VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 453 



or ], 000,000,000,000th, root, and multiply the said root by b, or 1,000,000, 

 000,000; and lastly, from the product we must subtract b, or 1,000,000,000,000: 

 and the remainder thus obtained will be nearly equal to the proposed infinite series 

 X + -^r.r + ^x^ + ^^ + ijT^ + &c. Q. E. I. 



As an example of this method of finding the value of the series x + -^x- + 

 \x^ + \x* + \x^ + &c. Mr. M. supposes x to be equal to -^•, whence 1 — x =■ 



-jL-, and— J- = 10. In this application he finds the value of the said series to 

 be nearly equal to 2.302585093, the hyperbolic logarithm of the number 10. 



XL 11. A Method of Extending Cardans Rule for resolving one Case of a 

 Cubic Equation of this Form, o:^ >fc — qx =. r, to the other Case of thfi same 

 Equation, tvhich it is not naturally fitted to solve, and which is therefore often 

 called the Irreducible Case. By Francis Maseres, Esq., F. R. S., &c. p. 902. 



It is well known that Cardan's rule, for resolving the cubic equation x^ — qx 

 = r, is only fitted to resolve it when -|-/'^ is equal to, or greater than, -^'^q^ ; and 

 that it is of no use in the resolution of the other case of this equation; for in 

 this case J-r'- — J4-9' becomes a negative quantity, and consequently its square 

 root becomes impossible, and the expression given by Cardan's rule for the value 

 of X involves in it the impossible quantity v* (ir^ — -aV?^)> ^nd therefore is un- 

 intelligible and useless. 



Yet it is possible, by the help of Sir Isaac Newton's binomial theorem, to 

 extend this rule to this latter case, in which ^r'^ is less than -^q^, and which it is 

 not of itself fitted to resolve; or, to speak with more accuracy, it is possible to 

 derive from the expression of the value of x given by Cardan's rule for the reso- 

 lution of the equation x^ — qx ^ r in the first case, in which J r- is greater than 

 ■^'-q^, another expression somewhat diiferent from the former, that shall exhibit 

 the true value of x in the 2d case, in which 4-r'^ is less than -aV?^' provided it be 

 not less than -rr?^; and this without any mention of either impossible, or nega- 

 tive, quantities. To show how this may be effected, is the design of this paper. 



Mr. M. then sets forth, at great length, the whole process of finding out 

 Cardan's rule; which, being given in most books on algebra, may well be 

 omitted in this place. This process brings out, for one form of Cardan's rule, 

 x=■^[■kr+^/ (-fr^- - Vt?')] + ^ [i'" - ^ (i'' - ^q')], or x = 4/[a + 

 \/(a* — b^)'] + -^[a — V{d- — A^)], where a = ^r, and b = ±-q. Or, if we 

 put V {a- — b^) = c, then that rule becomes j; = ■^{a + c) -\- ^ {a — c). Now 

 by expanding the two quantities 4^ {a -\- c) and ^ {a — c), or the equal forms 

 (a -|- c)^ and (a — c)j, into infinite series, by Newton's binomial theorem, and 

 adding the two series together, the value of x will be expressed in one series only. 

 And because the two quantities a -{- c and a — c have the same terms a and c, 

 but differing only in the sign of the latter, the terms of both the two series will 



