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The compass-box is divided from the north and south to the east and west, into 4 quadrants; each quadrant cut into degrees and half degrees. Sometimes having a vernier of five minutes, attached to the box. The upper circle, which is called the circumferentor, is divided from the north by the west into degrees and half degrees.
The two points in which the axis of the earth meets the surface, are called the poles.
A meridian is a line on the surface of the earth, if produced both ways, would pass through the poles. Therefore, all meridians intersect each other, at the two poles.
The meridians are in reality, curve lines, meeting as above in the north and south poles. No two meridians are parallel; but in small surveys their deviation is so very small, that in practice we may consider them as parallel. All the meridians passing through any small survey, may be considered as straight and parallel lines.
The bearing of a line, is the angle which it makes, with a meridian passing through one end of the line; reckoned from the north or south points of the horizon, toward the east or west point. (This is called a quartered compass bearing of a line.)
The circumferentor bearing of a line is the arc of the horizon from the north by the west to meet the line produced. This is the arc of the box from the north by the west to the south of the needle.
The meridian to be understood throughout a survey, is a magnetic meridian, taking the position of the needle at each station.
Example. Let N'S, fig. 7, represent the magnetic meridian; W and E, the west and east points. Let OB be the position of a chain line. The angle NOB is the quar tered compass bearing of said line; and NWSB the circumferentor bearing of the same line.
The reverse bearing of a line, is the bearing taken from the other end of the line, but directly in the opposite quadrant. The difference between the circumferentor bearing of a line back and forward, must always be 180 degrees.
As the bearing or course of a line, is necessary to find the latitude and departure of said line, it is, also, necessary to know the quadrant of the globe, in order to give the proper name to the latitude and departure of said line.
To find the bearing of OB fig. 7, with its proper quadrant. Put the north of the box towards the eye at A, the south will go towards B. Let the east and west change place on the box and the east will come to N, at right angles to AB, and the west to M: now the south end of the needle will rest on the N.E. quadrants of the box, and cut the angle AOS, which is = the bearing NOB.
From this it appears that by having the east and west inverted on the box, and keeping the north to eye when cutting an object, that the south end of the needle will rest upon the same quadrant of the box, that the object does on the globe. Also, that the degrees cut by the south end of the needle, are the number of degrees the object is from the next north and south pole. This is the reason why the east and west points of the compass are inverted.
The bearing of a line may be found by keeping either the north or south of box towards the eye, but to count to the south or north of the needle accordingly.
The difference of latitude or the northing or southing of a line, is the distance one end is further north or south than the other end; or the distance between the beginning of the line, and a perpendicular from the other end, on the meridian. Thus OQ is the northing of OB.
The departure or easting or westing of a line, is the dis
tance that one end is further east or west than the other end. Thus QB is the departure or easting of OB. It is plain form the last figure, the difference of latitude and departure form the sides of a right angle triangle, having the chain line the hypothenuse. Both latitude and departure are easily obtained, having the bearing and distance given.
Prob. 16. To find the number of degrees contained in a given angle by the circumferentor.
Rule. Take the difference between the bearing of both lines, and if less than 180 it will be the angle, but if more than 180, take it from 360, and it will be the given angle.
Proff. Let BOA be the angle required; suppose NS, the position of the needle or meridian at that station.
The bearing of OA, is the arc MNS, and the bearing of OB=NNS; their difference angle NOM= angle AOB. But if POA be the required angle, the bearing of OP = the arc or angle TOS, take this from the bearing of OA, it gives the arc MNT which is more than 180; take this difference from 360 and it will give the angle TOM = angle POA.
Note. The angle may, also, be found from the quartered compass bearing S.
SURVEYING WITH THE COMPASS.
În taking the surround of a tract of land, it is plain the whole northing and southing must be =, and the whole easting and westing. If the sum of the computed northing and southing are =, and, also, the easting and westing, or nearly so, we may conclude that both survey and calculation has been correctly made.
But when there are a considerable difference, an error has been committed, which must be corrected from the calculation, or by a re-survey. The following elegant solution was given by Mr. Bowditch and others, for correcting an error in the latitudes and departures of a survey, without going on the land. (But in this case the error must not be much.)
As the sum of all the distances is to each particular dis tance, so is the whole error in latitude or departure, to the correction corresponding to said latitude or departure. Apply these corrections to the latitudes and departures, by adding when of the same name, and substracting when of different name.
The length of each line is generally measured more than the geometrical line, in some cases perhaps less. Let us suppose for a moment the bearings to be correct.
In this case when there is an error in the chain line, both the latitude and departure of said line will be more than the truth. But the latitude is to the departure, as the error in the latitude is to the error in the departure. From this, it is evident when the bearing is high, suppose up to 80 degrees, little or no error falls into the latitude, but all into the departure. And on the contrary, when the bearing is small, the whole error falls into the latitude.
Again, allow an error in the bearing and the chain line. correct. Let AD or AB fig. 9, represent the line, and NS the meridian. Let the angle DAB be the error; let fall the perpendiculars DO, BC, and DN. Now it is evident DN and NB are the errors in latitude and departure.
Similar triangles we get AO: OD::BN: ND. Hence BN the error in departure, is =, great or less than ND, the error in latitude according as the bearing is =, less, greater than 45 degrees.
An expert surveyor, with a perfect knowledge of what has been said, aided by his own practical judgment, will correct his traverse without a second calculation, or the trouble of going on the land the second time.
Note. When the circumferentor is used, each bearing should be proved on the land. But a good theodolite is by far the best to use.
Prob. 17. A general rule to find the content of any piece of land, having the bearing and distance of each line given. (First find the latitude and departure of each line and correct them if required. Now allow the first meridian to pass through any station at pleasure and find the meridian distance to the middle of each line.)
Rule.-Meridian distance east, multiplied by a southing, and distance when west by a northing, will give areas. But, meridian distance east, multiplied into a northing, and when west into a southing, will give deductions. The difference between the area and deduction columns gives the area of the land.
Demonstration.-Let fig. 9 be the map proposed. A, the first station, and NS the first meridian. Now find the meridian distance to the middle of each line, as OX, RQ, FK, and HL.
NAXOX and KFXCS are deductions, being a northing by an easting, and a southing by a westing. QR BY and HLX DV, are both areas, these are southing by an easting, and a northing by a westing. It is plain the difference between the area and deduction columns will give the area of the map. To find the meridian distance to the middle of each line.
Rule.-Meridian distance and departure of the same name, their sum is a meridian distance of the same name;