VOL. XVIII.] PHILOSOPHICAL TRANSACTIONS. 645 



fied this desire, till M. de Lagney evinced, by what he has done in his book, 

 that the thing might be performed still more compendiously. Now my method 

 is this : 



Let the root z of any equation be supposed to be composed of two terms 

 a + or — e, of which a is assumed as near as may be to z ; which, however, 

 is not absolutely necessary, but only convenient ; then let the quantity a + or 

 — e be raised to all the powers of z found in the given equation, and affixing 

 to each the corresponding numeral co-efficients. Then let the power to be 

 resolved be subtracted from the sum of the given parts in the first column, 

 where e is not found, called the homogeneum comparationis, and let the dif- 

 ference be + b. Next, take the sum of all the co-efficients of e in the second 

 column, which put ^ s. Lastly, collect together all the co-efficients of e in the 

 third column, the sum of which call t. Then will the near value of the root z 



be thus, viz. in the rational form, z = a -\ ^-r- — -,, and z = a + '^^■^ — \ss^ t 



~ S S ± t t 



in the irrational form ; which it may be worth while to illustrate by some exam- 

 ples. But, as a convenient help, it may not be improper to have at hand a 

 general table, exhibiting all the powers oi a ± e, which may easily be conti- 

 nued further if necessary ; which table may justly be called a Qeneral Analytical 

 Speculum. The said powers arising from the continual multiplication of a -J- e 

 = z, are as follow, with their annexed co-efficients. 



Table of Powers, 

 s 

 cz=co-)- c e t 



dz'^=^da^-\-2daeJc d e e u 



/z'=/a^-f 3/a'e-j- 3/a e e -\- f ^ w 



gz* = g a* -{■ 4 g a^ e -\- 6 g a^ e e -{- 4 g a e^ -\- g e* x 



Az* = Aa*-f- 5 A a*e-f- 10Aa^ee+ lOAa'e^ 4- bha e* -|- h e* y 

 ;i z^ = ^ o* + 6 h.a^ e -|- 15 ^ a" e e -f- 20 ^ a" e^ -f- 15 A a^ e* + 6 /i a e^ -f /i e". 



But if it should be a — e = z, the table would still be composed of the same 

 terms, but only the odd powers of e must be negative, as e, e^, e', &c. and the 

 even powers, e% e*, e°, &c. must still be positive. Also let the sura of the co- 

 efficients of the first power e be called s ; those of the square e* = < ; of the 

 cube e" = M ; of the biquadrate e* = w ; of the sursolid e^ = x ; of the sixth 

 power e" = y ; and so on. And since e is supposed to be only a small part of 

 the root required, all the powers of e will become much less than the like 

 powers of a, therefore for a first process the higher powers may be rejected, as 

 has been shown in the pure powers ; then forming a new equation, by substi- 



