245. Logarithms, which are always expressed by decimals, are composed of two parts, namely, the units placed on the left of the comma, and the decimal figures found on the right. The

values of y for those of x between 0,001 and 0,009, between 0,0001 and 0,0009; thus we shall be furnished with the following table.

By means of this table, we may find the logarithm of any number whatever, by dividing it by 10 a sufficient number of times. To obtain, for example, the logarithm of 2549, we first divide this num

first of these is called the characteristic, because in the logarithms under consideration, which are adapted to the supposition of a = 10, and which are called common logarithms, this part shows,

ber by (10)3 or 1000, which is the greatest power of 10 it contains; we have then

2549 = (10)3 × 2,549;

we then seek in the table the power of 10 immediately below 2,549, and find

(10),42,511886432;

dividing 2,549 by this last number, we have

2,549 (10)0,4 X 1,014775177,

Again seeking in the table the power of 10 immediately below 1,014775177 we find

(10)0006, = 1,013911386;

then dividing the preceding quotient 1,014775177 by this number, we obtain a third quotient 1,000851742. This process is to be continued, until we arrive at a quotient, which differs from unity only in those decimal places we propose to neglect.

If we consider, in the present case, the third quotient as equal to unity, the proposed number will be resolved into factors, which will be powers of 10, for we shall have

2549 = (10)3 X (10),* X (10)0,006

= (10)3,406,

from which it is evident, that 3,406 is the logarithm of the number 2549. By extending the divisions to 7 in number, this logarithm will be found to be 3,406869.

The same table enables us with still more ease to find a number by means of its logarithm, as in the following example.

Let 2,547 be the given logarithm; the number sought will be (10)2,547 = (10) X (10)0,5 × (10) × (10) 0,007; it will, therefore, be equal to the product of the numbers

= 1,096478196,

(10)0,007 1,016248694,

taken from the table; and will, consequently, be

A table of the same kind with the above, but much more extended, has been published in England, by Dodson, the object of which is to furnish the means of finding the number answering to a given logarithm.

to what order of units the number corresponding to the logarithm belongs. The several logarithms of the numbers between 1 and 10, as they are between 0 and 1, have, necessarily, O for their characteristic; those of the numbers between 10 and 100 have 1 for their characteristic; those of the numbers between 100 and 1000 have 2; in general, the characteristic of a logarithm contains as many units, as the proposed number has figures, minus one.

246. It is important also to remark, that the decimal part of the logarithms of numbers, which are decuple the one of the other, is the same; for example,

the logarithm of

54360

is 4,7352794,

for, as each of these numbers is the quotient of that which precedes it, divided by 10, the logarithm of the one is found by taking an unit from the characteristic of that of the other (241,242).

247. According to what has been said in art. 240, the logarithms of fractional numbers are, upon our present hypothesis, negative; and we may easily deduce them from those of entire numbers, if we observe that a fraction represents the quotient arising from the division of the numerator by the denominator. When the numerator is less than the denominator, its logarithm is also less than that of the denominator, and, consequently, if we subtract the latter from the former, the result will be negative.

In order to obtain the logarithm of the fraction, for example, we subtract from 0, which denotes the logarithms of 1, the fraction 0,3010300, which represents that of 2; the result is

If we subtract from 0 the number 1,3010300, which is the loga rithm of 20, we have the logarithm of, equal to

The logarithm of 3 being 0,4771213, that of will be

0,3010300 0,4771213 =

0,1760913.

248. It is evident from the manner in which the logarithms of fractions are obtained, that, considered independently of their signs, they belong (241) to the quotients, arising from the division of the denominator by the numerator, and, consequently, an

swer to the number, by which it is necessary to divide unity in order to obtain the proposed fraction. Indeed, , for example, may be exhibited under the form, and 12 = 13—12=0,1760913.

It would be inconvenient, in order to find the value of a fraction, to which a given negative logarithm belongs, to employ the number to which the same logarithm answers when positive, since it would be necessary to divide unity by this number; but if we subtract this logarithm from 1, 2, 3, &c. units, the remainder will be the logarithm of a number, which expresses the fraction sought, when reduced to decimals, since this subtraction answers to the division of the numbers, 10, 100, 1000, &c. by the number to which the proposed logarithm belongs.

Let there be, for example, -0,3010300; if without regarding the sign, we take this logarithm from 1, or 1,0000000, the remainder 0, 6989700, being the logarithm of 5, shows, that the fraction sought is equal to 0,5, since we supposed unity to be composed of 10 parts.

If, in seeking the logarithm of a fraction, we conceive unity to be made up of 10, or 100, or 1000, &c. parts, or which amounts to the same thing, if we augment the characteristic of the logarithm of the numerator by a number of units sufficient to enable us to subtract that of the denominator from it, we obtain in this way a positive logarithm, which may be employed in the place of that indicated above.

In order to introduce uniformity into our calculations, we most frequently augment the characteristic of the logarithm of the numerator by 10 units. If we do this with respect to the fraction, for example, we have

10,3010300

It will be readily seen, that this logarithm exceeds the negative logarithm 0,1760913 by 10 units, and that, consequently, whenever we add it to others, we introduce 10 units too much into the result; but the subtraction of these ten units is easily performed, and by performing it we effect at the same time the subtraction of 0,1760913. Let N be the number, to which we add the positive logarithm 9,8239087; the result of the operation will be represented by

N100,1760913;

and if we subtract 10, we have simply N 0,1760913.

According to the preceding observations, we cause addition to take the place of subtraction, by employing, instead of the number to be subtracted, its arithmetical complement, that is, what remains, when this number is subtracted from one of the numbers, 10, 100, 1000, &c., a result which is obtained by taking the units of the proposed number from 10 and the several other figures from 9. We add this complement to the number, from which the proposed logarithm is to be subtracted, and from the sum subtract an unit of the same order as the complement.

It is evident, that if the complement is repeated several times, we must subtract, after the addition, as many units of the same order with the complement, as there are in the number, by which it is multiplied; and, for the same reason, if several complements are employed, we must subtract for each an unit of the same. order, or as many units as there are complements, if they are all of the same order.

Sometimes this subtraction cannot be effected; in this case, the result is the arithmetical complement of the logarithm of a fraction, and answers in the tables to the expression of this fraction reduced to decimals. If 10 units remain to be taken from the characteristic, as is most frequently the case, the result is the same as if we had multiplied by 10000000000, the numerator of the fraction sought, in order to render it divisible by the denominator; the characteristic of the logarithm of the quotient shows the highest order of the units contained in this quotient, considered with reference to those of the dividend. In 9,8239087, the characteristic 9 shows, that the quotient must have one figure less than the number, by which we have multiplied unity; and, consequently, if we separate 10 figures for decimals, the first significant figure on the left will be tenths; and we shall find only hundredths, thousandths, &c., for the numbers the arithmetical complements of which have 8, 7, &c. for their characteristics.

249. What has been said respecting the system of logarithms, in which a 10, brings into view the general principles necessary for understanding the nature of the tables; for more particular information the learner is referred to the tables themselves, which usually contain the requisite instruction relating to their arrangement and the method of using them. I will merely