found. For example; if it be required to know what number corresponds to the logarithm 0,5432725; as this logarithm falls between the logarithms of 3 and 4, and as the number to which it belongs is, consequently, much below 1500, we seek for this logarithm with three units added to its characteristic, that is, we seek for 3,5432725. This logarithm falls between the logarithms of 3493 and 3494, hence we infer, that the number sought is 3,493; to three places of decimals. But, if this approximation is not sufficient, we take 739, the difference between 3,5432725 and the logarithm of 3493; and also 1243, the difference between the logarithms of 3494 and 3493, and seek, by reasoning as in article (243,) the fourth term of a proportion, that commences with these three,

1243: 1 :: 739 :

The fourth term, in decimals, is 0,594; then the number sought is 3,493594. Finally, this second approximation is limited; for, the logarithms of the Table being exact only to half a decimal unit of the seventh order, the differences are affected by this slight defect. The approximation, however, may be extended with confidence to the third place of decimals, and we seldom have occasion to continue it further. This remark is also applicable to the use, that was made of the same pròportion in articles (239 and 243).

246. If the fraction be required, to which a given negative logarithm corresponds, we subtract from the logarithm 1, 2, 3, 4, &c. units, according to the extent of the Table; and after having found the number, that corresponds to the remaining logarithm, we separate upon the right of it, as many figures as there were units in the number whose logarithm was subtracted. For example, if it be required to know what fraction belongs to this logarithm- 1,532732, we subtract 1,532732 from 4, and there remains 2,467268, which in the Table is found between the logarithms of 293 and 294; hence we infer, that the fraction sought is between 0,0293 and 0,0294; that is, it is 0,0293, to four places of decimals. But, to subtract the given logarithm 1,532752 from 4, is (236) to multiply 10000 by the fraction to which this logarithm belongs; or, it is the same thing as to multiply this

fraction by 10000. Then, the number which has been found, is 10000 times too great; it must, therefore, be considered as composed of ten thousandths of an unit, and is expressed by four decimal figures.

We now proceed to show, by some examples, the advantages that logarithms afford for facility in calcula

tions.

EXAMPLE I.

The quotient of 17954 divided by 12836 is required to four places of decimals.

The logarithm of 17954 is
The logarithm of 12836 is

Remainder

4,254161

4,108430

0,145731

This remainder, with a characteristic augmented by four units, corresponds in the Table to 13987; then (238) the quotient sought is 1,3987.

EXAMPLE II.

The cube root of 53 is required to three places of decimals.

The logarithm of 53 is
Its third part (250) is

1,724276
0,574759

This last logarithm sought in the Table with a characteristic augmented by three units, corresponds to 3756; then (238) the root sought is 3,756.

To judge of the important use of logarithms, we need only seek for this root by the method given in article (156). That method, however, ought not to be considered useless; because it applies to an infinity of numbers, to which logarithms do not, by reason of the limits of the Tables.

EXAMPLE III..

The fifth root of the cube of 5736 is required to two places of decimals.

We triple 3,758609, the logarithm of 5736, and have 11,275827, for the logarithm of the cube of 5736. Tak

ing the fifth of this last logarithm, we have 2,255165 for the logarithm of the fifth root of the cube of 5736. This logarith, sought in the Table with a characteristic augmented by two units, corresponds to a number between 17995 and 17996; the root required is then 179,96, to two places of decimals.

EXAMPLE IV.

It is required to find four geometrical mean proportionals between 2 and 53.

