only; the characteristic in this and every other instance being left for the operator to supply. Thus in the present example, if the given number 2345 be a whole number, 3 must be prefixed as a characteristic to the logarithm; if the last figure 5 be a decimal, 2 must be prefixed; if there are two decimals, 1 must be prefixed; if three decimals, 0; if the whole be a decimal, -1, &c.

To find the logarithm of 66534. Find 6653 in the column marked N, and 4 at the top; then in the column under 4, and level with 6653, stands .8230436; to which prefixing 4 for a characteristic, the required logarithm is 4.8230436.

In like manner, the logarithm of 4056 is

3.6080979.

That of .88526 is -1.9470708.

The characteristic in each of these, and in every other instance, being one less than the number of integral places in the given natural number.

26. To find the natural number belonging to a logarithm. Look for the logarithm in the columns marked 0, 1, 2, &c. and having found it, the number standing opposite, with the figure at the top of the column subjoined, will be the number; then mark off from this number as many places of whole numbers, as are equal to one more than the index of the given logarithm, and it will be the number required,

. Thus, to find the natural number belonging to the logarithm 2.8230436, look in the columns marked 0, 1, 2, &c. for the deciinal part only of this logarithm, (rejecting the index,) and having found it, the number opposite, in the column marked N, is 6653, and the figure at the head

of the column containing the given logarithm, is 4, which must be subjoined to the above four figures and since 2 is the index of the given logarithm, three figures of this number must be pointed off for whole numbers; whence the natural number agreeing with the above logarithm is 663.34, as was required.

In like manner, the natural number belonging to the logarithm 1.7889104 is 61.505.

That belonging to the log. 3.9181562 is 8282.4.
That belonging to the log. 0.5410798 is 3.476.
That belonging to the log. 2.1682617 is 147.52.
That belonging to the log.-2.9187483 is .082937.

27. In the four former examples, the places of whole numbers pointed off are one more than the index of the respective logarithm; in the latter example, the index -2, shews that the left hand figure (viz. 8.) of its number must stand in the second place below units; a cipher must therefore be placed before it, and the whole will be a decimal.

There are frequently two additional columns in the tables, one for the differences of every two adjacent logarithms, and the other for the proportional parts of those differences; each difference being divided into nine parts in the ratio of the numbers 1, 2, 3, &c. to 9, for the purpose of finding the logarithm of any number, containing one or two places more than the numbers in the tables consist of, and likewise the number corresponding to any logarithm between two adjacent ones in the tables. Ample directions for these purposes are given with every collection of tables".

• Thus, to find the logarithm of a number, consisting of six figures. Find the decimal part of the logarithm for the first five figures, and take the difference between that, and the next greater logarithm. Find the difference in the column marked D, then under that difference in the column marked pts,

and against the figure occupying the sixth place, stands the part which must be added to the logarithm found.

To find the logarithm for seven figures. Find the logarithm for the first six, as before; then divide the number corresponding to the seventh figure (in the column of pts marked D,) by 10; add the quotient to the decimal part of the logarithm for six figures, observing to place the first figure on the right, in the eighth place of the logarithm.

To find a number to six, seven, or more figures, answering to any given logarithm. From the given logarithm subtract the next less; add as many ciphers to the right of the difference, as there are additional figures required; divide this quantity by the difference between the next greater and next less than the given logarithm; and the quotient will be the figures required.

And by a converse process, numbers consisting of six, seven, or eight places, answering to any intermediate logarithm, may be readily found. See Vince's Trigonometry, Hutton's Mathematical Tables, pp, 131, 132, 133, and 184. second edit.

LOGARITHMICAL ARITHMETIC.

28. Logarithmical Arithmetic teaches to perform arithmetical operations, by means of logarithms previously computed and arranged in tables for use.

MULTIPLICATION BY LOGARITHMS.

29. When the indices of the logarithms are affirmative, or +. RULE I. Seek in the table the logarithms of the factors, place them one under another, and add them together; their sum will be the logarithm of the product.

II. Seek this logarithm in the table, and the natural number answering to it will be the product required ».

part of their sum (viz. .3802112) in the table among the logarithms, opposite which, in the column marked N, I find 2400, which is the product, and 3 being the characteristic, I mark off 4 places for whole numbers.

2. Multiply 15.27. by 3.172.

OPERATION.

The log. of 15.27 = 1.1838390
The log. of 3.172 = 0.5013332
The product 48.437 = 1.6851722

Explanation.

Having looked out the logarithms, placed them under each other with their proper indices, and added them together, I find the number 48437 is the nearest in the table, which answers to

their sum; from this I mark off 2 places of whole numbers, because the index of the sum is 1.

3. Multiply 1.2345 ... 20.517, and 5.4321 together.

The log. of 1.2345 = 0.0914911
The log. of 20.517 = 1.3111139
The log. of 5.4321 = 0.7349678
The product=137.27 = 2.1375728

The truth of this rule is plain from the nature of logarithms, which has

been fully explained in the Introduction.

The log. of 13.954 1.1446987

The log. of 2.3456 = 0.3702540
The product=319.49 = 2.5044608

30. When any of the indices are negative, or —.

RULE I. Find the sum of the decimals as before, then add what is carried, and the affirmative indices into one sum, and the negative indices into another.

II. Subtract the less of these sums from the greater, and to the remainder prefix the sign of the greater, and it will be the index to be prefixed to the decimal part of the sum o.

5. Multiply 18.32 ... 2.405, and .61245 together.

7. Multiply .703 ... .0918, and 47.345 together.

OPERATION.

Log. of 703 =-1.8469553
Log. of .0918 = −2.9628427
Log. of 47.345 = · 1.6752741
Product 3.0554= 0.4850721

Explanation.

Here 2 being carried from the decimals, the sum of the affirmative indices is equal to that of the negative ones, each being 3, whence 0 is the index to be supplied.

We have before shewn, that the decimal part of a logarithm is always affirmative, and therefore the number carried from that decimal will evidently be affirmative. The reason why the sum of two numbers, having different signs, is found by subtraction, is explained in the notes on Addition of Algebra.