equations of all orders , by continuous approximation. 333 
Wherefore the root is 2.638605803327, correct to the 
J2th decimal, and capable, like the former results, of being 
verified by simple inspection. The other roots are imaginary; 
for when — #=.4, the fourth derivee vanishes between 
two affirmative results, and when — x = . 7 &c., the second 
disappears under similar circumstances. (Arts. 21, 22.) 
It appears to me, that no explanation of this solution can 
be offered, which has not been abundantly anticipated in this 
Essay; and the student who peruses it in connection with the 
General Synopsis, and Arts. 23, 27, will acquire an indelible 
impression of the whole algorithm. 
Ex. VI. If it were proposed to obtain a very accurate solu- 
tion of an equation of very high dimensions, or of the irra- 
tional or transcendental kind, a plan similar to the following 
might be adopted. Suppose, for example, the root of 
x* — 100, or x log x = 2 
were required correct to 60 decimal places. By an easy ex- 
periment we find x — 3.6 nearly ; and thence, by a process 
of the third order, # = 3.597285 more accurately. 
Now, 3597286 = 98 x 7 1 * 47 x 11, whose logarithms, 
found to 61 decimals in Sharpe's Tables, give R log R = 
2.00000096658, &c. correct to 7 figures ; whence the subse- 
quent functions need be taken out to 55 figures only. They 
are 
a = Mod 4- log R = .990269449408, &c. 
b = Mod 2R = 0*60364, &c. 
c = — b ~3R = — -o i4 55, &c. 
&c. The significant part disappears after the 8th derivee ; 
consequently, the process will at first be of the eighth order. 
If the root is now made to advance by single digits, the first 
X x 
mdcccxix. 
