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mediately 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
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 comer is found 20 links to the left hand; *hať 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 ! .
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. 44° 26'.E.
Note. The different variations do not affect the area in the calculation, as they are similar in every part of the survey.
* 57.3 Is the radius of a circle (nearly) in such parts as the circumference contains 360.
LOGARITHMS OF NUMBERS.
GARITHMS are a series of numbers so contrived, that the sum of the Logarithms of any two numbers, is the logarithm of the product of these numbers. Hence it is inferred, that if a rank, or series of numbers in arithmetical progression, be adapted to a series of numbers in geometrical progression, any term in the arithmetical progression will be the logarithm of the corresponding term in the geometrical progression.
This table contains the common logarithms of all the natural numbers from 0 to 10000, calculated to six decimal places ; such, on account of their superior accuracy, being preferable to those, that are computed only to five places of decimals.
In this form, the logarithm of 1 is 0, of 10, 1; of 100, 2 ; of 1000, 3 &c. Whence the logarithm of any term between 1 and 10, being greater than o, but less than 1, is a proper fraction, and is expressed decimally. The logarithm of each term between 10 and 100, is I, with a decimal fraction annexed; the logarithm of each term between 100 and 1000 is 2, with a decimal annexed, and so on.
The integral part of the logarithm is called the Index, and the other the decimal part.-. Except in the first hundred logarithms of this Table, the Indexes are not printed, being so readily supplied by the operator from this general rule; the Index of a Logarithm is always one less than the number of figures contained in its corresponding natural 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, of integers, is affirmative ; but when the value of a number is
: less than unity, or 1, the index is negative, and is usually marked by the sign, - placed either before, or above the index. If the first significant figure of the decimal fraction be adjacent to the decimal point, the index is 1, or its arithmetical complement 9 ; if there is one cipher between the decimal point and the first significant figure in the decimal, the index is 2, or its arith. comp. 8; if two ciphers, the index is - 3, or 7, and so on ; but the arithmetical complements, 9, 8, 7&c. are rather more conveniently used in trigonometrical calculations.
The decimal parts of the logarithms of numbers, consisting of the same figures, are the same, whether the number be integral, fractional, or mixed : thus,
of the natural
N. B. The arithmetical complement of the logarithm of any number, is found by subtracting the given logarithm from that of the radius, or by subtracting each of its figures from 9, except the last, or right-band figure, which is to be taken from 10. The arithmetical complement of an index is found by subtracting it from 10.
To find the logarithm of any given number.
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 forst 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 logarithm of 37.68 is 1.5761||--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 ibere are in the multiplier, and the remainder, add:
ed to the logarithm, answering to the first four figures, will be the fee quired logarithm, nearly.
Thus; to find the logarithm of 738582 ; the log of the first four figures, viz. 7385 .868350 the next greater logarithm
= 868409 Dif.
59 to be multiplied by the remaining figures
then to .868350
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 logarithm 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 1;
from the log. of 7 = 0.845098 subtract the log. of 8 =
Remainder = 9.942008 = the required log.
To find the number answering to any given logarithm.
1. Find the next less logarithm to that given in the column marked o 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 tbe 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. 3.152580 gives the Nat. Numb. 1357 2.132580 gives
135.7 1.132580 gives
13.57 0.132580 gives
1.357 9,132580 gives
.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 loga
rithi's, to which annex as many ciphers as there are figures required above four in the natural number ; which divide by the difference be tween the next less, and next greater logarithnıs, and the quotient an pexed 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
4 therefore 1.4 being annexed to 6735, the required natural number, 67351.4, is now obtained.
LOGARITHMS OF NUMBERS.
1.908486 1.913814 1.919078 1.924279
1.934498 1.939519 1.944483 1.949390
0.301030 3 0.477121 4 0.602060 50.698970 6 0.778151 7 0.845098 8
0.903090 9 0.954243 TO 1.000000 II 1.041393 12 1.079181 13 1113943 14 1.146128 IS 1.176091 16
1.204120 17 1.230449 18
No. Log. No. Log. No. Log. No. Log
84 25 1.397940 45 1.65321365 1.812913 85
1.414973 | 46 | 1.662758 66 1.819544 1 86 27 1.431364 47 1.67 2098 67 1.826075 87 28 1.447158 48 1.681241 68 1.832509
88 29 | 1.462398 49 1.690196 69 1.838849 89 30 1.477121 50 1.698970 70 1.845098 90 1.954243 31 1.491362
51 1.707570 71 1.851258 91 32 1.505150 52 1.716003 72 1.857332
92 33 1.518514 53 1.724276 73 1.863323 93 34 1.531479 54 1.732394 74 1.869232 94 35 1.544068 55 1.740363 75 1.875061 95
1.977724 36 1.556302 56 1.748188 76 1.880814
90 37 1.568202 57 1.755875 77 1.886491 97 38 1.57978458 1,76342878 1.892095
98 39 1.591065 59 1.770852 / 79 1.897627 99 40 1.602060 60 1.778551 80 1.903090 1100
1.959041 1.963788 1.968483 1.973128