equations of all orders , by continuous approximation. 323 
the last or n — 1 th derivee, and being conveyed to the first 
through a regular system of preparatory addends dependent on 
the last quotient-correction, and of closing addends dependent 
on the new one. The overt and registered manner of con- 
ducting the whole calculation, enables us to derive important 
advantage from anticipated corrections of the divisors, not only 
at the first step, but, if requisite, through the whole perform- 
ance, and also, without the necessity of a minute's bye- calcu- 
lation, communicates, with the result, its verification. 
25. Let us trace the operation of the theorem as far as 
may be requisite, through the ascending scale of equations. 
1 . In Simple equations , the reduced equation may be repre- 
sented by A = az ; whence % = Now the theorem directs 
us to proceed thus : 
a A (r + r ' + 
— ar 
— 
— ar' 
&c. 
precisely the common arithmetical process of division. 
2. In Quadratics, we have A = az + z % and proceed in this 
manner : 
a A (r-|- r' + . . 
r_ — Ar 
A A' 
r —A v 
r' A" 
A' &C. 
&C. = 
1 
• • * 
