446 Mr. Graves's Explanation of a remarkable Paradox 



supposition which it is necessary to make if we would prove» 

 c- = 1), without making vj; itself change the form it had when 



\J/ X was equal to c vj/ — ; in other words, (1.) and (2.) are es- 

 sentially non-simultaneous equations in the illustrative in- 

 stances before us, for the ;:■ of x, of whatever sign it be or be 

 considered to be, though such z be infinitesimal or zero, as in 

 the case oi x real, is always or must always be considered to 



he. of a different sign from the ^ of — . When ^ x = c 



equation (1.) will not be satisfied for all real quantitative va- 

 lues of a:, unless zero be considered positive, nor again when 



^ x = c "^ % unless zero be considered negative. Vice versa 

 with respect to equation (2.). One supposition excludes the 

 other : zero may be considered either positive or negative, but 

 not both together. Hence, even in the case of x real, where 

 the solutions would appear on first view to be concurrent, they 

 are, in truth, alternative. We are bound to consider x the 

 same in state as well as quaiitity on both sides of the equa- 

 tions (1.) and (2.), and here obscurity arises from the symbols 

 of algebra not expressing to the eye a difference of state be- 

 tween reals having the same quantity. Such difference of 

 state in things denoted by algebraic symbols is in most cases 

 immaterial, unless no quantity remain in either of their con- 

 stituents; but we know that it is of importance in the case of 

 vanishing fractions, and we perceive that it may become so in 

 certain other fine circumstances, such as those which we have 

 just discussed. 



We have shown therefore by a particular example (or ra- 

 ther by two correlated examples) that the paradox noticed by 

 Mr. Babbage is only a remarkably subtle instance of the fol- 

 lowing general proposition which is not a priori improbable. 

 Though we may prove it to be impossible to find one Jixed 

 form 4/, such that the equation -i^ x ■= FiJ/a.r(F and « being 

 given functions) shall hold good simultaneously in different 

 cases where particular values of x are assumed (the term 

 " value" including state as well as quantity), we are not there- 

 fore to despair of finding distinct forms of xj/, absolute or al- 

 ternative, which for certain values of x, within appropriate 

 limits, shall severally satisfy the equation \{/ ar = F ^^ a .r. Such 

 a partial form of 4/ j: and the corresponding partial form of 

 FvJ/ ax taken with it may be likened to two curves which co- 



