols (sin co and cosoc) to be indeterminate in value, 

 which is strictly true as far as a priori considerations are con- 

 cerned ; for a periodic function of A' cannot be said to be in one 

 part of its period rather than another, when x is infinite. If, 

 however, we assume ^ ix) to stand for x terms of the series 

 1 — 1 + 1 — 14- &C.5 we might equally conclude that <$> [x] is in- 

 determinate when X is infinite, no reason existing to prefer 

 to 1 or 1 to 0; nevertheless there exists no doubt that this 

 series represents half a unit" (vide Library of Useful Know- 

 ledge, The Differential and Integral Calculus, p. 640). 



If the true nature of the above series had been distinctly 

 recognised, we should not now have to discuss the meaning of 

 sin 00 and cos 00. The true value of that series continued to 

 infinity is not half a unit, but is either 1 or indifferently. 



If we put S = 1 — ] + 1 — 1 -f &c. in inf., 



we have S= 1 - 1 + 1 — &c. ...... 



and .-. 2 S = 1. 



