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ELEMENTS OF SURVEYING.
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Definitions and Introductory Remarks,
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Of Logarithms. 1. The nature and properties of the logarithms in common use, will be readily understood, by considering attentively the different powers of the number 10. They are,
&c. It is plain, that the indices or exponents 0, 1, 2, 3, 4, 5, &c. form an arithmetical series of which the common difference is 1; and that the numbers 1, 10, 100, 1000, 10000, 100000, &c. form a geometrical series of which the common ratio is 10. The number 10, is called the base of the system of logarithms; and the indices, 0, 1, 2, 3, 4, 5, &c., are the logarithms of the numbers which are produced by raising 10 to the powers denoted by those indices.
2. Let a denote the base of the system of logarithms, m any exponent, and M the corresponding number: we shall then have,
a" =M in which m is the logarithm of M.
If we take a second exponent n, and let N denote the corresponding number, we shall have,
a"=N in which n is the logarithm of N.
If now, we multiply the first of these equations by the second, member by member, we have
but since a is the base of the system, mtn is the logarithm MxN'; hence,
The sum of the logarithms of any two numbers is equal to the logarithm of their product.
Therefore, the addition of logarithms corresponds to the multiplication of their numbers.
3. If we divide the equations by each other, member by member, we have,
N but since a is the base of the system, m-n is the logarithm
M of hence:
If one number be divided by another, the logarithm of the quotient will be equal to the logarithm of the dividend diminished by that of the divisor.
Therefore, the subtraction of logarithms corresponds to the division of their numbers. 4. Let us examine further the equations
&c. &c. It is plain that the logarithm of 1 is 0, and that the logarithms of all the numbers between 1 and 10, are greater than 0 and less than 1. They are generally expressed by decimal fractions : thus,
log 2=0.301030. The logarithms of all numbers greater than 10 and less than 100, are greater than 1 and less than 2, and are generally expressed by 1 and a decimal fraction : thus,
log 50=1.698970. The logarithms of numbers greater than 100 and less than 1000, are greater than 2 and less than 3, and are generally expressed by uniting 2 with a decimal fraction ; thus,
log 126=2.100371. The part of the logarithm which stands on the left of the
The characteristic is always one less than the places of integer figures in the number whose logarithm is taken.
Thus, in the first case, for numbers between 1 and 10, there is but one place of figures, and the characteristic is 0. For numbers between 10 and 100, there are two places of figures, and the characteristic is 1; and similarly for other numbers.
TABLE OF LOGARITHMS.
5. A table of logarithms, is a table in which are written the logarithms of all numbers between 1 and some other given number. The logarithms of all numbers between 1 and 10,000 are written in the annexed table.
6. The first column on the left of each page of the table, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed directly opposite them, and on the same horizontal line.
To find, from the table, the logarithm of any number. 7. If the number is less than 100, look on the first page of the table, along the column of numbers under N, until the number is found: the number directly opposite, in the column designated log, is the logarithm sought. Thus,
log 9=0.954243. When the number is greater than 100, and less than 10,000.
8. Since the characteristic of every logarithm is less by unity than the places of integer figures in its corresponding number (Art. 4), its value is known by a simple inspection of the number whose logarithm is sought. Hence, it has not been deemed necessary to write the characteristics in the table.
To obtain the decimal part of the logarithm, find, in the column of numbers, the first three figures of the given number. Then, pass across the page, along a horizontal line, into the columns marked 0, 1, 2, 3, 4, 5, &c., until you come to the column which is designated by the fourth figure of the given number: at this place there are four figures of the required logarithm. To the four figures so found, two figures taken from the column marked 0, are to be prefixed. If the four figures thus found, stand opposite to a row of six figures in the column marked 0, the two figures from this column, which are to be prefixed, are the first two on the left hand: but if
the four figures found are opposite a line of only four figures, you are then to ascend the column till you come to the line of six figures; the two figures, at the left hand, are to be prefixed, and then the decimal part of the logarithm is obtained ; to which prefix the characteristic, and you have the entire logarithm sought. For example,
log 8979=3.953228 In several of the columns, designated 0, 1, 2, 3, 4, &c., small dots are found. When the logarithm falls at such places, a cipher must be written for each of the dots, and the two figures, from the column 0, which are to be prefixed, are then found in the horizontal line directly below. Thus,
log 2188=3.340047 the two dots being changed into two ciphers, and the 34 to be taken from the column 0, is found in the horizontal line directly below.
The two figures from the column 0, must also be taken from the horizontal line below, if any dots shall have been passed over, in passing along the horizontal line: thus,
log 3098=3.491081 the 49 from the column 0, being taken from the line 310. When the number exceeds 10,000, or is expressed by five or
more figures. 9. Consider all the figures, after the fourth from the left hand, as ciphers. Find from the table the logarithm of the first four figures, and to it prefix a characteristic less by unity than all the places of figures in the given number. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the figures that have been considered as 'ciphers: then cut off from the right hand as many places for decimals as there are figures in the multiplier, and add the product so obtained to the first logarithm, and the sum will be the logarithm sought.
Let it be required, for example, to find the logarithm of 672887.
log 672800=5.827886 the characteristic being 5, since there are six places of figures. The corresponding number, in the column D is 65, which