Multiply the II. Bring down the next two terms for a dividend. Divide it by twice the root just found, and add the quotient, both to the root and to the divisor. divisor thus increased, into the term last root, and subtract the product from the dividend. placed in the III. Bring down two or three additional terms and proceed as before. EXAMPLES. 1. What is the square root of ́a2+2ab+b2+2(a+b)c+c2+2(a+b+c)d+d2? OPERATION. ROOT. (a+b+c+d. a2+2ab+b2+2(a+b)c+c2+2(a+b+c)d+d2 4x+12x+5xa—2x3 +7x2-2x+1? 3. What is the square root of x1—2x2y2—2x2+y1+2y2+1? Ans. x2-y2-1. 4. What is the square root of being useless, they have lent their aid in the solution of questions, which required the most refined and delicate analysis. (103.) Before closing this chapter, we will show the interpretation of the following symbols. We know from the nature of multiplication, that O multiplied by a finite quantity, that is, O repeated a finite number of times, must still remain equal to 0; hence, we have this condition OX A=0. Dividing both members of (1) by A, we find A (1) (2) Therefore the symbol will always be equal to 0, as as long as A is a finite quantity. (104.) Since the quotient arising from dividing one quantity by another, becomes greater in proportion as the divisor is diminished, it follows that when the divisor becomes less than any assignable quantity, then the quotient will exceed any assignable quantity. it is usual for mathematicians to say, that A 0 Hence, is the re presentation of an infinite quantity. The symbol employed to represent infinity is ∞, so that we have (105.) Dividing both members of (1) by 0, we find This being true for all values of A, shows that is the symbol of an indeterminate quantity. To illustrate this last symbol, we will take several If, before substituting a for x, we divide both numerator and denominator of the given fraction, by x-a, (Art. 42,) we find x2 -a2 x+a bx-ab Now, substituting a for x, in this reduced form, we find |