The first three of these methods are easily deduced from the Exam ple before given. To demonstrate the fourth, let the first three approximations be denoted by

From the manner in which the third approximation is derived from the second and first, as before stated, we shall then have

N""CN"+N' ;

D"-cD"+D'.

Now it is evident that the third approximation passes into the fourth,

The second members of these last equations are equal to

In the numerators of these fractions, we have, by decomposing the terms containing d, and recurring to the first and second equations.

cdN"'+dN'=d(cN"+N')=dN''' ;
cdD"+dD'=d(cD"+D')=dD'''.

Hence, by substituting these values in the numerators of the preceding fractions, and omitting the common denominator d (81), we find

This shows that the fourth approximation may be found from the third and second, in the same manner in which the third is obtained from the second and first; and it is evident that a similar demonstration is applicable to the fifth and subsequent approximations.

(303.) Any one of the approximating fractions differs in value from the fraction developed, by less than a unit the product of the denom inators of that and the succeeding approximation.

To establish this proposition, it must first be shown, that

If any two consecutive approximations be reduced to a common denominator, the difference of the resulting numerators will be a unit.

This is readily shown for the first and second approximations: thus, from the Example already given, we have

For a general demonstration, let the first three approximations be represented by

By reducing to a common denominator, and subtracting, we have

But we have already seen that

N""CN"+N', and D""-cD"+D' (302... III);

and by substituting these values of N'"' and D'"' in the numerator of the second member of the second equation, and simplifying, we have

The numerator of N'D'-N'D', of this second difference is the same as the numerator N'D"-N"D', of the first difference, with the signs changed.

Now, the numerator of the first difference has already been found to be ab+1-ab=+1; hence, the numerator of the second difference is equal to -1; and by proceeding in the same manner with the secona third, and fourth approximations, we shall find the successive differences to be alternately +1 and 1.

The difference between any two consecutive approximations, is therefore equal to a unit by their common denominator, that is, the product of their denominators; and since the value of the fraction developed is greater than the one, and less than the other, of any two consecutive approximations (301), it differs from either of these less than they differ from each other.

These considerations establish the foregoing proposition. It will hence follow, that—

(304.) Any particular approximation differs in value from the fraction, or ratio, developed, by less than a unit the square of the denominator of that approximation.

For the denominators successively increase, and the square of any denominator is therefore less than the product of that denominator x the next denominator.

4. Find approximate values of 3.14159, which is the ratio, nearly, of the circumference to the diameter of a circle.

The pyramid of Cheops in Egypt, which is the loftiest structure. ever raised by man, is 543 feet; and the altitude of the loftiest mountain is 29,000 feet. What are approximate ratios of these altitudes ?

6. Find approximate ratios of the altitude of the loftiest mountain, 29000 feet, to the semi-diameter of Earth, taken at 4000 miles.

251

CHAPTER XIV.

LOGARITHMS AND THEIR APPLICATIONS.

LOGARITHMS.

(305.) LOGARITHMS of numbers are the exponents of the powers to which some assumed number, called the base of the logarithms, must be raised, to produce the given numbers.

Thus, in the Equations

an and ay=m,

a may represent the constant base, and then the exponents x and y will represent the logarithms of the numbers, n and m, respectively.

Unity could not be taken for the base of a system of logarithms, since the series of numbers could not be obtained from the powers of unity (199). Any other number might be used for this purpose, and hence there may be

Different Systems of Logarithms.

(306.) The Common System of Logarithms, that is, the system in common use, is based on the number 10; all numbers being regarded. in this system, as powers of 10.

Commencing with the 0 power of the base 10, we have

(10)=1;.(10)1-10; (10)2=100; (10)3-1000; (10)=10000, &c.; the exponents, 0, 1, 2, 3, 4, &c., being, respectively, the logarithms of the numbers 1, 10, 100, 1000, 10000, &c.

Common Logarithms, also called Briggs's Logarithms, from the name of their author, are used to facilitate tedious operations in the multiplication, division, involution, and evolution of numbers.

Another System of Logarithms is that of Napier, the inventor of logarithms. The base of this system will be shown hereafter.

Naperian Logarithms are employed in some of the applications of the Differential and Integral Calculus: they also facilitate the computation of common logarithms.

GENERAL PROPERTIES OF LOGARITHMS.

The General Theory of Logarithms is contained in the following propositions, on which the methods of logarithmic calculation are mainly dependent.

Logarithm of Unity, &c.

(307.) The logarithm of unity is always 0, and the logarithm of the base of logarithms is always unity.

For whatever be the value of the base a, we have

ao=1(48); and a1: =a.

In the first equation, O is the logarithm of 1, and in the second, 1 is the logarithm of a.

Logarithm of a Product.

(308.) The Logarithm of the product of two or more numbers, is equal to the sum of the logarithms of those numbers.

In the equations aan and am, the exponents, x and y, are the logarithms of n and m, for the base a; and by multiplying the equations together, we have

a2+y=n m (41),

in which x+y is the logarithm of the product n m.

Logarithm of a Quotient.

(309.) The logarithm of the quotient of two numbers, is equal to the logarithm of the dividend minus the logarithm of the divisor.

Dividing the Equation an by the Equation am, we have

ax-y=n÷÷m (47),

in which x-y is the logarithm of the quotient n÷m, for the base a; while x and y are the logarithms of n and m.