whatever, has been already pointed out, (Art. 289); and it has also been observed (Art. 450), that the series will always terminate, wheu m is a whole number: But when m is a negative number, or a fraction, then the series expressing the value of (a+x)m does not terminate. Let m=, and substitute for m in the series (Art. 450); then () + nar n(n n(n-r) a2 (=2 ) + &c. = u2 { i + " 24.2 (~~)+ &c., which is a general ex 2r3 α 2 pression for finding the approximate value of any n binomial surd quantity, being either positive or negative, n and r any whole numbers whatever. Ex. 1. Find the approximate value of √(b3+c3) or (b3+c3)3. Here a=b3 •*. a* =3/b3=b; 3 2r2 2.32 n(n-r).(n−2r)(x 1 (1-2). (1-6)(x (c = 69 2.3.33 ; 34699 2.3r3 456. Now let n=1, (a+x)==(a+x)1=√(a+x); and aa; hence the series (Art. 454) is transformed into √/(a+x)=√⁄a(1 + ! ( ~ ) + 1 = Let a=1, b=1; then ;/2=1+i+2=" then +&c. 210 +&c. And if r=3, () (A). (B). 1 1 5 2.5 2.11 2.7.11 2=1+3 - + + By means of the series marked A, the rth root of many other numbers may be found; if a and x be so assumed, that x is a small number with respect to a, anda, a whole number. Ex. 3. It is required to convert 5, or its equal ✓(4+1), into an infinite series. Here a=4, x=1, r=2; then a=√4=2, and we 1: have √(4+1)=2(1+2-27+210-215+&c.) Ex. 4. It is required to convert 9, or its equal (8+1), into an infinite series. Here a=8, x=1, r=3; then ་ we obtain 2/(8+1)=2/9=2 {1+ a=8=2, and 457. The several terms of these series are found by substituting for a, x, and r, their values in the general series marked (A) or (B), and then rejecting the factors common to both the numerators and denominators of the fractions. Thus, for instance, to find the 5th term of the series expressing the approximate value of 2/9, we take the 5th term of the general series marked (A), which is where a=8, (1--r). (1—2r). (1—3r) (~+), 3 3 4r4 x=1, and r=3; .. the value of the fraction is 2.5 In this manner each term of the series 35 84° is calculated; and the law which they observe is, that the numerators of the fractions consist of certain combinations of prime numbers, and the denominators of combinations of certain powers of a and r. 3 Ex. 5. Find the value of (c2-x)in a series. 3x3 3x4 5x6 Ans./c3 9 (1 1 -&c. 92.c2 25.c 27.c 6 zc.) Ex. 6. It is required to convert /6, or its equal (8-2), into an infinite series. Ans. 21 2 (1 1 3.4 32.42 34.43 .) Ex. 7. It is required to extract the square root os 10, in an infinite series. Ans. 3+ + -&c. I 2.3 2.4.33 2.4.6.35 Ex. 8. To expand a2(a2 — x) ̄1 ́ in a series. + &c. 1 3 3.5 Ans. a +(-) + 2() +2.4.6) 2 2. 3 05 Ex. 9. To find the values of (a3+x3) in a series. 6x15 25a9 125α14 -&c. 3 Ex. 10. Find the cube root of 1-x3, in a series. x 3 x 6 5x9 10x12 Ans. 1. CHAPTER XIII. ON -&c. PROPORTION AND PROGRESSION. § I. ARITHMETICAL PROPORTION AND PROGRESSION. 458. ARITHMETICAL PROPORTION is the relation which two numbers, or quantities, of the same kind, have to two others, when the difference of the first pair is equal to that of the second.. 459. Hence, three quantities are in arithmetical proportion, when the difference of the first and second is equal to the difference of the second and third. Thus, 2, 4, 6; and a, a+b, a+2b, are quantities in arithmetical proportion. 460. And four quantities are in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth. Thus, 3, 7, 12, 16; and a, a+b, c, c+b, are quantities in arithmetical proportion. 461. ARITHMETICAL PROGRESSION is, when a series of numbers or quantities increase or decrease by the same common difference. Thus 1, 3, 5, 7, 9, &c. and a, a+d, a+2d, a+3d, &c. are an increasing series in arithmetical progression, the common differences of which are 2 and d. And 15, 12, 9, 6, &c. and a, a-d, a-2d, a—3d, &c. are decreasing series in arithmetical progression, the cominon differences of which are 3 and d. 462 It may be observed, that GARNIER, and other European writers on Algebra, at present, treat of arithmetical proportion and progression under the denomination of equi-differences, which they consider, as BONNYCASTLE justly observes, not without reason, as a more appropriate appellation than the former, as the term arithmetical conveys no idea of the nature of the subject to which it is applied. 463. They also represent the relations of these quantities under the form of an equation, instead of by points, as is usually done; so that if a, b, c, d, taken in the order in which they stand, be four quantities in arithmetical proportion, this relation will be expressed by a-b-c-d; where it is evident that all the properties of this kind of proportion can be obtained by the mere transposition of the terms of the equation. 464. Thus, by transposition, a+d=b+c. From which it appears, that the sum of the two extremes is equal to the sum of the two means: And if the third term in this case be the same as the second, or c=b, the equi-difference is said to be continued, and we have a+d=2b; or b= (a+d); where it is evident, that the sum of the extremes is double the mean; or the mean equal to half the sum of the extremes. 465. In like manner, by transposing all the terms of the original equation, a-b-c-d, we shall have b-a=d-c; which shows that the consequents b, d, |