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ADAPTED TO THE
Use of Schools, Private Students, and Practical Men:
LOGARITHMIC ARITHMETIC, PRACTICAL GEOMETRY, PLANE TRIGONOMETRY,
BY ALEXANDER INGRAM,
Author of A Concise System of Mathematics, The Principles of Arithmetic, &c.
WITH MANY IMPORTANT ADDITIONS AND IMPROVEMENTS,
BY JAMES TROTTER,
Of the Scottish Naval and Military Academy; Author of A Manual of Logarithms,
OLIVER & BOYD, TWEEDDALE COURT.
LONDON: SIMPKIN, MARSHALL, & CO.
[Price Two Shillings.]
183. C. 33.
The references throughout the Work are to the complete edition of
DEFINITION AND NOTATION OF LOGARITHMS.
LOGARITHMS are a set of artificial numbers invented and formed into Tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers in such a manner, that by their aid Addition supplies the place of Multiplication, Subtraction that of Division, Multiplication that of Involution, and Division that of Evolution or the Extraction of Roots.
Let a series of Numbers in Arithmetical Progression be adapted to another in Geometrical Progression, so that the least term of the one may correspond with the least term of the other, and the rest in order; thus,—
Arithmetical Progression, 0, 1, 2, 3, 4, 5, 6, 7, &c. Geometrical Progression, 1, 4, 16, 64, 256, 1024, 4096, 16384, &c.
Now, let it be required to multiply together any two terms of the Geometrical series, as 16 and 64. This may be done by adding 2 and 3, the corresponding terms of the Arithmetical series, for the sum 5 is the term corresponding to 1024, the product. Again, if we wish to divide any term of the Geometrical series by any other, as 1024 by 16, we have only to subtract 2 from 5, the corresponding terms of the Arithmetical series, for the difference 3 is the term corresponding to the quotient 64.
Hence, the use of such an adaptation is manifest; but it is very limited in the present state of the series. We may, however, easily extend it by interpolating an Arithmetical mean between every two terms of the Arithmetical series, and a Geometrical mean between every two terms of the Geometrical series, when the number of terms will be doubled; thus,—
Arith. Prog. 0, 0.5, 1, 1.5, 2, 2.5, 3, 35, 4, 45, 5, 5.5, Geom. Prog. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048,
6, 6.5, 7, 7.5, &c. 4096, 8192, 16384, 32768, &c.
These progressions may again be interpolated in the same manner by new terms, and the process carried on continually, till at length every integer shall occur in the Geometrical series, or a number so near to it, that the difference may be neglected without sensible error, and then the numbers in the Arithmetical series corresponding to these integers may be called their Logarithms.
In forming the common Tables of Logarithms, the progressions first assumed were,―
Arith. Prog. 0, 1, 2, 3, 4', 5, 6, &c. Geom. Prog. 1, 10, 100, 1000, 10,000, 100,000, 1,000,000, &c. New terms were continually interpolated, in the manner shown in the former series, until the natural numbers 1, 2, 3, 4, &c. occurred in the Geometrical series; and then the numbers in the Arithmetical series, corresponding to these natural numbers, were taken to compose the Tables of Logarithms.
In this system of Logarithms it is manifest that the Logarithm of any number and that of another, 10, 100, 1000, &c. times greater or less will consist of the same decimal fraction called the mantissa, and differ only in the integral part; so that all numbers, whether they are integers, decimals, or partly integral and partly decimal, have the same positive quantity for the decimal part of their logarithm; thus,
APPLICATION OF LOGARITHMS.†
The index or integral part of the logarithm of any whole or mixed number, is always one less than the number of integral figures of which the number consists; and in decimal fractions, the index which is negative is that number which points out the distance of the first significant figure from the place of units. Instead of negative indices, their arithmetical complements to 10 are often used. Thus, if there is no cipher between the decimal point and the first significant figure of the decimal, the index is 1 or 9; if there is one cipher be
* When the index or characteristic of the log. is negative, the sign is generally put above it, in order to distinguish the index from the mantissa, which must always be considered + or affirmative. + The Problems and Rules given under this head apply only to Tables of Logarithms of numbers from 1 to 10,000, such tables being of more common use than those which are more extensive.
tween them, the index is 2 or 8; if two ciphers are between them, it is 3 or 7, and so on.
The indices being thus readily found are generally omitted in the common logarithmic tables, and the mantissa only of the logarithm is inserted.
PROBLEM I. To find the logarithm of any given number from the table.
1. If the given number is not greater than 100, the logarithm, with its index prefixed, will, in ordinary tables, be found in the first page, immediately opposite the number; thus, the log. of 65 is 1·812913, and that of 88 is 1.944483.
2. If the number consists of three figures, it will be found in the margin on the left hand side of the page, and the decimal part of its log. immediately opposite, in the column under 0; thus, the log. of 536 is 2.729165, and that of 760 is 2.880814.
3. If the number consists of four figures, the first three will be found in the margin on the left hand side of the page, and immediately opposite, and under the fourth figure found at the top of the page, will be got the mantissa of the logarithm; thus, the log. of 7846 is 3.894648, and that of 37.56 is 1.574726.
4. If the given number consists of more than four figures, find the logarithm of the first four figures, as before directed; multiply the difference between this logarithm and the next higher in the table by the remaining figures of the given number, and cut off as many figures from the product as are in the multiplier; the remaining figures added to the logarithm of the first four figures will give the logarithm required.
NOTE. The mean difference given under D, in the right hand column, may be used, except in the first three pages of the table, where they vary rapidly.
To find the logarithm of 476.384. Opposite the number 476 in the margin, and under 3 at the top, is 677881, the difference in the column D, on the right hand side of the page, is 91, which, multiplied by 84, and two figures cut off, gives 76 to be added to 677881, and since there are three integral figures the index is 2, the logarithm of 476-384 is therefore 2.677957. Again, to find the logarithm of 1056-472. The logarithm of the first four figures is 023664, the difference between this and the next higher in the table is 411 (that in the column marked D being 412). 411 x 472, and three figures cut off, gives 194, which added to the logarithm of the first four figures, and the proper index prefixed, gives 3.023858 as the required logarithm. If the number had been 105647-2, the decimal part of the logarithm would have been the same, but the index would have been 5.