160 A “Treatise of Algebra.'

All which will appear evident from the Theoremin p. 40, t 3 V. from whence these Equations, or The-orems may be continued at pleasure; the Value heretaken being nothing more than the two first Terms ofthe Quantity thence arising. But now to apply thisto the Purpose above ment ioned, let there be giventhe Equation 4/1 x z -j- /2 -4- x z -f- /3 -f- x z == 10,to find the Root of it; then, since it appears that * musthere be nearly equal to 3, let 3 -j- e he substituted for asin the foregoing Examples, rejecting all the Powers of ;,above the first, as inconside rable, an d th en the g ivenEquation will stand thus, /10 -j- 6; -f- /ii 6 e -j-/iz -f- 6; z= 10 : But, by Theorem 2 0 , /io -j- 6 e

Will be = -^ I °— nearly ; for, in this Cafe,

A^B

A is — xo, and B — 6;, and therefore A 1 -j- —

2A

/io 1‘kcManner is /n -{- t>; = /n

3/11 x ; , . , — , 3/10 x«

-j- — IX —, £sV. and consequently / 10 + -

10

. 34/11 X; , — . ;/i2 x, .

+ /i I + — -b \/ ,2 + ~“|2~ =l0i whlch >

contracted, gives 9-944- ~b 2.718# = 10 • whence 2.718;= .056 and , — 0205 j consequently x — 3.0205 near-ly. Wherefore, in order to repeat the Operation, let3.0205 e be now substituted for x; then, by pro-ceeding inthefameManner, we shall have_

/io. 1234.2 -j- 6.04.1c -j- /11. 1234.2 6.041; -f-

/12.12342 -f- 6.041; =10 ■, whence, by Theorem zd,_______ 6.041;

/10.12342 +

6.041;

2/11.12342

2/10.12342

+ /xi.12342 +

+ /I2.X2342 -}-

6.041;2/12.12342

— IO, of

9.9987814