30 DISSERTATION SECOND. [paktu. 



The second case of this first division is, when the given 

 function is a fraction having a binomial or multinomial de- 

 nominator, the terms of which contain any powers whatever 

 of the variable magnitude, but without involving the radical 

 sign. If the denominator contain only the simple power of 

 the variable quantity, the integral is easily found by loga- 

 rithms ; if it be complex, it must be resolved either into sim- 

 ple or quadratic divisors, which, granting the solution of 

 equations, is always possible, at least by approximation, and 

 the given fraction is then found equal to an aggregate of sim- 

 ple fractions, having these divisors for their denominators, 

 and of which the fluents can always be exhibited in algebraic 

 terms, or in terms of logarithms and circular arches. This 

 very general and important problem was resolved by J. Ber- 

 noulli as early as the year 1 702. 



The denominator is in this last case supposed rational ; but 

 if it be irrational, the integration requires other means to be 

 employed. Here Leibnitz and Bernoulli both taught, how, 

 by substitutions, as in Diophantine problems, the irrationality 

 might be removed, and the integration of course reduced to 

 the former case. Newton employed a different method, and, 

 in his Quadrature of Curves, found the fluents, by comparing 

 the given fluxion with the formulas immediately derived from 

 the expression of circular or hyperbolic areas. The integra- 

 tions of these irrational formula?, whichever of the methods 

 be employed, often admit of being effected with singular 

 elegance and simplicity ; but a general integration of all the 

 formulae of this kind, except by approximation, is not yet 

 within the power of analysis. 



The second general division of the problem of integration, 

 viz. when the two variable quantities and their differentials 

 arc mixed together on each side of the equation, is a more 

 dillicult subject of inquiry than the preceding. It may in- 

 deed happen, that an equation, which at first presents itself 



