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and by knowing the variation, and allowing it upon every bearing, and having the distances, you would have notes sufficient for a trace. But a true meridian line is seldom to be met with, therefore we are obliged to have recourse to the foregoing method. It is therefore advised to lay out a true meridian line upon every map.

To find the difference between the present variation, and that at a time, when a tract was formerly surveyed, in order to trace or run out the original lines.

If the old variation be specified in the map or writings, and the present be known, by calculation or otherwise, then the difference is immediately seen by inspection; but as it more frequently happens, that neither is certainly known, and as the variation of different instruments is not always alike at the same time, the following practical method will be found to answer every purpose.

Go to any part of the premises where any two adjacent corners are known; and, if one can be seen from the other, take their bearing; which, compared with that of the same line in the former survey, shows the difference. But if trees, hills, &c. obstruct the view of the object, run the line according to the given bearing, and observe the nearest distance between the line so run and the corner, then,

As the length of the whole line

Is to 57.3 degrees,*

So is the said distance

To the difference of variation required.

EXAMPLE.

Suppose it be required to run a line which some years ago bore NE. 45°, distance 80 perches, and in running this line by the given bearing, the corner is found 20 links to the left hand; what allowance must be made on each bearing to trace the old lines, and what is the present bearing of this particular line by the compass?

* 57.3 Is the radius of a circle (nearly) in such parts as the circum » ference contains 360.

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Answer, 34 minutes, or a little better than half a degree to the left hand, is the allowance required, and the line in question bears N. 44o 26'. E.

Note. The different variations do not affect the area in the calculation, as they are similar in every part of the survey.

FINIS.

LOGARITHMS OF NUMBERS.

EXPLANATION.

OGARITHMS are a series of numbers so contrived, that the sum of e Logarithms of any two numbers, is the logarithm of the product of ese numbers. Hence it is inferred, that if a rank, or series of numbers arithmetical progression, be adapted to a series of numbers in geometrical ogression, any term in the arithmetical progression will be the logarithm the corresponding term in the geometrical progression.

This table contains the common logarithms of all the natural numbers m 0 to 10000, calculated to six decimal places; such, on account of their perior accuracy, being preferable to those, that are computed only to e places of decimals.

In this form, the logarithm of 1 is 0, of 10, 1; of 100, 2; of 1000, 3 &c. hence the logarithm of any term between 1 and 10, being greater than but less than 1, is a proper fraction, and is expressed decimally. The garithm of each term between 10 and 100, is 1, with a decimal fraction nexed; the logarithm of each term between 100 and 1000 is 2, with a cimal annexed, and so on. The integral part of the logarithm is called e Index, and the other the decimal part. Except in the first hundred garithms of this table, the Indexes are not printed, being so readily suped by the operator from this general rule; the Index of a Logarithm is ways one less than the number of figures contained in its corresponding natu I number-exclusive of fractions, when there are any in that number. The Index of the logarithm of a number, consisting in whole, or in parts, integers, is affirmative; but when the value of a number is less than nity, or 1, the index is negative, and is usually marked by the sign, -, aced either before, or above the index. If the first significant figure of e decimal fraction be adjacent to the decimal point, the index is 1, or arithmetical complement 9; if there is one cipher between the demal point and the first significant figure in the decimal, the index is- 2, its arith. comp. 8; if two ciphers, the index is 3, or 7, and so on;

ut the arithmetical compliments, 9, 8, 7, &c. are rather more convenientused in trigonometrical calculations.

The decimal parts of the logarithms of numbers, consisting of the same gures, are the same, whether the number be integral, fractional, or mixd: thus,

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N. B. The arithmetical complement of the logarithm of any number, is Sund by subtracting the given logarithm from that of the radius, or by sub

tracting each of its figures from 9, except the last, or right-hand figure, which is to be taken from 10. The arithmetical complement of an index is found by subtracting it from 10.

PROBLEM I.

To find the Logarithm of any given number.

RULES.

1. If the number is under 100, its logarithm is found in the first page of the table, immediately opposite thereto.

Thus the Log. of 53, is 1.724276.

2. If the number consists of three figures, find it in the first column of the following part of the table, opposite to which, and under 0, is its logarithm. Thus the Log. of 384 is 2.584331-prefixing the index 2, because the natural number contains 3 figures.

Again the log. of 65.7 is 1.817565-prefixing the index 1, because there are two figures only in the integral part of the given number.

3. If the given number contains four figures, the three first are to be found, as before, in the side column, and under the fourth at the top of the table is the logarithm required.

Thus the log. of 8735 is 3.941263-for against 873, the three first figures found in the left side column, and under 5, the fourth figure found at the top, stands the decimal part of the logarithm, viz. .941263, to which prefixing the index, 3, because there are four figures in the natural number, the proper logarithm is obtained.

Again the log. of 37.68 is 1.576111-Here the decimal part of the logarithm is found, as before, for the four figures; but the index is 1, because there are two integral places only in the natural number.

4. If the given number exceeds four figures, find the difference between the logarithms answering to the first four figures of the given number, and the next following logarithm; multiply this difference by the remaining figures in the given number, point off as many figures to the right-hand as there are in the multiplier, and the remainder, added to the logarithm, answering to the first four figures, will be the required logarithm, nearly. Thus; to find the logarithm of 738582;

the log. of the first four figures, viz. 7385 .868350 the next greater logarithm

to be multiplied by the remaining figures

.868409

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82

118

472

48|38

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the sum 5.868398, with the proper index prefixed is the required logarithm.

5. The logarithm of a vulgar fraction is found by subtracting the logaithm of the denominator from that of the numerator; and that of a mixed quantity is found by reducing it to an improper fraction, and proceeding as before.

Thus to find the Logarithm of 7;

from the log. of 7 =0.845098 subtract the log. of 8 = 0.903090

Remainder = 9.942008 the required log.

EXAMPLES.

Answer 2.077004.
Ans. 3.322012.
Ans. 0.755875.

1. Required the log. of 119.4?
2. Required the leg. of 2099?
3. Required the log. of 5.7?
4. Required the log. of .279?
5. Required the log. of .008297? Ans. 3.918921.
6. Required the log. of 94238 ? Ans. 4.974226.

Ans. 1.445604.

PROBLEM II.

To find the number answering to any given Logarithm.

RULES.

1. Find the next less logarithm to that given in the column marked 0 at the top, and continue the sight along that horizontal line, and a logarithm the same as that given, or very near it, will be found; then the three first figures of the corresponding natural number will be found opposite thereto in the side column, and the fourth figure immediately above it, at the top of the page. If the index of the given logarithm is 3, the four figures thus found are integers; if the index is 2, the three first figures are integers, and the fourth is a decimal, and so on.

Thus the log. 3.132580 gives the Nat Numb. 1357.

2.132580 gives
1.132580 gives
0.132580 gives
7.132580 gives

135.7

13.57

1.357

.1357 &c.

2. If the given logarithm cannot be exactly found in the table, and if more than four figures be wanted in the corresponding natural number; then find the difference between the given and the next less logarithms, to which annex as many ciphers as there are figures required above four in the natural number; which divide by the difference between the next less, and next greater logarithms, and the quotient annexed to the four figures formerly found, will give the required natural number.

Thus to find the natural number of the log. 4.828991;
the next less log. is .828982 which gives 6735;
the next greater log. is .829046

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