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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1826 Excerpt:...is negative, that is, the term will be come negative. Thus, the literal part--of the third term, (I in the above example, is naturally +, the coefficient of it, 1 & short, if any coefficient, found as before, have the same sign as the term to which it belongs naturally has, that term will be positive; but if it have a different sign, the term will be negative. Ex. 3. Expand j-into an infinite series. however, comes out--s, and hence the term is--. In From the preceding example, it appears, that the Binomial Theorem may not only be employed in extracting imperfect roots, but also in finding quotients. Thus, j-, evidently expresses the quotient of a divided by the cube root of (--y). If, by means of this theorem, the student find the quotients of a divided by a+x, and of a divided by a--x, or, which is the same thing, expand a(a±x), and a(a--x)" into infinite series, he will find the results to agree with those stated on page 68, as found by actual division. The Binomial Theorem may be very conveniently employed for extracting imperfect roots of numbers, more particularly when the number exceeds the complete power, whose root is to be extracted by a very little. Ex. Required the cube root of 29. i i i 1 2 1 22, 9s 2 4 40 &c.) = 3(1+-_+3595523-&c.) = (rejecting the last fraction) 3(1+.0240817) = 3.0722451 nearly. In all examples of this kind, when the second term of the binomial is small in comparison of the first, it is obvious that a few of the first terms of the expanded form will be sufficiently accurate for most purposes. EXAMPLES. 1 Ex. 1. Convert (a--x)3 into an infinite series. I Ex. 2. Convert (a2+£2) into an infinite series. Ex. 3. Required the quotient of a2 divided by the cube root a of the square of j+u2. That is, requir...