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5. Half the sum of these four logarithms will be the sine of an arch, which doubled, will be the Sun's true azimuth; to be reckoned from the south in north latitude, and from the north in south latitude towards the east in the morning, and towards the west in the afternoon.
6. Then if the true and observed azimuths be both on the east, or both on the west side of the meridian, their difference is the variation: but if one be on the east and the other on the west side of the meridian, their sum is the variation; and to know if it be east or west, suppose the observer looking towards that point of the compass representing the magnetic azimuth; then if the true azimuth be to the right of the magnetic, the variation is east, but if the true be to the left of the magnetic, the variation is west.
November 2, 1812, in latitude 259 32' N. and longitude 75° W. the altitude of the Sun's lower limb was observed to be 15° 36', about 4h. 10m. P. M. his magnetic azimuth at that time being S. 58° 32/ W. and the height of the eye 18 feet; required the variation of the compass.
Sun's de. Nov. 2, at n. 14° 48'S.
Obs. alt. Sun's lower limb 15° 66/
Mag. azimuth S. 58 32 W.
5 56 east, because the true azimuth is to the
Variation right of the magnetic.
To draw a true meridian line to a map, having the variation and magnetical meridian given.
On any magnetical meridian or parallel, upon which the map is protracted, set off an angle from the north towards the east, equal to the degrees or quantity of variation, if it be westerly, or from the north towards the west if it be easterly, and the line which consti- • tutes such an angle with the magnetical meridian, will be a truc meridian line.
For if the variation be westerly, the magnetical meridian will be the quantity of variation of the west side of the true meridian, but if easterly on the east side, therefore the true mcridian must be a like quantity on the east side of the magnetical one, when the variation is westerly, and on the west side when it is easterly.
To lay out a true meridian line by the circumferentor.
If the variation be westerly, turn the box about till the north of the needle points as many degrees from the flower-de-luce towards the east of the box, or till the south of the needle points the like number of degrees from the south towards the west, as are the number of degrees contained in the variation, and the Index will be then due north. and south therefore if a line be struck out in the direction thereof, it will be a true meridian line.
If the variation was casterly, let the north of the needle point as many degrees from the flower-de-luce towards the west of the box, or let the south of the needle point as many degrees towards the east, as are the number of degrees contained in the variation, and then the north and south of the box will coincide with the north and south points of the horizon, and consequently a line being laid out by the direction of the index, will be a true meridian line.
This will be found to be very useful in setting an horizontal dial, for if you lay the edge of the index by the base of the stile of the dial, and keep the angular point of the stile toward the south of the box, and allow the variation as before, the dial will then be due north and south, and in its proper situation, provided the plane upon which it is fixed be duly horizontal, and the sun be south at noon; but in places where it is north at noon, the angular point of the index must be turned to the north.
How maps may be traced by the help of a true meridian line.
If all maps had a true meridian line laid out upon them, it would be easy by producing it, and drawing parallels, to make out field-notes; 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 as 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 im
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
Is to 57.3 degrees,*
So is the said distance
To the difference of variation required.
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?
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.
OGARITHMS 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 0, but less than 1, is a proper fraction, and is expressed decimally. The logarithm of each term between 10 and 100, is 1, 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, Y&c. are rather more conveniently used in trigonometrical calculations. A
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,
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-hand 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 columa 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 logarithm 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 be tween 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, add