ELISHA MITCHELL SCIENTIFIC SOCIETY. II 



indefinitely, that y increases indefinitely, and this is the 



a 

 meaning of the notation — =00, which has no sense by 



o 

 itself. Although the limit of x above is zero, the limit 

 of y is not infinity, since \{ y had a limit it could be made 

 to differ from it by as small a quantity as we wish, whereas 

 any finite quantity {y) will always differ from infinity by 

 infinity. 



Thus the principle of limits, "if two variables are equal 

 and each approaches a limit^ their limits are equal," does 

 not apply, as both variables do not approach limits. 



Therefore the singular forms mentioned must always be 

 regarded as abbreviations, having the meaning attributed 

 to them above and not as meaning anything in themselves. 

 We have an illustration of such forms in trigonometry. 

 Thus, 



sin X 



tan .r= 



cos X 



As X approaches 90°, sin x approaches i, cos .r, 6>, and 

 the left member, though always finite, increases indefi- 

 nitely. The latter is said to be infinite for x = 90°, 

 though strictly, according to the usual definition, there is 

 no tangent of 90°, as the moving radius produced, being 

 parallel to the tangent, can never intersect it. As parallel 

 lines are everywhere the same distance apart, they cannot 

 meet, however far produced, so that the statement that two 

 parallel lines meet at infinity is essentially false. 



Similarly we can reason for all the functions that increase 

 indefinitely, without ever ceasing to be finite, where the 

 angle approaches some limit, or fixed value it can never 

 attain, with any meaning corresponding to the functions. 

 The above is still more evident when we regard the ratio 

 definitions first given in trigonometry, for then, there can 

 be no function without a right triangle can be formed and 



