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Now for a first approximation neglect the second and the following terms, and take simply

b=mam-12 whence we derive



This value of 2, substituted in the second term of (A), gives

mm—1) cm-2xx_


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2mat(m-16 In this value substitute the value of b=E(N-am) and it becomes +2a(N-am)

+2a(N-am) 2mam +(m-1)(N-am) (m-1)N+(m+1)am Finally, substituting this value of z, we have


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With this formula we may, by successive substitutions, approximate to any degree of accuracy.

For example, find the square root of 8. In the formula make m=2, we have

3N+a N?=ax


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In this let N=8, and substitute as a first approximation, a=3. Then we have

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Then substituting a=2.829, we have

24+8.003241 82=2.829 X


which agrees

with the root found in Art. 50.

In the same way we may approximate very rapidly to any root of a number. The general formula will become for the particular

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This very convenient method of approximation was first investigated in a general manner by Hutton. See his " Tracts," Vol. I. p. 45, et seq.

A similar method, however, had been before employed in the extraction of the cube root by Mr. J. Dodson, and also by Dr. HALLEY.

52. An important facilitation in the ordinary mode of extracting the square root is also suggested by considering the binomial square. Let N represent the number whose square root is sought, and a an approximate root correct to n places of decimals; and let x be the correction to be added to a, or the difference between a and the true root; then the first significant figure of x must be in the (n+1)th place, and a+x represents the true root of N; that is,

N=(a +x)=a2+2x+x8. Now since the first significant figure of x is in the (n+1)th place, the first significant figure of x® is at least in the (2n+1)th place. Hence, if we desire the root to 2n places only we may neglect *, and the above expression becomes


* A first approximation may be found to five places of figures with the aid of logarithms, (for which see Chapter VI.,) and by the above methods we may then extend it to any required number of decimals.

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That is, having found an approximate root a, subtract its square from the given number, and divide the remainder by 2a for the remaining figures of the root, which will be accurate to twice as many places as there are in a.

8 ,

To illustrate this, take the above example, viz: to find the square root of 8.

8 | 2.82842,71247,

48 400


384 562 | 1600

1124 5648 | 47600

45184 56564 | 241600


565682 1534389


5,6,5,6,8,4 | 4030361,71247, Here N=8


7057 a=2.82842


1400 N-a=.00004,03036

1131 2a=5.65684


226 x=.00000,71247


39 a+x=2.82842,71247 (true in the last place.) 4



53. An exponential equation is one in which the unknown quantity is an exponent. Thus,

ax=b is an exponential equation from which, a and b being known, the value of x may be found.

It is evident that if b is some exact power or root of a, the value of x is the exponent of this power or root. For example, if a=2, b=8, the equation becomes

2*=8, in which x is evidently =3, since 23=8. If a=16,b=2, then 16x=2; from which we find x= since

4' 161=2, or 16=2.


from which we find x=-6, for


1 If a=2, b=

then 2x= 64'

64' 1 1 2-6=

(Art. 20.) 26 64'

If a=;

1 1000

b=10, then

(1000) *=10; from which we find Good)


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54. When b is not an exact power or root of a, the solution of the equation is not so readily effected. There are methods, however, of approximating to the value of x as nearly as we please. The following is that of Lagrange. Take the equation

10x=500. Now 10o=100 and 103=1000; that is, 10% is too small and 108 is too great. Therefore the value of x is between 2 and 3, and is

equal to 2+ some fraction. Represent this fractional part of a by 1 ; that is, let

x=2+ then our equation becomes


10 or (Arts. 5 and 21,)



10% X 10%=500, whence we have

10%! =5.

Involving to the power denoted by x', this becomes

10=5x', which is another exponential equation, from which we are to find the value of x'. Now 5-=5<10, and 59=25 >10. Therefore x' is between 1 and 2. Let

1 x'=l+


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then we have in the same manner as above
2=10, or 5x52"=10, or 521 _10

5=2, or

5=2a", from which we are to find ac". We have *2=4<5, 28=8>5; therefore x is between 2 and 3. Let

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125 128

< 64


or 2,

625 512 and

> or 2;

256 256 therefore x'"' is between 3 and 4.

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