326 Mr. Horner’s new method of solving numerical 
problem ; and I say this to invite comparison, not so much 
with his mode of treating it, as with theirs. 
Ex. II. What is the value of x in the equation x 3 — 22=5. 
The root is manifestly a very little greater than 2. Make 
it x = 2 + z, and the equation becomes 
1 = 10 z + 6z 2 -|- z 3 ’ 
Hence, arranging the derivees, 
6. 10 . . 1.000 ( 
6 
The first digit will obviously be so nearly 1, that by antici- 
pating its effect on the divisor, we are sure this will be very 
nearly 106. Hence 
10.6) l.ooof.o^ first correction 
The square is 94 s = 8836. 
Hence we have 
6... 10 i.ooooooooo(.094, 
094 ^--572836 ^ ^993 8 4A584 
‘ 
6094x94='' 10572836' 6153416 
188 581672 
3 
The first digit of the next correction will evidently be 5 ; 
the effect of which we have as before anticipated as far as 
one digit. The divisor will therefore be 1 1 158 correct to the 
last figure. Hence 
11158)6153416(55148, second correction. 
The square is 30413, &c. to 10 digits. 
Hence, 
6094 10572836 
18855148 581672 6153416 
— — - ^ -^346470 1 490 1 904 ^-61533987854178101 
628255148x5 &c.= _ “ 
110296. 111579727014901204 172145821898 
34650056 
1 
