(54 PHILOSOPHICAL TRANSACTIONS. [aNNO 1786. 



often converged slowly, or, if the truth may he confessed, commonly not at all, 

 to deduce the area of a curve contained hetween two values a and b of the ab- 

 sciss, or fluent of a fluxion between two values a and /; of the variable quantity 

 a:, interpolated the series or area between a and b\ that is, found the area or 

 fluent contained between the abscissae a and a -\- a, then between the abscissae 

 a -}- a and a -\- 2a, and then between the abscissae a + 2a and a -j- 3a, and so 

 on, till they came to the area between b — a and b. M. Euler observed, that 

 when the ordinate became O or infinite, the series expressing the area converges 

 slowly; and therefore, in order to investigate the area near the points of the 

 absciss where the ordinates become O or infinite, he transforms the equation, and 

 finds serieses expressing the area near those points, in which serieses the abscissae 

 or unknown quantities begin from those points. 



In the Meditationes it is asserted, that in a series proceeding according to the 

 dimensions of x^ if any root f the above-mentioned equations be situated be- 

 tween the beginning of the absciss O and its end x, the series will not converge; 

 it is therefore necessary to transform the absciss so that it may begin or end at 

 each of the roots of the above-mentioned equations, and consequently where the 

 ordinates become O or infinite, &c.; those cases excepted where the ordinate be- 

 comes O, and its correspondent abscissa is a root of a rational function w of a;- 

 without a denominator, and /wp.r is equal to the given series ; and in general 

 the abscissae ought to begin from the above-mentioned points ; for if they end 

 there, the series will converge very slow, if at all. It is further asserted, that if 

 a and /;, the values of the abscissse between which the area is required, be both 

 more near to one root of the above-mentioned equations than to any other, and 

 rz, serieses are to be found, whose sum expresses the area contained between a and 

 b\ then that the sum of the n serieses may converge the swiftest, the distances 

 of the beginnings of each of the n abscissae from the adjacent root will form a 

 geometrical progression. 



Mr. Craig found the fluent of any fluxion of the formula (a -}- bx^ -\- cx^" -j- 

 &c.)".r'^~^r by a series of the following kind (a -f bx"" -\- cx^" -f &c.)'"+^ X x^ X 

 (a 4- pa;" -f yx^" -\- &c. in infinitum). Sir Isaac Newton, by serieses of the 

 same kind, found the fluents of fluxions of this formula {a -f bx^ -j- cx'^" -\- 

 &c.y X (e+/i" + g^'"-" -\- &C.)'" X hc.x^-\v; the same principle is extended 

 somewhat more general in the Meditationes. Mr. John Bernoulli found the 

 fluent of any fluxion fnz = nz — 77^ + o T-l — ^^' f^^m the principles which 

 Mr. Craig published for finding the fluents effluxions involving fluents. In the 

 Meditationes something is added of the convergency of these series; and also, 

 in them a new method is given of finding approximations. Let some terms in 

 the given quantity be much less or greater than the rest; then reduce the quan- 

 tity into terms proceeding according to the dimensions of the small quantities, or 



