On the Principles of Motion, <^c. 301 



logic, and derogatory to the purity, and evidence of the 

 mathematics. For an illustration of our remarks, suppose 

 e to be an increment of a uniformly varying quantity x ; 

 then x-\-e will be the quantity varied by the increment, this 

 variation will be uniform in all values of x, but the variation 

 of the variables dependent on x-\-e, or what is denominated 

 the functions of this new value of x, as x-\-e n ; will not be 

 uniform, but may be easily investigated by the development 

 of x-\-e n ; for greater simplicity suppose the function to be 

 x-{-e*=x 2 -\-%xe-{-ec, the increment, or variation of this 

 from its first value, when it had no increment is 2-re-f-ee, 

 which is to the uniform increment of x, or e, as 2xe-\-ee to 

 to e, or as 2x+e to 1. 



Here the ratio of the increment of the function to its base, 

 or root, is ascertained very readily by its algebraic develop- 

 ment ; and if this were truly its differential or fluxion, there 

 would be no ground of questioning the legitimacy of the 

 logic of this science, but the objection to it rests entirely on 

 the casting away of the increment e, from the expression 

 2x-\-e of the ratio of the whole increment of the function, 

 since e, must ever constitute a part of it while it has any 

 finite value. It may be said that the ratio of %x to 1 , or what 

 is called the differential co-efficient, is independent of e, and 

 has a real value when e vanishes ; which is true, but it is then 

 at its limit and the ratio is that of the limit, and not of the 

 increment, consequently no new discovery is made by this 

 mode of conception. If the second term of the develop- 

 ment be assumed as the true differential, this will be apetitio 

 principii, or taking for granted what is to be proved. In 

 short, we perceive no logical principles in LaGrange's Ana- 

 lytical demonstrations, which are not common to the geo- 

 metrical. It must, however, be allowed, that the former af- 

 ford facilities of operation, which are peculiar to analysis, 

 but the latter are more remarkable for their clearness, and ir- 

 resistable evidence. Both should go hand in hand, as each 

 contributes to the other great advantages. A predilection 

 for analysis, which term is improperly applied to algebra 

 only, has led some modern mathematicians into extremes by 

 imagining this instrument of discovery immeasurably potent. 

 That it is very much so, will not be denied ; but the abstract 

 reasonings from symbols, do not always discover truth. 

 There must be a reference to its kindred science, or great 



