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rithms for all numbers are derived solely from the principle of an arithmetical progression, applied to a geometrical one, which has been stated and explained above. It is likewise proper, that the learner should know how to examine the accuracy of any logarithm ; but it would not be worth his while to undertake the laborious task of computing a system of those numbers, since the tables of logarithms already in print are sufficiently accurate and extensive for every practical purpose. 24. The logarithms are placed in tables opposite their correspondent natural numbers, for the convenience of practice; so that a number being given, its logarithm may be readily found, and in like manner the number may be found from its logarithm being given. The following are the most usual plan and arrangement of the tables.—A page is divided from top to bottom into eleven columns, marked at top and bottom, N, O, 1, 2, 3, 4, 5, 6, 7, 8, 9. The left hand column, marked N, contains the four left hand figures of the natural number, to which for the right hand figure the proper one from the top or bottom must be taken; the remaining ten columns contain logarithms. 25. To find the logarithm of a natural number consisting of four figures. Look for the proposed number in the column marked N, and in a line with it in the next column stands its logarithm. 26. To find the logarithm of a number consisting offive figures. Look for the first four figures in the column marked N, and for the fifth figure at top or bottom; then in a line with the former, and in the same column with the latter, stands the proper logarithm. Thus, to find the logarithm of 2345; find this number in the column marked N, and opposite thereto, in the next column, stands the logarithm .870.1428, which is the decimal part 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, I must be prefixed; if three decimals, O; 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 391 is 2.592.1768. That of 3.366 is 0.5271141. That of 63.519 is 1.8029037. That of .88526 is – 19470708. 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 inder 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 decimal 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 665.34, 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 147.32. That belonging to the log.—2.9187488 is .082987. 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 sir, seven, or more figures, nnswering 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 Pince's Trigonometry, Hutton's Mathematical Tables, pp. 131, 132, 133, and 134. second edit.
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”.
ExAMPLEs. 1. Multiply 200 and 12 together.
Operation. I first find the logarithms of 200 rst find the logarithms or .
The log. of 200 = 2.3010300 and 13, prefixing to that of the for.
The log. of 12 = 1,0791812 mer 2 for a characteristic, and to that
5-TETTI, of the latter l; I place the logarithms The product 2400... 3.3S()2]. 19 one under the other, add them toge
ther, and then look for the decimal 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 of 4 places for whole numbers.
2. Multiply 1527. by 3.172.
OPERATIon. H looked he l - aving looked out the logaThe log. of 15.27 = 1.1838390 rithms, placed them under each
The log. of 3.172 = 0.50.13332 other with their proper indices, The product 48.437 = 1.6551722 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 12345... 20.517, and 5.4321 together.
P The truth of this rule is plain from the nature of logarithms, which has been fully explained in the Introduction.