According to the method given in article (215,) the ratio between the terms of this progression must be found by dividing 53 by 23, and extracting the fifth root of the quotient. But by logarithms this operation is very simple. We find by the tables, that the logarithm of 5 or 23 is 0,759668. In like manner, we find, that the logarithm of 2 is 0,425969. We then subtract (231) this logarithm from the first, and have 0,333699. Then taking the fifth of this last, have 0,066740 for the logarithm of the ratio sought. This logarithm, found in the tables, with a characteristic augmented by 4 units, for 4 decimals, corresponds to 11661 in whole numbers. The ratio, then, is 1,1661, to four places of decimals. Now, to obtain the required mean proportionals, we multiply 23, the first term, by 1,1661; then, the product of that multiplication by 1,1661; and so on. But these operations may be more readily performed by the aid of Logarithms, by adding to 0,425969, the logarithm of 23, the first term, the logarithm 0,066740 of the ratio; then its double, triple, and quadruple; hence we have 0,492709; 0,559449; 0,626189; 0,692929; for the logarithms of the four required geometrical mean proportionals. And if we seek for these logarithms in the Tables; with three units for their characteristic, we shall find, that these four mean proportionals are 3,109; 3,626; 4,288; 4,931.

REMARK.

In an operation performed by logarithms, where subtraction is required, the operation may be simplified by the following method.

When any number is to be subtracted from another, which is an unit followed by as many ciphers as there are figures in the first; the operation may be performed by taking the difference between 9 and each of the figures of the given number, except the last; for which, we write the difference between 10 and that figure. For example, if 526927 is to be subtracted from 1000000; we subtract successively the figures 5, 2, 6, 9, 2, from 9, and the last figure, from 10, and have 473073, for a remainder. This remainder is called the arithmetical complement of the given number.

As this method of subtraction is too simple to be called an operation, it follows, when a result is to be obtained from the addition and subtraction of many numbers, that the operation may be reduced to addition. For example, if it be required to add these two numbers 672736, 426452, and subtract from their sum these other two numbers 432752, 18675; instead of one addition and two subtractions, which would be required by the common method, we substitute the following, 672736

426452

Arithmetical complement of 432752
Arithmetical complement of 18675

567248

981325

Sum

2647761

That is, we add together the two first given numbers, and the arithmetical complements of the two last, and their sum is 2647761. The first figure 2 must be suppressed; and the remaining figures 647761 is the result sought.

The reason of this operation is plain; for if, instead of subtracting 432752, we add its arithmetical complement, that is, 1000000 minus 432752, we at the same time perform the required subtraction, and augment the sum by 1000000; that is, by one tenth foo much in the first figure of the result. Then, for each arithmetical complement used in the operation, we shall have one tenth too much in the first figure of the result.

We will now show the application of this principle to logarithms. For example, if it be required to divide 3760 by 79; by the ordinary method, the logarithm of 79 would be subtracted from that of 3760. But instead

of that operation we, substitute the following, and write the logarithm of 3760

3,575188

and the arith. complt. of the log. of 79 8,102373

Sum

11,677561

Thus, 1,677561 is the logarithm of the quotient, and corresponds to 47, 59, to two places of decimals.

675

952

377

For a second example, let it be required to multiply 879 by 344. According to the usual method (106), 675 must be multiplied by 952; and 527 by 377, then the first product divided by the second.

By logarithms, the operation is as follows.

The logarithm of the product is 0,509789, which with three units for a characteristic, corresponds to 3,234.

We can also use the arithmetical complement, in order to give to the logarithms of fractions the same form as those of whole numbers, and employing them in the same manner, avoid, in making calculations, the distinction of positive and negative logarithms. It may be well to recollect, that the characteristic of the logarithms of fractions, properly so called, is too great by 10 units. For example, to obtain the logarithm of 3, which is no other (96) than 3 divided 4; instead of subtracting the logarithm of 4 from that of 3, that is, the logarithm of 3 from that of 4, and affixing the sign minus to the difference (235); we add to the logarithm of 3, the arithmetical complement of the logarithm of 4; thus,

The logarithm of 3 is

0,477121

Arith. complt. of the log of 4 9,397940

This sum is the logarithm of 3, the characteristic of which is too large by 10 units. But we need not actually diminish it by 10 units; for so many may be reject